GNU Octave

A high-level interactive language for numerical computations

Edition 3 for Octave version 3.8.2 December 2013

Free Your Numbers

John W. Eaton David Bateman Søren Hauberg Rik Wehbring

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1996, 1997, 1999, 2000, 2001, 2002, 2005, 2006, 2007, 2011, 2013 John W. Eaton. This is the third edition of the Octave documentation, and is consistent with version 3.8.2 of Octave. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the same conditions as for modified versions. Portions of this document have been adapted from the gawk, readline, gcc, and C library manuals, published by the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301–1307, USA.

Table of Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Citing Octave in Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 How You Can Contribute to Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1 A Brief Introduction to Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1 Running Octave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Simple Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Elementary Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Creating a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.4 Solving Systems of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.5 Integrating Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.6 Producing Graphical Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.7 Editing What You Have Typed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.8 Help and Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Evaluation Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.3 Printing Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.4 Error Messages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.5 Format of Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.5.1 A Sample Function Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.5.2 A Sample Command Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Invoking Octave from the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Command Line Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Startup Files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Quitting Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Commands for Getting Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Command Line Editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Cursor Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.2 Killing and Yanking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.3 Commands For Changing Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.4 Letting Readline Type For You. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.5 Commands For Manipulating The History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.6 Customizing readline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.7 Customizing the Prompt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.8 Diary and Echo Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 How Octave Reports Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Executable Octave Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.7 Comments in Octave Programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.1 Single Line Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.7.2 Block Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.7.3 Comments and the Help System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1 Built-in Data Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.1 Numeric Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.2 Missing Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.3 String Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.4 Data Structure Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.5 Cell Array Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 User-defined Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Object Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Numeric Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.1 Empty Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Single Precision Data Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4 Integer Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4.1 Integer Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.5 Bit Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.6 Logical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.7 Promotion and Demotion of Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.8 Predicates for Numeric Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1 Escape Sequences in String Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Character Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Creating Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3.1 Concatenating Strings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.3.2 Converting Numerical Data to Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.4 Comparing Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.5 Manipulating Strings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.6 String Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.7 Character Class Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Data Containers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.1 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.1.1 Basic Usage and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.1.2 Structure Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.1.3 Creating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.1.4 Manipulating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.1.5 Processing Data in Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2 Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.2.1 Basic Usage of Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.2.2 Creating Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2.3 Indexing Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.2.4 Cell Arrays of Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.2.5 Processing Data in Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3 Comma Separated Lists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.3.1 Comma Separated Lists Generated from Cell Arrays. . . . . . . . . . . . . . . . . . . . 119 6.3.2 Comma Separated Lists Generated from Structure Arrays . . . . . . . . . . . . . . 120

7 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.1 Global Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2 Persistent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.3 Status of Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.1 Index Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.1.1 Advanced Indexing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.2 Calling Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.2.1 Call by Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.2.2 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.3 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.4 Comparison Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.5 Boolean Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.5.1 Element-by-element Boolean Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8.5.2 Short-circuit Boolean Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.6 Assignment Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.7 Increment Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.8 Operator Precedence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9 Evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.1 Calling a Function by its Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.2 Evaluation in a Different Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

10 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.1 The if Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 10.2 The switch Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

10.2.1 Notes for the C Programmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 10.3 The while Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 10.4 The do-until Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 10.5 The for Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

10.5.1 Looping Over Structure Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 10.6 The break Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 10.7 The continue Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.8 The unwind protect Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 10.9 The try Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 10.10 Continuation Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

11 Functions and Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

11.1 Introduction to Function and Script Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 11.2 Defining Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 11.3 Multiple Return Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 11.4 Variable-length Argument Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 11.5 Ignoring Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 11.6 Variable-length Return Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 11.7 Returning from a Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 11.8 Default Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 11.9 Function Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

11.9.1 Manipulating the Load Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 11.9.2 Subfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 11.9.3 Private Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 11.9.4 Nested Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 11.9.5 Overloading and Autoloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 11.9.6 Function Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 11.9.7 Function Precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

11.10 Script Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 11.11 Function Handles, Anonymous Functions, Inline Functions . . . . . . . . . . . . . . . . 190

11.11.1 Function Handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 11.11.2 Anonymous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 11.11.3 Inline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

11.12 Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 11.13 Organization of Functions Distributed with Octave . . . . . . . . . . . . . . . . . . . . . . . . 194

12 Errors and Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

12.1 Handling Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 12.1.1 Raising Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 12.1.2 Catching Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 12.1.3 Recovering From Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

12.2 Handling Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 12.2.1 Issuing Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 12.2.2 Enabling and Disabling Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

13 Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

13.1 Entering Debug Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 13.2 Leaving Debug Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 13.3 Breakpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 13.4 Debug Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 13.5 Call Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 13.6 Profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 13.7 Profiler Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

14 Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

14.1 Basic Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

14.1.1 Terminal Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

14.1.1.1 Paging Screen Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 14.1.2 Terminal Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 14.1.3 Simple File I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

14.1.3.1 Saving Data on Unexpected Exits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 14.2 C-Style I/O Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

14.2.1 Opening and Closing Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 14.2.2 Simple Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 14.2.3 Line-Oriented Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 14.2.4 Formatted Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 14.2.5 Output Conversion for Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 14.2.6 Output Conversion Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 14.2.7 Table of Output Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 14.2.8 Integer Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 14.2.9 Floating-Point Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 14.2.10 Other Output Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 14.2.11 Formatted Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 14.2.12 Input Conversion Syntax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 14.2.13 Table of Input Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 14.2.14 Numeric Input Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 14.2.15 String Input Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 14.2.16 Binary I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 14.2.17 Temporary Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 14.2.18 End of File and Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 14.2.19 File Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

15 Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

15.1 Introduction to Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 15.2 High-Level Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

15.2.1 Two-Dimensional Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 15.2.1.1 Axis Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 15.2.1.2 Two-dimensional Function Plotting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 15.2.1.3 Two-dimensional Geometric Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

15.2.2 Three-Dimensional Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 15.2.2.1 Aspect Ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 15.2.2.2 Three-dimensional Function Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

15.2.2.3 Three-dimensional Geometric Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 15.2.3 Plot Annotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 15.2.4 Multiple Plots on One Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 15.2.5 Multiple Plot Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 15.2.6 Manipulation of Plot Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 15.2.7 Use of the interpreter Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 15.2.8 Printing and Saving Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 15.2.9 Interacting with Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 15.2.10 Test Plotting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

15.3 Graphics Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 15.3.1 Introduction to Graphics Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 15.3.2 Graphics Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

15.3.2.1 Creating Graphics Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 15.3.2.2 Handle Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

15.3.3 Graphics Object Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 15.3.3.1 Root Figure Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 15.3.3.2 Figure Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 15.3.3.3 Axes Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 15.3.3.4 Line Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 15.3.3.5 Text Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 15.3.3.6 Image Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 15.3.3.7 Patch Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 15.3.3.8 Surface Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

15.3.4 Searching Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 15.3.5 Managing Default Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

15.4 Advanced Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 15.4.1 Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 15.4.2 Line Styles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 15.4.3 Marker Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 15.4.4 Callbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 15.4.5 Application-defined Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 15.4.6 Object Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

15.4.6.1 Data Sources in Object Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 15.4.6.2 Area Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 15.4.6.3 Bar Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 15.4.6.4 Contour Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 15.4.6.5 Error Bar Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 15.4.6.6 Line Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 15.4.6.7 Quiver Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 15.4.6.8 Scatter Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 15.4.6.9 Stair Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 15.4.6.10 Stem Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 15.4.6.11 Surface Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

15.4.7 Graphics Toolkits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 15.4.7.1 Customizing Toolkit Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

16 Matrix Manipulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

16.1 Finding Elements and Checking Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 16.2 Rearranging Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 16.3 Special Utility Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 16.4 Famous Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

17 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

17.1 Exponents and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 17.2 Complex Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 17.3 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 17.4 Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 17.5 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 17.6 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 17.7 Rational Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 17.8 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 17.9 Mathematical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

18 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

18.1 Techniques Used for Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 18.2 Basic Matrix Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 18.3 Matrix Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 18.4 Functions of a Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 18.5 Specialized Solvers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

19 Vectorization and Faster Code Execution . . . . . . . . . . . . 457

19.1 Basic Vectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 19.2 Broadcasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

19.2.1 Broadcasting and Legacy Code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 19.3 Function Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 19.4 Accumulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 19.5 JIT Compiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 19.6 Miscellaneous Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 19.7 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

20 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

20.1 Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 20.2 Minimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

21 Diagonal and Permutation Matrices . . . . . . . . . . . . . . . . . . 479

21.1 Creating and Manipulating Diagonal/Permutation Matrices . . . . . . . . . . . . . . . . 479 21.1.1 Creating Diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 21.1.2 Creating Permutation Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 21.1.3 Explicit and Implicit Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

21.2 Linear Algebra with Diagonal/Permutation Matrices . . . . . . . . . . . . . . . . . . . . . . . 481 21.2.1 Expressions Involving Diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 21.2.2 Expressions Involving Permutation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

21.3 Functions That Are Aware of These Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 21.3.1 Diagonal Matrix Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 21.3.2 Permutation Matrix Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

21.4 Examples of Usage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 21.5 Differences in Treatment of Zero Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

22 Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

22.1 Creation and Manipulation of Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 22.1.1 Storage of Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 22.1.2 Creating Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 22.1.3 Finding Information about Sparse Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 22.1.4 Basic Operators and Functions on Sparse Matrices . . . . . . . . . . . . . . . . . . . . 497

22.1.4.1 Sparse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 22.1.4.2 Return Types of Operators and Functions . . . . . . . . . . . . . . . . . . . . . . . . 498 22.1.4.3 Mathematical Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

22.2 Linear Algebra on Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 22.3 Iterative Techniques Applied to Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 22.4 Real Life Example using Sparse Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

23 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

23.1 Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 23.2 Orthogonal Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 23.3 Functions of Multiple Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

24 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

24.1 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 24.2 Differential-Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

25 Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

25.1 Linear Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 25.2 Quadratic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 25.3 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 25.4 Linear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

26 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

26.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 26.2 Basic Statistical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 26.3 Statistical Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 26.4 Correlation and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 26.5 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 26.6 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 26.7 Random Number Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

27 Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

27.1 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

28 Polynomial Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

28.1 Evaluating Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 28.2 Finding Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 28.3 Products of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 28.4 Derivatives / Integrals / Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 28.5 Polynomial Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 28.6 Miscellaneous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

29 Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

29.1 One-dimensional Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 29.2 Multi-dimensional Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

30 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

30.1 Delaunay Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 30.1.1 Plotting the Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 30.1.2 Identifying Points in Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630

30.2 Voronoi Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 30.3 Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 30.4 Interpolation on Scattered Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

31 Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

32 Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

32.1 Loading and Saving Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 32.2 Displaying Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 32.3 Representing Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 32.4 Plotting on top of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 32.5 Color Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

33 Audio Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 34 Object Oriented Programming . . . . . . . . . . . . . . . . . . . . . . . . 673

34.1 Creating a Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 34.2 Manipulating Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 34.3 Indexing Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

34.3.1 Defining Indexing And Indexed Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 34.3.2 Indexed Assignment Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

34.4 Overloading Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 34.4.1 Function Overloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 34.4.2 Operator Overloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 34.4.3 Precedence of Objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684

34.5 Inheritance and Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686

35 GUI Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

35.1 I/O Dialogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 35.2 Progress Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 35.3 GUI Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 35.4 User-Defined Preferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

36 System Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

36.1 Timing Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 36.2 Filesystem Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 36.3 File Archiving Utilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 36.4 Networking Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

36.4.1 FTP Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 36.4.2 URL Manipulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 36.4.3 Base64 and Binary Data Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720

36.5 Controlling Subprocesses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 36.6 Process, Group, and User IDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 36.7 Environment Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 36.8 Current Working Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 36.9 Password Database Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 36.10 Group Database Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 36.11 System Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 36.12 Hashing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736

37 Java Interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737

37.1 Java Interface Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 37.2 Dialog Box Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 37.3 FAQ - Frequently asked Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

37.3.1 How to distinguish between Octave and Matlab? . . . . . . . . . . . . . . . . . . . . . . 745 37.3.2 How to make Java classes available to Octave? . . . . . . . . . . . . . . . . . . . . . . . . 745 37.3.3 How to create an instance of a Java class? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 37.3.4 How can I handle memory limitations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 37.3.5 Which TEX symbols are implemented in dialog functions? . . . . . . . . . . . . . 748

38 Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749

38.1 Installing and Removing Packages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 38.2 Using Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 38.3 Administrating Packages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 38.4 Creating Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

38.4.1 The DESCRIPTION File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 38.4.2 The INDEX File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 38.4.3 PKG ADD and PKG DEL Directives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758 38.4.4 Missing Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758

Appendix A External Code Interface. . . . . . . . . . . . . . . . . . . . 759

A.1 Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 A.1.1 Getting Started with Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 A.1.2 Matrices and Arrays in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 A.1.3 Character Strings in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766 A.1.4 Cell Arrays in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 A.1.5 Structures in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 A.1.6 Sparse Matrices in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770

A.1.6.1 Array and Sparse Class Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770 A.1.6.2 Creating Sparse Matrices in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 A.1.6.3 Using Sparse Matrices in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774

A.1.7 Accessing Global Variables in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775 A.1.8 Calling Octave Functions from Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 A.1.9 Calling External Code from Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 A.1.10 Allocating Local Memory in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780 A.1.11 Input Parameter Checking in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780 A.1.12 Exception and Error Handling in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 A.1.13 Documentation and Test of Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783

A.2 Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 A.2.1 Getting Started with Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 A.2.2 Working with Matrices and Arrays in Mex-Files. . . . . . . . . . . . . . . . . . . . . . . . 786 A.2.3 Character Strings in Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788 A.2.4 Cell Arrays with Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788 A.2.5 Structures with Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789 A.2.6 Sparse Matrices with Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 A.2.7 Calling Other Functions in Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

A.3 Standalone Programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

Appendix B Test and Demo Functions . . . . . . . . . . . . . . . . . . 799

B.1 Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 B.2 Demonstration Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

Appendix C Tips and Standards. . . . . . . . . . . . . . . . . . . . . . . . . 809

C.1 Writing Clean Octave Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 C.2 Tips on Writing Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 C.3 Conventional Headers for Octave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 C.4 Tips for Documentation Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811

Appendix D Contributing Guidelines. . . . . . . . . . . . . . . . . . . . 819

D.1 How to Contribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 D.2 Building the Development Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 D.3 Basics of Generating a Changeset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 D.4 General Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 D.5 Octave Sources (m-files) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822 D.6 C++ Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 D.7 Other Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824

Appendix E Obsolete Functions . . . . . . . . . . . . . . . . . . . . . . . . . 825

Appendix F Known Causes of Trouble . . . . . . . . . . . . . . . . . . 829

F.1 Actual Bugs We Haven’t Fixed Yet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 F.2 Reporting Bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829

F.2.1 Have You Found a Bug? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 F.2.2 Where to Report Bugs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 F.2.3 How to Report Bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 F.2.4 Sending Patches for Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831

F.3 How To Get Help with Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832

Appendix G Installing Octave. . . . . . . . . . . . . . . . . . . . . . . . . . . . 833

G.1 Build Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 G.1.1 Obtaining the Dependencies Automatically . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 G.1.2 Build Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 G.1.3 External Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834

G.2 Running Configure and Make. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836 G.3 Compiling Octave with 64-bit Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840 G.4 Installation Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843

Appendix H Emacs Octave Support . . . . . . . . . . . . . . . . . . . . . 847

H.1 Installing EOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 H.2 Using Octave Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 H.3 Running Octave from Within Emacs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851 H.4 Using the Emacs Info Reader for Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852

Appendix I Grammar and Parser . . . . . . . . . . . . . . . . . . . . . . . 855

I.1 Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 I.2 Parser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855

Appendix J GNU GENERAL PUBLIC LICENSE …. 857

Concept Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869

Function Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875

Operator Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889

Preface

Octave was originally intended to be companion software for an undergraduate-level textbook on chemical reactor design being written by James B. Rawlings of the University of Wisconsin-Madison and John G. Ekerdt of the University of Texas.

Clearly, Octave is now much more than just another ‘courseware’ package with limited utility beyond the classroom. Although our initial goals were somewhat vague, we knew that we wanted to create something that would enable students to solve realistic problems, and that they could use for many things other than chemical reactor design problems. We find that most students pick up the basics of Octave quickly, and are using it confidently in just a few hours.

Although it was originally intended to be used to teach reactor design, it has been used in several other undergraduate and graduate courses in the Chemical Engineering Department at the University of Texas, and the math department at the University of Texas has been using it for teaching differential equations and linear algebra as well. More recently, Octave has been used as the primary computational tool for teaching Stanford’s online Machine Learning class (ml-class.org) taught by Andrew Ng. Tens of thousands of students participated in the course.

If you find Octave useful, please let us know. We are always interested to find out how Octave is being used.

Virtually everyone thinks that the name Octave has something to do with music, but it is actually the name of one of John W. Eaton’s former professors who wrote a famous textbook on chemical reaction engineering, and who was also well known for his ability to do quick ‘back of the envelope’ calculations. We hope that this software will make it possible for many people to do more ambitious computations just as easily.

Everyone is encouraged to share this software with others under the terms of the GNU General Public License (see Appendix J [Copying], page 857). You are also encouraged to help make Octave more useful by writing and contributing additional functions for it, and by reporting any problems you may have.

Acknowledgements

Many people have contributed to Octave’s development. The following people have helped code parts of Octave or aided in various other ways (listed alphabetically).

Ben Abbott  Andy Adler  Adam H. Aitkenhead
Giles Anderson  Joel Andersson  Muthiah Annamalai
Markus Appel  Branden Archer  Marco Atzeri
Shai Ayal  Roger Banks  Ben Barrowes
Alexander Barth  David Bateman  Heinz Bauschke
Julien Bect  Roman Belov  Karl Berry
David Billinghurst  Don Bindner  Jakub Bogusz
Moritz Borgmann  Paul Boven  Richard Bovey
John Bradshaw  Marcus Brinkmann  Max Brister
Remy Bruno  Clemens Buchacher  Ansgar Burchard
Marco Caliari  Daniel Calvelo  John C. Campbell
Juan Pablo Carbajal  Jean-Francois Cardoso  Joao Cardoso

Larrie Carr  David Castelow  Vincent Cautaerts
Clinton Chee  Albert Chin-A-Young  Carsten Clark
Catalin Codreanu  J. D. Cole  Martin Costabel
Michael Creel  Richard Crozier  Jeff Cunningham
Martin Dalecki  Jacob Dawid  Jorge Barros de Abreu
Carlo de Falco  Thomas D. Dean  Philippe Defert
Bill Denney  Fabian Deutsch  Christos Dimitrakakis
Pantxo Diribarne  Vivek Dogra  John Donoghue
David M. Doolin  Carn Draug  Pascal A. Dupuis
John W. Eaton  Dirk Eddelbuettel  Pieter Eendebak
Paul Eggert  Stephen Eglen  Peter Ekberg
Rolf Fabian  Gunnar Farnebck  Stephen Fegan
Ramon Garcia Fernandez  Torsten Finke  Jose Daniel Munoz Frias
Brad Froehle  Castor Fu  Eduardo Gallestey
Walter Gautschi  Klaus Gebhardt  Driss Ghaddab
Nicolo Giorgetti  Arun Giridhar  Michael D. Godfrey
Michael Goffioul  Glenn Golden  Tomislav Goles
Keith Goodman  Brian Gough  Steffen Groot
Etienne Grossmann  David Grundberg  Kyle Guinn
Peter Gustafson  Kai Habel  Patrick Hcker
William P. Y. Hadisoeseno  Jaroslav Hajek  Benjamin Hall
Kim Hansen  Sren Hauberg  Dave Hawthorne
Daniel Heiserer  Martin Helm  Stefan Hepp
Martin Hepperle  Jordi Gutirrez Hermoso  Yozo Hida
Ryan Hinton  Roman Hodek  A. Scottedward Hodel
Richard Allan Holcombe  Tom Holroyd  David Hoover
Kurt Hornik  Christopher Hulbert  Cyril Humbert
John Hunt  Teemu Ikonen  Alan W. Irwin
Geoff Jacobsen  Mats Jansson  Cai Jianming
Steven G. Johnson  Heikki Junes  Matthias Jschke
Atsushi Kajita  Jarkko Kaleva  Mohamed Kamoun
Lute Kamstra  Fotios Kasolis  Thomas Kasper
Joel Keay  Mumit Khan  Paul Kienzle
Aaron A. King  Erik Kjellson  Arno J. Klaassen
Alexander Klein  Geoffrey Knauth  Heine Kolltveit
Ken Kouno  Kacper Kowalik  Daniel Kraft
Nir Krakauer  Aravindh Krishnamoorthy Oyvind Kristiansen
Artem Krosheninnikov  Piotr Krzyzanowski  Volker Kuhlmann
Tetsuro Kurita  Philipp Kutin  Miroslaw Kwasniak
Rafael Laboissiere  Kai Labusch  Claude Lacoursiere
Walter Landry  Bill Lash  Dirk Laurie
Maurice LeBrun  Friedrich Leisch  Thorsten Liebig
Jyh-miin Lin  Timo Lindfors  Benjamin Lindner
Ross Lippert  David Livings  Sebastien Loisel
Erik de Castro Lopo  Massimo Lorenzin  Emil Lucretiu
Hoxide Ma  Colin Macdonald  James Macnicol
Jens-Uwe Mager  Rob Mahurin  Alexander Mamonov

Ricardo Marranita  Orestes Mas  Axel Mathi
Makoto Matsumoto  Tatsuro Matsuoka  Christoph Mayer
Laurent Mazet  G. D. McBain  Ronald van der Meer
Jlio Hoffimann Mendes  Ed Meyer  Thorsten Meyer
Petr Mikulik  Mike Miller  Stefan Monnier
Antoine Moreau  Kai P. Mueller  Hannes Mller
Victor Munoz  Iain Murray  Carmen Navarrete
Todd Neal  Philip Nienhuis  Al Niessner
Felipe G. Nievinski  Rick Niles  Takuji Nishimura
Kai Noda  Patrick Noffke  Eric Norum
Krzesimir Nowak  Michael O’Brien  Peter O’Gorman
Thorsten Ohl  Arno Onken  Valentin Ortega-Clavero
Luis F. Ortiz  Carl Osterwisch  Janne Olavi Paanajrvi
Scott Pakin  Gabriele Pannocchia  Sylvain Pelissier
Per Persson  Primozz Peterlin  Jim Peterson
Danilo Piazzalunga  Nicholas Piper  Elias Pipping
Robert Platt  Hans Ekkehard Plesser  Tom Poage
Orion Poplawski  Ondrej Popp  Jef Poskanzer
Francesco Potort  Konstantinos Poulios  Jarno Rajahalme
James B. Rawlings  Eric S. Raymond  Balint Reczey
Joshua Redstone  Lukas Reichlin  Michael Reifenberger
Jens Restemeier  Anthony Richardson  Jason Riedy
E. Joshua Rigler  Sander van Rijn  Petter Risholm
Matthew W. Roberts  Peter Rosin  Andrew Ross
Fabio Rossi  Mark van Rossum  Joe Rothweiler
Kevin Ruland  Kristian Rumberg  Ryan Rusaw
Olli Saarela  Toni Saarela  Juhani Saastamoinen
Radek Salac  Ben Sapp  Aleksej Saushev
Alois Schlgl  Michel D. Schmid  Julian Schnidder
Nicol N. Schraudolph  Sebastian Schubert  Ludwig Schwardt
Thomas L. Scofield  Daniel J. Sebald  Dmitri A. Sergatskov
Vanya Sergeev  Baylis Shanks  Andriy Shinkarchuck
Robert T. Short  Joseph P. Skudlarek  John Smith
Julius Smith  Shan G. Smith  Peter L. Sondergaard
Joerg Specht  Quentin H. Spencer  Christoph Spiel
Richard Stallman  Russell Standish  Brett Stewart
Doug Stewart  Jonathan Stickel  Judd Storrs
Thomas Stuart  Ivan Sutoris  John Swensen
Daisuke Takago  Ariel Tankus  Falk Tannhuser
Duncan Temple Lang  Matthew Tenny  Kris Thielemans
Georg Thimm  Corey Thomasson  Olaf Till
Christophe Tournery  Thomas Treichl  Karsten Trulsen
David Turner  Frederick Umminger  Utkarsh Upadhyay
Stefan van der Walt  Peter Van Wieren  James R. Van Zandt
Risto Vanhanen  Gregory Vanuxem  Mihas Varantsou
Ivana Varekova  Sbastien Villemot  Daniel Wagenaar
Thomas Walter  Andreas Weber  Olaf Weber

Thomas Weber  Rik Wehbring  Bob Weigel
Andreas Weingessel  Martin Weiser  Michael Weitzel
David Wells  Fook Fah Yap  Sean Young
Michael Zeising  Federico Zenith  Alex Zvoleff

Special thanks to the following people and organizations for supporting the development of Octave:

• The United States Department of Energy, through grant number DE-FG02-04ER25635.

• Ashok Krishnamurthy, David Hudak, Juan Carlos Chaves, and Stanley C. Ahalt of the Ohio Supercomputer Center.

• The National Science Foundation, through grant numbers CTS-0105360, CTS-9708497, CTS-9311420, CTS-8957123, and CNS-0540147.

• The industrial members of the Texas-Wisconsin Modeling and Control Consortium

  (TWMCC (http://www.che.utexas.edu/twmcc)).

• The Paul A. Elfers Endowed Chair in Chemical Engineering at the University of Wisconsin-Madison.

• Digital Equipment Corporation, for an equipment grant as part of their External Research Program.

• Sun Microsystems, Inc., for an Academic Equipment grant.

• International Business Machines, Inc., for providing equipment as part of a grant to the University of Texas College of Engineering.

• Texaco Chemical Company, for providing funding to continue the development of this software.

• The University of Texas College of Engineering, for providing a Challenge for Excellence Research Supplement, and for providing an Academic Development Funds grant.

• The State of Texas, for providing funding through the Texas Advanced Technology Program under Grant No. 003658-078.

• Noel Bell, Senior Engineer, Texaco Chemical Company, Austin Texas.

• John A. Turner, Group Leader, Continuum Dynamics (CCS-2), Los Alamos National Laboratory, for registering the octave.org domain name.

• James B. Rawlings, Professor, University of Wisconsin-Madison, Department of Chemical and Biological Engineering.

• Richard Stallman, for writing GNU.

This project would not have been possible without the GNU software used in and to produce Octave.

Citing Octave in Publications

In view of the many contributions made by numerous developers over many years it is common courtesy to cite Octave in publications when it has been used during the course of research or the preparation of figures. The citation function can automatically generate a recommended citation text for Octave or any of its packages. See the help text below on how to use citation.

[Command] citation

[Command] citation package

Display instructions for citing GNU Octave or its packages in publications.

When called without an argument, display information on how to cite the core GNU Octave system. When given a package name package, display information on citing the specific named package. Note that some packages may not yet have instructions on how to cite them. The GNU Octave developers and its active community of package authors have invested a lot of time and effort in creating GNU Octave as it is today. Please give credit where credit is due and cite GNU Octave and its packages when you use them.

How You Can Contribute to Octave

There are a number of ways that you can contribute to help make Octave a better system. Perhaps the most important way to contribute is to write high-quality code for solving new problems, and to make your code freely available for others to use. See Appendix D [Contributing Guidelines], page 819, for detailed information on contributing new code.

If you find Octave useful, consider providing additional funding to continue its development. Even a modest amount of additional funding could make a significant difference in the amount of time that is available for development and support.

Donations supporting Octave development may be made on the web at

https://my.fsf.org/donate/working-together/octave. These donations also help to

support the Free Software Foundation

If you’d prefer to pay by check or money order, you can do so by sending a check to the FSF at the following address:

Free Software Foundation 51 Franklin Street, Suite 500 Boston, MA 02110-1335 USA

If you pay by check, please be sure to write “GNU Octave” in the memo field of your check.

If you cannot provide funding or contribute code, you can still help make Octave better and more reliable by reporting any bugs you find and by offering suggestions for ways to improve Octave. See Appendix F [Trouble], page 829, for tips on how to write useful bug reports.

Distribution

Octave is free software. This means that everyone is free to use it and free to redistribute it on certain conditions. Octave is not, however, in the public domain. It is copyrighted and there are restrictions on its distribution, but the restrictions are designed to ensure that others will have the same freedom to use and redistribute Octave that you have. The precise conditions can be found in the GNU General Public License that comes with Octave and that also appears in Appendix J [Copying], page 857.

To download a copy of Octave, please visit http://www.octave.org/download.html.

1 A Brief Introduction to Octave

GNU Octave is a high-level language, primarily intended for numerical computations. It is typically used for such problems as solving linear and nonlinear equations, numerical linear algebra, statistical analysis, and for performing other numerical experiments. It may also be used as a batch-oriented language for automated data processing.

Until recently GNU Octave provided a command-line interface with graphical results displayed in separate windows. The current version (version 3.8, released in late 2013) also provides, by default, a graphical user interface.

GNU Octave is freely redistributable software. You may redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. The GPL is included in this manual, see Appendix J [Copying], page 857.

This manual provides comprehensive documentation on how to install, run, use, and extend GNU Octave. Additional chapters describe how to report bugs and help contribute code.

This document corresponds to Octave version 3.8.2.

1.1 Running Octave

On most systems, Octave is started with the shell command ‘octave’. This, by default, starts the graphical user interface (GUI). The central window in the GUI is the Octave command-line interface. Octave displays an initial message and then a prompt indicating it is ready to accept input. If you have chosen the traditional command-line interface only the command prompt appears. In any case, you can immediately begin typing Octave commands.

If you get into trouble, you can usually interrupt Octave by typing Control-C (written C-c for short). C-c gets its name from the fact that you type it by holding down CTRL and then pressing C. Doing this will normally return you to Octave’s prompt.

To exit Octave, type quit, or exit at the Octave prompt.

On systems that support job control, you can suspend Octave by sending it a SIGTSTP signal, usually by typing C-z.

1.2 Simple Examples

The following chapters describe all of Octave’s features in detail, but before doing that, it might be helpful to give a sampling of some of its capabilities.

If you are new to Octave, I recommend that you try these examples to begin learning Octave by using it. Lines marked like so, ‘octave:13>’, are lines you type, ending each with a carriage return. Octave will respond with an answer, or by displaying a graph.

1.2.1 Elementary Calculations

Octave can easily be used for basic numerical calculations. Octave knows about arithmetic operations (+,-,*,/), exponentiation (^), natural logarithms/exponents (log, exp), and the trigonometric functions (sin, cos, …). Moreover, Octave calculations work on real or imaginary numbers (i,j). In addition, some mathematical constants such as the base of the natural logarithm (e) and the ratio of a circle’s circumference to its diameter (pi) are pre-defined. For example, to verify Euler’s Identity,

e^{ıπ} = −1

type the following which will evaluate to -1 within the tolerance of the calculation.

     octave:1> exp (i*pi)

1.2.2 Creating a Matrix

Vectors and matrices are the basic building blocks for numerical analysis. To create a new matrix and store it in a variable so that you can refer to it later, type the command

     octave:1> A = [ 1, 1, 2; 3, 5, 8; 13, 21, 34 ]

Octave will respond by printing the matrix in neatly aligned columns. Octave uses a comma or space to separate entries in a row, and a semicolon or carriage return to separate one row from the next. Ending a command with a semicolon tells Octave not to print the result of the command. For example,

     octave:2> B = rand (3, 2);

will create a 3 row, 2 column matrix with each element set to a random value between zero and one.

To display the value of a variable, simply type the name of the variable at the prompt. For example, to display the value stored in the matrix B, type the command

     octave:3> B

1.2.3 Matrix Arithmetic

Octave has a convenient operator notation for performing matrix arithmetic. For example, to multiply the matrix A by a scalar value, type the command

     octave:4> 2 * A

To multiply the two matrices A and B, type the command

     octave:5> A * B

and to form the matrix product A^{T} A, type the command

     octave:6> A' * A

1.2.4 Solving Systems of Linear Equations

Systems of linear equations are ubiquitous in numerical analysis. To solve the set of linear equations Ax = b, use the left division operator, ‘\’:

     x = A \ b

This is conceptually equivalent to A^{−1}b, but avoids computing the inverse of a matrix directly.

If the coefficient matrix is singular, Octave will print a warning message and compute a minimum norm solution.

A simple example comes from chemistry and the need to obtain balanced chemical equations. Consider the burning of hydrogen and oxygen to produce water.

H_{2} + O_{2} → H_{2}O

The equation above is not accurate. The Law of Conservation of Mass requires that the number of molecules of each type balance on the left- and right-hand sides of the equation. Writing the variable overall reaction with individual equations for hydrogen and oxygen one finds:

x_{1}H_{2} + x_{2}O_{2} → H_{2}O

H : 2x_{1} + 0x_{2} → 2
O : 0x_{1} + 2x_{2} → 1

The solution in Octave is found in just three steps.

octave:1> A = [ 2, 0; 0, 2 ];
octave:2> b = [ 2; 1 ];
octave:3> x = A \ b

1.2.5 Integrating Differential Equations

Octave has built-in functions for solving nonlinear differential equations of the form

\frac{dx}{dt} = f(x, t), x(t = t_{0}) = x_{0}

For Octave to integrate equations of this form, you must first provide a definition of the function f(x, t). This is straightforward, and may be accomplished by entering the function body directly on the command line. For example, the following commands define the righthand side function for an interesting pair of nonlinear differential equations. Note that while you are entering a function, Octave responds with a different prompt, to indicate that it is waiting for you to complete your input.

octave:1> function xdot = f (x, t)
>
>  r = 0.25;
>  k = 1.4;
>  a = 1.5;
>  b = 0.16;
>  c = 0.9;
>  d = 0.8;
>
>  xdot(1) = r*x(1)*(1 - x(1)/k) - a*x(1)*x(2)/(1 + b*x(1));
>  xdot(2) = c*a*x(1)*x(2)/(1 + b*x(1)) - d*x(2);
>
> endfunction

Given the initial condition

octave:2> x0 = [1; 2];

and the set of output times as a column vector (note that the first output time corresponds to the initial condition given above)

     octave:3> t = linspace (0, 50, 200)';

it is easy to integrate the set of differential equations:

     octave:4> x = lsode ("f", x0, t);

The function lsode uses the Livermore Solver for Ordinary Differential Equations, described in A. C. Hindmarsh, ODEPACK, a Systematized Collection of ODE Solvers, in: Scientific Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam, 1983, pages 55–64.

1.2.6 Producing Graphical Output

To display the solution of the previous example graphically, use the command

     octave:1> plot (t, x)

If you are using a graphical user interface, Octave will automatically create a separate window to display the plot.

To save a plot once it has been displayed on the screen, use the print command. For example,

     print -deps foo.eps

will create a file called ‘foo.eps’ that contains a rendering of the current plot in Encapsulated PostScript format. The command

     help print

explains more options for the print command and provides a list of additional output file formats.

1.2.7 Editing What You Have Typed

At the Octave prompt, you can recall, edit, and reissue previous commands using Emacsor vi-style editing commands. The default keybindings use Emacs-style commands. For example, to recall the previous command, press Control-p (written C-p for short). Doing this will normally bring back the previous line of input. C-n will bring up the next line of input, C-b will move the cursor backward on the line, C-f will move the cursor forward on the line, etc.

A complete description of the command line editing capability is given in this manual, see Section 2.4 [Command Line Editing], page 24.

1.2.8 Help and Documentation

Octave has an extensive help facility. The same documentation that is available in printed form is also available from the Octave prompt, because both forms of the documentation are created from the same input file.

In order to get good help you first need to know the name of the command that you want to use. This name of the function may not always be obvious, but a good place to start is to type help --list. This will show you all the operators, keywords, built-in functions, and loadable functions available in the current session of Octave. An alternative is to search the documentation using the lookfor function. This function is described in Section 2.3 [Getting Help], page 20.

Once you know the name of the function you wish to use, you can get more help on the function by simply including the name as an argument to help. For example,

     help plot

will display the help text for the plot function.

Octave sends output that is too long to fit on one screen through a pager like less or more. Type a RET to advance one line, a SPC to advance one page, and Q to exit the pager.

The part of Octave’s help facility that allows you to read the complete text of the printed manual from within Octave normally uses a separate program called Info. When you invoke Info you will be put into a menu driven program that contains the entire Octave manual. Help for using Info is provided in this manual, see Section 2.3 [Getting Help], page 20.

1.3 Conventions

This section explains the notational conventions that are used in this manual. You may want to skip this section and refer back to it later.

1.3.1 Fonts

Examples of Octave code appear in this font or form: svd (a). Names that represent variables or function arguments appear in this font or form: first-number. Commands that you type at the shell prompt appear in this font or form: ‘octave --no-init-file’. Commands that you type at the Octave prompt sometimes appear in this font or form: foo --bar --baz. Specific keys on your keyboard appear in this font or form: ANY.

1.3.2 Evaluation Notation

In the examples in this manual, results from expressions that you evaluate are indicated with ‘⇒’. For example:

     sqrt (2) ⇒ 1.4142

You can read this as “sqrt (2) evaluates to 1.4142”.

In some cases, matrix values that are returned by expressions are displayed like this

     [1, 2; 3, 4] == [1, 3; 2, 4] ⇒ [ 1, 0; 0, 1 ]

and in other cases, they are displayed like this

     eye (3) ⇒ 1  0  0 0  1  0 0  0  1

in order to clearly show the structure of the result.

Sometimes to help describe one expression, another expression is shown that produces identical results. The exact equivalence of expressions is indicated with ‘ ≡ ’. For example:

     rot90 ([1, 2; 3, 4], -1)

≡

     rot90 ([1, 2; 3, 4], 3)

≡

     rot90 ([1, 2; 3, 4], 7)

1.3.3 Printing Notation

Many of the examples in this manual print text when they are evaluated. In this manual the printed text resulting from an example is indicated by ‘ ⊣ ’. The value that is returned by evaluating the expression is displayed with ‘⇒’ (1 in the next example) and follows on a separate line.

     printf ("foo %s\n", "bar")
          ⊣ foo bar

          ⇒ 1

1.3.4 Error Messages

Some examples signal errors. This normally displays an error message on your terminal. Error messages are shown on a line beginning with error:.

     fieldnames ([1, 2; 3, 4])
     error: fieldnames: Invalid input argument

1.3.5 Format of Descriptions

Functions and commands are described in this manual in a uniform format. The first line of a description contains the name of the item followed by its arguments, if any. The category— function, command, or whatever—is printed next to the right margin. The description follows on succeeding lines, sometimes with examples.

1.3.5.1 A Sample Function Description

In a function description, the name of the function being described appears first. It is followed on the same line by a list of parameters. The names used for the parameters are also used in the body of the description.

Here is a description of an imaginary function foo:

[Function File] foo (x) [Function File] foo (x, y) [Function File] foo (x, y, …)

The function foo subtracts x from y, then adds the remaining arguments to the result. If y is not supplied, then the number 19 is used by default.

          foo (1, [3, 5], 3, 9)
               ⇒ [ 14, 16 ]
          foo (5)
               ⇒ 14

More generally,

          foo (w, x, y, ...)

≡

          x - w + y + ...

Any parameter whose name contains the name of a type (e.g., integer or matrix) is expected to be of that type. Parameters named object may be of any type. Parameters with other sorts of names (e.g., new file) are discussed specifically in the description of the function. In some sections, features common to parameters of several functions are described at the beginning.

Functions in Octave may be defined in several different ways. The category name for functions may include another name that indicates the way that the function is defined. These additional tags include

Function File The function described is defined using Octave commands stored in a text file. See Section 11.9 [Function Files], page 178.

Built-in Function The function described is written in a language like C++, C, or Fortran, and is part of the compiled Octave binary.

Loadable Function The function described is written in a language like C++, C, or Fortran. On systems that support dynamic linking of user-supplied functions, it may be automatically linked while Octave is running, but only if it is needed. See Appendix A [External Code Interface], page 759.

Mapping Function The function described works element-by-element for matrix and vector arguments.

1.3.5.2 A Sample Command Description

Command descriptions have a format similar to function descriptions, except that the word ‘Function’ is replaced by ‘Command’. Commands are functions that may be called without surrounding their arguments in parentheses. For example, here is the description for Octave’s cd command:

[Command] cd dir

[Command] chdir dir

Change the current working directory to dir. For example, cd ~/octave changes the current working directory to ‘~/octave’. If the directory does not exist, an error message is printed and the working directory is not changed.

2 Getting Started

This chapter explains some of Octave’s basic features, including how to start an Octave session, get help at the command prompt, edit the command line, and write Octave programs that can be executed as commands from your shell.

2.1 Invoking Octave from the Command Line

Normally, Octave is used interactively by running the program ‘octave’ without any arguments. Once started, Octave reads commands from the terminal until you tell it to exit.

You can also specify the name of a file on the command line, and Octave will read and execute the commands from the named file and then exit when it is finished.

You can further control how Octave starts by using the command-line options described in the next section, and Octave itself can remind you of the options available. Type ‘octave --help’ to display all available options and briefly describe their use (‘octave -h’ is a shorter equivalent).

2.1.1 Command Line Options

Here is a complete list of the command line options that Octave accepts.

--built-in-docstrings-file filename

Specify the name of the file containing documentation strings for the built-in functions of Octave. This value is normally correct and should only need to specified in extraordinary situations.

--debug

-d Enter parser debugging mode. Using this option will cause Octave’s parser to print a lot of information about the commands it reads, and is probably only useful if you are actually trying to debug the parser.

--debug-jit

Enable JIT compiler debugging and tracing.

--doc-cache-file filename

Specify the name of the doc cache file to use. The value of filename specified on the command line will override any value of OCTAVE_DOC_CACHE_FILE found in the environment, but not any commands in the system or user startup files that use the doc_cache_file function.

--echo-commands

-x Echo commands as they are executed.

--eval code

Evaluate code and exit when finished unless ‘--persist’ is also specified.

--exec-path path

Specify the path to search for programs to run. The value of path specified on the command line will override any value of OCTAVE_EXEC_PATH found in the environment, but not any commands in the system or user startup files that set the built-in variable EXEC_PATH.

--force-gui

Force the graphical user interface (GUI) to start.

--help
-h

-? Print short help message and exit.

--image-path path

Add path to the head of the search path for images. The value of path specified on the command line will override any value of OCTAVE_IMAGE_PATH found in the environment, but not any commands in the system or user startup files that set the built-in variable IMAGE_PATH.

--info-file filename

Specify the name of the info file to use. The value of filename specified on the command line will override any value of OCTAVE_INFO_FILE found in the environment, but not any commands in the system or user startup files that use the info_file function.

--info-program program

Specify the name of the info program to use. The value of program specified on the command line will override any value of OCTAVE_INFO_PROGRAM found in the environment, but not any commands in the system or user startup files that use the info_program function.

--interactive

-i Force interactive behavior. This can be useful for running Octave via a remote shell command or inside an Emacs shell buffer. For another way to run Octave within Emacs, see Appendix H [Emacs Octave Support], page 847.

--jit-compiler

Enable the JIT compiler used for accelerating loops.

--line-editing

Force readline use for command-line editing.

--no-gui Disable the graphical user interface (GUI) and use the command line interface (CLI) instead.

--no-history

-H Disable recording of command-line history.

--no-init-file

Don’t read the initialization files ‘~/.octaverc’ and ‘.octaverc’.

--no-init-path

Don’t initialize the search path for function files to include default locations.

--no-line-editing

Disable command-line editing.

--no-site-file

Don’t read the site-wide ‘octaverc’ initialization files.

--no-window-system

-W Disable use of a windowing system including graphics. This forces a strictly terminal-only environment.

--norc

-f Don’t read any of the system or user initialization files at startup. This is equivalent to using both of the options ‘--no-init-file’ and ‘--no-site-file’.

--path path

-p path Add path to the head of the search path for function files. The value of path specified on the command line will override any value of OCTAVE_PATH found in the environment, but not any commands in the system or user startup files that set the internal load path through one of the path functions.

--persist

Go to interactive mode after ‘--eval’ or reading from a file named on the command line.

--silent
--quiet

-q Don’t print the usual greeting and version message at startup.

--texi-macros-file filename

Specify the name of the file containing Texinfo macros for use by makeinfo.

--traditional
--braindead

For compatibility with matlab, set initial values for user preferences to the following values

PS1                             = ">> "
PS2                             = ""
allow_noninteger_range_as_index = true
beep_on_error                   = true
confirm_recursive_rmdir         = false
crash_dumps_octave_core         = false
save_default_options            = "-mat-binary"
do_braindead_shortcircuit_evaluation = true
fixed_point_format              = true
history_timestamp_format_string = "%%-- %D %I:%M %p --%%"
page_screen_output              = false
print_empty_dimensions          = false

and disable the following warnings

Octave:abbreviated-property-match
Octave:fopen-file-in-path
Octave:function-name-clash
Octave:load-file-in-path

Note that this does not enable the Octave:matlab-incompatible warning, which you might want if you want to be told about writing code that works in Octave but not matlab (see [warning], page 204, [warning ids], page 205).

--verbose

-V Turn on verbose output.

--version

-v Print the program version number and exit.

file Execute commands from file. Exit when done unless ‘--persist’ is also specified.

Octave also includes several functions which return information about the command line, including the number of arguments and all of the options.

[Built-in Function] argv ()

Return the command line arguments passed to Octave. For example, if you invoked Octave using the command

     octave --no-line-editing --silent

argv would return a cell array of strings with the elements ‘--no-line-editing’ and

‘--silent’.

If you write an executable Octave script, argv will return the list of arguments passed to the script. See Section 2.6 [Executable Octave Programs], page 35, for an example of how to create an executable Octave script.

[Built-in Function] program_name ()

Return the last component of the value returned by program_invocation_name.

See also: [program invocation name], page 18.

[Built-in Function] program_invocation_name ()

Return the name that was typed at the shell prompt to run Octave.

If executing a script from the command line (e.g., octave foo.m) or using an executable Octave script, the program name is set to the name of the script. See Section 2.6 [Executable Octave Programs], page 35, for an example of how to create an executable Octave script.

See also: [program name], page 18.

Here is an example of using these functions to reproduce the command line which invoked Octave.

printf ("%s", program_name ());
arg_list = argv ();
for i = 1:nargin
  printf (" %s", arg_list{i});
endfor
printf ("\n");

See Section 6.2.3 [Indexing Cell Arrays], page 114, for an explanation of how to retrieve objects from cell arrays, and Section 11.2 [Defining Functions], page 167, for information about the variable nargin.

2.1.2 Startup Files

When Octave starts, it looks for commands to execute from the files in the following list. These files may contain any valid Octave commands, including function definitions.

octave-home/share/octave/site/m/startup/octaverc

where octave-home is the directory in which Octave is installed (the default is ‘/usr/local’). This file is provided so that changes to the default Octave environment can be made globally for all users at your site for all versions of Octave you have installed. Care should be taken when making changes to this file since all users of Octave at your site will be affected. The default file may be overridden by the environment variable OCTAVE_SITE_INITFILE.

octave-home/share/octave/version/m/startup/octaverc

where octave-home is the directory in which Octave is installed (the default is ‘/usr/local’), and version is the version number of Octave. This file is provided so that changes to the default Octave environment can be made globally for all users of a particular version of Octave. Care should be taken when making changes to this file since all users of Octave at your site will be affected. The default file may be overridden by the environment variable

OCTAVE_VERSION_INITFILE.
~/.octaverc

This file is used to make personal changes to the default Octave environment.

.octaverc

This file can be used to make changes to the default Octave environment for a particular project. Octave searches for this file in the current directory after it reads ‘~/.octaverc’. Any use of the cd command in the ‘~/.octaverc’ file will affect the directory where Octave searches for ‘.octaverc’. If you start Octave in your home directory, commands from the file ‘~/.octaverc’ will only be executed once.

A message will be displayed as each of the startup files is read if you invoke Octave with the ‘--verbose’ option but without the ‘--silent’ option.

The dump_prefs function is useful for determining what customizations to Octave are possible and which are in effect.

[Function File] dump_prefs () [Function File] dump_prefs (fid)

Dump all of the current user preference variables in a format that can be parsed by Octave later. fid is a file descriptor as returned by fopen. If file is omitted, the listing is printed to stdout.

2.2 Quitting Octave

[Built-in Function] exit (status) [Built-in Function] quit (status)

Exit the current Octave session. If the optional integer value status is supplied, pass that value to the operating system as the Octave’s exit status. The default value is zero.

[Built-in Function] atexit (fcn) [Built-in Function] atexit (fcn, flag)

Register a function to be called when Octave exits. For example,

     function last_words ()
       disp ("Bye bye");
     endfunction
     atexit ("last_words");

will print the message "Bye bye" when Octave exits.

The additional argument flag will register or unregister fcn from the list of functions to be called when Octave exits. If flag is true, the function is registered, and if flag is false, it is unregistered. For example, after registering the function last_words above,

     atexit ("last_words", false);

will remove the function from the list and Octave will not call last_words when it exits.

Note that atexit only removes the first occurrence of a function from the list, so if a function was placed in the list multiple times with atexit, it must also be removed from the list multiple times.

2.3 Commands for Getting Help

The entire text of this manual is available from the Octave prompt via the command doc. In addition, the documentation for individual user-written functions and variables is also available via the help command. This section describes the commands used for reading the manual and the documentation strings for user-supplied functions and variables. See Section 11.9 [Function Files], page 178, for more information about how to document the functions you write.

[Command] help name

[Command] help --list

[Command] help .

Display the help text for name. For example, the command help help prints a short message describing the help command.

Given the single argument --list, list all operators, keywords, built-in functions, and loadable functions available in the current session of Octave.

Given the single argument ., list all operators available in the current session of Octave.

If invoked without any arguments, help display instructions on how to access help from the command line.

The help command can give you information about operators, but not the comma and semicolons that are used as command separators. To get help for those, you must

type help comma or help semicolon.

See also: [doc], page 21, [lookfor], page 21, [which], page 130.

[Command] doc function_name

Display documentation for the function function name directly from an online version of the printed manual, using the GNU Info browser. If invoked without any arguments, the manual is shown from the beginning. For example, the command doc rand starts the GNU Info browser at the rand node in the online version of the manual. Once the GNU Info browser is running, help for using it is available using the command C-h.

See also: [help], page 20.

[Command] lookfor str

[Command] lookfor -all str

[Function File] [func, helpstring] = lookfor (str)
[Function File] [func, helpstring] = lookfor ("-all", str)

Search for the string str in all functions found in the current function search path. By default, lookfor searches for str in the first sentence of the help string of each function found. The entire help text of each function can be searched if the "-all" argument is supplied. All searches are case insensitive. Called with no output arguments, lookfor prints the list of matching functions to the terminal. Otherwise, the output arguments func and helpstring define the matching functions and the first sentence of each of their help strings. The ability of lookfor to correctly identify the first sentence of the help text is dependent on the format of the function’s help. All Octave core functions are correctly formatted, but the same can not be guaranteed for external packages and user-supplied functions. Therefore, the use of the "-all" argument may be necessary to find related functions that are not a part of Octave.

See also: [help], page 20, [doc], page 21, [which], page 130.

To see what is new in the current release of Octave, use the news function.

[Command] news

[Command] news package

Display the current NEWS file for Octave or an installed package. When called without an argument, display the NEWS file for Octave. When given a package name package, display the current NEWS file for that package.

[Function File] info ()

Display contact information for the GNU Octave community.

[Built-in Function] warranty ()

Describe the conditions for copying and distributing Octave.

The following functions can be used to change which programs are used for displaying the documentation, and where the documentation can be found.

[Built-in Function] val = info_file ()

[Built-in Function] old_val = info_file (new_val)

[Built-in Function] info_file (new_val, "local")

Query or set the internal variable that specifies the name of the Octave info file. The default value is ‘octave-home/info/octave.info’, in which octave-home is the root directory of the Octave installation. The default value may be overridden by the environment variable OCTAVE_INFO_FILE, or the command line argument ‘--info-file

FNAME’.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [info program], page 22, [doc], page 21, [help], page 20, [makeinfo program], page 22.

[Built-in Function] val = info_program ()

[Built-in Function] old_val = info_program (new_val)

[Built-in Function] info_program (new_val, "local")

Query or set the internal variable that specifies the name of the info program to run.

The default value is ‘octave-home/libexec/octave/version/exec/arch/info’

in which octave-home is the root directory of the Octave installation, version is the Octave version number, and arch is the system type (for example, i686-pc-linux-gnu). The default value may be overridden by the environment variable OCTAVE_INFO_PROGRAM, or the command line argument ‘--info-program

NAME’.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [info file], page 21, [doc], page 21, [help], page 20, [makeinfo program], page 22.

[Built-in Function] val = makeinfo_program ()
[Built-in Function] old_val = makeinfo_program (new_val)

[Built-in Function] makeinfo_program (new_val, "local")

Query or set the internal variable that specifies the name of the program that Octave runs to format help text containing Texinfo markup commands. The default value is

makeinfo.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [texi macros file], page 22, [info file], page 21, [info program], page 22, [doc], page 21, [help], page 20.

[Built-in Function] val = texi_macros_file ()
[Built-in Function] old_val = texi_macros_file (new_val)

[Built-in Function] texi_macros_file (new_val, "local")

Query or set the internal variable that specifies the name of the file containing Texinfo macros that are prepended to documentation strings before they are passed to makeinfo. The default value is ‘octave-

home/share/octave/version/etc/macros.texi’, in which octave-home is

the root directory of the Octave installation, and version is the Octave version number. The default value may be overridden by the environment variable

OCTAVE_TEXI_MACROS_FILE, or the command line argument ‘--texi-macros-file
FNAME’.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [makeinfo program], page 22.

[Built-in Function] val = doc_cache_file ()

[Built-in Function] old_val = doc_cache_file (new_val)

[Built-in Function] doc_cache_file (new_val, "local")

Query or set the internal variable that specifies the name of the Octave documentation cache file. A cache file significantly improves the performance of the lookfor com-

mand. The default value is ‘octave-home/share/octave/version/etc/doc-cache’,

in which octave-home is the root directory of the Octave installation, and version is the Octave version number. The default value may be overridden by the environment vari-

able OCTAVE_DOC_CACHE_FILE, or the command line argument ‘--doc-cache-file
FNAME’.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [doc cache create], page 24, [lookfor], page 21, [info program], page 22, [doc], page 21, [help], page 20, [makeinfo program], page 22.

[Built-in Function] val = built_in_docstrings_file ()
[Built-in Function] old_val = built_in_docstrings_file (new_val)
[Built-in Function] built_in_docstrings_file (new_val, "local")

Query or set the internal variable that specifies the name of the file containing docstrings for built-in Octave functions. The default value is ‘octave-

home/share/octave/version/etc/built-in-docstrings’, in which octave-home

is the root directory of the Octave installation, and version is the Octave version number. The default value may be overridden by the environment variable OCTAVE_BUILT_IN_DOCSTRINGS_FILE, or the command line argument

‘--built-in-docstrings-file FNAME’.

Note: This variable is only used when Octave is initializing itself. Modifying it during a running session of Octave will have no effect.

[Built-in Function] val = suppress_verbose_help_message ()
[Built-in Function] old_val = suppress_verbose_help_message (new_val)
[Built-in Function] suppress_verbose_help_message (new_val, "local")

Query or set the internal variable that controls whether Octave will add additional help information to the end of the output from the help command and usage messages for built-in commands.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

The following functions are principally used internally by Octave for generating the documentation. They are documented here for completeness and because they may occasionally be useful for users.

[Function File] doc_cache_create (out_file, directory)

Generate documentation caches for all functions in a given directory. A documentation cache is generated for all functions in directory. The resulting cache is saved in the file out file. The cache is used to speed up lookfor. If no directory is given (or it is the empty matrix), a cache for built-in operators, etc. is generated.

See also: [doc cache file], page 23, [lookfor], page 21, [path], page 181.

[Built-in Function] [text, format] = get_help_text (name)

Return the raw help text of function name. The raw help text is returned in text and the format in format The format is a string

which is one of "texinfo", "html", or "plain text".
[Built-in Function] [text, format] = get_help_text_from_file (fname)

Return the raw help text from the file fname. The raw help text is returned in text and the format in format The format is a string

which is one of "texinfo", "html", or "plain text".
[Function File] [text, status] = get_first_help_sentence (name)
[Function File] [text, status] = get_first_help_sentence (name, max_len)

Return the first sentence of a function’s help text. The first sentence is defined as the text after the function declaration until either the first period (".") or the first appearance of two consecutive newlines ("\n\n"). The text is truncated to a maximum length of max len, which defaults to 80. The optional output argument status returns the status reported by makeinfo. If only one output argument is requested, and status is non-zero, a warning is displayed. As an example, the first sentence of this help text is

     get_first_help_sentence ("get_first_help_sentence")
     ⊣ ans = Return the first sentence of a function's help text.

2.4 Command Line Editing

Octave uses the GNU Readline library to provide an extensive set of command-line editing and history features. Only the most common features are described in this manual. In addition, all of the editing functions can be bound to different key strokes at the user’s discretion. This manual assumes no changes from the default Emacs bindings. See the GNU Readline Library manual for more information on customizing Readline and for a complete feature list.

To insert printing characters (letters, digits, symbols, etc.), simply type the character. Octave will insert the character at the cursor and advance the cursor forward.

Many of the command-line editing functions operate using control characters. For example, the character Control-a moves the cursor to the beginning of the line. To type C-a, hold down CTRL and then press A. In the following sections, control characters such as Control-a are written as C-a.

Another set of command-line editing functions use Meta characters. To type M-u, hold down the META key and press U. Depending on the keyboard, the META key may be labeled ALT or even WINDOWS. If your terminal does not have a META key, you can still type Meta characters using two-character sequences starting with ESC. Thus, to enter M-u, you would type ESC U. The ESC character sequences are also allowed on terminals with real Meta keys. In the following sections, Meta characters such as Meta-u are written

as M-u.

2.4.1 Cursor Motion

The following commands allow you to position the cursor.

C-b Move back one character.

C-f Move forward one character.

BACKSPACE Delete the character to the left of the cursor.

DEL Delete the character underneath the cursor.

C-d Delete the character underneath the cursor.

M-f Move forward a word.

M-b Move backward a word.

C-a Move to the start of the line.

C-e Move to the end of the line.

C-l Clear the screen, reprinting the current line at the top.

C-_

C-/ Undo the last action. You can undo all the way back to an empty line.

M-r Undo all changes made to this line. This is like typing the ‘undo’ command enough times to get back to the beginning.

The above table describes the most basic possible keystrokes that you need in order to do editing of the input line. On most terminals, you can also use the left and right arrow keys in place of C-f and C-b to move forward and backward.

Notice how C-f moves forward a character, while M-f moves forward a word. It is a loose convention that control keystrokes operate on characters while meta keystrokes operate on words.

The function clc will allow you to clear the screen from within Octave programs.

[Built-in Function] clc () [Built-in Function] home ()

Clear the terminal screen and move the cursor to the upper left corner.

2.4.2 Killing and Yanking

Killing text means to delete the text from the line, but to save it away for later use, usually by yanking it back into the line. If the description for a command says that it ‘kills’ text, then you can be sure that you can get the text back in a different (or the same) place later.

Here is the list of commands for killing text.

C-k Kill the text from the current cursor position to the end of the line.

M-d Kill from the cursor to the end of the current word, or if between words, to the end of the next word.

M-DEL Kill from the cursor to the start of the previous word, or if between words, to the start of the previous word.

C-w Kill from the cursor to the previous whitespace. This is different than M-DEL because the word boundaries differ.

And, here is how to yank the text back into the line. Yanking means to copy the most-recently-killed text from the kill buffer.

C-y Yank the most recently killed text back into the buffer at the cursor.

M-y Rotate the kill-ring, and yank the new top. You can only do this if the prior command is C-y or M-y.

When you use a kill command, the text is saved in a kill-ring. Any number of consecutive kills save all of the killed text together, so that when you yank it back, you get it in one clean sweep. The kill ring is not line specific; the text that you killed on a previously typed line is available to be yanked back later, when you are typing another line.

2.4.3 Commands For Changing Text

The following commands can be used for entering characters that would otherwise have a special meaning (e.g., TAB, C-q, etc.), or for quickly correcting typing mistakes.

C-q

C-v Add the next character that you type to the line verbatim. This is how to insert things like C-q for example.

M-TAB Insert a tab character.

C-t Drag the character before the cursor forward over the character at the cursor, also moving the cursor forward. If the cursor is at the end of the line, then transpose the two characters before it.

M-t Drag the word behind the cursor past the word in front of the cursor moving the cursor over that word as well.

M-u Uppercase the characters following the cursor to the end of the current (or following) word, moving the cursor to the end of the word.

M-l Lowercase the characters following the cursor to the end of the current (or following) word, moving the cursor to the end of the word.

M-c Uppercase the character following the cursor (or the beginning of the next word if the cursor is between words), moving the cursor to the end of the word.

2.4.4 Letting Readline Type For You

The following commands allow Octave to complete command and variable names for you.

TAB Attempt to do completion on the text before the cursor. Octave can complete the names of commands and variables.

M-? List the possible completions of the text before the cursor.

[Built-in Function] val = completion_append_char ()
[Built-in Function] old_val = completion_append_char (new_val)
[Built-in Function] completion_append_char (new_val, "local")

Query or set the internal character variable that is appended to successful commandline completion attempts. The default value is " " (a single space). When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

[Built-in Function] completion_matches (hint)

Generate possible completions given hint.

This function is provided for the benefit of programs like Emacs which might be controlling Octave and handling user input. The current command number is not incremented when this function is called. This is a feature, not a bug.

2.4.5 Commands For Manipulating The History

Octave normally keeps track of the commands you type so that you can recall previous commands to edit or execute them again. When you exit Octave, the most recent commands you have typed, up to the number specified by the variable history_size, are saved in a file. When Octave starts, it loads an initial list of commands from the file named by the

variable history_file.

Here are the commands for simple browsing and searching the history list.

LFD RET Accept the current line regardless of where the cursor is. If the line is non-

empty, add it to the history list. If the line was a history line, then restore the history line to its original state.

C-p Move ‘up’ through the history list.

C-n Move ‘down’ through the history list.

M-< Move to the first line in the history.

M-> Move to the end of the input history, i.e., the line you are entering!

C-r Search backward starting at the current line and moving ‘up’ through the history as necessary. This is an incremental search.

C-s Search forward starting at the current line and moving ‘down’ through the history as necessary.

On most terminals, you can also use the up and down arrow keys in place of C-p and C-n to move through the history list.

In addition to the keyboard commands for moving through the history list, Octave provides three functions for viewing, editing, and re-running chunks of commands from the history list.

[Command] history

[Command] history opt1 … [Built-in Function] h = history () [Built-in Function] h = history (opt1, …)

If invoked with no arguments, history displays a list of commands that you have executed. Valid options are:

n

-n Display only the most recent n lines of history.

-c Clear the history list.

-q Don’t number the displayed lines of history. This is useful for cutting and pasting commands using the X Window System.

-r file Read the file file, appending its contents to the current history list. If the name is omitted, use the default history file (normally ‘~/.octave_hist’).

-w file Write the current history to the file file. If the name is omitted, use the default history file (normally ‘~/.octave_hist’).

For example, to display the five most recent commands that you have typed without displaying line numbers, use the command history -q 5. If invoked with a single output argument, the history will be saved to that argument as a cell string and will not be output to screen.

[Command] edit_history
[Command] edit_history cmd_number
[Command] edit_history first last

Edit the history list using the editor named by the variable EDITOR. The commands to be edited are first copied to a temporary file. When you exit the editor, Octave executes the commands that remain in the file. It is often more convenient to use edit_history to define functions rather than attempting to enter them directly on the command line. The block of commands is executed as soon as you exit the editor. To avoid executing any commands, simply delete all the lines from the buffer before leaving the editor. When invoked with no arguments, edit the previously executed command; With one argument, edit the specified command cmd number; With two arguments, edit the list of commands between first and last. Command number specifiers may also be negative where -1 refers to the most recently executed command. The following are equivalent and edit the most recently executed command.

     edit_history
     edit_history -1

When using ranges, specifying a larger number for the first command than the last command reverses the list of commands before they are placed in the buffer to be edited.

See also: [run history], page 29.

[Command] run_history
[Command] run_history cmd_number
[Command] run_history first last

Run commands from the history list. When invoked with no arguments, run the previously executed command; With one argument, run the specified command cmd number; With two arguments, run the list of commands between first and last. Command number specifiers may also be negative where -1 refers to the most recently executed command. For example, the command

     run_history
          OR
     run_history -1

executes the most recent command again. The command

     run_history 13 169

executes commands 13 through 169. Specifying a larger number for the first command than the last command reverses the list of commands before executing them. For example:

     disp (1)
     disp (2)
     run_history -1 -2

⇒

      2
      1

See also: [edit history], page 28.

Octave also allows you customize the details of when, where, and how history is saved.

[Built-in Function] val = history_save ()

[Built-in Function] old_val = history_save (new_val)

[Built-in Function] history_save (new_val, "local")

Query or set the internal variable that controls whether commands entered on the command line are saved in the history file. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [history control], page 29, [history file], page 30, [history size], page 30, [history timestamp format string], page 30.

[Built-in Function] val = history_control ()

[Built-in Function] old_val = history_control (new_val)

Query or set the internal variable that specifies how commands are saved to the history list. The default value is an empty character string, but may be overridden by the environment variable OCTAVE_HISTCONTROL. The value of history_control is a colon-separated list of values controlling how commands are saved on the history list. If the list of values includes ignorespace, lines which begin with a space character are not saved in the history list. A value of ignoredups causes lines matching the previous history entry to not be saved. A value

of ignoreboth is shorthand for ignorespace and ignoredups. A value of erasedups

causes all previous lines matching the current line to be removed from the history list before that line is saved. Any value not in the above list is ignored. If history_ control is the empty string, all commands are saved on the history list, subject to

the value of history_save.

See also: [history file], page 30, [history size], page 30, [history timestamp format string], page 30, [history save], page 29.

[Built-in Function] val = history_file ()

[Built-in Function] old_val = history_file (new_val)

Query or set the internal variable that specifies the name of the file used to store command history. The default value is ‘~/.octave_hist’, but may be overridden by the environment variable OCTAVE_HISTFILE.

See also: [history size], page 30, [history save], page 29, [history timestamp format string], page 30.

[Built-in Function] val = history_size ()

[Built-in Function] old_val = history_size (new_val)

Query or set the internal variable that specifies how many entries to store in the history file. The default value is 1000, but may be overridden by the environment

variable OCTAVE_HISTSIZE.

See also: [history file], page 30, [history timestamp format string], page 30, [history save], page 29.

[Built-in Function] val = history_timestamp_format_string ()
[Built-in Function] old_val = history_timestamp_format_string (new_val)
[Built-in Function] history_timestamp_format_string (new_val, "local")

Query or set the internal variable that specifies the format string for the comment line that is written to the history file when Octave exits. The format string is passed to strftime. The default value is

     "# Octave VERSION, %a %b %d %H:%M:%S %Y %Z <USER@HOST>"

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [strftime], page 699, [history file], page 30, [history size], page 30, [history save], page 29.

[Built-in Function] val = EDITOR ()

[Built-in Function] old_val = EDITOR (new_val)

[Built-in Function] EDITOR (new_val, "local")

Query or set the internal variable that specifies the default text editor. The default value is taken from the environment variable EDITOR when Octave starts. If the environment variable is not initialized, EDITOR will be set to "emacs".

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [edit], page 179, [edit history], page 28.

2.4.6 Customizing readline

Octave uses the GNU Readline library for command-line editing and history features. Readline is very flexible and can be modified through a configuration file of commands (See the GNU Readline library for the exact command syntax). The default configuration file is normally ‘~/.inputrc’.

Octave provides two commands for initializing Readline and thereby changing the command line behavior.

[Built-in Function] readline_read_init_file (file)

Read the readline library initialization file file. If file is omitted, read the default initialization file (normally ‘~/.inputrc’). See Section “Readline Init File” in GNU Readline Library, for details.

See also: [readline re read init file], page 31.

[Built-in Function] readline_re_read_init_file ()

Re-read the last readline library initialization file that was read. See Section “Readline Init File” in GNU Readline Library, for details.

See also: [readline read init file], page 31.

2.4.7 Customizing the Prompt

The following variables are available for customizing the appearance of the command-line prompts. Octave allows the prompt to be customized by inserting a number of backslashescaped special characters that are decoded as follows:

‘\t’ The time.

‘\d’ The date.

‘\n’ Begins a new line by printing the equivalent of a carriage return followed by a line feed.

‘\s’ The name of the program (usually just ‘octave’).

‘\w’ The current working directory.

‘\W’ The basename of the current working directory.

‘\u’ The username of the current user.

‘\h’ The hostname, up to the first ‘.’.

‘\H’ The hostname.

‘\#’ The command number of this command, counting from when Octave starts.

‘\!’ The history number of this command. This differs from ‘\#’ by the number of commands in the history list when Octave starts.

‘\$’ If the effective UID is 0, a ‘#’, otherwise a ‘$’.

‘\nnn’ The character whose character code in octal is nnn.

‘\\’ A backslash.

[Built-in Function] val = PS1 () [Built-in Function] old_val = PS1 (new_val) [Built-in Function] PS1 (new_val, "local")

Query or set the primary prompt string. When executing interactively, Octave displays the primary prompt when it is ready to read a command.

The default value of the primary prompt string is "octave:\#> ". To change it, use a command like

     PS1 ("\\u@\\H> ")

which will result in the prompt ‘boris@kremvax> ’ for the user ‘boris’ logged in on the host ‘kremvax.kgb.su’. Note that two backslashes are required to enter a backslash into a double-quoted character string. See Chapter 5 [Strings], page 65.

You can also use ANSI escape sequences if your terminal supports them. This can be useful for coloring the prompt. For example,

     PS1 ("\\[\\033[01;31m\\]\\s:\\#> \\[\\033[0m\\]")

will give the default Octave prompt a red coloring.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [PS2], page 32, [PS4], page 32.

[Built-in Function] val = PS2 () [Built-in Function] old_val = PS2 (new_val) [Built-in Function] PS2 (new_val, "local")

Query or set the secondary prompt string. The secondary prompt is printed when Octave is expecting additional input to complete a command. For example, if you are typing a for loop that spans several lines, Octave will print the secondary prompt at the beginning of each line after the first. The default value of the secondary prompt string is "> ".

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [PS1], page 32, [PS4], page 32.

[Built-in Function] val = PS4 () [Built-in Function] old_val = PS4 (new_val) [Built-in Function] PS4 (new_val, "local")

Query or set the character string used to prefix output produced when echoing commands is enabled. The default value is "+ ". See Section 2.4.8 [Diary and Echo Commands], page 33, for a description of echoing commands.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [echo], page 33, [echo executing commands], page 33, [PS1], page 32, [PS2], page 32.

2.4.8 Diary and Echo Commands

Octave’s diary feature allows you to keep a log of all or part of an interactive session by recording the input you type and the output that Octave produces in a separate file.

[Command] diary

[Command] diary on

[Command] diary off

[Command] diary filename

Record a list of all commands and the output they produce, mixed together just as they appear on the terminal. Valid options are:

on Start recording a session in a file called ‘diary’ in the current working directory.

off Stop recording the session in the diary file.

filename Record the session in the file named filename.

With no arguments, diary toggles the current diary state.

See also: [history], page 28.

Sometimes it is useful to see the commands in a function or script as they are being evaluated. This can be especially helpful for debugging some kinds of problems.

[Command] echo options Control whether commands are displayed as they are executed. Valid options are:

on Enable echoing of commands as they are executed in script files.

off Disable echoing of commands as they are executed in script files.

on all Enable echoing of commands as they are executed in script files and functions.

off all Disable echoing of commands as they are executed in script files and functions.

With no arguments, echo toggles the current echo state.

[Built-in Function] val = echo_executing_commands ()
[Built-in Function] old_val = echo_executing_commands (new_val)
[Built-in Function] echo_executing_commands (new_val, "local")

Query or set the internal variable that controls the echo state. It may be the sum of the following values:

1 Echo commands read from script files.

2 Echo commands from functions.

4 Echo commands read from command line.

More than one state can be active at once. For example, a value of 3 is equivalent to the command echo on all. The value of echo_executing_commands may be set by the echo command or the command line option ‘--echo-commands’. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

2.5 How Octave Reports Errors

Octave reports two kinds of errors for invalid programs.

A parse error occurs if Octave cannot understand something you have typed. For example, if you misspell a keyword,

     octave:13> function y = f (x) y = x***2; endfunction

Octave will respond immediately with a message like this:

     parse error:
       syntax error
     >>> function y = f (x) y = x***2; endfunction
                                   ^

For most parse errors, Octave uses a caret (‘^’) to mark the point on the line where it was unable to make sense of your input. In this case, Octave generated an error message because the keyword for exponentiation (**) was misspelled. It marked the error at the third ‘*’ because the code leading up to this was correct but the final ‘*’ was not understood.

Another class of error message occurs at evaluation time. These errors are called run-time errors, or sometimes evaluation errors, because they occur when your program is being run, or evaluated. For example, if after correcting the mistake in the previous function definition, you type

     octave:13> f ()

Octave will respond with

     error: `x' undefined near line 1 column 24
     error: called from:
     error:   f at line 1, column 22

This error message has several parts, and gives quite a bit of information to help you locate the source of the error. The messages are generated from the point of the innermost error, and provide a traceback of enclosing expressions and function calls.

In the example above, the first line indicates that a variable named ‘x’ was found to be undefined near line 1 and column 24 of some function or expression. For errors occurring within functions, lines are counted from the beginning of the file containing the function definition. For errors occurring outside of an enclosing function, the line number indicates the input line number, which is usually displayed in the primary prompt string.

The second and third lines of the error message indicate that the error occurred within the function f. If the function f had been called from within another function, for example, g, the list of errors would have ended with one more line:

     error:   g at line 1, column 17

These lists of function calls make it fairly easy to trace the path your program took before the error occurred, and to correct the error before trying again.

2.6 Executable Octave Programs

Once you have learned Octave, you may want to write self-contained Octave scripts, using the ‘#!’ script mechanism. You can do this on GNU systems and on many Unix systems^{1}.

Self-contained Octave scripts are useful when you want to write a program which users can invoke without knowing that the program is written in the Octave language. Octave scripts are also used for batch processing of data files. Once an algorithm has been developed and tested in the interactive portion of Octave, it can be committed to an executable script and used again and again on new data files.

As a trivial example of an executable Octave script, you might create a text file named ‘hello’, containing the following lines:

     #! octave-interpreter-name -qf
     # a sample Octave program
     printf ("Hello, world!\n");

(where octave-interpreter-name should be replaced with the full path and name of your Octave binary). Note that this will only work if ‘#!’ appears at the very beginning of the file. After making the file executable (with the chmod command on Unix systems), you can simply type:

     hello

at the shell, and the system will arrange to run Octave as if you had typed:

     octave hello

The line beginning with ‘#!’ lists the full path and filename of an interpreter to be run, and an optional initial command line argument to pass to that interpreter. The operating system then runs the interpreter with the given argument and the full argument list of the executed program. The first argument in the list is the full file name of the Octave executable. The rest of the argument list will either be options to Octave, or data files, or both. The ‘-qf’ options are usually specified in stand-alone Octave programs to prevent them from printing the normal startup message, and to keep them from behaving differently depending on the contents of a particular user’s ‘~/.octaverc’ file. See Section 2.1 [Invoking Octave from the Command Line], page 15.

Note that some operating systems may place a limit on the number of characters that are recognized after ‘#!’. Also, the arguments appearing in a ‘#!’ line are parsed differently by various shells/systems. The majority of them group all the arguments together in one string and pass it to the interpreter as a single argument. In this case, the following script:

1 The ‘#!’ mechanism works on Unix systems derived from Berkeley Unix, System V Release 4, and some

System V Release 3 systems.

     #! octave-interpreter-name -q -f # comment

is equivalent to typing at the command line:

     octave "-q -f # comment"

which will produce an error message. Unfortunately, it is not possible for Octave to determine whether it has been called from the command line or from a ‘#!’ script, so some care is needed when using the ‘#!’ mechanism.

Note that when Octave is started from an executable script, the built-in function argv returns a cell array containing the command line arguments passed to the executable Octave script, not the arguments passed to the Octave interpreter on the ‘#!’ line of the script. For example, the following program will reproduce the command line that was used to execute the script, not ‘-qf’.

     #! /bin/octave -qf
     printf ("%s", program_name ());
     arg_list = argv ();
     for i = 1:nargin
       printf (" %s", arg_list{i});
     endfor
     printf ("\n");

2.7 Comments in Octave Programs

A comment is some text that is included in a program for the sake of human readers, and which is NOT an executable part of the program. Comments can explain what the program does, and how it works. Nearly all programming languages have provisions for comments, because programs are typically hard to understand without them.

2.7.1 Single Line Comments

In the Octave language, a comment starts with either the sharp sign character, ‘#’, or the percent symbol ‘%’ and continues to the end of the line. Any text following the sharp sign or percent symbol is ignored by the Octave interpreter and not executed. The following example shows whole line and partial line comments.

     function countdown
       # Count down for main rocket engines
       disp (3);
       disp (2);
       disp (1);
       disp ("Blast Off!");  # Rocket leaves pad
     endfunction

2.7.2 Block Comments

Entire blocks of code can be commented by enclosing the code between matching ‘#{’ and ‘#}’ or ‘%{’ and ‘%}’ markers. For example,

     function quick_countdown
       # Count down for main rocket engines
       disp (3);
      #{
       disp (2);
       disp (1);
      #}
       disp ("Blast Off!");  # Rocket leaves pad
     endfunction

will produce a very quick countdown from '3' to "Blast Off" as the lines "disp (2);" and "disp (1);" won’t be executed.

The block comment markers must appear alone as the only characters on a line (excepting whitespace) in order to be parsed correctly.

2.7.3 Comments and the Help System

The help command (see Section 2.3 [Getting Help], page 20) is able to find the first block of comments in a function and return those as a documentation string. This means that the same commands used to get help on built-in functions are available for properly formatted user-defined functions. For example, after defining the function f below,

     function xdot = f (x, t)
     # usage: f (x, t)
     #
     # This function defines the right-hand
     # side functions for a set of nonlinear
     # differential equations.
       r = 0.25;
       ...
     endfunction

the command help f produces the output

      usage: f (x, t)
      This function defines the right-hand
      side functions for a set of nonlinear
      differential equations.

Although it is possible to put comment lines into keyboard-composed, throw-away Octave programs, it usually isn’t very useful because the purpose of a comment is to help you or another person understand the program at a later time.

The help parser currently only recognizes single line comments (see Section 2.7.1 [Single Line Comments], page 36) and not block comments for the initial help text.

3 Data Types

All versions of Octave include a number of built-in data types, including real and complex scalars and matrices, character strings, a data structure type, and an array that can contain all data types.

It is also possible to define new specialized data types by writing a small amount of C++ code. On some systems, new data types can be loaded dynamically while Octave is running, so it is not necessary to recompile all of Octave just to add a new type. See Appendix A [External Code Interface], page 759, for more information about Octave’s dynamic linking capabilities. Section 3.2 [User-defined Data Types], page 43 describes what you must do to define a new data type for Octave.

[Built-in Function] typeinfo () [Built-in Function] typeinfo (expr)

Return the type of the expression expr, as a string. If expr is omitted, return a cell array of strings containing all the currently installed data types.

See also: [class], page 39, [isa], page 39.

3.1 Built-in Data Types

The standard built-in data types are real and complex scalars and matrices, ranges, character strings, a data structure type, and cell arrays. Additional built-in data types may be added in future versions. If you need a specialized data type that is not currently provided as a built-in type, you are encouraged to write your own user-defined data type and contribute it for distribution in a future release of Octave.

The data type of a variable can be determined and changed through the use of the following functions.

[Function File] classname = class (obj)

[Function File] class (s, id) [Function File] class (s, id, p, …)

Return the class of the object obj or create a class with fields from structure s and name (string) id. Additional arguments name a list of parent classes from which the new class is derived.

See also: [typeinfo], page 39, [isa], page 39.

[Function File] isa (obj, classname)

Return true if obj is an object from the class classname. classname may also be one of the following class categories:

"float" Floating point value comprising classes "double" and "single".

     "integer"

Integer value comprising classes (u)int8, (u)int16, (u)int32, (u)int64.

     "numeric"

Numeric value comprising either a floating point or integer value.

See also: [class], page 39, [typeinfo], page 39.

[Function File] cast (val, type)

Convert val to data type type.

See also: [int8], page 54, [uint8], page 54, [int16], page 54, [uint16], page 54, [int32], page 54, [uint32], page 55, [int64], page 55, [uint64], page 55, [double], page 47.

[Built-in Function] typecast (x, class)

Return a new array y resulting from interpreting the data of x in memory as data of the numeric class class. Both the class of x and class must be one of the built-in numeric classes:

     "logical"
     "char"
     "int8"
     "int16"
     "int32"
     "int64"
     "uint8"
     "uint16"
     "uint32"
     "uint64"
     "double"
     "single"
     "double complex"
     "single complex"

the last two are reserved for class; they indicate that a complex-valued result is requested. Complex arrays are stored in memory as consecutive pairs of real numbers. The sizes of integer types are given by their bit counts. Both logical and char are typically one byte wide; however, this is not guaranteed by C++. If your system is IEEE conformant, single and double should be 4 bytes and 8 bytes wide, respectively. "logical" is not allowed for class. If the input is a row vector, the return value is a row vector, otherwise it is a column vector. If the bit length of x is not divisible by that of class, an error occurs.

An example of the use of typecast on a little-endian machine is

     x = uint16 ([1, 65535]);
     typecast (x, "uint8")
     ⇒ [   1,   0, 255, 255]

See also: [cast], page 40, [bitunpack], page 41, [bitpack], page 41, [swapbytes], page 40.

[Function File] swapbytes (x)

Swap the byte order on values, converting from little endian to big endian and vice versa. For example:

     swapbytes (uint16 (1:4))
     ⇒ [   256   512   768  1024]

See also: [typecast], page 40, [cast], page 40.

[Built-in Function] y = bitpack (x, class)

Return a new array y resulting from interpreting an array x as raw bit patterns for data of the numeric class class. class must be one of the built-in numeric classes:

"char"
"int8"
"int16"
"int32"
"int64"
"uint8"
"uint16"
"uint32"
"uint64"
"double"
"single"

The number of elements of x should be divisible by the bit length of class. If it is not, excess bits are discarded. Bits come in increasing order of significance, i.e., x(1) is bit 0, x(2) is bit 1, etc. The result is a row vector if x is a row vector, otherwise it is a column vector.

See also: [bitunpack], page 41, [typecast], page 40.

[Built-in Function] y = bitunpack (x)

Return an array y corresponding to the raw bit patterns of x. x must belong to one of the built-in numeric classes:

"char"
"int8"
"int16"
"int32"
"int64"
"uint8"
"uint16"
"uint32"
"uint64"
"double"
"single"

The result is a row vector if x is a row vector; otherwise, it is a column vector.

See also: [bitpack], page 41, [typecast], page 40.

3.1.1 Numeric Objects

Octave’s built-in numeric objects include real, complex, and integer scalars and matrices. All built-in floating point numeric data is currently stored as double precision numbers. On systems that use the IEEE floating point format, values in the range of approximately 2.2251 × 10^{−308} to 1.7977 × 10^{308} can be stored, and the relative precision is approximately 2.2204 × 10^{−16}. The exact values are given by the variables realmin, realmax, and eps, respectively.

Matrix objects can be of any size, and can be dynamically reshaped and resized. It is easy to extract individual rows, columns, or submatrices using a variety of powerful indexing features. See Section 8.1 [Index Expressions], page 131.

See Chapter 4 [Numeric Data Types], page 47, for more information.

3.1.2 Missing Data

It is possible to represent missing data explicitly in Octave using NA (short for “Not Available”). Missing data can only be represented when data is represented as floating point numbers. In this case missing data is represented as a special case of the representation of

NaN.

[Built-in Function] NA [Built-in Function] NA (n) [Built-in Function] NA (n, m) [Built-in Function] NA (n, m, k, …) [Built-in Function] NA (…, class)

Return a scalar, matrix, or N-dimensional array whose elements are all equal to the special constant used to designate missing values. Note that NA always compares not equal to NA (NA != NA). To find NA values, use the isna function. When called with no arguments, return a scalar with the value ‘NA’. When called with a single argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The optional argument class specifies the return type and may be either

"double" or "single".

See also: [isna], page 42.

[Mapping Function] isna (x)

Return a logical array which is true where the elements of x are NA (missing) values and false where they are not. For example:

     isna ([13, Inf, NA, NaN])
          ⇒ [ 0, 0, 1, 0 ]

See also: [isnan], page 380, [isinf], page 380, [isfinite], page 380.

3.1.3 String Objects

A character string in Octave consists of a sequence of characters enclosed in either doublequote or single-quote marks. Internally, Octave currently stores strings as matrices of characters. All the indexing operations that work for matrix objects also work for strings.

See Chapter 5 [Strings], page 65, for more information.

3.1.4 Data Structure Objects

Octave’s data structure type can help you to organize related objects of different types. The current implementation uses an associative array with indices limited to strings, but the syntax is more like C-style structures.

See Section 6.1 [Structures], page 97, for more information.

3.1.5 Cell Array Objects

A Cell Array in Octave is general array that can hold any number of different data types. See Section 6.2 [Cell Arrays], page 110, for more information.

3.2 User-defined Data Types

Someday I hope to expand this to include a complete description of Octave’s mechanism for managing user-defined data types. Until this feature is documented here, you will have to make do by reading the code in the ‘ov.h’, ‘ops.h’, and related files from Octave’s ‘src’ directory.

3.3 Object Sizes

The following functions allow you to determine the size of a variable or expression. These functions are defined for all objects. They return −1 when the operation doesn’t make sense. For example, Octave’s data structure type doesn’t have rows or columns, so the rows and columns functions return −1 for structure arguments.

[Built-in Function] ndims (a)

Return the number of dimensions of a. For any array, the result will always be larger than or equal to 2. Trailing singleton dimensions are not counted.

     ndims (ones (4, 1, 2, 1))
         ⇒ 3

See also: [size], page 44.

[Built-in Function] columns (a)

Return the number of columns of a.

See also: [rows], page 43, [size], page 44, [length], page 44, [numel], page 43, [isscalar], page 62, [isvector], page 62, [ismatrix], page 62.

[Built-in Function] rows (a)

Return the number of rows of a.

See also: [columns], page 43, [size], page 44, [length], page 44, [numel], page 43, [isscalar], page 62, [isvector], page 62, [ismatrix], page 62.

[Built-in Function] numel (a) [Built-in Function] numel (a, idx1, idx2, …)

Return the number of elements in the object a. Optionally, if indices idx1, idx2, … are supplied, return the number of elements that would result from the indexing

     a(idx1, idx2, ...)

Note that the indices do not have to be numerical. For example,

     a = 1;
     b = ones (2, 3);
     numel (a, b)

will return 6, as this is the number of ways to index with b.

This method is also called when an object appears as lvalue with cs-list indexing, i.e.,

object{...} or object(...).field.

See also: [size], page 44.

[Built-in Function] length (a)

Return the length of the object a. The length is 0 for empty objects, 1 for scalars, and the number of elements for vectors. For matrix objects, the length is the number of rows or columns, whichever is greater (this odd definition is used for compatibility with matlab).

See also: [numel], page 43, [size], page 44.

[Built-in Function] size (a) [Built-in Function] size (a, dim)

Return the number of rows and columns of a. With one input argument and one output argument, the result is returned in a row vector. If there are multiple output arguments, the number of rows is assigned to the first, and the number of columns to the second, etc. For example:

     size ([1, 2; 3, 4; 5, 6])
        ⇒ [ 3, 2 ]
     [nr, nc] = size ([1, 2; 3, 4; 5, 6])
         ⇒ nr = 3

         ⇒ nc = 2

If given a second argument, size will return the size of the corresponding dimension. For example,

     size ([1, 2; 3, 4; 5, 6], 2) ⇒ 2

returns the number of columns in the given matrix.

See also: [numel], page 43, [ndims], page 43, [length], page 44, [rows], page 43, [columns], page 43.

[Built-in Function] isempty (a)

Return true if a is an empty matrix (any one of its dimensions is zero). Otherwise, return false.

See also: [isnull], page 44, [isa], page 39.

[Built-in Function] isnull (x)

Return true if x is a special null matrix, string, or single quoted string. Indexed assignment with such a value on the right-hand side should delete array elements. This function should be used when overloading indexed assignment for user-defined classes instead of isempty, to distinguish the cases:

A(I) = [] This should delete elements if I is nonempty.

X = []; A(I) = X

This should give an error if I is nonempty.

See also: [isempty], page 44, [isindex], page 134.

[Built-in Function] sizeof (val)

Return the size of val in bytes.

See also: [whos], page 126.

[Built-in Function] size_equal (a, b, …)

Return true if the dimensions of all arguments agree. Trailing singleton dimensions are ignored. Called with a single or no argument, size equal returns true.

See also: [size], page 44, [numel], page 43, [ndims], page 43.

[Built-in Function] squeeze (x)

Remove singleton dimensions from x and return the result. Note that for compatibility with matlab, all objects have a minimum of two dimensions and row vectors are left unchanged.

See also: [reshape], page 385.

4 Numeric Data Types

A numeric constant may be a scalar, a vector, or a matrix, and it may contain complex values.

The simplest form of a numeric constant, a scalar, is a single number that can be an integer, a decimal fraction, a number in scientific (exponential) notation, or a complex number. Note that by default numeric constants are represented within Octave in doubleprecision floating point format (complex constants are stored as pairs of double-precision floating point values). It is, however, possible to represent real integers as described in Section 4.4 [Integer Data Types], page 54. Here are some examples of real-valued numeric constants, which all have the same value:

     105 1.05e+2 1050e-1

To specify complex constants, you can write an expression of the form

     3 + 4i
     3.0 + 4.0i
     0.3e1 + 40e-1i

all of which are equivalent. The letter ‘i’ in the previous example stands for the pure imaginary constant, defined as \sqrt{−1}.

For Octave to recognize a value as the imaginary part of a complex constant, a space must not appear between the number and the ‘i’. If it does, Octave will print an error message, like this:

     octave:13> 3 + 4 i
     parse error:
       syntax error
     >>> 3 + 4 i
               ^

You may also use ‘j’, ‘I’, or ‘J’ in place of the ‘i’ above. All four forms are equivalent.

[Built-in Function] double (x)

Convert x to double precision type.

See also: [single], page 53.

[Built-in Function] complex (x) [Built-in Function] complex (re, im)

Return a complex result from real arguments. With 1 real argument x, return the complex result x + 0i. With 2 real arguments, return the complex result re + im. complex can often be more convenient than expressions such as a + i*b. For example:

          complex ([1, 2], [3, 4])
            ⇒ [ 1 + 3i   2 + 4i ]

See also: [real], page 409, [imag], page 409, [iscomplex], page 62, [abs], page 409, [arg], page 409.

4.1 Matrices

It is easy to define a matrix of values in Octave. The size of the matrix is determined automatically, so it is not necessary to explicitly state the dimensions. The expression

     a = [1, 2; 3, 4]

results in the matrix

a = [ \begin{matrix}1 & 2 \\ 3 & 4 \end{matrix} ]

Elements of a matrix may be arbitrary expressions, provided that the dimensions all make sense when combining the various pieces. For example, given the above matrix, the expression

     [ a, a ]

produces the matrix

     ans = 1  2  1  2 3  4  3  4

but the expression

     [ a, 1 ]

produces the error

     error: number of rows must match (1 != 2) near line 13, column 6

(assuming that this expression was entered as the first thing on line 13, of course).

Inside the square brackets that delimit a matrix expression, Octave looks at the surrounding context to determine whether spaces and newline characters should be converted into element and row separators, or simply ignored, so an expression like

     a = [ 1 2 3 4 ]

will work. However, some possible sources of confusion remain. For example, in the expression

     [ 1 - 1 ]

the ‘-’ is treated as a binary operator and the result is the scalar 0, but in the expression

     [ 1 -1 ]

the ‘-’ is treated as a unary operator and the result is the vector [ 1, -1 ]. Similarly, the expression

     [ sin (pi) ]

will be parsed as

     [ sin, (pi) ]

and will result in an error since the sin function will be called with no arguments. To get around this, you must omit the space between sin and the opening parenthesis, or enclose the expression in a set of parentheses:

     [ (sin (pi)) ]

Whitespace surrounding the single quote character (‘'’, used as a transpose operator and for delimiting character strings) can also cause confusion. Given a = 1, the expression

[ 1 a' ]

results in the single quote character being treated as a transpose operator and the result is the vector [ 1, 1 ], but the expression

[ 1 a ' ]

produces the error message

parse error:
  syntax error
>>> [ 1 a ' ]
              ^

because not doing so would cause trouble when parsing the valid expression

[ a 'foo' ]

For clarity, it is probably best to always use commas and semicolons to separate matrix elements and rows.

The maximum number of elements in a matrix is fixed when Octave is compiled. The allowable number can be queried with the function sizemax. Note that other factors, such as the amount of memory available on your machine, may limit the maximum size of matrices to something smaller.

[Built-in Function] sizemax ()

Return the largest value allowed for the size of an array. If Octave is compiled with 64-bit indexing, the result is of class int64, otherwise it is of class int32. The maximum array size is slightly smaller than the maximum value allowable for the relevant class as reported by intmax.

See also: [intmax], page 55.

When you type a matrix or the name of a variable whose value is a matrix, Octave responds by printing the matrix in with neatly aligned rows and columns. If the rows of the matrix are too large to fit on the screen, Octave splits the matrix and displays a header before each section to indicate which columns are being displayed. You can use the following variables to control the format of the output.

[Built-in Function] val = output_max_field_width ()
[Built-in Function] old_val = output_max_field_width (new_val)
[Built-in Function] output_max_field_width (new_val, "local")

Query or set the internal variable that specifies the maximum width of a numeric output field.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [format], page 224, [fixed point format], page 50, [output precision], page 50.

[Built-in Function] val = output_precision ()
[Built-in Function] old_val = output_precision (new_val)

[Built-in Function] output_precision (new_val, "local")

Query or set the internal variable that specifies the minimum number of significant figures to display for numeric output.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [format], page 224, [fixed point format], page 50, [output max field width], page 49.

It is possible to achieve a wide range of output styles by using different values of output_ precision and output_max_field_width. Reasonable combinations can be set using the format function. See Section 14.1 [Basic Input and Output], page 223.

[Built-in Function] val = split_long_rows ()

[Built-in Function] old_val = split_long_rows (new_val)

[Built-in Function] split_long_rows (new_val, "local")

Query or set the internal variable that controls whether rows of a matrix may be split when displayed to a terminal window. If the rows are split, Octave will display the matrix in a series of smaller pieces, each of which can fit within the limits of your terminal width and each set of rows is labeled so that you can easily see which columns are currently being displayed. For example:

     octave:13> rand (2,10)
     ans =
      Columns 1 through 6:
       0.75883  0.93290  0.40064  0.43818  0.94958  0.16467
       0.75697  0.51942  0.40031  0.61784  0.92309  0.40201
      Columns 7 through 10:
       0.90174  0.11854  0.72313  0.73326
       0.44672  0.94303  0.56564  0.82150

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [format], page 224.

Octave automatically switches to scientific notation when values become very large or very small. This guarantees that you will see several significant figures for every value in a matrix. If you would prefer to see all values in a matrix printed in a fixed point format, you can set the built-in variable fixed_point_format to a nonzero value. But doing so is not recommended, because it can produce output that can easily be misinterpreted.

[Built-in Function] val = fixed_point_format ()
[Built-in Function] old_val = fixed_point_format (new_val)
[Built-in Function] fixed_point_format (new_val, "local")

Query or set the internal variable that controls whether Octave will use a scaled format to print matrix values.

The scaled format prints a scaling factor on the first line of output chosen such that the largest matrix element can be written with a single leading digit. For example:

     logspace (1, 7, 5)'
     ans =
       1.0e+07  *
       0.00000
       0.00003
       0.00100
       0.03162
       1.00000

Notice that the first value appears to be 0 when it is actually 1. Because of the possibilty for confusion you should be careful about enabling fixed_point_format.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [format], page 224, [output max field width], page 49, [output precision], page 50.

4.1.1 Empty Matrices

A matrix may have one or both dimensions zero, and operations on empty matrices are handled as described by Carl de Boor in An Empty Exercise, SIGNUM, Volume 25, pages 2-6, 1990 and C. N. Nett and W. M. Haddad, in A System-Theoretic Appropriate Realization of the Empty Matrix Concept, IEEE Transactions on Automatic Control, Volume 38, Number 5, May 1993. Briefly, given a scalar s, an m×n matrix M_{m×n}, and an m×n empty matrix [ ]_{m×n} (with either one or both dimensions equal to zero), the following are true:

\begin{aligned}s · [ ]_{m×n} = [ ]_{m×n} · s & = [ ]_{m×n} \\ [ ]_{m×n} + [ ]_{m×n} & = [ ]_{m×n} \\ [ ]_{0×m} · M_{m×n} & = [ ]_{0×n} \\ M_{m×n} · [ ]_{n×0} & = [ ]_{m×0} \\ [ ]_{m×0} · [ ]_{0×n} & = 0_{m×n} \end{aligned}

By default, dimensions of the empty matrix are printed along with the empty matrix symbol, ‘[]’. The built-in variable print_empty_dimensions controls this behavior.

[Built-in Function] val = print_empty_dimensions ()
[Built-in Function] old_val = print_empty_dimensions (new_val)

[Built-in Function] print_empty_dimensions (new_val, "local")

Query or set the internal variable that controls whether the dimensions of empty matrices are printed along with the empty matrix symbol, ‘[]’. For example, the expression

          zeros (3, 0)

will print

          ans = [](3x0)

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [format], page 224.

Empty matrices may also be used in assignment statements as a convenient way to delete rows or columns of matrices. See Section 8.6 [Assignment Expressions], page 145.

When Octave parses a matrix expression, it examines the elements of the list to determine whether they are all constants. If they are, it replaces the list with a single matrix constant.

4.2 Ranges

A range is a convenient way to write a row vector with evenly spaced elements. A range expression is defined by the value of the first element in the range, an optional value for the increment between elements, and a maximum value which the elements of the range will not exceed. The base, increment, and limit are separated by colons (the ‘:’ character) and may contain any arithmetic expressions and function calls. If the increment is omitted, it is assumed to be 1. For example, the range

     1 : 5

defines the set of values ‘[ 1, 2, 3, 4, 5 ]’, and the range

     1 : 3 : 5

defines the set of values ‘[ 1, 4 ]’.

Although a range constant specifies a row vector, Octave does not convert range constants to vectors unless it is necessary to do so. This allows you to write a constant like ‘1 : 10000’ without using 80,000 bytes of storage on a typical 32-bit workstation.

A common example of when it does become necessary to convert ranges into vectors occurs when they appear within a vector (i.e., inside square brackets). For instance, whereas

     x = 0 : 0.1 : 1;

defines x to be a variable of type range and occupies 24 bytes of memory, the expression

     y = [ 0 : 0.1 : 1];

defines y to be of type matrix and occupies 88 bytes of memory.

Note that the upper (or lower, if the increment is negative) bound on the range is not always included in the set of values, and that ranges defined by floating point values can produce surprising results because Octave uses floating point arithmetic to compute the values in the range. If it is important to include the endpoints of a range and the number of elements is known, you should use the linspace function instead (see Section 16.3 [Special Utility Matrices], page 391).

When adding a scalar to a range, subtracting a scalar from it (or subtracting a range from a scalar) and multiplying by scalar, Octave will attempt to avoid unpacking the range and keep the result as a range, too, if it can determine that it is safe to do so. For instance, doing

     a = 2*(1:1e7) - 1;

will produce the same result as ‘1:2:2e7-1’, but without ever forming a vector with ten million elements.

Using zero as an increment in the colon notation, as ‘1:0:1’ is not allowed, because a division by zero would occur in determining the number of range elements. However, ranges with zero increment (i.e., all elements equal) are useful, especially in indexing, and Octave allows them to be constructed using the built-in function ones. Note that because a range must be a row vector, ‘ones (1, 10)’ produces a range, while ‘ones (10, 1)’ does not.

When Octave parses a range expression, it examines the elements of the expression to determine whether they are all constants. If they are, it replaces the range expression with a single range constant.

4.3 Single Precision Data Types

Octave includes support for single precision data types, and most of the functions in Octave accept single precision values and return single precision answers. A single precision variable is created with the single function.

[Built-in Function] single (x)

Convert x to single precision type.

See also: [double], page 47.

for example:

     sngl = single (rand (2, 2))
          ⇒ sngl =
             0.37569   0.92982
             0.11962   0.50876
     class (sngl)
         ⇒ single

Many functions can also return single precision values directly. For example

     ones (2, 2, "single")
     zeros (2, 2, "single")
     eye (2, 2,  "single")
     rand (2, 2, "single")
     NaN (2, 2, "single")
     NA (2, 2, "single")
     Inf (2, 2, "single")

will all return single precision matrices.

4.4 Integer Data Types

Octave supports integer matrices as an alternative to using double precision. It is possible to use both signed and unsigned integers represented by 8, 16, 32, or 64 bits. It should be noted that most computations require floating point data, meaning that integers will often change type when involved in numeric computations. For this reason integers are most often used to store data, and not for calculations.

In general most integer matrices are created by casting existing matrices to integers. The following example shows how to cast a matrix into 32 bit integers.

float = rand (2, 2)
     ⇒ float = 0.37569   0.92982
                0.11962   0.50876
integer = int32 (float)
     ⇒ integer = 0  1
                  0  1

As can be seen, floating point values are rounded to the nearest integer when converted.

[Built-in Function] isinteger (x)

Return true if x is an integer object (int8, uint8, int16, etc.). Note that isinteger (14) is false because numeric constants in Octave are double precision floating point values.

See also: [isfloat], page 61, [ischar], page 66, [islogical], page 63, [isnumeric], page 61, [isa], page 39.

[Built-in Function] int8 (x)

Convert x to 8-bit integer type.

See also: [uint8], page 54, [int16], page 54, [uint16], page 54, [int32], page 54, [uint32], page 55, [int64], page 55, [uint64], page 55.

[Built-in Function] uint8 (x)

Convert x to unsigned 8-bit integer type.

See also: [int8], page 54, [int16], page 54, [uint16], page 54, [int32], page 54, [uint32], page 55, [int64], page 55, [uint64], page 55.

[Built-in Function] int16 (x)

Convert x to 16-bit integer type.

See also: [int8], page 54, [uint8], page 54, [uint16], page 54, [int32], page 54, [uint32], page 55, [int64], page 55, [uint64], page 55.

[Built-in Function] uint16 (x)

Convert x to unsigned 16-bit integer type.

See also: [int8], page 54, [uint8], page 54, [int16], page 54, [int32], page 54, [uint32], page 55, [int64], page 55, [uint64], page 55.

[Built-in Function] int32 (x)

Convert x to 32-bit integer type.

See also: [int8], page 54, [uint8], page 54, [int16], page 54, [uint16], page 54, [uint32], page 55, [int64], page 55, [uint64], page 55.

[Built-in Function] uint32 (x)

Convert x to unsigned 32-bit integer type. See also: [int8], page 54, [uint8], page 54, [int16], page 54, [uint16], page 54, [int32], page 54, [int64], page 55, [uint64], page 55.

[Built-in Function] int64 (x)

Convert x to 64-bit integer type. See also: [int8], page 54, [uint8], page 54, [int16], page 54, [uint16], page 54, [int32], page 54, [uint32], page 55, [uint64], page 55.

[Built-in Function] uint64 (x)

Convert x to unsigned 64-bit integer type. See also: [int8], page 54, [uint8], page 54, [int16], page 54, [uint16], page 54, [int32], page 54, [uint32], page 55, [int64], page 55.

[Built-in Function] intmax (type)

Return the largest integer that can be represented in an integer type. The variable type can be int8 signed 8-bit integer. int16 signed 16-bit integer. int32 signed 32-bit integer. int64 signed 64-bit integer. uint8 unsigned 8-bit integer. uint16 unsigned 16-bit integer. uint32 unsigned 32-bit integer. uint64 unsigned 64-bit integer. The default for type is int32. See also: [intmin], page 55, [flintmax], page 56, [bitmax], page 57.

[Built-in Function] intmin (type)

Return the smallest integer that can be represented in an integer type. The variable type can be int8 signed 8-bit integer. int16 signed 16-bit integer. int32 signed 32-bit integer. int64 signed 64-bit integer. uint8 unsigned 8-bit integer. uint16 unsigned 16-bit integer. uint32 unsigned 32-bit integer. uint64 unsigned 64-bit integer.

The default for type is int32.

See also: [intmax], page 55, [flintmax], page 56, [bitmax], page 57.

[Built-in Function] flintmax () [Built-in Function] flintmax ("double") [Built-in Function] flintmax ("single")

Return the largest integer that can be represented consecutively in a floating point value. The default class is "double", but "single" is a valid option. On IEEE-754 compatible systems, flintmax is 2^{5}3 for "double" and 2^{2}4 for "single".

See also: [bitmax], page 57, [intmax], page 55, [realmax], page 432, [realmin], page 432.

4.4.1 Integer Arithmetic

While many numerical computations can’t be carried out in integers, Octave does support basic operations like addition and multiplication on integers. The operators +, -, .*, and ./ work on integers of the same type. So, it is possible to add two 32 bit integers, but not to add a 32 bit integer and a 16 bit integer.

When doing integer arithmetic one should consider the possibility of underflow and overflow. This happens when the result of the computation can’t be represented using the chosen integer type. As an example it is not possible to represent the result of 10 − 20 when using unsigned integers. Octave makes sure that the result of integer computations is the integer that is closest to the true result. So, the result of 10 − 20 when using unsigned integers is zero.

When doing integer division Octave will round the result to the nearest integer. This is different from most programming languages, where the result is often floored to the nearest integer. So, the result of int32 (5) ./ int32 (8) is 1.

[Function File] idivide (x, y, op)

Integer division with different rounding rules. The standard behavior of integer division such as a ./ b is to round the result to the nearest integer. This is not always the desired behavior and idivide permits integer element-by-element division to be performed with different treatment for the fractional part of the division as determined by the op flag. op is a string with one of the values:

"fix" Calculate a ./ b with the fractional part rounded towards zero.

"round" Calculate a ./ b with the fractional part rounded towards the nearest integer.

"floor" Calculate a ./ b with the fractional part rounded towards negative infinity.

"ceil" Calculate a ./ b with the fractional part rounded towards positive infinity.

If op is not given it defaults to "fix". An example demonstrating these rounding rules is

          idivide (int8 ([-3, 3]), int8 (4), "fix")
            ⇒ int8 ([0, 0])
          idivide (int8 ([-3, 3]), int8 (4), "round")
            ⇒ int8 ([-1, 1])
          idivide (int8 ([-3, 3]), int8 (4), "floor")
            ⇒ int8 ([-1, 0])
          idivide (int8 ([-3, 3]), int8 (4), "ceil")
            ⇒ int8 ([0, 1])

See also: [ldivide], page 139, [rdivide], page 140.

4.5 Bit Manipulations

Octave provides a number of functions for the manipulation of numeric values on a bit by bit basis. The basic functions to set and obtain the values of individual bits are bitset

and bitget.

[Function File] C = bitset (A, n) [Function File] C = bitset (A, n, val)

Set or reset bit(s) n of unsigned integers in A. val = 0 resets and val = 1 sets the bits. The lowest significant bit is: n = 1. All variables must be the same size or scalars.

          dec2bin (bitset (10, 1))
            ⇒ 1011

See also: [bitand], page 58, [bitor], page 58, [bitxor], page 58, [bitget], page 57, [bitcmp], page 58, [bitshift], page 58, [bitmax], page 57.

[Function File] c = bitget (A, n)

Return the status of bit(s) n of unsigned integers in A the lowest significant bit is n = 1.

          bitget (100, 8:-1:1) ⇒ 0  1  1  0  0  1  0  0

See also: [bitand], page 58, [bitor], page 58, [bitxor], page 58, [bitset], page 57, [bitcmp], page 58, [bitshift], page 58, [bitmax], page 57.

The arguments to all of Octave’s bitwise operations can be scalar or arrays, except for bitcmp, whose k argument must a scalar. In the case where more than one argument is an array, then all arguments must have the same shape, and the bitwise operator is applied to each of the elements of the argument individually. If at least one argument is a scalar and one an array, then the scalar argument is duplicated. Therefore

     bitget (100, 8:-1:1)

is the same as

     bitget (100 * ones (1, 8), 8:-1:1)

It should be noted that all values passed to the bit manipulation functions of Octave are treated as integers. Therefore, even though the example for bitset above passes the floating point value 10, it is treated as the bits [1, 0, 1, 0] rather than the bits of the native floating point format representation of 10.

As the maximum value that can be represented by a number is important for bit manipulation, particularly when forming masks, Octave supplies the function bitmax.

[Built-in Function] bitmax () [Built-in Function] bitmax ("double") [Built-in Function] bitmax ("single")

Return the largest integer that can be represented within a floating point value. The default class is "double", but "single" is a valid option. On IEEE-754 compatible systems, bitmax is 2^{53} − 1 for "double" and 2^{24} − 1 for "single".

See also: [flintmax], page 56, [intmax], page 55, [realmax], page 432, [realmin], page 432.

This is the double precision version of the function intmax, previously discussed.

Octave also includes the basic bitwise ’and’, ’or’, and ’exclusive or’ operators.

[Built-in Function] bitand (x, y)

Return the bitwise AND of non-negative integers. x, y must be in the range [0,bitmax]

See also: [bitor], page 58, [bitxor], page 58, [bitset], page 57, [bitget], page 57, [bitcmp], page 58, [bitshift], page 58, [bitmax], page 57.

[Built-in Function] bitor (x, y)

Return the bitwise OR of non-negative integers. x, y must be in the range [0,bitmax]

See also: [bitor], page 58, [bitxor], page 58, [bitset], page 57, [bitget], page 57, [bitcmp], page 58, [bitshift], page 58, [bitmax], page 57.

[Built-in Function] bitxor (x, y)

Return the bitwise XOR of non-negative integers. x, y must be in the range [0,bitmax]

See also: [bitand], page 58, [bitor], page 58, [bitset], page 57, [bitget], page 57, [bitcmp], page 58, [bitshift], page 58, [bitmax], page 57.

The bitwise ’not’ operator is a unary operator that performs a logical negation of each of the bits of the value. For this to make sense, the mask against which the value is negated must be defined. Octave’s bitwise ’not’ operator is bitcmp.

[Function File] bitcmp (A, k)

Return the k-bit complement of integers in A. If k is omitted k = log2 (bitmax) + 1 is assumed.

     bitcmp (7,4)
       ⇒ 8
     dec2bin (11)
       ⇒ 1011
     dec2bin (bitcmp (11, 6))
       ⇒ 110100

See also: [bitand], page 58, [bitor], page 58, [bitxor], page 58, [bitset], page 57, [bitget], page 57, [bitcmp], page 58, [bitshift], page 58, [bitmax], page 57.

Octave also includes the ability to left-shift and right-shift values bitwise.

[Built-in Function] bitshift (a, k) [Built-in Function] bitshift (a, k, n)

Return a k bit shift of n-digit unsigned integers in a. A positive k leads to a left shift; A negative value to a right shift. If n is omitted it defaults to log2(bitmax)+1. n must be in the range [1,log2(bitmax)+1] usually [1,33].

          bitshift (eye (3), 1)

⇒

          2 0 0 0 2 0 0 0 2

          bitshift (10, [-2, -1, 0, 1, 2])
          ⇒ 2   5  10  20  40

See also: [bitand], page 58, [bitor], page 58, [bitxor], page 58, [bitset], page 57, [bitget], page 57, [bitcmp], page 58, [bitmax], page 57.

Bits that are shifted out of either end of the value are lost. Octave also uses arithmetic shifts, where the sign bit of the value is kept during a right shift. For example:

     bitshift (-10, -1)
     ⇒ -5
     bitshift (int8 (-1), -1)
     ⇒ -1

Note that bitshift (int8 (-1), -1) is -1 since the bit representation of -1 in the int8

data type is [1, 1, 1, 1, 1, 1, 1, 1].

4.6 Logical Values

Octave has built-in support for logical values, i.e., variables that are either true or false. When comparing two variables, the result will be a logical value whose value depends on whether or not the comparison is true.

The basic logical operations are &, |, and !, which correspond to “Logical And”, “Logical Or”, and “Logical Negation”. These operations all follow the usual rules of logic.

It is also possible to use logical values as part of standard numerical calculations. In this case true is converted to 1, and false to 0, both represented using double precision floating point numbers. So, the result of true*22 - false/6 is 22.

Logical values can also be used to index matrices and cell arrays. When indexing with a logical array the result will be a vector containing the values corresponding to true parts of the logical array. The following example illustrates this.

     data = [ 1, 2; 3, 4 ];
     idx = (data <= 2);
     data(idx)
          ⇒ ans = [ 1; 2 ]

Instead of creating the idx array it is possible to replace data(idx) with data( data <= 2 ) in the above code.

Logical values can also be constructed by casting numeric objects to logical values, or by using the true or false functions.

[Built-in Function] logical (x)

Convert the numeric object x to logical type. Any non-zero values will be converted to true (1) while zero values will be converted to false (0). The non-numeric value NaN cannot be converted and will produce an error. Compatibility Note: Octave accepts complex values as input, whereas matlab issues an error.

See also: [double], page 47, [single], page 53, [char], page 69.

[Built-in Function] true (x) [Built-in Function] true (n, m) [Built-in Function] true (n, m, k, …)

Return a matrix or N-dimensional array whose elements are all logical 1. If invoked with a single scalar integer argument, return a square matrix of the specified size. If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with given dimensions.

See also: [false], page 60.

[Built-in Function] false (x) [Built-in Function] false (n, m) [Built-in Function] false (n, m, k, …)

Return a matrix or N-dimensional array whose elements are all logical 0. If invoked with a single scalar integer argument, return a square matrix of the specified size. If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with given dimensions.

See also: [true], page 60.

4.7 Promotion and Demotion of Data Types

Many operators and functions can work with mixed data types. For example,

uint8 (1) + 1 ⇒ 2

where the above operator works with an 8-bit integer and a double precision value and returns an 8-bit integer value. Note that the type is demoted to an 8-bit integer, rather than promoted to a double precision value as might be expected. The reason is that if Octave promoted values in expressions like the above with all numerical constants would need to be explicitly cast to the appropriate data type like

uint8 (1) + uint8 (1)
    ⇒ 2

which becomes difficult for the user to apply uniformly and might allow hard to find bugs to be introduced. The same applies to single precision values where a mixed operation such as

single (1) + 1 ⇒ 2

returns a single precision value. The mixed operations that are valid and their returned data types are

Mixed Operation  Result
double OP single  single
double OP integer  integer
double OP char  double
double OP logical  double
single OP integer  integer
single OP char  single
single OP logical  single

The same logic applies to functions with mixed arguments such as

min (single (1), 0)
   ⇒ 0

where the returned value is single precision.

In the case of mixed type indexed assignments, the type is not changed. For example,

x = ones (2, 2);
x(1, 1) = single (2)
   ⇒ x = 2   1
          1   1

where x remains of the double precision type.

4.8 Predicates for Numeric Objects

Since the type of a variable may change during the execution of a program, it can be necessary to do type checking at run-time. Doing this also allows you to change the behavior of a function depending on the type of the input. As an example, this naive implementation of abs returns the absolute value of the input if it is a real number, and the length of the input if it is a complex number.

function a = abs (x)
  if (isreal (x))
    a = sign (x) .* x;
  elseif (iscomplex (x))
    a = sqrt (real(x).^2 + imag(x).^2);
  endif
endfunction

The following functions are available for determining the type of a variable.

[Built-in Function] isnumeric (x)

Return true if x is a numeric object, i.e., an integer, real, or complex array. Logical and character arrays are not considered to be numeric.

See also: [isinteger], page 54, [isfloat], page 61, [isreal], page 62, [iscomplex], page 62, [islogical], page 63, [ischar], page 66, [iscell], page 111, [isstruct], page 105, [isa], page 39.

[Built-in Function] isfloat (x)

Return true if x is a floating-point numeric object. Objects of class double or single are floating-point objects.

See also: [isinteger], page 54, [ischar], page 66, [islogical], page 63, [isnumeric], page 61, [isa], page 39.

[Built-in Function] isreal (x)

Return true if x is a non-complex matrix or scalar. For compatibility with matlab, this includes logical and character matrices.

See also: [iscomplex], page 62, [isnumeric], page 61, [isa], page 39.

[Built-in Function] iscomplex (x)

Return true if x is a complex-valued numeric object.

See also: [isreal], page 62, [isnumeric], page 61, [islogical], page 63, [ischar], page 66, [isfloat], page 61, [isa], page 39.

[Built-in Function] ismatrix (a)

Return true if a is a numeric, logical, or character matrix. Scalars (1x1 matrices) and vectors (1xN or Nx1 matrices) are subsets of the more general N-dimensional matrix and ismatrix will return true for these objects as well.

See also: [isscalar], page 62, [isvector], page 62, [iscell], page 111, [isstruct], page 105, [issparse], page 493, [isa], page 39.

[Function File] isvector (x)

Return true if x is a vector. A vector is a 2-D array where one of the dimensions is equal to 1. As a consequence a 1x1 array, or scalar, is also a vector.

See also: [isscalar], page 62, [ismatrix], page 62, [size], page 44, [rows], page 43, [columns], page 43, [length], page 44.

[Function File] isrow (x)

Return true if x is a row vector.

See also: [iscolumn], page 62, [isscalar], page 62, [isvector], page 62, [ismatrix], page 62.

[Function File] iscolumn (x)

Return true if x is a column vector.

See also: [isrow], page 62, [isscalar], page 62, [isvector], page 62, [ismatrix], page 62.

[Function File] isscalar (x)

Return true if x is a scalar.

See also: [isvector], page 62, [ismatrix], page 62.

[Function File] issquare (x)

Return true if x is a square matrix.

See also: [isscalar], page 62, [isvector], page 62, [ismatrix], page 62, [size], page 44.

[Function File] issymmetric (x) [Function File] issymmetric (x, tol)

Return true if x is a symmetric matrix within the tolerance specified by tol. The default tolerance is zero (uses faster code). Matrix x is considered symmetric if norm

(x - x.', Inf) / norm (x, Inf) < tol.

See also: [ishermitian], page 63, [isdefinite], page 63.

[Function File] ishermitian (x) [Function File] ishermitian (x, tol)

Return true if x is Hermitian within the tolerance specified by tol. The default tolerance is zero (uses faster code). Matrix x is considered symmetric if norm (x -

x', Inf) / norm (x, Inf) < tol.

See also: [issymmetric], page 62, [isdefinite], page 63.

[Function File] isdefinite (x) [Function File] isdefinite (x, tol)

Return 1 if x is symmetric positive definite within the tolerance specified by tol or 0 if x is symmetric positive semidefinite. Otherwise, return -1. If tol is omitted, use a

tolerance of 100 * eps * norm (x, "fro")

See also: [issymmetric], page 62, [ishermitian], page 63.

[Built-in Function] islogical (x) [Built-in Function] isbool (x)

Return true if x is a logical object.

See also: [isfloat], page 61, [isinteger], page 54, [ischar], page 66, [isnumeric], page 61, [isa], page 39.

[Function File] isprime (x)

Return a logical array which is true where the elements of x are prime numbers and false where they are not. If the maximum value in x is very large, then you should be using special purpose factorization code.

     isprime (1:6) ⇒ [0, 1, 1, 0, 1, 0]

See also: [primes], page 421, [factor], page 420, [gcd], page 420, [lcm], page 420.

If instead of knowing properties of variables, you wish to know which variables are defined and to gather other information about the workspace itself, see Section 7.3 [Status of Variables], page 126.

5 Strings

A string constant consists of a sequence of characters enclosed in either double-quote or single-quote marks. For example, both of the following expressions

     "parrot" 'parrot'

represent the string whose contents are ‘parrot’. Strings in Octave can be of any length.

Since the single-quote mark is also used for the transpose operator (see Section 8.3 [Arithmetic Ops], page 137) but double-quote marks have no other purpose in Octave, it is best to use double-quote marks to denote strings.

Strings can be concatenated using the notation for defining matrices. For example, the expression

     [ "foo" , "bar" , "baz" ]

produces the string whose contents are ‘foobarbaz’. See Chapter 4 [Numeric Data Types], page 47, for more information about creating matrices.

5.1 Escape Sequences in String Constants

In double-quoted strings, the backslash character is used to introduce escape sequences that represent other characters. For example, ‘\n’ embeds a newline character in a double-quoted string and ‘\"’ embeds a double quote character. In single-quoted strings, backslash is not a special character. Here is an example showing the difference:

     toascii ("\n")
         ⇒ 10
     toascii ('\n')
         ⇒ [ 92 110 ]

Here is a table of all the escape sequences used in Octave (within double quoted strings). They are the same as those used in the C programming language. \\ Represents a literal backslash, ‘\’. \" Represents a literal double-quote character, ‘"’. \' Represents a literal single-quote character, ‘'’. \0 Represents the null character, control-@, ASCII code 0. \a Represents the “alert” character, control-g, ASCII code 7. \b Represents a backspace, control-h, ASCII code 8. \f Represents a formfeed, control-l, ASCII code 12. \n Represents a newline, control-j, ASCII code 10. \r Represents a carriage return, control-m, ASCII code 13. \t Represents a horizontal tab, control-i, ASCII code 9. \v Represents a vertical tab, control-k, ASCII code 11. \nnn Represents the octal value nnn, where nnn are one to three digits between 0

and 7. For example, the code for the ASCII ESC (escape) character is ‘\033’.

\xhh... Represents the hexadecimal value hh, where hh are hexadecimal digits (‘0’ through ‘9’ and either ‘A’ through ‘F’ or ‘a’ through ‘f’). Like the same construct in ansi C, the escape sequence continues until the first non-hexadecimal digit is seen. However, using more than two hexadecimal digits produces undefined results.

In a single-quoted string there is only one escape sequence: you may insert a single quote character using two single quote characters in succession. For example,

     'I can''t escape'
         ⇒ I can't escape

In scripts the two different string types can be distinguished if necessary by using is_

dq_string and is_sq_string.

[Built-in Function] is_dq_string (x)

Return true if x is a double-quoted character string.

See also: [is sq string], page 66, [ischar], page 66.

[Built-in Function] is_sq_string (x)

Return true if x is a single-quoted character string.

See also: [is dq string], page 66, [ischar], page 66.

5.2 Character Arrays

The string representation used by Octave is an array of characters, so internally the string "dddddddddd" is actually a row vector of length 10 containing the value 100 in all places (100 is the ASCII code of "d"). This lends itself to the obvious generalization to character matrices. Using a matrix of characters, it is possible to represent a collection of same-length strings in one variable. The convention used in Octave is that each row in a character matrix is a separate string, but letting each column represent a string is equally possible.

The easiest way to create a character matrix is to put several strings together into a matrix.

     collection = [ "String #1"; "String #2" ];

This creates a 2-by-9 character matrix.

The function ischar can be used to test if an object is a character matrix.

[Built-in Function] ischar (x)

Return true if x is a character array.

See also: [isfloat], page 61, [isinteger], page 54, [islogical], page 63, [isnumeric], page 61, [iscellstr], page 117, [isa], page 39.

To test if an object is a string (i.e., a character vector and not a character matrix) you can use the ischar function in combination with the isvector function as in the following example:

ischar (collection)
     ⇒ 1
ischar (collection) && isvector (collection)
     ⇒ 0
ischar ("my string") && isvector ("my string")
     ⇒ 1

One relevant question is, what happens when a character matrix is created from strings of different length. The answer is that Octave puts blank characters at the end of strings shorter than the longest string. It is possible to use a different character than the blank character using the string_fill_char function.

[Built-in Function] val = string_fill_char ()
[Built-in Function] old_val = string_fill_char (new_val)

[Built-in Function] string_fill_char (new_val, "local")

Query or set the internal variable used to pad all rows of a character matrix to the same length; It must be a single character. The default value is " " (a single space). For example:

     string_fill_char ("X");
     [ "these"; "are"; "strings" ]
           ⇒ "theseXX"
               "areXXXX"
               "strings"

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

This shows a problem with character matrices. It simply isn’t possible to represent strings of different lengths. The solution is to use a cell array of strings, which is described in Section 6.2.4 [Cell Arrays of Strings], page 116.

5.3 Creating Strings

The easiest way to create a string is, as illustrated in the introduction, to enclose a text in double-quotes or single-quotes. It is however possible to create a string without actually writing a text. The function blanks creates a string of a given length consisting only of blank characters (ASCII code 32).

[Function File] blanks (n)

Return a string of n blanks, for example:

     blanks (10);
     whos ans

⇒

           Attr Name        Size                     Bytes  Class
           ==== ====        ====                     =====  =====
                ans         1x10                        10  char

See also: [repmat], page 392.

5.3.1 Concatenating Strings

Strings can be concatenated using matrix notation (see Chapter 5 [Strings], page 65, Section 5.2 [Character Arrays], page 66) which is often the most natural method. For example:

     fullname = [fname ".txt"];
     email = ["<" user "@" domain ">"];

In each case it is easy to see what the final string will look like. This method is also the most efficient. When using matrix concatenation the parser immediately begins joining the strings without having to process the overhead of a function call and the input validation of the associated function.

Nevertheless, there are several other functions for concatenating string objects which can be useful in specific circumstances: char, strvcat, strcat, and cstrcat. Finally, the general purpose concatenation functions can be used: see [cat], page 384, [horzcat], page 384, and [vertcat], page 385.

• All string concatenation functions except cstrcat convert numerical input into character data by taking the corresponding ASCII character for each element, as in the following example:

         char ([98, 97, 110, 97, 110, 97])
            ⇒ banana

• char and strvcat concatenate vertically, while strcat and cstrcat concatenate horizontally. For example:

         char ("an apple", "two pears")
             ⇒ an apple
                two pears
         strcat ("oc", "tave", " is", " good", " for you")
              ⇒ octave is good for you

• char generates an empty row in the output for each empty string in the input. strvcat, on the other hand, eliminates empty strings.

         char ("orange", "green", "", "red")
             ⇒ orange
                green
                red
         strvcat ("orange", "green", "", "red")
             ⇒ orange
                green
                red

• All string concatenation functions except cstrcat also accept cell array data (see Section 6.2 [Cell Arrays], page 110). char and strvcat convert cell arrays into character arrays, while strcat concatenates within the cells of the cell arrays:

    char ({"red", "green", "", "blue"})
         ⇒ red
            green
            blue
    strcat ({"abc"; "ghi"}, {"def"; "jkl"})

⇒

            {
              [1,1] = abcdef
              [2,1] = ghijkl
            }

• strcat removes trailing white space in the arguments (except within cell arrays), while cstrcat leaves white space untouched. Both kinds of behavior can be useful as can be seen in the examples:

    strcat (["dir1";"directory2"], ["/";"/"], ["file1";"file2"])
         ⇒ dir1/file1
            directory2/file2
    cstrcat (["thirteen apples"; "a banana"], [" 5$";" 1$"])
          ⇒ thirteen apples 5$
             a banana        1$

Note that in the above example for cstrcat, the white space originates from the internal representation of the strings in a string array (see Section 5.2 [Character Arrays], page 66).

[Built-in Function] char (x) [Built-in Function] char (x, …) [Built-in Function] char (s1, s2, …) [Built-in Function] char (cell_array)

Create a string array from one or more numeric matrices, character matrices, or cell arrays. Arguments are concatenated vertically. The returned values are padded with blanks as needed to make each row of the string array have the same length. Empty input strings are significant and will concatenated in the output. For numerical input, each element is converted to the corresponding ASCII character. A range error results if an input is outside the ASCII range (0-255). For cell arrays, each element is concatenated separately. Cell arrays converted through char can mostly be converted back with cellstr. For example:

     char ([97, 98, 99], "", {"98", "99", 100}, "str1", ["ha", "lf"])
        ⇒ ["abc    "
            "       "
            "98     "
            "99     "
            "d      "
            "str1   "
            "half   "]

See also: [strvcat], page 70, [cellstr], page 117.

[Built-in Function] strvcat (x) [Built-in Function] strvcat (x, …) [Built-in Function] strvcat (s1, s2, …) [Built-in Function] strvcat (cell_array)

Create a character array from one or more numeric matrices, character matrices, or cell arrays. Arguments are concatenated vertically. The returned values are padded with blanks as needed to make each row of the string array have the same length. Unlike char, empty strings are removed and will not appear in the output.

For numerical input, each element is converted to the corresponding ASCII character. A range error results if an input is outside the ASCII range (0-255).

For cell arrays, each element is concatenated separately. Cell arrays converted through strvcat can mostly be converted back with cellstr. For example:

     strvcat ([97, 98, 99], "", {"98", "99", 100}, "str1", ["ha", "lf"])
           ⇒ ["abc    "
               "98     "
               "99     "
               "d      "
               "str1   "
               "half   "]

See also: [char], page 69, [strcat], page 70, [cstrcat], page 71.

[Function File] strcat (s1, s2, …)

Return a string containing all the arguments concatenated horizontally. If the arguments are cell strings, strcat returns a cell string with the individual cells concatenated. For numerical input, each element is converted to the corresponding ASCII character. Trailing white space for any character string input is eliminated before the strings are concatenated. Note that cell string values do not have whitespace trimmed.

For example:

     strcat ("|", " leading space is preserved", "|")
         ⇒ | leading space is preserved|
     strcat ("|", "trailing space is eliminated ", "|")
         ⇒ |trailing space is eliminated|
     strcat ("homogeneous space |", "  ", "| is also eliminated")
         ⇒ homogeneous space || is also eliminated
     s = [ "ab"; "cde" ];
     strcat (s, s, s)

⇒

             "ababab   "
             "cdecdecde"

     s = { "ab"; "cd " };
     strcat (s, s, s)

⇒

             {
               [1,1] = ababab
               [2,1] = cd cd cd
             }

See also: [cstrcat], page 71, [char], page 69, [strvcat], page 70.

[Function File] cstrcat (s1, s2, …)

Return a string containing all the arguments concatenated horizontally. Trailing white space is preserved. For example:

     cstrcat ("ab   ", "cd")
           ⇒ "ab   cd"
     s = [ "ab"; "cde" ];
     cstrcat (s, s, s)
           ⇒ "ab ab ab "
              "cdecdecde"

See also: [strcat], page 70, [char], page 69, [strvcat], page 70.

5.3.2 Converting Numerical Data to Strings

Apart from the string concatenation functions (see Section 5.3.1 [Concatenating Strings], page 68) which cast numerical data to the corresponding ASCII characters, there are several functions that format numerical data as strings. mat2str and num2str convert real or complex matrices, while int2str converts integer matrices. int2str takes the real part of complex values and round fractional values to integer. A more flexible way to format numerical data as strings is the sprintf function (see Section 14.2.4 [Formatted Output], page 243, [sprintf], page 244).

[Function File] s = mat2str (x, n) [Function File] s = mat2str (x, n, "class")

Format real, complex, and logical matrices as strings. The returned string may be used to reconstruct the original matrix by using the eval function.

The precision of the values is given by n. If n is a scalar then both real and imaginary parts of the matrix are printed to the same precision. Otherwise n(1) defines the precision of the real part and n(2) defines the precision of the imaginary part. The default for n is 15.

If the argument "class" is given then the class of x is included in the string in such a way that eval will result in the construction of a matrix of the same class.

mat2str ([ -1/3 + i/7; 1/3 - i/7 ], [4 2])
     ⇒ "[-0.3333+0.14i;0.3333-0.14i]"
mat2str ([ -1/3 +i/7; 1/3 -i/7 ], [4 2])
     ⇒ "[-0.3333+0i 0+0.14i;0.3333+0i -0-0.14i]"
mat2str (int16 ([1 -1]), "class")
     ⇒ "int16([1 -1])"
mat2str (logical (eye (2)))
     ⇒ "[true false;false true]"
isequal (x, eval (mat2str (x)))
     ⇒ 1

See also: [sprintf], page 244, [num2str], page 72, [int2str], page 73.

[Function File] num2str (x)

[Function File] num2str (x, precision)

[Function File] num2str (x, format)

Convert a number (or array) to a string (or a character array). The optional second argument may either give the number of significant digits (precision) to be used in the output or a format template string (format) as in sprintf (see Section 14.2.4 [Formatted Output], page 243). num2str can also handle complex numbers. Examples:

num2str (123.456)
     ⇒ "123.46"
num2str (123.456, 4)
     ⇒ "123.5"
s = num2str ([1, 1.34; 3, 3.56], "%5.1f")
     ⇒ s =
        1.0  1.3
        3.0  3.6
whos s

⇒

      Attr Name        Size                     Bytes  Class
      ==== ====        ====                     =====  =====
           s           2x8                         16  char
num2str (1.234 + 27.3i)
     ⇒ "1.234+27.3i"

Notes: For matlab compatibility, leading spaces are stripped before returning the string. The num2str function is not very flexible. For better control over the results, use sprintf (see Section 14.2.4 [Formatted Output], page 243).

For complex x, the format string may only contain one output conversion specification and nothing else. Otherwise, results will be unpredictable.

See also: [sprintf], page 244, [int2str], page 73, [mat2str], page 71.

[Function File] int2str (n)

Convert an integer (or array of integers) to a string (or a character array).

     int2str (123)
          ⇒ "123"
     s = int2str ([1, 2, 3; 4, 5, 6])
          ⇒ s =
             1  2  3
             4  5  6
     whos s

⇒

           Attr Name        Size                     Bytes  Class
           ==== ====        ====                     =====  =====
                s           2x7                         14  char

This function is not very flexible. For better control over the results, use sprintf (see Section 14.2.4 [Formatted Output], page 243).

See also: [sprintf], page 244, [num2str], page 72, [mat2str], page 71.

5.4 Comparing Strings

Since a string is a character array, comparisons between strings work element by element as the following example shows:

GNU = "GNU's Not UNIX";
spaces = (GNU == " ")
     ⇒ spaces =
       0   0   0   0   0   1   0   0   0   1   0   0   0   0

To determine if two strings are identical it is necessary to use the strcmp function. It compares complete strings and is case sensitive. strncmp compares only the first N characters (with N given as a parameter). strcmpi and strncmpi are the corresponding functions for case-insensitive comparison.

[Built-in Function] strcmp (s1, s2)

Return 1 if the character strings s1 and s2 are the same, and 0 otherwise. If either s1 or s2 is a cell array of strings, then an array of the same size is returned, containing the values described above for every member of the cell array. The other argument may also be a cell array of strings (of the same size or with only one element), char matrix or character string. Caution: For compatibility with matlab, Octave’s strcmp function returns 1 if the character strings are equal, and 0 otherwise. This is just the opposite of the corresponding C library function.

See also: [strcmpi], page 74, [strncmp], page 74, [strncmpi], page 74.

[Built-in Function] strncmp (s1, s2, n)

Return 1 if the first n characters of strings s1 and s2 are the same, and 0 otherwise.

     strncmp ("abce", "abcd", 3)
           ⇒ 1

If either s1 or s2 is a cell array of strings, then an array of the same size is returned, containing the values described above for every member of the cell array. The other argument may also be a cell array of strings (of the same size or with only one element), char matrix or character string.

     strncmp ("abce", {"abcd", "bca", "abc"}, 3)
          ⇒ [1, 0, 1]

Caution: For compatibility with matlab, Octave’s strncmp function returns 1 if the character strings are equal, and 0 otherwise. This is just the opposite of the corresponding C library function.

See also: [strncmpi], page 74, [strcmp], page 73, [strcmpi], page 74.

[Built-in Function] strcmpi (s1, s2)

Return 1 if the character strings s1 and s2 are the same, disregarding case of alphabetic characters, and 0 otherwise. If either s1 or s2 is a cell array of strings, then an array of the same size is returned, containing the values described above for every member of the cell array. The other argument may also be a cell array of strings (of the same size or with only one element), char matrix or character string. Caution: For compatibility with matlab, Octave’s strcmp function returns 1 if the character strings are equal, and 0 otherwise. This is just the opposite of the corresponding C library function. Caution: National alphabets are not supported.

See also: [strcmp], page 73, [strncmp], page 74, [strncmpi], page 74.

[Built-in Function] strncmpi (s1, s2, n)

Return 1 if the first n character of s1 and s2 are the same, disregarding case of alphabetic characters, and 0 otherwise. If either s1 or s2 is a cell array of strings, then an array of the same size is returned, containing the values described above for every member of the cell array. The other argument may also be a cell array of strings (of the same size or with only one element), char matrix or character string. Caution: For compatibility with matlab, Octave’s strncmpi function returns 1 if the character strings are equal, and 0 otherwise. This is just the opposite of the corresponding C library function. Caution: National alphabets are not supported.

See also: [strncmp], page 74, [strcmp], page 73, [strcmpi], page 74.

[Function File] validstr = validatestring (str, strarray)
[Function File] validstr = validatestring (str, strarray, funcname)
[Function File] validstr = validatestring (str, strarray, funcname,
   varname)

[Function File] validstr = validatestring (. . ., position)

Verify that str is an element, or substring of an element, in strarray.

When str is a character string to be tested, and strarray is a cellstr of valid values, then validstr will be the validated form of str where validation is defined as str being a member or substring of validstr. This is useful for both verifying and expanding short options, such as "r", to their longer forms, such as "red". If str is a substring of validstr, and there are multiple matches, the shortest match will be returned if all matches are substrings of each other. Otherwise, an error will be raised because the expansion of str is ambiguous. All comparisons are case insensitive.

The additional inputs funcname, varname, and position are optional and will make any generated validation error message more specific.

Examples:

      validatestring ("r", {"red", "green", "blue"})
      ⇒ "red"
      validatestring ("b", {"red", "green", "blue", "black"})
      ⇒ error: validatestring: multiple unique matches were found for 'b':
         blue, black

See also: [strcmp], page 73, [strcmpi], page 74.

5.5 Manipulating Strings

Octave supports a wide range of functions for manipulating strings. Since a string is just a matrix, simple manipulations can be accomplished using standard operators. The following example shows how to replace all blank characters with underscores.

quote = ...
  "First things first, but not necessarily in that order";
quote( quote == " " ) = "_"
⇒ quote =
    First_things_first,_but_not_necessarily_in_that_order

For more complex manipulations, such as searching, replacing, and general regular expressions, the following functions come with Octave.

[Function File] deblank (s)

Remove trailing whitespace and nulls from s. If s is a matrix, deblank trims each row to the length of longest string. If s is a cell array of strings, operate recursively on each string element.

Examples:

     deblank ("    abc  ")
          ⇒ "    abc"
     deblank ([" abc   "; "   def   "])
          ⇒ [" abc  " ; "   def"]

See also: [strtrim], page 76.

[Function File] strtrim (s)

Remove leading and trailing whitespace from s. If s is a matrix, strtrim trims each row to the length of longest string. If s is a cell array of strings, operate recursively on each string element. For example:

     strtrim ("    abc  ")
          ⇒ "abc"
     strtrim ([" abc   "; "   def   "])
          ⇒ ["abc  "  ; "  def"]

See also: [deblank], page 75.

[Function File] strtrunc (s, n)

Truncate the character string s to length n. If s is a character matrix, then the number of columns is adjusted. If s is a cell array of strings, then the operation is performed on each cell element and the new cell array is returned.

[Function File] findstr (s, t) [Function File] findstr (s, t, overlap)

Return the vector of all positions in the longer of the two strings s and t where an occurrence of the shorter of the two starts. If the optional argument overlap is true, the returned vector can include overlapping positions (this is the default). For example:

     findstr ("ababab", "a")
          ⇒ [1, 3, 5];
     findstr ("abababa", "aba", 0)
          ⇒ [1, 5]

Caution: findstr is scheduled for deprecation. Use strfind in all new code.

See also: [strfind], page 77, [strmatch], page 78, [strcmp], page 73, [strncmp], page 74, [strcmpi], page 74, [strncmpi], page 74, [find], page 381.

[Function File] idx = strchr (str, chars)
[Function File] idx = strchr (str, chars, n)
[Function File] idx = strchr (str, chars, n, direction)

[Function File] [i, j] = strchr (…)

Search for the string str for occurrences of characters from the set chars. The return value(s), as well as the n and direction arguments behave identically as in find. This will be faster than using regexp in most cases.

See also: [find], page 381.

[Function File] index (s, t) [Function File] index (s, t, direction)

Return the position of the first occurrence of the string t in the string s, or 0 if no occurrence is found. s may also be a string array or cell array of strings. For example:

     index ("Teststring", "t")
         ⇒ 4

If direction is "first", return the first element found. If direction is "last", return the last element found.

See also: [find], page 381, [rindex], page 77.

[Function File] rindex (s, t)

Return the position of the last occurrence of the character string t in the character string s, or 0 if no occurrence is found. s may also be a string array or cell array of strings. For example:

     rindex ("Teststring", "t")
          ⇒ 6

The rindex function is equivalent to index with direction set to "last".

See also: [find], page 381, [index], page 76.

[Built-in Function] idx = strfind (str, pattern)
[Built-in Function] idx = strfind (cellstr, pattern)

[Built-in Function] idx = strfind (…, "overlaps", val)

Search for pattern in the string str and return the starting index of every such occurrence in the vector idx. If there is no such occurrence, or if pattern is longer than str, then idx is the empty array []. The optional argument "overlaps" determines whether the pattern can match at every position in str (true), or only for unique occurrences of the complete pattern (false). The default is true. If a cell array of strings cellstr is specified then idx is a cell array of vectors, as specified above. Examples:

     strfind ("abababa", "aba")
          ⇒ [1, 3, 5]
     strfind ("abababa", "aba", "overlaps", false)
          ⇒ [1, 5]
     strfind ({"abababa", "bebebe", "ab"}, "aba")

⇒

             {
               [1,1] =
                  1   3   5
               [1,2] = [](1x0)
               [1,3] = [](1x0)
             }

See also: [findstr], page 76, [strmatch], page 78, [regexp], page 84, [regexpi], page 87, [find], page 381.

[Function File] str = strjoin (cstr)

[Function File] str = strjoin (cstr, delimiter)

Join the elements of the cell string array, cstr, into a single string.

If no delimiter is specified, the elements of cstr separated by a space.

If delimiter is specified as a string, the cell string array is joined using the string. Escape sequences are supported.

If delimiter is a cell string array whose length is one less than cstr, then the elements of cstr are joined by interleaving the cell string elements of delimiter. Escape sequences are not supported.

     strjoin ({'Octave','Scilab','Lush','Yorick'}, '*')
           ⇒ 'Octave*Scilab*Lush*Yorick'

See also: strsplit.

[Function File] strmatch (s, A) [Function File] strmatch (s, A, "exact")

Return indices of entries of A which begin with the string s. The second argument A must be a string, character matrix, or a cell array of strings. If the third argument "exact" is not given, then s only needs to match A up to the length of s. Trailing spaces and nulls in s and A are ignored when matching.

For example:

     strmatch ("apple", "apple juice")
          ⇒ 1
     strmatch ("apple", ["apple  "; "apple juice"; "an apple"])
          ⇒ [1; 2]
     strmatch ("apple", ["apple  "; "apple juice"; "an apple"], "exact")
          ⇒ [1]

Caution: strmatch is scheduled for deprecation. Use strncmp (normal case), or strcmp ("exact" case), or regexp in all new code.

See also: [strfind], page 77, [findstr], page 76, [strcmp], page 73, [strncmp], page 74, [strcmpi], page 74, [strncmpi], page 74, [find], page 381.

[Function File] [tok, rem] = strtok (str)
[Function File] [tok, rem] = strtok (str, delim)

Find all characters in the string str up to, but not including, the first character which is in the string delim. If rem is requested, it contains the remainder of the string, starting at the first delimiter. Leading delimiters are ignored. If delim is not specified, whitespace is assumed. str may also be a cell array of strings in which case the function executes on every individual string and returns a cell array of tokens and remainders.

Examples:

strtok ("this is the life")
⇒ "this"
[tok, rem] = strtok ("14*27+31", "+-*/")

⇒

tok = 14
rem = *27+31

See also: [index], page 76, [strsplit], page 79, [strchr], page 76, [isspace], page 94.

[Function File] [cstr] = strsplit (s)
[Function File] [cstr] = strsplit (s, del)
[Function File] [cstr] = strsplit (. . ., name, value)
[Function File] [cstr, matches] = strsplit (. . .)

Split the string s using the delimiters specified by del and return a cell string array of substrings. If a delimiter is not specified the string, s, is split at whitespace. The delimiter, del may be a string, a scalar cell string, or cell string array. By default, consecutive delimiters in the input string s are collapsed into one. The second output, matches, returns the delimiters which were matched in the original string. Example:

strsplit ("a b c")

⇒

{
[1,1] = a
[1,2] = b
[1,3] = c
}
strsplit ("a,b,c", ",")

⇒

{
[1,1] = a
[1,2] = b
[1,3] = c
}
strsplit ("a foo b,bar c", {"\s", "foo", "bar"})

⇒

{
[1,1] = a
[1,2] = b
[1,3] = c
}
strsplit ("a,,b, c", {",", " "}, false)

⇒

{
[1,1] = a
[1,2] =
[1,3] = b
[1,4] =
[1,5] = c
}

Supported name/value pair arguments are; • collapsedelimiters may take the value of true or false with the default being false.

• delimitertype may take the value of simple or regularexpression. The default is delimitertype is simple.

Example:

strsplit ("a foo b,bar c", ",|\\s|foo|bar", "delimitertype", "regularexpression")

⇒

{
[1,1] = a
[1,2] = b
[1,3] = c
}
strsplit ("a,,b, c", "[, ]", false, "delimitertype", "regularexpression")

⇒

{
[1,1] = a
[1,2] =
[1,3] = b
[1,4] =
[1,5] = c
}
strsplit ("a,\t,b, c", {',', '\s'}, "delimitertype", "regularexpression")

⇒

{
[1,1] = a
[1,2] = b
[1,3] = c
}
strsplit ("a,\t,b, c", {',', ' ', '\t'}, "collapsedelimiters", false)

⇒

{
[1,1] = a
[1,2] =
[1,3] =
[1,4] = b

                 [1,5] =
                 [1,6] = c
               }

See also: [ostrsplit], page 81, [strjoin], page 78, [strtok], page 78, [regexp], page 84.

[Function File] [cstr] = ostrsplit (s, sep)
[Function File] [cstr] = ostrsplit (s, sep, strip_empty)

Split the string s using one or more separators sep and return a cell array of strings. Consecutive separators and separators at boundaries result in empty strings, unless strip empty is true. The default value of strip empty is false. 2-D character arrays are split at separators and at the original column boundaries. Example:

     ostrsplit ("a,b,c", ",")

⇒

               {
                 [1,1] = a
                 [1,2] = b
                 [1,3] = c
               }
     ostrsplit (["a,b" ; "cde"], ",")

⇒

               {
                 [1,1] = a
                 [1,2] = b
                 [1,3] = cde
               }

See also: [strsplit], page 79, [strtok], page 78.

[Function File] [a, ...] = strread (str)
[Function File] [a, ...] = strread (str, format)
[Function File] [a, ...] = strread (str, format, format_repeat)
[Function File] [a, ...] = strread (str, format, prop1, value1, . . .)
[Function File] [a, ...] = strread (str, format, format_repeat, prop1,
   value1, . . .)

Read data from a string. The string str is split into words that are repeatedly matched to the specifiers in format. The first word is matched to the first specifier, the second to the second specifier and so forth. If there are more words than specifiers, the process is repeated until all words have been processed. The string format describes how the words in str should be parsed. It may contain any combination of the following specifiers:

%s The word is parsed as a string.

%f

%n The word is parsed as a number and converted to double.

%d

%u The word is parsed as a number and converted to int32.

%*', '%*f', '%*s

The word is skipped.

For %s and %d, %f, %n, %u and the associated %*s … specifiers an optional width can be specified as %Ns, etc. where N is an integer > 1. For %f, format specifiers like %N.Mf are allowed.

literals In addition the format may contain literal character strings; these will be skipped during reading.

Parsed word corresponding to the first specifier are returned in the first output argument and likewise for the rest of the specifiers. By default, format is "%f", meaning that numbers are read from str. This will do if str contains only numeric fields. For example, the string

     str = "\
     Bunny Bugs   5.5\n\
     Duck Daffy  -7.5e-5\n\
     Penguin Tux   6"

can be read using

     [a, b, c] = strread (str, "%s %s %f");

Optional numeric argument format repeat can be used for limiting the number of items read:

-1 (default) read all of the string until the end.

N Read N times nargout items. 0 (zero) is an acceptable value for format repeat.

The behavior of strread can be changed via property-value pairs. The following properties are recognized:

"commentstyle"

Parts of str are considered comments and will be skipped. value is the comment style and can be any of the following.

• "shell" Everything from # characters to the nearest end-of-line is skipped.

• "c" Everything between /* and */ is skipped.

• "c++" Everything from // characters to the nearest end-of-line is skipped.

• "matlab" Everything from % characters to the nearest end-of-line is skipped.

• user-supplied. Two options: (1) One string, or 1x1 cell string: Skip everything to the right of it; (2) 2x1 cell string array: Everything between the left and right strings is skipped.

"delimiter"

Any character in value will be used to split str into words (default value = any whitespace).

"emptyvalue":

Value to return for empty numeric values in non-whitespace delimited data. The default is NaN. When the data type does not support NaN (int32 for example), then default is zero.

"multipledelimsasone"

Treat a series of consecutive delimiters, without whitespace in between, as a single delimiter. Consecutive delimiter series need not be vertically

"aligned".
"treatasempty"

Treat single occurrences (surrounded by delimiters or whitespace) of the string(s) in value as missing values.

"returnonerror"

If value true (1, default), ignore read errors and return normally. If false (0), return an error.

"whitespace"

Any character in value will be interpreted as whitespace and trimmed; the string defining whitespace must be enclosed in double quotes for proper processing of special characters like \t. The default value for whitespace = " \b\r\n\t" (note the space). Unless whitespace is set to ” (empty) AND at least one "%s" format conversion specifier is supplied, a space is always part of whitespace.

When the number of words in str doesn’t match an exact multiple of the number of format conversion specifiers, strread’s behavior depends on the last character of str:

last character = "\n" Data columns are padded with empty fields or Nan so that all columns have equal length

last character is not "\n" Data columns are not padded; strread returns columns of unequal length

See also: [textscan], page 236, [textread], page 236, [load], page 232, [dlmread], page 235, [fscanf], page 248.

[Built-in Function] newstr = strrep (str, ptn, rep)
[Built-in Function] newstr = strrep (cellstr, ptn, rep)

[Built-in Function] newstr = strrep (…, "overlaps", val)

Replace all occurrences of the pattern ptn in the string str with the string rep and return the result. The optional argument "overlaps" determines whether the pattern can match at every position in str (true), or only for unique occurrences of the complete pattern (false). The default is true. s may also be a cell array of strings, in which case the replacement is done for each element and a cell array is returned. Example:

     strrep ("This is a test string", "is", "&%$")
         ⇒ "Th&%$ &%$ a test string"

See also: [regexprep], page 87, [strfind], page 77, [findstr], page 76.

[Function File] substr (s, offset) [Function File] substr (s, offset, len)

Return the substring of s which starts at character number offset and is len characters long. Position numbering for offsets begins with 1. If offset is negative, extraction starts that far from the end of the string. If len is omitted, the substring extends to the end of S. A negative value for len extracts to within len characters of the end of the string Examples:

     substr ("This is a test string", 6, 9)
          ⇒ "is a test"
     substr ("This is a test string", -11)
          ⇒ "test string"
     substr ("This is a test string", -11, -7)
          ⇒ "test"

This function is patterned after the equivalent function in Perl.

[Built-in Function] [s, e, te, m, t, nm, sp] = regexp (str, pat)

[Built-in Function] [...] = regexp (str, pat, "opt1", …)

Regular expression string matching. Search for pat in str and return the positions and substrings of any matches, or empty values if there are none. The matched pattern pat can include any of the standard regex operators, including:

. Match any character

* + ? {} Repetition operators, representing

* Match zero or more times

+ Match one or more times

? Match zero or one times

{n} Match exactly n times

{n,} Match n or more times

{m,n} Match between m and n times

[...] [^...]

List operators. The pattern will match any character listed between "[" and "]". If the first character is "^" then the pattern is inverted and any character except those listed between brackets will match. Escape sequences defined below can also be used inside list operators. For example, a template for a floating point number might be [-+.\d]+.

() (?:) Grouping operator. The first form, parentheses only, also creates a token. | Alternation operator. Match one of a choice of regular expressions. The

alternatives must be delimited by the grouping operator () above.

^$ Anchoring operators. Requires pattern to occur at the start (^) or end

($) of the string.

In addition, the following escaped characters have special meaning. \d Match any digit \D Match any non-digit \s Match any whitespace character \S Match any non-whitespace character \w Match any word character \W Match any non-word character \< Match the beginning of a word \> Match the end of a word \B Match within a word Implementation Note: For compatibility with matlab, ordinary escape sequences (e.g., "\n" => newline) are processed in pat regardless of whether pat has been defined within single quotes. Use a second backslash to stop interpolation of the escape sequence (e.g., "\\n") or use the regexptranslate function. The outputs of regexp default to the order given below s The start indices of each matching substring e The end indices of each matching substring te The extents of each matched token surrounded by (...) in pat m A cell array of the text of each match t A cell array of the text of each token matched nm A structure containing the text of each matched named token, with

the name being used as the fieldname. A named token is denoted by

          (?<name>...).

sp A cell array of the text not returned by match, i.e., what remains if you

split the string based on pat.

Particular output arguments, or the order of the output arguments, can be selected by additional opt arguments. These are strings and the correspondence between the output arguments and the optional argument are

                 'start'  s
                 'end'  e
                 'tokenExtents'  te
                 'match'  m

       'tokens'  t
       'names'  nm
       'split'  sp

Additional arguments are summarized below.

‘once’ Return only the first occurrence of the pattern.

‘matchcase’

Make the matching case sensitive. (default) Alternatively, use (?-i) in the pattern.

‘ignorecase’

Ignore case when matching the pattern to the string. Alternatively, use (?i) in the pattern.

‘stringanchors’

Match the anchor characters at the beginning and end of the string. (default) Alternatively, use (?-m) in the pattern.

‘lineanchors’

Match the anchor characters at the beginning and end of the line. Alternatively, use (?m) in the pattern.

‘dotall’ The pattern . matches all characters including the newline character. (default) Alternatively, use (?s) in the pattern.

‘dotexceptnewline’

The pattern . matches all characters except the newline character. Alternatively, use (?-s) in the pattern.

‘literalspacing’

All characters in the pattern, including whitespace, are significant and are used in pattern matching. (default) Alternatively, use (?-x) in the pattern.

‘freespacing’

The pattern may include arbitrary whitespace and also comments beginning with the character ‘#’. Alternatively, use (?x) in the pattern.

‘noemptymatch’

Zero-length matches are not returned. (default)

‘emptymatch’

Return zero-length matches.

regexp ('a', 'b*', 'emptymatch') returns [1 2] because there are

zero or more 'b' characters at positions 1 and end-of-string.

See also: [regexpi], page 87, [strfind], page 77, [regexprep], page 87.

[Built-in Function] [s, e, te, m, t, nm, sp] = regexpi (str, pat)

[Built-in Function] [...] = regexpi (str, pat, "opt1", …)

Case insensitive regular expression string matching. Search for pat in str and return the positions and substrings of any matches, or empty values if there are none. See [regexp], page 84, for details on the syntax of the search pattern.

See also: [regexp], page 84.

[Built-in Function] outstr = regexprep (string, pat, repstr)
[Built-in Function] outstr = regexprep (string, pat, repstr, "opt1", . . .)

Replace occurrences of pattern pat in string with repstr. The pattern is a regular expression as documented for regexp. See [regexp], page 84. The replacement string may contain $i, which substitutes for the ith set of parentheses in the match string. For example,

     regexprep ("Bill Dunn", '(\w+) (\w+)', '$2, $1')

returns "Dunn, Bill" Options in addition to those of regexp are

‘once’ Replace only the first occurrence of pat in the result.

‘warnings’

This option is present for compatibility but is ignored.

Implementation Note: For compatibility with matlab, ordinary escape sequences (e.g., "\n" => newline) are processed in both pat and repstr regardless of whether they were defined within single quotes. Use a second backslash to stop interpolation of the escape sequence (e.g., "\\n") or use the regexptranslate function.

See also: [regexp], page 84, [regexpi], page 87, [strrep], page 83.

[Function File] regexptranslate (op, s)

Translate a string for use in a regular expression. This may include either wildcard replacement or special character escaping. The behavior is controlled by op which can take the following values

"wildcard"

The wildcard characters ., *, and ? are replaced with wildcards that are appropriate for a regular expression. For example:

               regexptranslate ("wildcard", "*.m")
                    ⇒ ".*\.m"

"escape" The characters $.?[], that have special meaning for regular expressions

are escaped so that they are treated literally. For example:

               regexptranslate ("escape", "12.5")
                    ⇒ "12\.5"

See also: [regexp], page 84, [regexpi], page 87, [regexprep], page 87.

[Function File] untabify (t) [Function File] untabify (t, tw)

[Function File] untabify (t, tw, deblank)

Replace TAB characters in t, with spaces. The tab width is specified by tw, or defaults to eight. The input, t, may be either a 2-D character array, or a cell array of character strings. The output is the same class as the input. If the optional argument deblank is true, then the spaces will be removed from the end of the character data. The following example reads a file and writes an untabified version of the same file with trailing spaces stripped.

     fid = fopen ("tabbed_script.m");
     text = char (fread (fid, "uchar")');
     fclose (fid);
     fid = fopen ("untabified_script.m", "w");
     text = untabify (strsplit (text, "\n"), 8, true);
     fprintf (fid, "%s\n", text{:});
     fclose (fid);

See also: [strjust], page 91, [strsplit], page 79, [deblank], page 75.

5.6 String Conversions

Octave supports various kinds of conversions between strings and numbers. As an example, it is possible to convert a string containing a hexadecimal number to a floating point number.

hex2dec ("FF")
      ⇒ 255

[Function File] bin2dec (s)

Return the decimal number corresponding to the binary number represented by the string s. For example:

     bin2dec ("1110") ⇒ 14

Spaces are ignored during conversion and may be used to make the binary number more readable.

     bin2dec ("1000 0001") ⇒ 129

If s is a string matrix, return a column vector with one converted number per row of s; Invalid rows evaluate to NaN. If s is a cell array of strings, return a column vector with one converted number per cell element in s.

See also: [dec2bin], page 88, [base2dec], page 90, [hex2dec], page 89.

[Function File] dec2bin (d, len)

Return a binary number corresponding to the non-negative integer d, as a string of ones and zeros. For example:

     dec2bin (14) ⇒ "1110"

If d is a matrix or cell array, return a string matrix with one row per element in d, padded with leading zeros to the width of the largest value. The optional second argument, len, specifies the minimum number of digits in the result.

See also: [bin2dec], page 88, [dec2base], page 89, [dec2hex], page 89.

[Function File] dec2hex (d, len)

Return the hexadecimal string corresponding to the non-negative integer d. For example:

     dec2hex (2748)
          ⇒ "ABC"

If d is a matrix or cell array, return a string matrix with one row per element in d, padded with leading zeros to the width of the largest value. The optional second argument, len, specifies the minimum number of digits in the result.

See also: [hex2dec], page 89, [dec2base], page 89, [dec2bin], page 88.

[Function File] hex2dec (s)

Return the integer corresponding to the hexadecimal number represented by the string s. For example:

     hex2dec ("12B")
           ⇒ 299
     hex2dec ("12b")
           ⇒ 299

If s is a string matrix, return a column vector with one converted number per row of s; Invalid rows evaluate to NaN. If s is a cell array of strings, return a column vector with one converted number per cell element in s.

See also: [dec2hex], page 89, [base2dec], page 90, [bin2dec], page 88.

[Function File] dec2base (d, base) [Function File] dec2base (d, base, len)

Return a string of symbols in base base corresponding to the non-negative integer d.

     dec2base (123, 3) ⇒ "11120"

If d is a matrix or cell array, return a string matrix with one row per element in d, padded with leading zeros to the width of the largest value. If base is a string then the characters of base are used as the symbols for the digits of d. Space (’ ’) may not be used as a symbol.

     dec2base (123, "aei")
        ⇒ "eeeia"

The optional third argument, len, specifies the minimum number of digits in the result.

See also: [base2dec], page 90, [dec2bin], page 88, [dec2hex], page 89.

[Function File] base2dec (s, base)

Convert s from a string of digits in base base to a decimal integer (base 10).

     base2dec ("11120", 3) ⇒ 123

If s is a string matrix, return a column vector with one value per row of s. If a row contains invalid symbols then the corresponding value will be NaN. If s is a cell array of strings, return a column vector with one value per cell element in s. If base is a string, the characters of base are used as the symbols for the digits of s. Space (’ ’) may not be used as a symbol.

     base2dec ("yyyzx", "xyz")
        ⇒ 123

See also: [dec2base], page 89, [bin2dec], page 88, [hex2dec], page 89.

[Built-in Function] s = num2hex (n)

Typecast a double or single precision number or vector to a 8 or 16 character hexadecimal string of the IEEE 754 representation of the number. For example:

     num2hex ([-1, 1, e, Inf])
     ⇒ "bff0000000000000
         3ff0000000000000
         4005bf0a8b145769
         7ff0000000000000"

If the argument n is a single precision number or vector, the returned string has a length of 8. For example:

     num2hex (single ([-1, 1, e, Inf]))
     ⇒ "bf800000
         3f800000
         402df854
         7f800000"

See also: [hex2num], page 90, [hex2dec], page 89, [dec2hex], page 89.

[Built-in Function] n = hex2num (s) [Built-in Function] n = hex2num (s, class)

Typecast the 16 character hexadecimal character string to an IEEE 754 double precision number. If fewer than 16 characters are given the strings are right padded with '0' characters. Given a string matrix, hex2num treats each row as a separate number.

     hex2num (["4005bf0a8b145769"; "4024000000000000"])
        ⇒ [2.7183; 10.000]

The optional argument class can be passed as the string "single" to specify that the given string should be interpreted as a single precision number. In this case, s should be an 8 character hexadecimal string. For example:

     hex2num (["402df854"; "41200000"], "single")
        ⇒ [2.7183; 10.000]

See also: [num2hex], page 90, [hex2dec], page 89, [dec2hex], page 89.

[Built-in Function] str2double (s)

Convert a string to a real or complex number.

The string must be in one of the following formats where a and b are real numbers and the complex unit is 'i' or 'j':

• a + bi

• a + b*i

• a + i*b

• bi + a

• b*i + a

• i*b + a

If present, a and/or b are of the form [+-]d[,.]d[[eE][+-]d] where the brackets indicate optional arguments and 'd' indicates zero or more digits. The special input values Inf, NaN, and NA are also accepted.

s may be a character string, character matrix, or cell array. For character arrays the conversion is repeated for every row, and a double or complex array is returned. Empty rows in s are deleted and not returned in the numeric array. For cell arrays each character string element is processed and a double or complex array of the same dimensions as s is returned.

For unconvertible scalar or character string input str2double returns a NaN. Similarly, for character array input str2double returns a NaN for any row of s that could not be converted. For a cell array, str2double returns a NaN for any element of s for which conversion fails. Note that numeric elements in a mixed string/numeric cell array are not strings and the conversion will fail for these elements and return NaN.

str2double can replace str2num, and it avoids the security risk of using eval on unknown data.

See also: [str2num], page 92.

[Function File] strjust (s) [Function File] strjust (s, pos)

Return the text, s, justified according to pos, which may be "left", "center", or "right". If pos is omitted it defaults to "right".

Null characters are replaced by spaces. All other character data are treated as nonwhite space.

Example:

     strjust (["a"; "ab"; "abc"; "abcd"])

⇒

             "   a"
             "  ab"
             " abc"
             "abcd"

See also: [deblank], page 75, [strrep], page 83, [strtrim], page 76, [untabify], page 87.

[Function File] x = str2num (s)

[Function File] [x, state] = str2num (s)

Convert the string (or character array) s to a number (or an array). Examples:

     str2num ("3.141596")
           ⇒ 3.141596
     str2num (["1, 2, 3"; "4, 5, 6"])
           ⇒ 1  2  3
              4  5  6

The optional second output, state, is logically true when the conversion is successful. If the conversion fails the numeric output, x, is empty and state is false. Caution: As str2num uses the eval function to do the conversion, str2num will execute any code contained in the string s. Use str2double for a safer and faster conversion. For cell array of strings use str2double.

See also: [str2double], page 91, [eval], page 151.

[Mapping Function] toascii (s)

Return ASCII representation of s in a matrix. For example:

     toascii ("ASCII")
          ⇒ [ 65, 83, 67, 73, 73 ]

See also: [char], page 69.

[Mapping Function] tolower (s) [Mapping Function] lower (s)

Return a copy of the string or cell string s, with each uppercase character replaced by the corresponding lowercase one; non-alphabetic characters are left unchanged. For example:

     tolower ("MiXeD cAsE 123")
           ⇒ "mixed case 123"

See also: [toupper], page 92.

[Mapping Function] toupper (s) [Mapping Function] upper (s)

Return a copy of the string or cell string s, with each lowercase character replaced by the corresponding uppercase one; non-alphabetic characters are left unchanged. For example:

     toupper ("MiXeD cAsE 123")
           ⇒ "MIXED CASE 123"

See also: [tolower], page 92.

[Built-in Function] do_string_escapes (string)

Convert special characters in string to their escaped forms.

[Built-in Function] undo_string_escapes (s)

Convert special characters in strings back to their escaped forms. For example, the expression

     bell = "\a";

assigns the value of the alert character (control-g, ASCII code 7) to the string variable bell. If this string is printed, the system will ring the terminal bell (if it is possible). This is normally the desired outcome. However, sometimes it is useful to be able to print the original representation of the string, with the special characters replaced by their escape sequences. For example,

     octave:13> undo_string_escapes (bell)
     ans = \a

replaces the unprintable alert character with its printable representation.

5.7 Character Class Functions

Octave also provides the following character class test functions patterned after the functions in the standard C library. They all operate on string arrays and return matrices of zeros and ones. Elements that are nonzero indicate that the condition was true for the corresponding character in the string array. For example:

isalpha ("!Q@WERT^Y&")
     ⇒ [ 0, 1, 0, 1, 1, 1, 1, 0, 1, 0 ]

[Mapping Function] isalnum (s)

Return a logical array which is true where the elements of s are letters or digits and false where they are not. This is equivalent to (isalpha (s) | isdigit (s)).

See also: [isalpha], page 93, [isdigit], page 94, [ispunct], page 94, [isspace], page 94, [iscntrl], page 94.

[Mapping Function] isalpha (s)

Return a logical array which is true where the elements of s are letters and false where they are not. This is equivalent to (islower (s) | isupper (s)).

See also: [isdigit], page 94, [ispunct], page 94, [isspace], page 94, [iscntrl], page 94, [isalnum], page 93, [islower], page 93, [isupper], page 94.

[Function File] isletter (s)

Return a logical array which is true where the elements of s are letters and false where they are not. This is an alias for the isalpha function.

See also: [isalpha], page 93, [isdigit], page 94, [ispunct], page 94, [isspace], page 94, [iscntrl], page 94, [isalnum], page 93.

[Mapping Function] islower (s)

Return a logical array which is true where the elements of s are lowercase letters and false where they are not.

See also: [isupper], page 94, [isalpha], page 93, [isletter], page 93, [isalnum], page 93.

[Mapping Function] isupper (s)

Return a logical array which is true where the elements of s are uppercase letters and false where they are not.

See also: [islower], page 93, [isalpha], page 93, [isletter], page 93, [isalnum], page 93.

[Mapping Function] isdigit (s)

Return a logical array which is true where the elements of s are decimal digits (0-9) and false where they are not.

See also: [isxdigit], page 94, [isalpha], page 93, [isletter], page 93, [ispunct], page 94, [isspace], page 94, [iscntrl], page 94.

[Mapping Function] isxdigit (s)

Return a logical array which is true where the elements of s are hexadecimal digits (0-9 and a-fA-F).

See also: [isdigit], page 94.

[Mapping Function] ispunct (s)

Return a logical array which is true where the elements of s are punctuation characters and false where they are not.

See also: [isalpha], page 93, [isdigit], page 94, [isspace], page 94, [iscntrl], page 94.

[Mapping Function] isspace (s)

Return a logical array which is true where the elements of s are whitespace characters (space, formfeed, newline, carriage return, tab, and vertical tab) and false where they are not.

See also: [iscntrl], page 94, [ispunct], page 94, [isalpha], page 93, [isdigit], page 94.

[Mapping Function] iscntrl (s)

Return a logical array which is true where the elements of s are control characters and false where they are not.

See also: [ispunct], page 94, [isspace], page 94, [isalpha], page 93, [isdigit], page 94.

[Mapping Function] isgraph (s)

Return a logical array which is true where the elements of s are printable characters (but not the space character) and false where they are not.

See also: [isprint], page 94.

[Mapping Function] isprint (s)

Return a logical array which is true where the elements of s are printable characters (including the space character) and false where they are not.

See also: [isgraph], page 94.

[Mapping Function] isascii (s)

Return a logical array which is true where the elements of s are ASCII characters (in the range 0 to 127 decimal) and false where they are not.

[Function File] isstrprop (str, prop)

Test character string properties. For example:

     isstrprop ("abc123", "alpha")
     ⇒ [1, 1, 1, 0, 0, 0]

If str is a cell array, isstrpop is applied recursively to each element of the cell array. Numeric arrays are converted to character strings. The second argument prop must be one of "alpha" True for characters that are alphabetic (letters).

"alnum"
"alphanum"

True for characters that are alphabetic or digits.

"lower" True for lowercase letters. "upper" True for uppercase letters. "digit" True for decimal digits (0-9). "xdigit" True for hexadecimal digits (a-fA-F0-9).

"space"

"wspace" True for whitespace characters (space, formfeed, newline, carriage return,

tab, vertical tab).

"punct" True for punctuation characters (printing characters except space or letter

or digit).

"cntrl" True for control characters.

"graph"
"graphic"

True for printing characters except space.

"print" True for printing characters including space. "ascii" True for characters that are in the range of ASCII encoding. See also: [isalpha], page 93, [isalnum], page 93, [islower], page 93, [isupper], page 94, [isdigit], page 94, [isxdigit], page 94, [isspace], page 94, [ispunct], page 94, [iscntrl], page 94, [isgraph], page 94, [isprint], page 94, [isascii], page 94.

6 Data Containers

Octave includes support for two different mechanisms to contain arbitrary data types in the same variable. Structures, which are C-like, and are indexed with named fields, and cell arrays, where each element of the array can have a different data type and or shape. Multiple input arguments and return values of functions are organized as another data container, the comma separated list.

6.1 Structures

Octave includes support for organizing data in structures. The current implementation uses an associative array with indices limited to strings, but the syntax is more like C-style structures.

6.1.1 Basic Usage and Examples

Here are some examples of using data structures in Octave.

Elements of structures can be of any value type. For example, the three expressions

     x.a = 1;
     x.b = [1, 2; 3, 4];
     x.c = "string";

create a structure with three elements. The ‘.’ character separates the structure name from the field name and indicates to Octave that this variable is a structure. To print the value of the structure you can type its name, just as for any other variable:

     x
          ⇒ x =
             {
               a = 1
               b =
                 1  2
                 3  4
               c = string
             }

Note that Octave may print the elements in any order.

Structures may be copied just like any other variable:

y = x
⇒ y =
{
a = 1
b =
  1  2
  3  4
c = string
}

Since structures are themselves values, structure elements may reference other structures. The following statements change the value of the element b of the structure x to be a data structure containing the single element d, which has a value of 3.

x.b.d = 3;
x.b
⇒ ans =
{
d = 3
}
x
⇒ x =
{
a = 1
b =
{
  d = 3
}
c = string
}

Note that when Octave prints the value of a structure that contains other structures, only a few levels are displayed. For example:

a.b.c.d.e = 1;
a
     ⇒ a =
        {
          b =
          {
            c =
            {
              1x1 struct array containing the fields:
              d: 1x1 struct
            }
          }
        }

This prevents long and confusing output from large deeply nested structures. The number of levels to print for nested structures may be set with the function struct_levels_to_ print, and the function print_struct_array_contents may be used to enable printing of the contents of structure arrays.

[Built-in Function] val = struct_levels_to_print ()
[Built-in Function] old_val = struct_levels_to_print (new_val)
[Built-in Function] struct_levels_to_print (new_val, "local")

Query or set the internal variable that specifies the number of structure levels to display.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [print struct array contents], page 99.

[Built-in Function] val = print_struct_array_contents ()
[Built-in Function] old_val = print_struct_array_contents (new_val)
[Built-in Function] print_struct_array_contents (new_val, "local")

Query or set the internal variable that specifies whether to print struct array contents.

If true, values of struct array elements are printed. This variable does not affect scalar structures whose elements are always printed. In both cases, however, printing will be limited to the number of levels specified by struct levels to print.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [struct levels to print], page 99.

Functions can return structures. For example, the following function separates the real and complex parts of a matrix and stores them in two elements of the same structure variable.

function y = f (x)
y.re = real (x);
y.im = imag (x);
endfunction

When called with a complex-valued argument, f returns the data structure containing the real and imaginary parts of the original function argument.

f (rand (2) + rand (2) * I)
⇒ ans =
{
im =
0.26475  0.14828
0.18436  0.83669
re =
0.040239  0.242160
0.238081  0.402523
}

Function return lists can include structure elements, and they may be indexed like any other variable. For example:

[ x.u, x.s(2:3,2:3), x.v ] = svd ([1, 2; 3, 4]);
x
⇒ x =
{
u =
-0.40455  -0.91451
-0.91451   0.40455
s =
 0.00000   0.00000   0.00000
 0.00000   5.46499   0.00000
 0.00000   0.00000   0.36597
v =
-0.57605   0.81742
-0.81742  -0.57605
}

It is also possible to cycle through all the elements of a structure in a loop, using a special form of the for statement (see Section 10.5.1 [Looping Over Structure Elements], page 161).

6.1.2 Structure Arrays

A structure array is a particular instance of a structure, where each of the fields of the structure is represented by a cell array. Each of these cell arrays has the same dimensions. Conceptually, a structure array can also be seen as an array of structures with identical fields. An example of the creation of a structure array is

     x(1).a = "string1";
     x(2).a = "string2";
     x(1).b = 1;
     x(2).b = 2;

which creates a 2-by-1 structure array with two fields. Another way to create a structure array is with the struct function (see Section 6.1.3 [Creating Structures], page 102). As previously, to print the value of the structure array, you can type its name:

     x
          ⇒ x =
             {
               1x2 struct array containing the fields:
                 a
                 b
             }

Individual elements of the structure array can be returned by indexing the variable like x(1), which returns a structure with two fields:

     x(1)
          ⇒ ans =
             {
               a = string1
               b =  1
             }

Furthermore, the structure array can return a comma separated list of field values (see Section 6.3 [Comma Separated Lists], page 118), if indexed by one of its own field names. For example:

     x.a

⇒

             ans = string1
             ans = string2

Here is another example, using this comma separated list on the left-hand side of an assignment:

     [x.a] = deal ("new string1", "new string2");
      x(1).a
          ⇒ ans = new string1
      x(2).a
          ⇒ ans = new string2

Just as for numerical arrays, it is possible to use vectors as indices (see Section 8.1 [Index Expressions], page 131):

x(3:4) = x(1:2);
[x([1,3]).a] = deal ("other string1", "other string2");
x.a

⇒

        ans = other string1
        ans = new string2
        ans = other string2
        ans = new string2

The function size will return the size of the structure. For the example above

size (x) ⇒ ans = 1   4

Elements can be deleted from a structure array in a similar manner to a numerical array, by assigning the elements to an empty matrix. For example

in = struct ("call1", {x, Inf, "last"},
             "call2", {x, Inf, "first"})
     ⇒ in =
        {
          1x3 struct array containing the fields:
            call1
            call2
        }
in(1) = [];
in.call1

⇒

       ans = Inf
       ans = last

6.1.3 Creating Structures

Besides the index operator ".", Octave can use dynamic naming "(var)" or the struct function to create structures. Dynamic naming uses the string value of a variable as the field name. For example:

a = "field2";
x.a = 1;
x.(a) = 2;
x
     ⇒ x =
        {
          a =  1
          field2 =  2
        }

Dynamic indexing also allows you to use arbitrary strings, not merely valid Octave identifiers (note that this does not work on matlab):

a = "long field with spaces (and funny char$)";
x.a = 1;
x.(a) = 2;
x
     ⇒ x =
        {
          a =  1
          long field with spaces (and funny char$) =  2
        }

The warning id Octave:matlab-incompatible can be enabled to warn about this usage. See [warning ids], page 205.

More realistically, all of the functions that operate on strings can be used to build the correct field name before it is entered into the data structure.

names = ["Bill"; "Mary"; "John"];
ages  = [37; 26; 31];
for i = 1:rows (names)
  database.(names(i,:)) = ages(i);
endfor
database
     ⇒ database =
        {
          Bill =  37
          Mary =  26
          John =  31
        }

The third way to create structures is the struct command. struct takes pairs of arguments, where the first argument in the pair is the fieldname to include in the structure and the second is a scalar or cell array, representing the values to include in the structure or structure array. For example:

struct ("field1", 1, "field2", 2)
⇒ ans =
      {
        field1 =  1
        field2 =  2
      }

If the values passed to struct are a mix of scalar and cell arrays, then the scalar arguments are expanded to create a structure array with a consistent dimension. For example:

s = struct ("field1", {1, "one"}, "field2", {2, "two"},
        "field3", 3);
s.field1

⇒

        ans =  1
        ans = one
s.field2

⇒

        ans =  2
        ans = two
s.field3

⇒

        ans =  3
        ans =  3

If you want to create a struct which contains a cell array as an individual field, you must wrap it in another cell array as shown in the following example:

struct ("field1", {{1, "one"}}, "field2", 2)
     ⇒ ans =
        {
          field1 =
        {
          [1,1] =  1
          [1,2] = one
        }
          field2 =  2
        }

[Built-in Function] s = struct ()

[Built-in Function] s = struct (field1, value1, field2, value2, . . .)

[Built-in Function] s = struct (obj)

Create a scalar or array structure and initialize its values. The field1, field2, … variables are strings specifying the names of the fields and the value1, value2, … variables can be of any type.

If the values are cell arrays, create a structure array and initialize its values. The dimensions of each cell array of values must match. Singleton cells and non-cell values are repeated so that they fill the entire array. If the cells are empty, create an empty structure array with the specified field names.

If the argument is an object, return the underlying struct.

Observe that the syntax is optimized for struct arrays. Consider the following examples:

     struct ("foo", 1)
       ⇒ scalar structure containing the fields:
         foo =  1
     struct ("foo", {})
       ⇒ 0x0 struct array containing the fields:
         foo
     struct ("foo", { {} })
       ⇒ scalar structure containing the fields:
         foo = {}(0x0)
     struct ("foo", {1, 2, 3})
       ⇒ 1x3 struct array containing the fields:
         foo

The first case is an ordinary scalar struct—one field, one value. The second produces an empty struct array with one field and no values, since s being passed an empty cell array of struct array values. When the value is a cell array containing a single entry, this becomes a scalar struct with that single entry as the value of the field. That single entry happens to be an empty cell array.

Finally, if the value is a non-scalar cell array, then struct produces a struct array.

See also: [cell2struct], page 118, [fieldnames], page 105, [orderfields], page 107, [getfield], page 107, [setfield], page 106, [rmfield], page 107, [structfun], page 466.

The function isstruct can be used to test if an object is a structure or a structure array.

[Built-in Function] isstruct (x)

Return true if x is a structure or a structure array.

See also: [ismatrix], page 62, [iscell], page 111, [isa], page 39.

6.1.4 Manipulating Structures

Other functions that can manipulate the fields of a structure are given below.

[Built-in Function] nfields (s)

Return the number of fields of the structure s.

See also: [fieldnames], page 105.

[Function File] names = fieldnames (struct)
[Function File] names = fieldnames (obj)
[Function File] names = fieldnames (javaobj)
[Function File] names = fieldnames ("jclassname")

Return a cell array of strings with the names of the fields in the specified input.

When the input is a structure struct, the names are the elements of the structure.

When the input is an Octave object obj, the names are the public properties of the object. When the input is a Java object javaobj or Java classname jclassname the name are the public data elements of the object or class.

See also: [nfields], page 105, [isfield], page 106, [orderfields], page 107, [struct], page 104, [methods], page 674.

[Built-in Function] isfield (x, "name") [Built-in Function] isfield (x, name)

Return true if the x is a structure and it includes an element named name. If name is a cell array of strings then a logical array of equal dimension is returned.

See also: [fieldnames], page 105.

[Function File] s = setfield (s, field, val)
[Function File] s = setfield (s, idx1, field1, idx2, field2, . . ., val)

Set a field member field in a structure s equal to val. For example:

     s = struct ();
     s = setfield (s, "foo bar", 42);

This is equivalent to

     s.("foo bar") = 42;

Note that ordinary structure syntax s.foo bar = 42 cannot be used here, as the field name is not a valid Octave identifier. Using arbitrary strings for field name is incompatible with matlab, so this usage will warn if the Octave:matlab-incompatible warning is set. See [XREFwarning ids], page 205. With the second calling form, set a field on a structure array, possibly nested, with successive nested indices idx1, idx2, … and fields field1, field2, … The indices must be cells containing the desired index at this nesting depth. Thus consider instead,

     s = struct ("baz", 42);
     setfield (s, {1}, "foo", {1}, "bar", 5)
         ⇒ ans =
         scalar structure containing the fields:
           baz =  42
           foo =
             scalar structure containing the fields:
               bar =  54

Here we first have an ordinary structure array with one field baz set to 42. Then we set another field in a nested scalar structure indexing with two single cells containing the unique desired indices. Finally an example with nested structure arrays,

     sa.foo = 1;
     sa = setfield (sa, {2}, "bar", {3}, "baz", 6);
     sa(2).bar(3)
          ⇒ ans =
          scalar structure containing the fields:
            baz =  6

Here sa is a structure array whose field fd at elements 1 and 2 field is in turn another structure array whose third element is a structure

Note that the same result as in the above example could be achieved by:

     SA.foo = 1;
     SA(2).bar(3).baz = 6

See also: [getfield], page 107, [rmfield], page 107, [isfield], page 106, [fieldnames], page 105, [isstruct], page 105, [struct], page 104.

[Function File] [val] = getfield (s, field)
[Function File] [val] = getfield (s, idx1, field1, idx2, field2, . . .)

Extract a field from a structure (or a nested structure). The syntax is the same as setfield, except it omits the final val argument, returning this value instead of setting it.

See also: [setfield], page 106, [rmfield], page 107, [isfield], page 106, [fieldnames], page 105, [isstruct], page 105, [struct], page 104.

[Built-in Function] s = rmfield (s, "f") [Built-in Function] s = rmfield (s, f)

Return a copy of the structure (array) s with the field f removed. If f is a cell array of strings or a character array, remove each of the named fields.

See also: [orderfields], page 107, [fieldnames], page 105.

[Function File] [t, p] = orderfields (s1)
[Function File] [t, p] = orderfields (s1, s2)

Return a copy of s1 with fields arranged alphabetically or as specified by s2.

Given one struct, arrange field names in s1 alphabetically.

If the second argument is a struct, arrange field names in s1 as they appear in s2. The second argument may also specify the order in a permutation vector or a cell array of strings containing the fieldnames of s1 in the desired order.

The optional second output argument p is assigned the permutation vector which converts the original name order into the new name order.

Examples:

     s = struct ("d", 4, "b", 2, "a", 1, "c", 3);
     t1 = orderfields (s)
          ⇒ t1 =
             {
               a =  1
               b =  2
               c =  3
               d =  4
             }

t = struct ("d", {}, "c", {}, "b", {}, "a", {});
t2 = orderfields (s, t)
⇒ t2 =
{
d =  4
c =  3
b =  2
a =  1
}
t3 = orderfields (s, [3, 2, 4, 1])
⇒ t3 =
{
a =  1
b =  2
c =  3
d =  4
}
[t4, p] = orderfields (s, {"d", "c", "b", "a"})
⇒ t4 =
{
d =  4
c =  3
b =  2
a =  1
}
p =
 1
 4
 2
 3

See also: [getfield], page 107, [rmfield], page 107, [isfield], page 106, [isstruct], page 105, [fieldnames], page 105, [struct], page 104.

[Function File] substruct (type, subs, …)

Create a subscript structure for use with subsref or subsasgn. For example:

idx = substruct ("()", {3, ":"})

⇒

       idx =
       {
         type = ()
         subs =
         {
           [1,1] =  3
           [1,2] = :
         }
       }
x = [1, 2, 3; 4, 5, 6; 7, 8, 9];
subsref (x, idx)
   ⇒ 7  8  9

See also: [subsref], page 678, [subsasgn], page 680.

6.1.5 Processing Data in Structures

The simplest way to process data in a structure is within a for loop (see Section 10.5.1 [Looping Over Structure Elements], page 161). A similar effect can be achieved with the structfun function, where a user defined function is applied to each field of the structure. See [structfun], page 466.

Alternatively, to process the data in a structure, the structure might be converted to another type of container before being treated.

[Built-in Function] c = struct2cell (s)

Create a new cell array from the objects stored in the struct object. If f is the number of fields in the structure, the resulting cell array will have a dimension vector corresponding to [f size(s)]. For example:

s = struct ("name", {"Peter", "Hannah", "Robert"},
           "age", {23, 16, 3});
c = struct2cell (s)
   ⇒ c = {2x1x3 Cell Array}
c(1,1,:)(:)

⇒

      {
        [1,1] = Peter
        [2,1] = Hannah
        [3,1] = Robert
      }
c(2,1,:)(:)

⇒

      {
        [1,1] = 23
        [2,1] = 16
        [3,1] = 3
      }

See also: [cell2struct], page 118, [fieldnames], page 105.

6.2 Cell Arrays

It can be both necessary and convenient to store several variables of different size or type in one variable. A cell array is a container class able to do just that. In general cell arrays work just like N-dimensional arrays with the exception of the use of ‘{’ and ‘}’ as allocation and indexing operators.

6.2.1 Basic Usage of Cell Arrays

As an example, the following code creates a cell array containing a string and a 2-by-2 random matrix

     c = {"a string", rand(2, 2)};

To access the elements of a cell array, it can be indexed with the { and } operators. Thus, the variable created in the previous example can be indexed like this:

     c{1}
          ⇒ ans = a string

As with numerical arrays several elements of a cell array can be extracted by indexing with a vector of indexes

     c{1:2}
          ⇒ ans = a string
          ⇒ ans =
                    0.593993   0.627732
                    0.377037   0.033643

The indexing operators can also be used to insert or overwrite elements of a cell array. The following code inserts the scalar 3 on the third place of the previously created cell array

     c{3} = 3
          ⇒ c =
              {
                [1,1] = a string
                [1,2] =
                   0.593993   0.627732
                   0.377037   0.033643
                [1,3] =  3
              }

Details on indexing cell arrays are explained in Section 6.2.3 [Indexing Cell Arrays], page 114.

In general nested cell arrays are displayed hierarchically as in the previous example. In some circumstances it makes sense to reference them by their index, and this can be performed by the celldisp function.

[Function File] celldisp (c) [Function File] celldisp (c, name)

Recursively display the contents of a cell array. By default the values are displayed with the name of the variable c. However, this name can be replaced with the variable name. For example:

     c = {1, 2, {31, 32}};
     celldisp (c, "b")

⇒

           b{1} =
            1
           b{2} =
            2
           b{3}{1} =
            31
           b{3}{2} =
            32

See also: [disp], page 223.

To test if an object is a cell array, use the iscell function. For example:

iscell (c) ⇒ ans = 1 iscell (3) ⇒ ans = 0

[Built-in Function] iscell (x)

Return true if x is a cell array object.

See also: [ismatrix], page 62, [isstruct], page 105, [iscellstr], page 117, [isa], page 39.

6.2.2 Creating Cell Arrays

The introductory example (see Section 6.2.1 [Basic Usage of Cell Arrays], page 110) showed how to create a cell array containing currently available variables. In many situations, however, it is useful to create a cell array and then fill it with data.

The cell function returns a cell array of a given size, containing empty matrices. This function is similar to the zeros function for creating new numerical arrays. The following example creates a 2-by-2 cell array containing empty matrices

c = cell (2,2)
     ⇒ c =
         {
           [1,1] = [](0x0)
           [2,1] = [](0x0)
           [1,2] = [](0x0)
           [2,2] = [](0x0)
         }

Just like numerical arrays, cell arrays can be multi-dimensional. The cell function accepts any number of positive integers to describe the size of the returned cell array. It is also possible to set the size of the cell array through a vector of positive integers. In the following example two cell arrays of equal size are created, and the size of the first one is displayed

c1 = cell (3, 4, 5);
c2 = cell ( [3, 4, 5] );
size (c1)
     ⇒ ans =
         3   4   5

As can be seen, the [size], page 44 function also works for cell arrays. As do other functions describing the size of an object, such as [length], page 44, [numel], page 43, [rows], page 43, and [columns], page 43.

[Built-in Function] cell (n) [Built-in Function] cell (m, n) [Built-in Function] cell (m, n, k, …) [Built-in Function] cell ([m n …])

Create a new cell array object.

If invoked with a single scalar integer argument, return a square NxN cell array. If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with the given dimensions.

See also: [cellstr], page 117, [mat2cell], page 113, [num2cell], page 112, [struct2cell], page 109.

As an alternative to creating empty cell arrays, and then filling them, it is possible to convert numerical arrays into cell arrays using the num2cell, mat2cell and cellslices functions.

[Built-in Function] C = num2cell (A) [Built-in Function] C = num2cell (A, dim)

Convert the numeric matrix A to a cell array. If dim is defined, the value C is of dimension 1 in this dimension and the elements of A are placed into C in slices. For example:

num2cell ([1,2;3,4])

⇒

{
  [1,1] =  1
  [2,1] =  3
  [1,2] =  2
  [2,2] =  4
}
num2cell ([1,2;3,4],1)

⇒

{
  [1,1] =
     1
     3
  [1,2] =
     2
     4
}

See also: [mat2cell], page 113.

[Built-in Function] C = mat2cell (A, m, n) [Built-in Function] C = mat2cell (A, d1, d2, …) [Built-in Function] C = mat2cell (A, r)

Convert the matrix A to a cell array. If A is 2-D, then it is required that sum (m) == size (A, 1) and sum (n) == size (A, 2). Similarly, if A is multi-dimensional and the number of dimensional arguments is equal to the dimensions of A, then it is

required that sum (di) == size (A, i).

Given a single dimensional argument r, the other dimensional arguments are assumed

to equal size (A,i).

An example of the use of mat2cell is

mat2cell (reshape (1:16,4,4), [3,1], [3,1])

⇒

{
[1,1] =
1   5   9
2   6  10
3   7  11
[2,1] =
4   8  12
[1,2] =

          14
          15
        [2,2] = 16
     }

See also: [num2cell], page 112, [cell2mat], page 117.

[Built-in Function] sl = cellslices (x, lb, ub, dim)

Given an array x, this function produces a cell array of slices from the array determined by the index vectors lb, ub, for lower and upper bounds, respectively. In other words, it is equivalent to the following code:

     n = length (lb);
     sl = cell (1, n);
     for i = 1:length (lb)
       sl{i} = x(:,...,lb(i):ub(i),...,:);
     endfor

The position of the index is determined by dim. If not specified, slicing is done along the first non-singleton dimension.

See also: [cell2mat], page 117, [cellindexmat], page 116, [cellfun], page 464.

6.2.3 Indexing Cell Arrays

As shown in see Section 6.2.1 [Basic Usage of Cell Arrays], page 110 elements can be extracted from cell arrays using the ‘{’ and ‘}’ operators. If you want to extract or access subarrays which are still cell arrays, you need to use the ‘(’ and ‘)’ operators. The following example illustrates the difference:

c = {"1", "2", "3"; "x", "y", "z"; "4", "5", "6"};
c{2,3}

     ⇒ ans = z

c(2,3)
     ⇒ ans =
        {
          [1,1] = z
        }

So with ‘{}’ you access elements of a cell array, while with ‘()’ you access a sub array of a cell array.

Using the ‘(’ and ‘)’ operators, indexing works for cell arrays like for multi-dimensional arrays. As an example, all the rows of the first and third column of a cell array can be set to 0 with the following command:

c(:, [1, 3]) = {0}
⇒ =
{
[1,1] = 0
[2,1] = 0
[3,1] = 0
[1,2] = 2
[2,2] =  10
[3,2] =  20
[1,3] = 0
[2,3] = 0
[3,3] = 0
}

Note, that the above can also be achieved like this:

c(:, [1, 3]) = 0;

Here, the scalar ‘0’ is automatically promoted to cell array ‘{0}’ and then assigned to the subarray of c.

To give another example for indexing cell arrays with ‘()’, you can exchange the first and the second row of a cell array as in the following command:

c = {1, 2, 3; 4, 5, 6};
c([1, 2], :) = c([2, 1], :)
⇒ =
{
[1,1] =  4
[2,1] =  1
[1,2] =  5
[2,2] =  2
[1,3] =  6
[2,3] =  3
}

Accessing multiple elements of a cell array with the ‘{’ and ‘}’ operators will result in a comma-separated list of all the requested elements (see Section 6.3 [Comma Separated Lists], page 118). Using the ‘{’ and ‘}’ operators the first two rows in the above example can be swapped back like this:

[c{[1,2], :}] = deal (c{[2, 1], :})
⇒ =
{
[1,1] =  1
[2,1] =  4
[1,2] =  2
[2,2] =  5
[1,3] =  3
[2,3] =  6
}

As for struct arrays and numerical arrays, the empty matrix ‘[]’ can be used to delete elements from a cell array:

     x = {"1", "2"; "3", "4"};
     x(1, :) = []
          ⇒ x =
             {
               [1,1] = 3
               [1,2] = 4
             }

The following example shows how to just remove the contents of cell array elements but not delete the space for them:

     x = {"1", "2"; "3", "4"};
     x{1, :} = []
     ⇒ x =
           {
             [1,1] = [](0x0)
             [2,1] = 3
             [1,2] = [](0x0)
             [2,2] = 4
           }

The indexing operations operate on the cell array and not on the objects within the cell array. By contrast, cellindexmat applies matrix indexing to the objects within each cell array entry and returns the requested values.

[Built-in Function] y = cellindexmat (x, varargin)

Given a cell array of matrices x, this function computes

          Y = cell (size (X));
          for i = 1:numel (X)
            Y{i} = X{i}(varargin{:});
          endfor

See also: [cellslices], page 114, [cellfun], page 464.

6.2.4 Cell Arrays of Strings

One common use of cell arrays is to store multiple strings in the same variable. It is also possible to store multiple strings in a character matrix by letting each row be a string. This, however, introduces the problem that all strings must be of equal length. Therefore, it is recommended to use cell arrays to store multiple strings. For cases, where the character matrix representation is required for an operation, there are several functions that convert a cell array of strings to a character array and back. char and strvcat convert cell arrays to a character array (see Section 5.3.1 [Concatenating Strings], page 68), while the function cellstr converts a character array to a cell array of strings:

     a = ["hello"; "world"];
     c = cellstr (a)
          ⇒ c =
              {
                [1,1] = hello
                [2,1] = world
              }

[Built-in Function] cstr = cellstr (strmat)

Create a new cell array object from the elements of the string array strmat.

Each row of strmat becomes an element of cstr. Any trailing spaces in a row are deleted before conversion.

To convert back from a cellstr to a character array use char.

See also: [cell], page 112, [char], page 69.

One further advantage of using cell arrays to store multiple strings is that most functions for string manipulations included with Octave support this representation. As an example, it is possible to compare one string with many others using the strcmp function. If one of the arguments to this function is a string and the other is a cell array of strings, each element of the cell array will be compared to the string argument:

     c = {"hello", "world"};
     strcmp ("hello", c)
          ⇒ ans =
             1   0

The following string functions support cell arrays of strings: char, strvcat, strcat (see Section 5.3.1 [Concatenating Strings], page 68), strcmp, strncmp, strcmpi, strncmpi (see Section 5.4 [Comparing Strings], page 73), str2double, deblank, strtrim, strtrunc, strfind, strmatch, , regexp, regexpi (see Section 5.5 [Manipulating Strings], page 75) and str2double (see Section 5.6 [String Conversions], page 88).

The function iscellstr can be used to test if an object is a cell array of strings.

[Built-in Function] iscellstr (cell)

Return true if every element of the cell array cell is a character string.

See also: [ischar], page 66.

6.2.5 Processing Data in Cell Arrays

Data that is stored in a cell array can be processed in several ways depending on the actual data. The simplest way to process that data is to iterate through it using one or more for loops. The same idea can be implemented more easily through the use of the cellfun function that calls a user-specified function on all elements of a cell array. See [cellfun], page 464.

An alternative is to convert the data to a different container, such as a matrix or a data structure. Depending on the data this is possible using the cell2mat and cell2struct functions.

[Function File] m = cell2mat (c)

Convert the cell array c into a matrix by concatenating all elements of c into a hyperrectangle. Elements of c must be numeric, logical, or char matrices; or cell arrays; or structs; and cat must be able to concatenate them together.

See also: [mat2cell], page 113, [num2cell], page 112.

[Built-in Function] cell2struct (cell, fields)
[Built-in Function] cell2struct (cell, fields, dim)

Convert cell to a structure. The number of fields in fields must match the number of elements in cell along dimension dim, that is numel (fields) == size (cell, dim). If dim is omitted, a value of 1 is assumed.

          A = cell2struct ({"Peter", "Hannah", "Robert";
                             185, 170, 168},
                           {"Name","Height"}, 1);
          A(1)

⇒

                {
                  Name   = Peter
                  Height = 185
                }

See also: [struct2cell], page 109, [cell2mat], page 117, [struct], page 104.

6.3 Comma Separated Lists

Comma separated lists^{1} are the basic argument type to all Octave functions - both for input and return arguments. In the example

     max (a, b)

‘a, b’ is a comma separated list. Comma separated lists can appear on both the right and left hand side of an assignment. For example

     x = [1 0 1 0 0 1 1; 0 0 0 0 0 0 7];
     [i, j] = find (x, 2, "last");

Here, ‘x, 2, "last"’ is a comma separated list constituting the input arguments of find. find returns a comma separated list of output arguments which is assigned element by element to the comma separated list ‘i, j’.

Another example of where comma separated lists are used is in the creation of a new array with [] (see Section 4.1 [Matrices], page 48) or the creation of a cell array with {} (see Section 6.2.1 [Basic Usage of Cell Arrays], page 110). In the expressions

     a = [1, 2, 3, 4]; c = {4, 5, 6, 7};

both ‘1, 2, 3, 4’ and ‘4, 5, 6, 7’ are comma separated lists.

Comma separated lists cannot be directly manipulated by the user. However, both structure arrays and cell arrays can be converted into comma separated lists, and thus used in place of explicitly written comma separated lists. This feature is useful in many ways, as will be shown in the following subsections.

1 Comma-separated lists are also sometimes informally referred to as cs-lists.

6.3.1 Comma Separated Lists Generated from Cell Arrays

As has been mentioned above (see Section 6.2.3 [Indexing Cell Arrays], page 114), elements of a cell array can be extracted into a comma separated list with the { and } operators. By surrounding this list with [ and ], it can be concatenated into an array. For example:

     a = {1, [2, 3], 4, 5, 6};
     b = [a{1:4}]
          ⇒ b =
              1   2   3   4   5

Similarly, it is possible to create a new cell array containing cell elements selected with {}. By surrounding the list with ‘{’ and ‘}’ a new cell array will be created, as the following example illustrates:

     a = {1, rand(2, 2), "three"};
     b = { a{ [1, 3] } }
          ⇒ b =
              {
                [1,1] =  1
                [1,2] = three
              }

Furthermore, cell elements (accessed by {}) can be passed directly to a function. The list of elements from the cell array will be passed as an argument list to a given function as if it is called with the elements as individual arguments. The two calls to printf in the following example are identical but the latter is simpler and can handle cell arrays of an arbitrary size:

     c = {"GNU", "Octave", "is", "Free", "Software"};
     printf ("%s ", c{1}, c{2}, c{3}, c{4}, c{5});
          ⊣ GNU Octave is Free Software
     printf ("%s ", c{:});
          ⊣ GNU Octave is Free Software

If used on the left-hand side of an assignment, a comma separated list generated with {} can be assigned to. An example is

in{1} = [10, 20, 30, 40, 50, 60, 70, 80, 90];
in{2} = inf;
in{3} = "last";
in{4} = "first";
out = cell (4, 1);
[out{1:3}] = find (in{1 : 3});
[out{4:6}] = find (in{[1, 2, 4]})
     ⇒ out =
        {
          [1,1] = 1
          [2,1] = 9
          [3,1] = 90
          [4,1] = 1
          [3,1] = 1
          [4,1] = 10
        }

6.3.2 Comma Separated Lists Generated from Structure Arrays

Structure arrays can equally be used to create comma separated lists. This is done by addressing one of the fields of a structure array. For example:

x = ceil (randn (10, 1));
in = struct ("call1", {x, 3, "last"},
             "call2", {x, inf, "first"});
out = struct ("call1", cell (2, 1), "call2", cell (2, 1));
[out.call1] = find (in.call1);
[out.call2] = find (in.call2);

7 Variables

Variables let you give names to values and refer to them later. You have already seen variables in many of the examples. The name of a variable must be a sequence of letters, digits and underscores, but it may not begin with a digit. Octave does not enforce a limit on the length of variable names, but it is seldom useful to have variables with names longer than about 30 characters. The following are all valid variable names

     x
     x15
     __foo_bar_baz__
     fucnrdthsucngtagdjb

However, names like __foo_bar_baz__that begin and end with two underscores are understood to be reserved for internal use by Octave. You should not use them in code you write, except to access Octave’s documented internal variables and built-in symbolic constants.

Case is significant in variable names. The symbols a and A are distinct variables.

A variable name is a valid expression by itself. It represents the variable’s current value. Variables are given new values with assignment operators and increment operators. See Section 8.6 [Assignment Expressions], page 145.

There is one built-in variable with a special meaning. The ans variable always contains the result of the last computation, where the output wasn’t assigned to any variable. The code a = cos (pi) will assign the value -1 to the variable a, but will not change the value of ans. However, the code cos (pi) will set the value of ans to -1.

Variables in Octave do not have fixed types, so it is possible to first store a numeric value in a variable and then to later use the same name to hold a string value in the same program. Variables may not be used before they have been given a value. Doing so results in an error.

[Automatic Variable] ans

The most recently computed result that was not explicitly assigned to a variable. For example, after the expression

          3^2 + 4^2

is evaluated, the value returned by ans is 25.

[Built-in Function] isvarname (name)

Return true if name is a valid variable name.

See also: [iskeyword], page 855, [exist], page 128, [who], page 126.

[Function File] varname = genvarname (str)
[Function File] varname = genvarname (str, exclusions)

Create unique variable(s) from str. If exclusions is given, then the variable(s) will be unique to each other and to exclusions (exclusions may be either a string or a cellstr). If str is a cellstr, then a unique variable is created for each cell in str.

          x = 3.141;
          genvarname ("x", who ())
            ⇒ x1

If wanted is a cell array, genvarname will make sure the returned strings are distinct:

     genvarname ({"foo", "foo"})

⇒

          {
            [1,1] = foo
            [1,2] = foo1
          }

Note that the result is a char array/cell array of strings, not the variables themselves. To define a variable, eval() can be used. The following trivial example sets x to 42.

     name = genvarname ("x");
     eval ([name " = 42"]);
       ⇒ x =  42

Also, this can be useful for creating unique struct field names.

     x = struct ();
     for i = 1:3
       x.(genvarname ("a", fieldnames (x))) = i;
     endfor
       ⇒ x =
          {
            a =  1
            a1 =  2
            a2 =  3
          }

Since variable names may only contain letters, digits and underscores, genvarname replaces any sequence of disallowed characters with an underscore. Also, variables may not begin with a digit; in this case an underscore is added before the variable name.

Variable names beginning and ending with two underscores "__" are valid but they are used internally by octave and should generally be avoided, therefore genvarname will not generate such names.

genvarname will also make sure that returned names do not clash with keywords such as "for" and "if". A number will be appended if necessary. Note, however, that this does not include function names, such as "sin". Such names should be included in avoid if necessary.

See also: [isvarname], page 121, [exist], page 128, [tmpnam], page 255, [eval], page 151.

[Function File] namelengthmax ()

Return the matlab compatible maximum variable name length. Octave is capable of storing strings up to 2^{31} −1 in length. However for matlab compatibility all variable, function, and structure field names should be shorter than the length supplied by namelengthmax. In particular variables stored to a matlab file format will have their names truncated to this length.

7.1 Global Variables

A variable that has been declared global may be accessed from within a function body without having to pass it as a formal parameter.

A variable may be declared global using a global declaration statement. The following statements are all global declarations.

global a
global a b
global c = 2
global d = 3 e f = 5

A global variable may only be initialized once in a global statement. For example, after executing the following code

global gvar = 1
global gvar = 2

the value of the global variable gvar is 1, not 2. Issuing a ‘clear gvar’ command does not change the above behavior, but ‘clear all’ does.

It is necessary declare a variable as global within a function body in order to access it. For example,

global x
function f ()
  x = 1;
endfunction
f ()

does not set the value of the global variable x to 1. In order to change the value of the global variable x, you must also declare it to be global within the function body, like this

function f ()
  global x;
  x = 1;
endfunction

Passing a global variable in a function parameter list will make a local copy and not modify the global value. For example, given the function

function f (x)
  x = 0
endfunction

and the definition of x as a global variable at the top level,

global x = 13

the expression

f (x)

will display the value of x from inside the function as 0, but the value of x at the top level remains unchanged, because the function works with a copy of its argument.

[Built-in Function] isglobal (name)

Return true if name is a globally visible variable. For example:

          global x
          isglobal ("x")
             ⇒ 1

See also: [isvarname], page 121, [exist], page 128.

7.2 Persistent Variables

A variable that has been declared persistent within a function will retain its contents in memory between subsequent calls to the same function. The difference between persistent variables and global variables is that persistent variables are local in scope to a particular function and are not visible elsewhere.

The following example uses a persistent variable to create a function that prints the number of times it has been called.

     function count_calls ()
       persistent calls = 0;
       printf ("'count_calls' has been called %d times\n",
               ++calls);
     endfunction
     for i = 1:3
       count_calls ();
     endfor
     ⊣ 'count_calls' has been called 1 times
     ⊣ 'count_calls' has been called 2 times
     ⊣ 'count_calls' has been called 3 times

As the example shows, a variable may be declared persistent using a persistent declaration statement. The following statements are all persistent declarations.

     persistent a
     persistent a b
     persistent c = 2
     persistent d = 3 e f = 5

The behavior of persistent variables is equivalent to the behavior of static variables in C. The command static in Octave is also recognized and is equivalent to persistent.

Like global variables, a persistent variable may only be initialized once. For example, after executing the following code

     persistent pvar = 1
     persistent pvar = 2

the value of the persistent variable pvar is 1, not 2.

If a persistent variable is declared but not initialized to a specific value, it will contain an empty matrix. So, it is also possible to initialize a persistent variable by checking whether it is empty, as the following example illustrates.

function count_calls ()
  persistent calls;
  if (isempty (calls))
    calls = 0;
  endif
  printf ("'count_calls' has been called %d times\n",
          ++calls);
endfunction

This implementation behaves in exactly the same way as the previous implementation of

count_calls.

The value of a persistent variable is kept in memory until it is explicitly cleared. Assuming that the implementation of count_calls is saved on disk, we get the following behavior.

for i = 1:2
  count_calls ();
endfor
⊣ 'count_calls' has been called 1 times
⊣ 'count_calls' has been called 2 times
clear
for i = 1:2
  count_calls ();
endfor
⊣ 'count_calls' has been called 3 times
⊣ 'count_calls' has been called 4 times
clear all
for i = 1:2
  count_calls ();
endfor
⊣ 'count_calls' has been called 1 times
⊣ 'count_calls' has been called 2 times
clear count_calls
for i = 1:2
  count_calls ();
endfor
⊣ 'count_calls' has been called 1 times
⊣ 'count_calls' has been called 2 times

That is, the persistent variable is only removed from memory when the function containing the variable is removed. Note that if the function definition is typed directly into the Octave prompt, the persistent variable will be cleared by a simple clear command as the entire function definition will be removed from memory. If you do not want a persistent variable to be removed from memory even if the function is cleared, you should use the mlock function (see Section 11.9.6 [Function Locking], page 187).

7.3 Status of Variables

When creating simple one-shot programs it can be very convenient to see which variables are available at the prompt. The function who and its siblings whos and whos_line_format will show different information about what is in memory, as the following shows.

str = "A random string";
who -variables
     ⊣ *** local user variables:

⊣

     ⊣ __nargin__str

[Command] who

[Command] who pattern …

[Command] who option pattern …

[Command] C = who ("pattern", …)

List currently defined variables matching the given patterns. Valid pattern syntax is the same as described for the clear command. If no patterns are supplied, all variables are listed. By default, only variables visible in the local scope are displayed. The following are valid options but may not be combined.

global List variables in the global scope rather than the current scope.

-regexp The patterns are considered to be regular expressions when matching the variables to display. The same pattern syntax accepted by the regexp function is used.

-file The next argument is treated as a filename. All variables found within the specified file are listed. No patterns are accepted when reading variables from a file.

If called as a function, return a cell array of defined variable names matching the given patterns.

See also: [whos], page 126, [isglobal], page 123, [isvarname], page 121, [exist], page 128, [regexp], page 84.

[Command] whos

[Command] whos pattern …

[Command] whos option pattern …

[Command] S = whos ("pattern", …)

Provide detailed information on currently defined variables matching the given patterns. Options and pattern syntax are the same as for the who command. Extended information about each variable is summarized in a table with the following default entries.

Attr Attributes of the listed variable. Possible attributes are:

blank Variable in local scope

a Automatic variable. An automatic variable is one created by the interpreter, for example argn.

c Variable of complex type. f Formal parameter (function argument). g Variable with global scope. p Persistent variable.

Name The name of the variable. Size The logical size of the variable. A scalar is 1x1, a vector is 1xN or Nx1,

a 2-D matrix is MxN.

Bytes The amount of memory currently used to store the variable. Class The class of the variable. Examples include double, single, char, uint16,

cell, and struct.

The table can be customized to display more or less information through the function

whos_line_format.

If whos is called as a function, return a struct array of defined variable names matching the given patterns. Fields in the structure describing each variable are: name, size, bytes, class, global, sparse, complex, nesting, persistent. See also: [who], page 126, [whos line format], page 127.

[Built-in Function] val = whos_line_format ()
[Built-in Function] old_val = whos_line_format (new_val)

[Built-in Function] whos_line_format (new_val, "local")

Query or set the format string used by the command whos. A full format string is:

      %[modifier]<command>[:width[:left-min[:balance]]];

The following command sequences are available: %a Prints attributes of variables (g=global, p=persistent, f=formal parame-

ter, a=automatic variable).

%b Prints number of bytes occupied by variables. %c Prints class names of variables. %e Prints elements held by variables. %n Prints variable names. %s Prints dimensions of variables. %t Prints type names of variables. Every command may also have an alignment modifier: l Left alignment. r Right alignment (default). c Column-aligned (only applicable to command %s).

The width parameter is a positive integer specifying the minimum number of columns used for printing. No maximum is needed as the field will auto-expand as required. The parameters left-min and balance are only available when the column-aligned modifier is used with the command ‘%s’. balance specifies the column number within the field width which will be aligned between entries. Numbering starts from 0 which indicates the leftmost column. left-min specifies the minimum field width to the left of the specified balance column.

The default format is " %a:4; %ln:6; %cs:16:6:1; %rb:12; %lc:-1;\n".

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [whos], page 126.

Instead of displaying which variables are in memory, it is possible to determine if a given variable is available. That way it is possible to alter the behavior of a program depending on the existence of a variable. The following example illustrates this.

if (! exist ("meaning", "var"))
  disp ("The program has no 'meaning'");
endif

[Built-in Function] exist (name, type)

Return 1 if the name exists as a variable, 2 if the name is an absolute file name, an ordinary file in Octave’s path, or (after appending ‘.m’) a function file in Octave’s path, 3 if the name is a ‘.oct’ or ‘.mex’ file in Octave’s path, 5 if the name is a built-in function, 7 if the name is a directory, or 103 if the name is a function not associated with a file (entered on the command line). Otherwise, return 0. This function also returns 2 if a regular file called name exists in Octave’s search path. If you want information about other types of files, you should use some combination of the functions file_in_path and stat instead. If the optional argument type is supplied, check only for symbols of the specified type. Valid types are

"var" Check only for variables.

"builtin"

Check only for built-in functions.

"file" Check only for files and directories.

"dir" Check only for directories.

See also: [file in loadpath], page 182, [file in path], page 714, [find dir in path], page 182, [stat], page 710.

Usually Octave will manage the memory, but sometimes it can be practical to remove variables from memory manually. This is usually needed when working with large variables that fill a substantial part of the memory. On a computer that uses the IEEE floating point format, the following program allocates a matrix that requires around 128 MB memory.

large_matrix = zeros (4000, 4000);

Since having this variable in memory might slow down other computations, it can be necessary to remove it manually from memory. The clear function allows this.

[Command] clear [options] pattern … Delete the names matching the given patterns from the symbol table. The pattern may contain the following special characters:

? Match any single character.

* Match zero or more characters.

[ list ] Match the list of characters specified by list. If the first character is ! or ^, match all characters except those specified by list. For example, the pattern ‘[a-zA-Z]’ will match all lowercase and uppercase alphabetic characters.

For example, the command

     clear foo b*r

clears the name foo and all names that begin with the letter b and end with the letter

r.

If clear is called without any arguments, all user-defined variables (local and global) are cleared from the symbol table. If clear is called with at least one argument, only the visible names matching the arguments are cleared. For example, suppose you have defined a function foo, and then hidden it by performing the assignment foo = 2. Executing the command clear foo once will clear the variable definition and restore the definition of foo as a function. Executing clear foo a second time will clear the function definition. The following options are available in both long and short form

-all, -a Clears all local and global user-defined variables and all functions from the symbol table.

-exclusive, -x

Clears the variables that don’t match the following pattern.

-functions, -f

Clears the function names and the built-in symbols names.

-global, -g

Clears the global symbol names.

-variables, -v

Clears the local variable names.

-classes, -c

Clears the class structure table and clears all objects.

-regexp, -r

The arguments are treated as regular expressions as any variables that match will be cleared.

With the exception of exclusive, all long options can be used without the dash as well.

[Function File] pack ()

Consolidate workspace memory in matlab. This function is provided for compatibility, but does nothing in Octave.

Information about a function or variable such as its location in the file system can also be acquired from within Octave. This is usually only useful during development of programs, and not within a program.

[Command] type name …

[Command] type -q name … [Function File] text = type ("name", …)

Display the contents of name which may be a file, function (m-file), variable, operator, or keyword. type normally prepends a header line describing the category of name such as function or variable; The ‘-q’ option suppresses this behavior. If no output variable is used the contents are displayed on screen. Otherwise, a cell array of strings is returned, where each element corresponds to the contents of each requested function.

[Command] which name …

Display the type of each name. If name is defined from a function file, the full name of the file is also displayed.

See also: [help], page 20, [lookfor], page 21.

[Command] what

[Command] what dir [Function File] w = what (dir)

List the Octave specific files in directory dir. If dir is not specified then the current directory is used. If a return argument is requested, the files found are returned in the structure w.

See also: [which], page 130.

8 Expressions

Expressions are the basic building block of statements in Octave. An expression evaluates to a value, which you can print, test, store in a variable, pass to a function, or assign a new value to a variable with an assignment operator.

An expression can serve as a statement on its own. Most other kinds of statements contain one or more expressions which specify data to be operated on. As in other languages, expressions in Octave include variables, array references, constants, and function calls, as well as combinations of these with various operators.

8.1 Index Expressions

An index expression allows you to reference or extract selected elements of a matrix or vector.

Indices may be scalars, vectors, ranges, or the special operator ‘:’, which may be used to select entire rows or columns.

Vectors are indexed using a single index expression. Matrices (2-D) and higher multidimensional arrays are indexed using either one index or N indices where N is the dimension of the array. When using a single index expression to index 2-D or higher data the elements of the array are taken in column-first order (like Fortran).

The output from indexing assumes the dimensions of the index expression. For example:

     a(2)       # result is a scalar
     a(1:2)     # result is a row vector
     a([1; 2])  # result is a column vector

As a special case, when a colon is used as a single index, the output is a column vector containing all the elements of the vector or matrix. For example:

     a(:)       # result is a column vector
     a(:)'      # result is a row vector

The above two code idioms are often used in place of reshape when a simple vector, rather than an arbitrarily sized array, is needed.

Given the matrix

     a = [1, 2; 3, 4]

all of the following expressions are equivalent and select the first row of the matrix.

     a(1, [1, 2])  # row 1, columns 1 and 2
     a(1, 1:2)     # row 1, columns in range 1-2
     a(1, :)       # row 1, all columns

In index expressions the keyword end automatically refers to the last entry for a particular dimension. This magic index can also be used in ranges and typically eliminates the needs to call size or length to gather array bounds before indexing. For example:

a = [1, 2, 3, 4];
a(1:end/2)        # first half of a => [1, 2]
a(end + 1) = 5;   # append element
a(end) = [];      # delete element
a(1:2:end)        # odd elements of a => [1, 3]
a(2:2:end)        # even elements of a => [2, 4]
a(end:-1:1)       # reversal of a => [4, 3, 2 , 1]

8.1.1 Advanced Indexing

An array with ‘n’ dimensions can be indexed using ‘m’ indices. More generally, the set of index tuples determining the result is formed by the Cartesian product of the index vectors (or ranges or scalars).

For the ordinary and most common case, m == n, and each index corresponds to its respective dimension. If m < n and every index is less than the size of the array in the i^{th} dimension, m(i) < n(i), then the index expression is padded with trailing singleton dimensions ([ones (m-n, 1)]). If m < n but one of the indices m(i) is outside the size of the current array, then the last n-m+1 dimensions are folded into a single dimension with an extent equal to the product of extents of the original dimensions. This is easiest to understand with an example.

a = reshape (1:8, 2, 2, 2)  # Create 3-D array
a =
ans(:,:,1) =
   1   3
   2   4
ans(:,:,2) =
   5   7
   6   8
a(2,1,2);   # Case (m == n): ans = 6
a(2,1);     # Case (m < n), idx within array:
            # equivalent to a(2,1,1), ans = 2
a(2,4);     # Case (m < n), idx outside array:
            # Dimension 2 & 3 folded into new dimension of size 2x2 = 4
            # Select 2nd row, 4th element of [2, 4, 6, 8], ans = 8

One advanced use of indexing is to create arrays filled with a single value. This can be done by using an index of ones on a scalar value. The result is an object with the dimensions of the index expression and every element equal to the original scalar. For example, the following statements

a = 13; a(ones (1, 4))

produce a vector whose four elements are all equal to 13.

Similarly, by indexing a scalar with two vectors of ones it is possible to create a matrix. The following statements

     a = 13;
     a(ones (1, 2), ones (1, 3))

create a 2x3 matrix with all elements equal to 13.

The last example could also be written as

     13(ones (2, 3))

It is more efficient to use indexing rather than the code construction scalar * ones (N, M, ...) because it avoids the unnecessary multiplication operation. Moreover, multiplication may not be defined for the object to be replicated whereas indexing an array is always defined. The following code shows how to create a 2x3 cell array from a base unit which is not itself a scalar.

     {"Hello"}(ones (2, 3))

It should be, noted that ones (1, n) (a row vector of ones) results in a range (with zero increment). A range is stored internally as a starting value, increment, end value, and total number of values; hence, it is more efficient for storage than a vector or matrix of ones whenever the number of elements is greater than 4. In particular, when ‘r’ is a row vector, the expressions

       r(ones (1, n), :)
       r(ones (n, 1), :)

will produce identical results, but the first one will be significantly faster, at least for ‘r’ and ‘n’ large enough. In the first case the index is held in compressed form as a range which allows Octave to choose a more efficient algorithm to handle the expression.

A general recommendation, for a user unaware of these subtleties, is to use the function repmat for replicating smaller arrays into bigger ones.

A second use of indexing is to speed up code. Indexing is a fast operation and judicious use of it can reduce the requirement for looping over individual array elements which is a slow operation.

Consider the following example which creates a 10-element row vector a containing the values a_{i} = \sqrt{i}.

     for i = 1:10
       a(i) = sqrt (i);
     endfor

It is quite inefficient to create a vector using a loop like this. In this case, it would have been much more efficient to use the expression

     a = sqrt (1:10);

which avoids the loop entirely.

In cases where a loop cannot be avoided, or a number of values must be combined to form a larger matrix, it is generally faster to set the size of the matrix first (pre-allocate storage), and then insert elements using indexing commands. For example, given a matrix

a,

[nr, nc] = size (a);
x = zeros (nr, n * nc);
for i = 1:n
  x(:,(i-1)*nc+1:i*nc) = a;
endfor

is considerably faster than

x = a;
for i = 1:n-1
  x = [x, a];
endfor

because Octave does not have to repeatedly resize the intermediate result.

[Function File] ind = sub2ind (dims, i, j) [Function File] ind = sub2ind (dims, s1, s2, …, sN)

Convert subscripts to a linear index. The following example shows how to convert the two-dimensional index (2,3) of a 3-by-3 matrix to a linear index. The matrix is linearly indexed moving from one column to next, filling up all rows in each column.

     linear_index = sub2ind ([3, 3], 2, 3)
     ⇒ 8

See also: [ind2sub], page 134.

[Function File] [s1, s2, ..., sN] = ind2sub (dims, ind)

Convert a linear index to subscripts. The following example shows how to convert the linear index 8 in a 3-by-3 matrix into a subscript. The matrix is linearly indexed moving from one column to next, filling up all rows in each column.

     [r, c] = ind2sub ([3, 3], 8)
         ⇒ r =  2

         ⇒ c =  3

See also: [sub2ind], page 134.

[Built-in Function] isindex (ind) [Built-in Function] isindex (ind, n)

Return true if ind is a valid index. Valid indices are either positive integers (although possibly of real data type), or logical arrays. If present, n specifies the maximum extent of the dimension to be indexed. When possible the internal result is cached so that subsequent indexing using ind will not perform the check again.

[Built-in Function] val = allow_noninteger_range_as_index ()
[Built-in Function] old_val = allow_noninteger_range_as_index (new_val)
[Built-in Function] allow_noninteger_range_as_index (new_val, "local")

Query or set the internal variable that controls whether non-integer ranges are allowed as indices. This might be useful for matlab compatibility; however, it is still not entirely compatible because matlab treats the range expression differently in different contexts.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

8.2 Calling Functions

A function is a name for a particular calculation. Because it has a name, you can ask for it by name at any point in the program. For example, the function sqrt computes the square root of a number.

A fixed set of functions are built-in, which means they are available in every Octave program. The sqrt function is one of these. In addition, you can define your own functions. See Chapter 11 [Functions and Scripts], page 167, for information about how to do this.

The way to use a function is with a function call expression, which consists of the function name followed by a list of arguments in parentheses. The arguments are expressions which give the raw materials for the calculation that the function will do. When there is more than one argument, they are separated by commas. If there are no arguments, you can omit the parentheses, but it is a good idea to include them anyway, to clearly indicate that a function call was intended. Here are some examples:

     sqrt (x^2 + y^2)      # One argument

ones (n, m) # Two arguments rand () # No arguments

Each function expects a particular number of arguments. For example, the sqrt function must be called with a single argument, the number to take the square root of:

     sqrt (argument)

Some of the built-in functions take a variable number of arguments, depending on the particular usage, and their behavior is different depending on the number of arguments supplied.

Like every other expression, the function call has a value, which is computed by the function based on the arguments you give it. In this example, the value of sqrt (argument) is the square root of the argument. A function can also have side effects, such as assigning the values of certain variables or doing input or output operations.

Unlike most languages, functions in Octave may return multiple values. For example, the following statement

     [u, s, v] = svd (a)

computes the singular value decomposition of the matrix a and assigns the three result matrices to u, s, and v.

The left side of a multiple assignment expression is itself a list of expressions, and is allowed to be a list of variable names or index expressions. See also Section 8.1 [Index Expressions], page 131, and Section 8.6 [Assignment Ops], page 145.

8.2.1 Call by Value

In Octave, unlike Fortran, function arguments are passed by value, which means that each argument in a function call is evaluated and assigned to a temporary location in memory before being passed to the function. There is currently no way to specify that a function parameter should be passed by reference instead of by value. This means that it is impossible to directly alter the value of a function parameter in the calling function. It can only change the local copy within the function body. For example, the function

     function f (x, n)
       while (n-- > 0)
         disp (x);
       endwhile
     endfunction

displays the value of the first argument n times. In this function, the variable n is used as a temporary variable without having to worry that its value might also change in the calling function. Call by value is also useful because it is always possible to pass constants for any function parameter without first having to determine that the function will not attempt to modify the parameter.

The caller may use a variable as the expression for the argument, but the called function does not know this: it only knows what value the argument had. For example, given a function called as

     foo = "bar";
     fcn (foo)

you should not think of the argument as being “the variable foo.” Instead, think of the argument as the string value, "bar".

Even though Octave uses pass-by-value semantics for function arguments, values are not copied unnecessarily. For example,

     x = rand (1000); f (x);

does not actually force two 1000 by 1000 element matrices to exist unless the function f modifies the value of its argument. Then Octave must create a copy to avoid changing the value outside the scope of the function f, or attempting (and probably failing!) to modify the value of a constant or the value of a temporary result.

8.2.2 Recursion

With some restrictions^{1}, recursive function calls are allowed. A recursive function is one which calls itself, either directly or indirectly. For example, here is an inefficient^{2} way to compute the factorial of a given integer:

     function retval = fact (n)
       if (n > 0)
         retval = n * fact (n-1);
       else
         retval = 1;
       endif
     endfunction

1 Some of Octave’s functions are implemented in terms of functions that cannot be called recursively. For

example, the ODE solver lsode is ultimately implemented in a Fortran subroutine that cannot be called

recursively, so lsode should not be called either directly or indirectly from within the user-supplied

function that lsode requires. Doing so will result in an error.

2 It would be much better to use prod (1:n), or gamma (n+1) instead, after first checking to ensure that

the value n is actually a positive integer.

This function is recursive because it calls itself directly. It eventually terminates because each time it calls itself, it uses an argument that is one less than was used for the previous call. Once the argument is no longer greater than zero, it does not call itself, and the recursion ends.

The built-in variable max_recursion_depth specifies a limit to the recursion depth and prevents Octave from recursing infinitely.

[Built-in Function] val = max_recursion_depth ()
[Built-in Function] old_val = max_recursion_depth (new_val)
[Built-in Function] max_recursion_depth (new_val, "local")

Query or set the internal limit on the number of times a function may be called recursively. If the limit is exceeded, an error message is printed and control returns to the top level. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

8.3 Arithmetic Operators

The following arithmetic operators are available, and work on scalars and matrices. The element-by-element operators and functions broadcast (see Section 19.2 [Broadcasting], page 459).

x + y Addition. If both operands are matrices, the number of rows and columns must both agree, or they must be broadcastable to the same shape.

x .+ y Element-by-element addition. This operator is equivalent to +.

x - y Subtraction. If both operands are matrices, the number of rows and columns of both must agree, or they must be broadcastable to the same shape.

x .- y Element-by-element subtraction. This operator is equivalent to -.

x * y Matrix multiplication. The number of columns of x must agree with the number of rows of y, or they must be broadcastable to the same shape.

x .* y Element-by-element multiplication. If both operands are matrices, the number of rows and columns must both agree, or they must be broadcastable to the same shape.

x / y Right division. This is conceptually equivalent to the expression

               (inverse (y') * x')'

but it is computed without forming the inverse of y’.

If the system is not square, or if the coefficient matrix is singular, a minimum norm solution is computed.

x ./ y Element-by-element right division.

x \ y Left division. This is conceptually equivalent to the expression

               inverse (x) * y

but it is computed without forming the inverse of x.

If the system is not square, or if the coefficient matrix is singular, a minimum norm solution is computed.

x .\ y Element-by-element left division. Each element of y is divided by each corresponding element of x.

x ^y x ** y Power operator. If x and y are both scalars, this operator returns x raised to

the power y. If x is a scalar and y is a square matrix, the result is computed using an eigenvalue expansion. If x is a square matrix, the result is computed by repeated multiplication if y is an integer, and by an eigenvalue expansion if y is not an integer. An error results if both x and y are matrices. The implementation of this operator needs to be improved.

x .^y x .** y Element-by-element power operator. If both operands are matrices, the number

of rows and columns must both agree, or they must be broadcastable to the same shape. If several complex results are possible, the one with smallest nonnegative argument (angle) is taken. This rule may return a complex root even when a real root is also possible. Use realpow, realsqrt, cbrt, or nthroot if a real result is preferred.

-x Negation.

+x Unary plus. This operator has no effect on the operand.

x’ Complex conjugate transpose. For real arguments, this operator is the same as the transpose operator. For complex arguments, this operator is equivalent to the expression

               conj (x.')

x.’ Transpose.

Note that because Octave’s element-by-element operators begin with a ‘.’, there is a possible ambiguity for statements like

     1./m

because the period could be interpreted either as part of the constant or as part of the operator. To resolve this conflict, Octave treats the expression as if you had typed

     (1) ./ m

and not

     (1.) / m

Although this is inconsistent with the normal behavior of Octave’s lexer, which usually prefers to break the input into tokens by preferring the longest possible match at any given point, it is more useful in this case.

[Built-in Function] ctranspose (x)

Return the complex conjugate transpose of x. This function and x' are equivalent.

See also: [transpose], page 140.

[Built-in Function] ldivide (x, y)

Return the element-by-element left division of x and y. This function and x .\ y are equivalent.

See also: [rdivide], page 140, [mldivide], page 139, [times], page 140, [plus], page 139.

[Built-in Function] minus (x, y)

This function and x - y are equivalent.

See also: [plus], page 139, [uminus], page 140.

[Built-in Function] mldivide (x, y)

Return the matrix left division of x and y. This function and x \ y are equivalent.

See also: [mrdivide], page 139, [ldivide], page 139, [rdivide], page 140.

[Built-in Function] mpower (x, y)

Return the matrix power operation of x raised to the y power. This function and x ^y are equivalent.

See also: [power], page 139, [mtimes], page 139, [plus], page 139, [minus], page 139.

[Built-in Function] mrdivide (x, y)

Return the matrix right division of x and y. This function and x / y are equivalent.

See also: [mldivide], page 139, [rdivide], page 140, [plus], page 139, [minus], page 139.

[Built-in Function] mtimes (x, y) [Built-in Function] mtimes (x1, x2, …)

Return the matrix multiplication product of inputs. This function and x * y are equivalent. If more arguments are given, the multiplication is applied cumulatively from left to right:

     (...((x1 * x2) * x3) * ...)

At least one argument is required.

See also: [times], page 140, [plus], page 139, [minus], page 139, [rdivide], page 140, [mrdivide], page 139, [mldivide], page 139, [mpower], page 139.

[Built-in Function] plus (x, y) [Built-in Function] plus (x1, x2, …)

This function and x + y are equivalent. If more arguments are given, the summation is applied cumulatively from left to right:

     (...((x1 + x2) + x3) + ...)

At least one argument is required.

See also: [minus], page 139, [uplus], page 140.

[Built-in Function] power (x, y)

Return the element-by-element operation of x raised to the y power. If several complex results are possible, returns the one with smallest non-negative argument (angle). Use realpow, realsqrt, cbrt, or nthroot if a real result is preferred. This function and x .^y are equivalent.

See also: [mpower], page 139, [realpow], page 408, [realsqrt], page 408, [cbrt], page 408, [nthroot], page 408.

[Built-in Function] rdivide (x, y)

Return the element-by-element right division of x and y. This function and x ./ y are equivalent.

See also: [ldivide], page 139, [mrdivide], page 139, [times], page 140, [plus], page 139.

[Built-in Function] times (x, y) [Built-in Function] times (x1, x2, …)

Return the element-by-element multiplication product of inputs. This function and x .* y are equivalent. If more arguments are given, the multiplication is applied cumulatively from left to right:

     (...((x1 .* x2) .* x3) .* ...)

At least one argument is required.

See also: [mtimes], page 139, [rdivide], page 140.

[Built-in Function] transpose (x)

Return the transpose of x. This function and x.' are equivalent.

See also: [ctranspose], page 138.

[Built-in Function] uminus (x)

This function and - x are equivalent.

See also: [uplus], page 140, [minus], page 139.

[Built-in Function] uplus (x)

This function and + x are equivalent.

See also: [uminus], page 140, [plus], page 139, [minus], page 139.

8.4 Comparison Operators

Comparison operators compare numeric values for relationships such as equality. They are written using relational operators.

All of Octave’s comparison operators return a value of 1 if the comparison is true, or 0 if it is false. For matrix values, they all work on an element-by-element basis. Broadcasting rules apply. See Section 19.2 [Broadcasting], page 459. For example:

[1, 2; 3, 4] == [1, 3; 2, 4]
     ⇒ 1  0
         0  1

According to broadcasting rules, if one operand is a scalar and the other is a matrix, the scalar is compared to each element of the matrix in turn, and the result is the same size as the matrix.

x < y True if x is less than y.

x <= y True if x is less than or equal to y.

x == y True if x is equal to y.

x >= y True if x is greater than or equal to y.

x > y True if x is greater than y.

x != y

x ~= y True if x is not equal to y.

For complex numbers, the following ordering is defined: z1 < z2 if and only if

  abs (z1) < abs (z2)
  || (abs (z1) == abs (z2) && arg (z1) < arg (z2))

This is consistent with the ordering used by max, min and sort, but is not consistent with matlab, which only compares the real parts.

String comparisons may also be performed with the strcmp function, not with the comparison operators listed above. See Chapter 5 [Strings], page 65.

[Built-in Function] eq (x, y)

Return true if the two inputs are equal. This function is equivalent to x == y.

See also: [ne], page 141, [isequal], page 141, [le], page 141, [ge], page 141, [gt], page 141, [ne], page 141, [lt], page 141.

[Built-in Function] ge (x, y)

This function is equivalent to x >= y.

See also: [le], page 141, [eq], page 141, [gt], page 141, [ne], page 141, [lt], page 141.

[Built-in Function] gt (x, y)

This function is equivalent to x > y.

See also: [le], page 141, [eq], page 141, [ge], page 141, [ne], page 141, [lt], page 141.

[Function File] isequal (x1, x2, …)

Return true if all of x1, x2, … are equal.

See also: [isequaln], page 141.

[Function File] isequaln (x1, x2, …)

Return true if all of x1, x2, … are equal under the additional assumption that NaN == NaN (no comparison of NaN placeholders in dataset).

See also: [isequal], page 141.

[Built-in Function] le (x, y)

This function is equivalent to x <= y.

See also: [eq], page 141, [ge], page 141, [gt], page 141, [ne], page 141, [lt], page 141.

[Built-in Function] lt (x, y)

This function is equivalent to x < y.

See also: [le], page 141, [eq], page 141, [ge], page 141, [gt], page 141, [ne], page 141.

[Built-in Function] ne (x, y)

Return true if the two inputs are not equal. This function is equivalent to x != y.

See also: [eq], page 141, [isequal], page 141, [le], page 141, [ge], page 141, [lt], page 141.

8.5 Boolean Expressions

8.5.1 Element-by-element Boolean Operators

An element-by-element boolean expression is a combination of comparison expressions using the boolean operators “or” (‘|’), “and” (‘&’), and “not” (‘!’), along with parentheses to control nesting. The truth of the boolean expression is computed by combining the truth values of the corresponding elements of the component expressions. A value is considered to be false if it is zero, and true otherwise.

Element-by-element boolean expressions can be used wherever comparison expressions can be used. They can be used in if and while statements. However, a matrix value used as the condition in an if or while statement is only true if all of its elements are nonzero.

Like comparison operations, each element of an element-by-element boolean expression also has a numeric value (1 if true, 0 if false) that comes into play if the result of the boolean expression is stored in a variable, or used in arithmetic.

Here are descriptions of the three element-by-element boolean operators.

boolean1 & boolean2

Elements of the result are true if both corresponding elements of boolean1 and boolean2 are true.

boolean1 | boolean2

Elements of the result are true if either of the corresponding elements of boolean1 or boolean2 is true.

! boolean
~ boolean

Each element of the result is true if the corresponding element of boolean is false.

These operators work on an element-by-element basis. For example, the expression

     [1, 0; 0, 1] & [1, 0; 2, 3]

returns a two by two identity matrix.

For the binary operators, broadcasting rules apply. See Section 19.2 [Broadcasting], page 459. In particular, if one of the operands is a scalar and the other a matrix, the operator is applied to the scalar and each element of the matrix.

For the binary element-by-element boolean operators, both subexpressions boolean1 and boolean2 are evaluated before computing the result. This can make a difference when the expressions have side effects. For example, in the expression

     a & b++

the value of the variable b is incremented even if the variable a is zero.

This behavior is necessary for the boolean operators to work as described for matrixvalued operands.

[Built-in Function] and (x, y) [Built-in Function] and (x1, x2, …)

Return the logical AND of x and y. This function is equivalent to x & y. If more arguments are given, the logical and is applied cumulatively from left to right:

(...((x1 & x2) & x3) & ...)

At least one argument is required.

See also: [or], page 143, [not], page 143, [xor], page 379.

[Built-in Function] not (x)

Return the logical NOT of x. This function is equivalent to ! x.

See also: [and], page 142, [or], page 143, [xor], page 379.

[Built-in Function] or (x, y) [Built-in Function] or (x1, x2, …)

Return the logical OR of x and y. This function is equivalent to x | y. If more arguments are given, the logical or is applied cumulatively from left to right:

(...((x1 | x2) | x3) | ...)

At least one argument is required.

See also: [and], page 142, [not], page 143, [xor], page 379.

8.5.2 Short-circuit Boolean Operators

Combined with the implicit conversion to scalar values in if and while conditions, Octave’s element-by-element boolean operators are often sufficient for performing most logical operations. However, it is sometimes desirable to stop evaluating a boolean expression as soon as the overall truth value can be determined. Octave’s short-circuit boolean operators work this way.

boolean1 && boolean2

The expression boolean1 is evaluated and converted to a scalar using the equivalent of the operation all (boolean1(:)). If it is false, the result of the overall expression is 0. If it is true, the expression boolean2 is evaluated and converted to a scalar using the equivalent of the operation all (boolean1(:)). If it is true, the result of the overall expression is 1. Otherwise, the result of the overall expression is 0. Warning: there is one exception to the rule of evaluating all (boolean1(:)), which is when boolean1 is the empty matrix. The truth value of an empty matrix is always false so [] && true evaluates to false even though all

([]) is true.
boolean1 || boolean2

The expression boolean1 is evaluated and converted to a scalar using the equivalent of the operation all (boolean1(:)). If it is true, the result of the overall expression is 1. If it is false, the expression boolean2 is evaluated and converted to a scalar using the equivalent of the operation all (boolean1(:)). If it is true, the result of the overall expression is 1. Otherwise, the result of the overall expression is 0. Warning: the truth value of an empty matrix is always false, see the previous list item for details.

The fact that both operands may not be evaluated before determining the overall truth value of the expression can be important. For example, in the expression

a && b++

the value of the variable b is only incremented if the variable a is nonzero.

This can be used to write somewhat more concise code. For example, it is possible write

function f (a, b, c)
  if (nargin > 2 && ischar (c))
    ...

instead of having to use two if statements to avoid attempting to evaluate an argument that doesn’t exist. For example, without the short-circuit feature, it would be necessary to write

function f (a, b, c)
  if (nargin > 2)
    if (ischar (c))
      ...

Writing

function f (a, b, c)
  if (nargin > 2 & ischar (c))
    ...

would result in an error if f were called with one or two arguments because Octave would be forced to try to evaluate both of the operands for the operator ‘&’.

matlab has special behavior that allows the operators ‘&’ and ‘|’ to short-circuit when used in the truth expression for if and while statements. The Octave parser may be instructed to behave in the same manner, but its use is strongly discouraged.

[Built-in Function] val = do_braindead_shortcircuit_evaluation ()
[Built-in Function] old_val = do_braindead_shortcircuit_evaluation
   (new_val)
[Built-in Function] do_braindead_shortcircuit_evaluation (new_val, "local")

Query or set the internal variable that controls whether Octave will do short-circuit evaluation of ‘|’ and ‘&’ operators inside the conditions of if or while statements. This feature is only provided for compatibility with matlab and should not be used unless you are porting old code that relies on this feature. To obtain short-circuit behavior for logical expressions in new programs, you should always use the ‘&&’ and ‘||’ operators. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

Finally, the ternary operator (?:) is not supported in Octave. If short-circuiting is not important, it can be replaced by the ifelse function.

[Built-in Function] merge (mask, tval, fval) [Built-in Function] ifelse (mask, tval, fval)

Merge elements of true val and false val, depending on the value of mask. If mask is a logical scalar, the other two arguments can be arbitrary values. Otherwise, mask must be a logical array, and tval, fval should be arrays of matching class, or cell arrays. In the scalar mask case, tval is returned if mask is true, otherwise fval is returned. In the array mask case, both tval and fval must be either scalars or arrays with dimensions equal to mask. The result is constructed as follows:

          result(mask) = tval(mask);
          result(! mask) = fval(! mask);

mask can also be arbitrary numeric type, in which case it is first converted to logical.

See also: [logical], page 59, [diff], page 380.

8.6 Assignment Expressions

An assignment is an expression that stores a new value into a variable. For example, the following expression assigns the value 1 to the variable z:

     z = 1

After this expression is executed, the variable z has the value 1. Whatever old value z had before the assignment is forgotten. The ‘=’ sign is called an assignment operator.

Assignments can store string values also. For example, the following expression would store the value "this food is good" in the variable message:

     thing = "food"
     predicate = "good"
     message = [ "this " , thing , " is " , predicate ]

(This also illustrates concatenation of strings.)

Most operators (addition, concatenation, and so on) have no effect except to compute a value. If you ignore the value, you might as well not use the operator. An assignment operator is different. It does produce a value, but even if you ignore the value, the assignment still makes itself felt through the alteration of the variable. We call this a side effect.

The left-hand operand of an assignment need not be a variable (see Chapter 7 [Variables], page 121). It can also be an element of a matrix (see Section 8.1 [Index Expressions], page 131) or a list of return values (see Section 8.2 [Calling Functions], page 135). These are all called lvalues, which means they can appear on the left-hand side of an assignment operator. The right-hand operand may be any expression. It produces the new value which the assignment stores in the specified variable, matrix element, or list of return values.

It is important to note that variables do not have permanent types. The type of a variable is simply the type of whatever value it happens to hold at the moment. In the following program fragment, the variable foo has a numeric value at first, and a string value later on:

     octave:13> foo = 1
     foo = 1
     octave:13> foo = "bar"
     foo = bar

When the second assignment gives foo a string value, the fact that it previously had a numeric value is forgotten.

Assignment of a scalar to an indexed matrix sets all of the elements that are referenced by the indices to the scalar value. For example, if a is a matrix with at least two columns,

     a(:, 2) = 5

sets all the elements in the second column of a to 5.

Assigning an empty matrix ‘[]’ works in most cases to allow you to delete rows or columns of matrices and vectors. See Section 4.1.1 [Empty Matrices], page 51. For example, given a 4 by 5 matrix A, the assignment

     A (3, :) = []

deletes the third row of A, and the assignment

     A (:, 1:2:5) = []

deletes the first, third, and fifth columns.

An assignment is an expression, so it has a value. Thus, z = 1 as an expression has the value 1. One consequence of this is that you can write multiple assignments together:

     x = y = z = 0

stores the value 0 in all three variables. It does this because the value of z = 0, which is 0, is stored into y, and then the value of y = z = 0, which is 0, is stored into x.

This is also true of assignments to lists of values, so the following is a valid expression

     [a, b, c] = [u, s, v] = svd (a)

that is exactly equivalent to

     [u, s, v] = svd (a)
     a = u
     b = s
     c = v

In expressions like this, the number of values in each part of the expression need not match. For example, the expression

     [a, b] = [u, s, v] = svd (a)

is equivalent to

     [u, s, v] = svd (a)
     a = u
     b = s

The number of values on the left side of the expression can, however, not exceed the number of values on the right side. For example, the following will produce an error.

     [a, b, c, d] = [u, s, v] = svd (a);
     ⊣ error: element number 4 undefined in return list

The symbol ~ may be used as a placeholder in the list of lvalues, indicating that the corresponding return value should be ignored and not stored anywhere:

     [~, s, v] = svd (a);

This is cleaner and more memory efficient than using a dummy variable. The nargout value for the right-hand side expression is not affected. If the assignment is used as an expression, the return value is a comma-separated list with the ignored values dropped.

A very common programming pattern is to increment an existing variable with a given value, like this

     a = a + 2;

This can be written in a clearer and more condensed form using the += operator

     a += 2;

Similar operators also exist for subtraction (-=), multiplication (*=), and division (/=). An expression of the form

     expr1 op= expr2

is evaluated as

     expr1 = (expr1) op (expr2)

where op can be either +, -, *, or /, as long as expr2 is a simple expression with no side effects. If expr2 also contains an assignment operator, then this expression is evaluated as

     temp = expr2
     expr1 = (expr1) op temp

where temp is a placeholder temporary value storing the computed result of evaluating expr2. So, the expression

     a *= b+1

is evaluated as

     a = a * (b+1)

and not

     a = a * b + 1

You can use an assignment anywhere an expression is called for. For example, it is valid to write x != (y = 1) to set y to 1 and then test whether x equals 1. But this style tends to make programs hard to read. Except in a one-shot program, you should rewrite it to get rid of such nesting of assignments. This is never very hard.

8.7 Increment Operators

Increment operators increase or decrease the value of a variable by 1. The operator to increment a variable is written as ‘++’. It may be used to increment a variable either before or after taking its value.

For example, to pre-increment the variable x, you would write ++x. This would add one to x and then return the new value of x as the result of the expression. It is exactly the same as the expression x = x + 1.

To post-increment a variable x, you would write x++. This adds one to the variable x, but returns the value that x had prior to incrementing it. For example, if x is equal to 2, the result of the expression x++ is 2, and the new value of x is 3.

For matrix and vector arguments, the increment and decrement operators work on each element of the operand.

Here is a list of all the increment and decrement expressions. ++x This expression increments the variable x. The value of the expression is the

new value of x. It is equivalent to the expression x = x + 1.

--x This expression decrements the variable x. The value of the expression is the

new value of x. It is equivalent to the expression x = x - 1.

x++ This expression causes the variable x to be incremented. The value of the expression is the old value of x. x-- This expression causes the variable x to be decremented. The value of the

expression is the old value of x.

8.8 Operator Precedence

Operator precedence determines how operators are grouped, when different operators appear close by in one expression. For example, ‘*’ has higher precedence than ‘+’. Thus, the expression a + b * c means to multiply b and c, and then add a to the product (i.e., a + (b

* c)).

You can overrule the precedence of the operators by using parentheses. You can think of the precedence rules as saying where the parentheses are assumed if you do not write parentheses yourself. In fact, it is wise to use parentheses whenever you have an unusual combination of operators, because other people who read the program may not remember what the precedence is in this case. You might forget as well, and then you too could make a mistake. Explicit parentheses will help prevent any such mistake.

When operators of equal precedence are used together, the leftmost operator groups first, except for the assignment operators, which group in the opposite order. Thus, the expression a - b + c groups as (a - b) + c, but the expression a = b = c groups as a = (b =

c).

The precedence of prefix unary operators is important when another operator follows the operand. For example, -x^2 means -(x^2), because ‘-’ has lower precedence than ‘^’.

Here is a table of the operators in Octave, in order of decreasing precedence. Unless noted, all operators group left to right.

function call and array indexing, cell array indexing, and structure element
indexing

‘()’ ‘{}’ ‘.’

postfix increment, and postfix decrement
          ‘++’ ‘--’

These operators group right to left.

transpose and exponentiation

‘'’ ‘.'’ ‘^’ ‘**’ ‘.^’ ‘.**’

unary plus, unary minus, prefix increment, prefix decrement, and logical "not"

‘+’ ‘-’ ‘++’ ‘--’ ‘~’ ‘!’

multiply and divide

‘*’ ‘/’ ‘\’ ‘.\’ ‘.*’ ‘./’

add, subtract

‘+’ ‘-’

colon  ‘:’
relational

‘<’ ‘<=’ ‘==’ ‘>=’ ‘>’ ‘!=’ ‘~=’

element-wise "and"

‘&’

element-wise "or"

‘|’

logical "and"
          ‘&&’
logical "or"
          ‘||’
assignment
          ‘=’ ‘+=’ ‘-=’ ‘*=’ ‘/=’ ‘\=’ ‘^=’ ‘.*=’ ‘./=’ ‘.\=’ ‘.^=’ ‘|=’ ‘&=’

These operators group right to left.

9 Evaluation

Normally, you evaluate expressions simply by typing them at the Octave prompt, or by asking Octave to interpret commands that you have saved in a file.

Sometimes, you may find it necessary to evaluate an expression that has been computed and stored in a string, which is exactly what the eval function lets you do.

[Built-in Function] eval (try) [Built-in Function] eval (try, catch)

Parse the string try and evaluate it as if it were an Octave program. If that fails, evaluate the optional string catch. The string try is evaluated in the current context, so any results remain available after eval returns. The following example creates the variable A with the approximate value of 3.1416 in the current workspace.

     eval ("A = acos(-1);");

If an error occurs during the evaluation of try then the catch string is evaluated, as the following example shows:

     eval ('error ("This is a bad example");',
           'printf ("This error occurred:\n%s\n", lasterr ());');
          ⊣ This error occurred:
             This is a bad example

Programming Note: if you are only using eval as an error-capturing mechanism, rather than for the execution of arbitrary code strings, Consider using try/catch blocks or unwind protect/unwind protect cleanup blocks instead. These techniques have higher performance and don’t introduce the security considerations that the evaluation of arbitrary code does.

See also: [evalin], page 154.

9.1 Calling a Function by its Name

The feval function allows you to call a function from a string containing its name. This is useful when writing a function that needs to call user-supplied functions. The feval function takes the name of the function to call as its first argument, and the remaining arguments are given to the function.

The following example is a simple-minded function using feval that finds the root of a user-supplied function of one variable using Newton’s method.

function result = newtroot (fname, x)
# usage: newtroot (fname, x)
#
#   fname : a string naming a function f(x).
#   x     : initial guess
  delta = tol = sqrt (eps);
  maxit = 200;

  fx = feval (fname, x);
  for i = 1:maxit
    if (abs (fx) < tol)
      result = x;
      return;
    else
      fx_new = feval (fname, x + delta);
      deriv = (fx_new - fx) / delta;
      x = x - fx / deriv;
      fx = fx_new;
    endif
  endfor
  result = x;
endfunction

Note that this is only meant to be an example of calling user-supplied functions and should not be taken too seriously. In addition to using a more robust algorithm, any serious code would check the number and type of all the arguments, ensure that the supplied function really was a function, etc. See Section 4.8 [Predicates for Numeric Objects], page 61, for a list of predicates for numeric objects, and see Section 7.3 [Status of Variables], page 126, for a description of the exist function.

[Built-in Function] feval (name, …)

Evaluate the function named name. Any arguments after the first are passed as inputs to the named function. For example,

     feval ("acos", -1)
          ⇒ 3.1416

calls the function acos with the argument ‘-1’.

The function feval can also be used with function handles of any sort (see Section 11.11.1 [Function Handles], page 190). Historically, feval was the only way to call user-supplied functions in strings, but function handles are now preferred due to the cleaner syntax they offer. For example,

     f = @exp;
     feval (f, 1)
         ⇒ 2.7183
     f (1)

         ⇒ 2.7183

are equivalent ways to call the function referred to by f. If it cannot be predicted beforehand whether f is a function handle, function name in a string, or inline function then feval can be used instead.

A similar function run exists for calling user script files, that are not necessarily on the user path

[Command] run script

[Function File] run ("script")

Run script in the current workspace. Scripts which reside in directories specified in Octave’s load path, and which end with the extension ‘".m"’, can be run simply by typing their name. For scripts not located on the load path, use run. The file name script can be a bare, fully qualified, or relative filename and with or without a file extension. If no extension is specified, Octave will first search for a script with the ‘".m"’ extension before falling back to the script name without an extension. Implementation Note: If script includes a path component, then run first changes the directory to the directory where script is found. run then executes the script, and returns to the original directory.

See also: [path], page 181, [addpath], page 180, [source], page 190.

9.2 Evaluation in a Different Context

Before you evaluate an expression you need to substitute the values of the variables used in the expression. These are stored in the symbol table. Whenever the interpreter starts a new function it saves the current symbol table and creates a new one, initializing it with the list of function parameters and a couple of predefined variables such as nargin. Expressions inside the function use the new symbol table.

Sometimes you want to write a function so that when you call it, it modifies variables in your own context. This allows you to use a pass-by-name style of function, which is similar to using a pointer in programming languages such as C.

Consider how you might write save and load as m-files. For example:

function create_data
  x = linspace (0, 10, 10);
  y = sin (x);
  save mydata x y
endfunction

With evalin, you could write save as follows:

function save (file, name1, name2)
  f = open_save_file (file);
  save_var (f, name1, evalin ("caller", name1));
  save_var (f, name2, evalin ("caller", name2));
endfunction

Here, ‘caller’ is the create_data function and name1 is the string "x", which evaluates simply as the value of x.

You later want to load the values back from mydata in a different context:

function process_data
  load mydata
  ... do work ...
endfunction

With assignin, you could write load as follows:

     function load (file)
       f = open_load_file (file);
       [name, val] = load_var (f);
       assignin ("caller", name, val);
       [name, val] = load_var (f);
       assignin ("caller", name, val);
     endfunction

Here, ‘caller’ is the process_data function.

You can set and use variables at the command prompt using the context ‘base’ rather than ‘caller’.

These functions are rarely used in practice. One example is the fail (`code', `pattern') function which evaluates ‘code’ in the caller’s context and checks that the error message it produces matches the given pattern. Other examples such as save and load are written in C++ where all Octave variables are in the ‘caller’ context and evalin is not needed.

[Built-in Function] evalin (context, try) [Built-in Function] evalin (context, try, catch)

Like eval, except that the expressions are evaluated in the context context, which

     may be either "caller" or "base".

See also: [eval], page 151, [assignin], page 154.

[Built-in Function] assignin (context, varname, value)

Assign value to varname in context context, which may be either "base" or "caller".

See also: [evalin], page 154.

10 Statements

Statements may be a simple constant expression or a complicated list of nested loops and conditional statements.

Control statements such as if, while, and so on control the flow of execution in Octave programs. All the control statements start with special keywords such as if and while, to distinguish them from simple expressions. Many control statements contain other statements; for example, the if statement contains another statement which may or may not be executed.

Each control statement has a corresponding end statement that marks the end of the control statement. For example, the keyword endif marks the end of an if statement, and endwhile marks the end of a while statement. You can use the keyword end anywhere a more specific end keyword is expected, but using the more specific keywords is preferred because if you use them, Octave is able to provide better diagnostics for mismatched or missing end tokens.

The list of statements contained between keywords like if or while and the corresponding end statement is called the body of a control statement.

10.1 The if Statement

The if statement is Octave’s decision-making statement. There are three basic forms of an if statement. In its simplest form, it looks like this:

     if (condition)
       then-body
     endif

condition is an expression that controls what the rest of the statement will do. The thenbody is executed only if condition is true.

The condition in an if statement is considered true if its value is non-zero, and false if its value is zero. If the value of the conditional expression in an if statement is a vector or a matrix, it is considered true only if it is non-empty and all of the elements are non-zero.

The second form of an if statement looks like this:

     if (condition)
       then-body
     else
       else-body
     endif

If condition is true, then-body is executed; otherwise, else-body is executed.

Here is an example:

     if (rem (x, 2) == 0)
       printf ("x is even\n");
     else
       printf ("x is odd\n");
     endif

In this example, if the expression rem (x, 2) == 0 is true (that is, the value of x is divisible by 2), then the first printf statement is evaluated, otherwise the second printf statement is evaluated.

The third and most general form of the if statement allows multiple decisions to be combined in a single statement. It looks like this:

     if (condition)
       then-body
     elseif (condition)
       elseif-body
     else
       else-body
     endif

Any number of elseif clauses may appear. Each condition is tested in turn, and if one is found to be true, its corresponding body is executed. If none of the conditions are true and the else clause is present, its body is executed. Only one else clause may appear, and it must be the last part of the statement.

In the following example, if the first condition is true (that is, the value of x is divisible by 2), then the first printf statement is executed. If it is false, then the second condition is tested, and if it is true (that is, the value of x is divisible by 3), then the second printf statement is executed. Otherwise, the third printf statement is performed.

     if (rem (x, 2) == 0)
       printf ("x is even\n");
     elseif (rem (x, 3) == 0)
       printf ("x is odd and divisible by 3\n");
     else
       printf ("x is odd\n");
     endif

Note that the elseif keyword must not be spelled else if, as is allowed in Fortran. If it is, the space between the else and if will tell Octave to treat this as a new if statement within another if statement’s else clause. For example, if you write

     if (c1)
       body-1
     else if (c2)
       body-2
     endif

Octave will expect additional input to complete the first if statement. If you are using Octave interactively, it will continue to prompt you for additional input. If Octave is reading this input from a file, it may complain about missing or mismatched end statements, or, if you have not used the more specific end statements (endif, endfor, etc.), it may simply produce incorrect results, without producing any warning messages.

It is much easier to see the error if we rewrite the statements above like this,

if (c1)
body-1
else
if (c2)
  body-2
endif

using the indentation to show how Octave groups the statements. See Chapter 11 [Functions and Scripts], page 167.

10.2 The switch Statement

It is very common to take different actions depending on the value of one variable. This is possible using the if statement in the following way

if (X == 1)
do_something ();
elseif (X == 2)
do_something_else ();
else
do_something_completely_different ();
endif

This kind of code can however be very cumbersome to both write and maintain. To overcome this problem Octave supports the switch statement. Using this statement, the above example becomes

switch (X)
case 1
  do_something ();
case 2
  do_something_else ();
otherwise
  do_something_completely_different ();
endswitch

This code makes the repetitive structure of the problem more explicit, making the code easier to read, and hence maintain. Also, if the variable X should change its name, only one line would need changing compared to one line per case when if statements are used.

The general form of the switch statement is

switch (expression)
case label
  command_list
case label
  command_list
...
otherwise
  command_list
endswitch

where label can be any expression. However, duplicate label values are not detected, and only the command list corresponding to the first match will be executed. For the switch statement to be meaningful at least one case label command_list clause must be present, while the otherwise command_list clause is optional.

If label is a cell array the corresponding command list is executed if any of the elements of the cell array match expression. As an example, the following program will print ‘Variable

is either 6 or 7’.
     A = 7;
     switch (A)
       case { 6, 7 }
         printf ("variable is either 6 or 7\n");
       otherwise
         printf ("variable is neither 6 nor 7\n");
     endswitch

As with all other specific end keywords, endswitch may be replaced by end, but you can get better diagnostics if you use the specific forms.

One advantage of using the switch statement compared to using if statements is that the labels can be strings. If an if statement is used it is not possible to write

     if (X == "a string") # This is NOT valid

since a character-to-character comparison between X and the string will be made instead of evaluating if the strings are equal. This special-case is handled by the switch statement, and it is possible to write programs that look like this

     switch (X)
       case "a string"
         do_something
       ...
     endswitch

10.2.1 Notes for the C Programmer

The switch statement is also available in the widely used C programming language. There are, however, some differences between the statement in Octave and C

• Cases are exclusive, so they don’t ‘fall through’ as do the cases in the switch statement of the C language.

• The command list elements are not optional. Making the list optional would have meant requiring a separator between the label and the command list. Otherwise, things like

         switch (foo)
           case (1) -2
           ...

would produce surprising results, as would

         switch (foo)
           case (1)
           case (2)
             doit ();
           ...

particularly for C programmers. If doit() should be executed if foo is either 1 or 2, the above code should be written with a cell array like this

         switch (foo)
           case { 1, 2 }
             doit ();
           ...

10.3 The while Statement

In programming, a loop means a part of a program that is (or at least can be) executed two or more times in succession.

The while statement is the simplest looping statement in Octave. It repeatedly executes a statement as long as a condition is true. As with the condition in an if statement, the condition in a while statement is considered true if its value is non-zero, and false if its value is zero. If the value of the conditional expression in a while statement is a vector or a matrix, it is considered true only if it is non-empty and all of the elements are non-zero.

Octave’s while statement looks like this:

     while (condition)
       body
     endwhile

Here body is a statement or list of statements that we call the body of the loop, and condition is an expression that controls how long the loop keeps running.

The first thing the while statement does is test condition. If condition is true, it executes the statement body. After body has been executed, condition is tested again, and if it is still true, body is executed again. This process repeats until condition is no longer true. If condition is initially false, the body of the loop is never executed.

This example creates a variable fib that contains the first ten elements of the Fibonacci sequence.

     fib = ones (1, 10);
     i = 3;
     while (i <= 10)
       fib (i) = fib (i-1) + fib (i-2);
       i++;
     endwhile

Here the body of the loop contains two statements.

The loop works like this: first, the value of i is set to 3. Then, the while tests whether i is less than or equal to 10. This is the case when i equals 3, so the value of the i-th element of fib is set to the sum of the previous two values in the sequence. Then the i++ increments the value of i and the loop repeats. The loop terminates when i reaches 11.

A newline is not required between the condition and the body; but using one makes the program clearer unless the body is very simple.

10.4 The do-until Statement

The do-until statement is similar to the while statement, except that it repeatedly executes a statement until a condition becomes true, and the test of the condition is at the end of the loop, so the body of the loop is always executed at least once. As with the condition in an if statement, the condition in a do-until statement is considered true if its value is non-zero, and false if its value is zero. If the value of the conditional expression in a do-until statement is a vector or a matrix, it is considered true only if it is non-empty and all of the elements are non-zero.

Octave’s do-until statement looks like this:

     do
       body
     until (condition)

Here body is a statement or list of statements that we call the body of the loop, and condition is an expression that controls how long the loop keeps running.

This example creates a variable fib that contains the first ten elements of the Fibonacci sequence.

     fib = ones (1, 10);
     i = 2;
     do
       i++;
       fib (i) = fib (i-1) + fib (i-2);
     until (i == 10)

A newline is not required between the do keyword and the body; but using one makes the program clearer unless the body is very simple.

10.5 The for Statement

The for statement makes it more convenient to count iterations of a loop. The general form of the for statement looks like this:

     for var = expression
       body
     endfor

where body stands for any statement or list of statements, expression is any valid expression, and var may take several forms. Usually it is a simple variable name or an indexed variable. If the value of expression is a structure, var may also be a vector with two elements. See Section 10.5.1 [Looping Over Structure Elements], page 161, below.

The assignment expression in the for statement works a bit differently than Octave’s normal assignment statement. Instead of assigning the complete result of the expression, it assigns each column of the expression to var in turn. If expression is a range, a row vector, or a scalar, the value of var will be a scalar each time the loop body is executed. If var is a column vector or a matrix, var will be a column vector each time the loop body is executed.

The following example shows another way to create a vector containing the first ten elements of the Fibonacci sequence, this time using the for statement:

     fib = ones (1, 10);
     for i = 3:10
       fib (i) = fib (i-1) + fib (i-2);
     endfor

This code works by first evaluating the expression 3:10, to produce a range of values from 3 to 10 inclusive. Then the variable i is assigned the first element of the range and the body of the loop is executed once. When the end of the loop body is reached, the next value in the range is assigned to the variable i, and the loop body is executed again. This process continues until there are no more elements to assign.

Within Octave is it also possible to iterate over matrices or cell arrays using the for statement. For example consider

     disp ("Loop over a matrix")
     for i = [1,3;2,4]
       i
     endfor
     disp ("Loop over a cell array")
     for i = {1,"two";"three",4}
       i
     endfor

In this case the variable i takes on the value of the columns of the matrix or cell matrix. So the first loop iterates twice, producing two column vectors [1;2], followed by [3;4], and likewise for the loop over the cell array. This can be extended to loops over multidimensional arrays. For example:

     a = [1,3;2,4]; c = cat (3, a, 2*a);
     for i = c
       i
     endfor

In the above case, the multi-dimensional matrix c is reshaped to a two-dimensional matrix as

reshape (c, rows (c), prod (size (c)(2:end))) and then the same behavior as a loop

over a two dimensional matrix is produced.

Although it is possible to rewrite all for loops as while loops, the Octave language has both statements because often a for loop is both less work to type and more natural to think of. Counting the number of iterations is very common in loops and it can be easier to think of this counting as part of looping rather than as something to do inside the loop.

10.5.1 Looping Over Structure Elements

A special form of the for statement allows you to loop over all the elements of a structure:

     for [ val, key ] = expression
       body
     endfor

In this form of the for statement, the value of expression must be a structure. If it is, key and val are set to the name of the element and the corresponding value in turn, until there are no more elements. For example:

x.a = 1
x.b = [1, 2; 3, 4]
x.c = "string"
for [val, key] = x
  key
  val
endfor
     ⊣ key = a
     ⊣ val = 1
     ⊣ key = b
     ⊣ val =

⊣

     ⊣ 1  2
     ⊣ 3  4

⊣

     ⊣ key = c
     ⊣ val = string

The elements are not accessed in any particular order. If you need to cycle through the list in a particular way, you will have to use the function fieldnames and sort the list yourself.

The key variable may also be omitted. If it is, the brackets are also optional. This is useful for cycling through the values of all the structure elements when the names of the elements do not need to be known.

10.6 The break Statement

The break statement jumps out of the innermost while, do-until, or for loop that encloses it. The break statement may only be used within the body of a loop. The following example finds the smallest divisor of a given integer, and also identifies prime numbers:

num = 103;
div = 2;
while (div*div <= num)
  if (rem (num, div) == 0)
    break;
  endif
  div++;
endwhile
if (rem (num, div) == 0)
  printf ("Smallest divisor of %d is %d\n", num, div)
else
  printf ("%d is prime\n", num);
endif

When the remainder is zero in the first while statement, Octave immediately breaks out of the loop. This means that Octave proceeds immediately to the statement following the loop and continues processing. (This is very different from the exit statement which stops the entire Octave program.)

Here is another program equivalent to the previous one. It illustrates how the condition of a while statement could just as well be replaced with a break inside an if:

num = 103;
div = 2;
while (1)
  if (rem (num, div) == 0)
    printf ("Smallest divisor of %d is %d\n", num, div);
    break;
  endif
  div++;
  if (div*div > num)
    printf ("%d is prime\n", num);
    break;
  endif
endwhile

10.7 The continue Statement

The continue statement, like break, is used only inside while, do-until, or for loops. It skips over the rest of the loop body, causing the next cycle around the loop to begin immediately. Contrast this with break, which jumps out of the loop altogether. Here is an example:

# print elements of a vector of random
# integers that are even.
# first, create a row vector of 10 random
# integers with values between 0 and 100:
vec = round (rand (1, 10) * 100);
# print what we're interested in:
for x = vec
  if (rem (x, 2) != 0)
    continue;
  endif
  printf ("%d\n", x);
endfor

If one of the elements of vec is an odd number, this example skips the print statement for that element, and continues back to the first statement in the loop.

This is not a practical example of the continue statement, but it should give you a clear understanding of how it works. Normally, one would probably write the loop like this:

     for x = vec
       if (rem (x, 2) == 0)
         printf ("%d\n", x);
       endif
     endfor

10.8 The unwind protect Statement

Octave supports a limited form of exception handling modeled after the unwind-protect form of Lisp.

The general form of an unwind_protect block looks like this:

     unwind_protect
       body
     unwind_protect_cleanup
       cleanup
     end_unwind_protect

where body and cleanup are both optional and may contain any Octave expressions or commands. The statements in cleanup are guaranteed to be executed regardless of how control exits body.

This is useful to protect temporary changes to global variables from possible errors. For example, the following code will always restore the original value of the global variable frobnosticate even if an error occurs in the first part of the unwind_protect block.

     save_frobnosticate = frobnosticate;
     unwind_protect
       frobnosticate = true;
       ...
     unwind_protect_cleanup
       frobnosticate = save_frobnosticate;
     end_unwind_protect

Without unwind_protect, the value of frobnosticate would not be restored if an error occurs while evaluating the first part of the unwind_protect block because evaluation would stop at the point of the error and the statement to restore the value would not be executed.

In addition to unwind protect, Octave supports another form of exception handling, the try block.

10.9 The try Statement

The original form of a try block looks like this:

     try
       body
     catch
       cleanup
     end_try_catch

where body and cleanup are both optional and may contain any Octave expressions or commands. The statements in cleanup are only executed if an error occurs in body.

No warnings or error messages are printed while body is executing. If an error does occur during the execution of body, cleanup can use the functions lasterr or lasterror to access the text of the message that would have been printed, as well as its identifier. The alternative form,

     try
       body
     catch err
       cleanup
     end_try_catch

will automatically store the output of lasterror in the structure err. See Chapter 12 [Errors and Warnings], page 197, for more information about the lasterr and lasterror functions.

10.10 Continuation Lines

In the Octave language, most statements end with a newline character and you must tell Octave to ignore the newline character in order to continue a statement from one line to the next. Lines that end with the characters ... are joined with the following line before they are divided into tokens by Octave’s parser. For example, the lines

     x = long_variable_name ...
         + longer_variable_name ...
         - 42

form a single statement.

Any text between the continuation marker and the newline character is ignored. For example, the statement

     x = long_variable_name ...    # comment one
         + longer_variable_name ...comment two
         - 42                      # last comment

is equivalent to the one shown above.

Inside double-quoted string constants, the character \ has to be used as continuation marker. The \ must appear at the end of the line just before the newline character:

     s = "This text starts in the first line \
     and is continued in the second line."

Input that occurs inside parentheses can be continued to the next line without having to use a continuation marker. For example, it is possible to write statements like

     if (fine_dining_destination == on_a_boat
         || fine_dining_destination == on_a_train)
       seuss (i, will, not, eat, them, sam, i, am, i,
              will, not, eat, green, eggs, and, ham);
     endif

without having to add to the clutter with continuation markers.

11 Functions and Scripts

Complicated Octave programs can often be simplified by defining functions. Functions can be defined directly on the command line during interactive Octave sessions, or in external files, and can be called just like built-in functions.

11.1 Introduction to Function and Script Files

There are seven different things covered in this section.

1. Typing in a function at the command prompt.

2. Storing a group of commands in a file — called a script file.

3. Storing a function in a file—called a function file.

4. Subfunctions in function files.

5. Multiple functions in one script file.

6. Private functions.

7. Nested functions.

Both function files and script files end with an extension of .m, for matlab compatibility. If you want more than one independent functions in a file, it must be a script file (see Section 11.10 [Script Files], page 189), and to use these functions you must execute the script file before you can use the functions that are in the script file.

11.2 Defining Functions

In its simplest form, the definition of a function named name looks like this:

     function name
       body
     endfunction

A valid function name is like a valid variable name: a sequence of letters, digits and underscores, not starting with a digit. Functions share the same pool of names as variables.

The function body consists of Octave statements. It is the most important part of the definition, because it says what the function should actually do.

For example, here is a function that, when executed, will ring the bell on your terminal (assuming that it is possible to do so):

     function wakeup
       printf ("\a");
     endfunction

The printf statement (see Chapter 14 [Input and Output], page 223) simply tells Octave to print the string "\a". The special character ‘\a’ stands for the alert character (ASCII 7). See Chapter 5 [Strings], page 65.

Once this function is defined, you can ask Octave to evaluate it by typing the name of the function.

Normally, you will want to pass some information to the functions you define. The syntax for passing parameters to a function in Octave is

     function name (arg-list)
       body
     endfunction

where arg-list is a comma-separated list of the function’s arguments. When the function is called, the argument names are used to hold the argument values given in the call. The list of arguments may be empty, in which case this form is equivalent to the one shown above.

To print a message along with ringing the bell, you might modify the wakeup to look like this:

     function wakeup (message)
       printf ("\a%s\n", message);
     endfunction

Calling this function using a statement like this

     wakeup ("Rise and shine!");

will cause Octave to ring your terminal’s bell and print the message ‘Rise and shine!’, followed by a newline character (the ‘\n’ in the first argument to the printf statement).

In most cases, you will also want to get some information back from the functions you define. Here is the syntax for writing a function that returns a single value:

     function ret-var = name (arg-list)
       body
     endfunction

The symbol ret-var is the name of the variable that will hold the value to be returned by the function. This variable must be defined before the end of the function body in order for the function to return a value.

Variables used in the body of a function are local to the function. Variables named in arg-list and ret-var are also local to the function. See Section 7.1 [Global Variables], page 123, for information about how to access global variables inside a function.

For example, here is a function that computes the average of the elements of a vector:

     function retval = avg (v)
       retval = sum (v) / length (v);
     endfunction

If we had written avg like this instead,

     function retval = avg (v)
       if (isvector (v))
         retval = sum (v) / length (v);
       endif
     endfunction

and then called the function with a matrix instead of a vector as the argument, Octave would have printed an error message like this:

     error: value on right hand side of assignment is undefined

because the body of the if statement was never executed, and retval was never defined. To prevent obscure errors like this, it is a good idea to always make sure that the return variables will always have values, and to produce meaningful error messages when problems are encountered. For example, avg could have been written like this:

     function retval = avg (v)
       retval = 0;
       if (isvector (v))
         retval = sum (v) / length (v);
       else
         error ("avg: expecting vector argument");
       endif
     endfunction

There is still one additional problem with this function. What if it is called without an argument? Without additional error checking, Octave will probably print an error message that won’t really help you track down the source of the error. To allow you to catch errors like this, Octave provides each function with an automatic variable called nargin. Each time a function is called, nargin is automatically initialized to the number of arguments that have actually been passed to the function. For example, we might rewrite the avg function like this:

     function retval = avg (v)
       retval = 0;
       if (nargin != 1)
         usage ("avg (vector)");
       endif
       if (isvector (v))
         retval = sum (v) / length (v);
       else
         error ("avg: expecting vector argument");
       endif
     endfunction

Although Octave does not automatically report an error if you call a function with more arguments than expected, doing so probably indicates that something is wrong. Octave also does not automatically report an error if a function is called with too few arguments, but any attempt to use a variable that has not been given a value will result in an error. To avoid such problems and to provide useful messages, we check for both possibilities and issue our own error message.

[Built-in Function] nargin () [Built-in Function] nargin (fcn)

Within a function, return the number of arguments passed to the function. At the top level, return the number of command line arguments passed to Octave.

If called with the optional argument fcn—a function name or handle— return the declared number of arguments that the function can accept.

If the last argument to fcn is varargin the returned value is negative. For example, the function union for sets is declared as

     function [y, ia, ib] = union (a, b, varargin)
     and
     nargin ("union")
     ⇒ -3

Programming Note: nargin does not work on built-in functions.

See also: [nargout], page 172, [varargin], page 174, [isargout], page 175, [varargout], page 174, [nthargout], page 171.

[Function File] inputname (n)

Return the name of the n-th argument to the calling function. If the argument is not a simple variable name, return an empty string.

[Built-in Function] val = silent_functions ()
[Built-in Function] old_val = silent_functions (new_val)

[Built-in Function] silent_functions (new_val, "local")

Query or set the internal variable that controls whether internal output from a function is suppressed. If this option is disabled, Octave will display the results produced by evaluating expressions within a function body that are not terminated with a semicolon.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

11.3 Multiple Return Values

Unlike many other computer languages, Octave allows you to define functions that return more than one value. The syntax for defining functions that return multiple values is

function [ret-list] = name (arg-list)
  body
endfunction

where name, arg-list, and body have the same meaning as before, and ret-list is a commaseparated list of variable names that will hold the values returned from the function. The list of return values must have at least one element. If ret-list has only one element, this form of the function statement is equivalent to the form described in the previous section.

Here is an example of a function that returns two values, the maximum element of a vector and the index of its first occurrence in the vector.

function [max, idx] = vmax (v)
  idx = 1;
  max = v (idx);
  for i = 2:length (v)
    if (v (i) > max)
      max = v (i);
      idx = i;
    endif
  endfor
endfunction

In this particular case, the two values could have been returned as elements of a single array, but that is not always possible or convenient. The values to be returned may not have compatible dimensions, and it is often desirable to give the individual return values distinct names.

It is possible to use the nthargout function to obtain only some of the return values or several at once in a cell array. See Section 3.1.5 [Cell Array Objects], page 43.

[Function File] nthargout (n, func, …) [Function File] nthargout (n, ntot, func, …)

Return the nth output argument of function given by the function handle or string func. Any arguments after func are passed to func. The total number of arguments to call func with can be passed in ntot; by default ntot is n. The input n can also be a vector of indices of the output, in which case the output will be a cell array of the requested output arguments. The intended use nthargout is to avoid intermediate variables. For example, when finding the indices of the maximum entry of a matrix, the following two compositions of nthargout

     m = magic (5);
     cell2mat (nthargout ([1, 2], @ind2sub, size (m),
                          nthargout (2, @max, m(:))))
     ⇒ 5   3

are completely equivalent to the following lines:

     m = magic (5);
     [~, idx] = max (M(:));
     [i, j] = ind2sub (size (m), idx);
     [i, j]
     ⇒ 5   3

It can also be helpful to have all output arguments in a single cell in the following manner:

     USV = nthargout ([1:3], @svd, hilb (5));

See also: [nargin], page 169, [nargout], page 172, [varargin], page 174, [varargout], page 174, [isargout], page 175.

In addition to setting nargin each time a function is called, Octave also automatically initializes nargout to the number of values that are expected to be returned. This allows you to write functions that behave differently depending on the number of values that the user of the function has requested. The implicit assignment to the built-in variable ans does not figure in the count of output arguments, so the value of nargout may be zero.

The svd and lu functions are examples of built-in functions that behave differently depending on the value of nargout.

It is possible to write functions that only set some return values. For example, calling the function

function [x, y, z] = f ()
  x = 1;
  z = 2;
endfunction

as

[a, b, c] = f ()

produces:

a = 1 b = [](0x0) c = 2

along with a warning.

[Built-in Function] nargout () [Built-in Function] nargout (fcn)

Within a function, return the number of values the caller expects to receive. If called with the optional argument fcn—a function name or handle—return the number of declared output values that the function can produce. If the final output argument is varargout the returned value is negative. For example,

     f ()

will cause nargout to return 0 inside the function f and

     [s, t] = f ()

will cause nargout to return 2 inside the function f. In the second usage,

     nargout (@histc) % or nargout ("histc")

will return 2, because histc has two outputs, whereas

     nargout (@imread)

will return -2, because imread has two outputs and the second is varargout. At the top level, nargout with no argument is undefined and will produce an error. nargout does not work for built-in functions and returns -1 for all anonymous functions. See also: [nargin], page 169, [varargin], page 174, [isargout], page 175, [varargout], page 174, [nthargout], page 171.

It is good practice at the head of a function to verify that it has been called correctly. In Octave the following idiom is seen frequently

if (nargin < min_#_inputs || nargin > max_#_inputs)
  print_usage ();
endif

which stops the function execution and prints a message about the correct way to call the function whenever the number of inputs is wrong.

For compatibility with matlab, nargchk, narginchk and nargoutchk are available which provide similar error checking.

[Function File] msgstr = nargchk (minargs, maxargs, nargs)
[Function File] msgstr = nargchk (minargs, maxargs, nargs, "string")
[Function File] msgstruct = nargchk (minargs, maxargs, nargs, "struct")

Return an appropriate error message string (or structure) if the number of inputs requested is invalid. This is useful for checking to see that the number of input arguments supplied to a function is within an acceptable range.

See also: [nargoutchk], page 173, [narginchk], page 173, [error], page 197, [nargin], page 169, [nargout], page 172.

[Function File] narginchk (minargs, maxargs)

Check for correct number of arguments or generate an error message if the number of arguments in the calling function is outside the range minargs and maxargs. Otherwise, do nothing. Both minargs and maxargs need to be scalar numeric values. Zero, Inf and negative values are all allowed, and minargs and maxargs may be equal. Note that this function evaluates nargin on the caller.

See also: [nargchk], page 173, [nargoutchk], page 173, [error], page 197, [nargout], page 172, [nargin], page 169.

[Function File] nargoutchk (minargs, maxargs)
[Function File] msgstr = nargoutchk (minargs, maxargs, nargs)
[Function File] msgstr = nargoutchk (minargs, maxargs, nargs, "string")
[Function File] msgstruct = nargoutchk (minargs, maxargs, nargs, "struct")

Check for correct number of output arguments. On the first form, returns an error unless the number of arguments in its caller is between the values of minargs and maxargs. It does nothing otherwise. Note that this function evaluates the value of nargout on the caller so its value must have not been tampered with. Both minargs and maxargs need to be a numeric scalar. Zero, Inf and negative are all valid, and they can have the same value. For backward compatibility reasons, the other forms return an appropriate error message string (or structure) if the number of outputs requested is invalid. This is useful for checking to see that the number of output arguments supplied to a function is within an acceptable range.

See also: [nargchk], page 173, [narginchk], page 173, [error], page 197, [nargout], page 172, [nargin], page 169.

11.4 Variable-length Argument Lists

Sometimes the number of input arguments is not known when the function is defined. As an example think of a function that returns the smallest of all its input arguments. For example:

     a = smallest (1, 2, 3);
     b = smallest (1, 2, 3, 4);

In this example both a and b would be 1. One way to write the smallest function is

     function val = smallest (arg1, arg2, arg3, arg4, arg5)
       body
     endfunction

and then use the value of nargin to determine which of the input arguments should be considered. The problem with this approach is that it can only handle a limited number of input arguments.

If the special parameter name varargin appears at the end of a function parameter list it indicates that the function takes a variable number of input arguments. Using varargin the function looks like this

     function val = smallest (varargin)
       body
     endfunction

In the function body the input arguments can be accessed through the variable varargin. This variable is a cell array containing all the input arguments. See Section 6.2 [Cell Arrays], page 110, for details on working with cell arrays. The smallest function can now be defined like this

     function val = smallest (varargin)
       val = min ([varargin{:}]);
     endfunction

This implementation handles any number of input arguments, but it’s also a very simple solution to the problem.

A slightly more complex example of varargin is a function print_arguments that prints all input arguments. Such a function can be defined like this

     function print_arguments (varargin)
       for i = 1:length (varargin)
         printf ("Input argument %d: ", i);
         disp (varargin{i});
       endfor
     endfunction

This function produces output like this

     print_arguments (1, "two", 3);
          ⊣ Input argument 1:  1
          ⊣ Input argument 2: two
          ⊣ Input argument 3:  3

[Function File] [reg, prop] = parseparams (params)
[Function File] [reg, var1, ...] = parseparams (params, name1, default1,

…) Return in reg the cell elements of param up to the first string element and in prop all remaining elements beginning with the first string element. For example:

     [reg, prop] = parseparams ({1, 2, "linewidth", 10})
     reg =
     {
       [1,1] = 1
       [1,2] = 2
     }
     prop =
     {
       [1,1] = linewidth
       [1,2] = 10
     }

The parseparams function may be used to separate regular numeric arguments from additional arguments given as property/value pairs of the varargin cell array.

In the second form of the call, available options are specified directly with their default values given as name-value pairs. If params do not form name-value pairs, or if an option occurs that does not match any of the available options, an error occurs. When called from an m-file function, the error is prefixed with the name of the caller function. The matching of options is case-insensitive.

See also: [varargin], page 174.

11.5 Ignoring Arguments

In the formal argument list, it is possible to use the dummy placeholder ~ instead of a name. This indicates that the corresponding argument value should be ignored and not stored to any variable.

function val = pick2nd (~, arg2)
  val = arg2;
endfunction

The value of nargin is not affected by using this declaration.

Return arguments can also be ignored using the same syntax. Functions may take advantage of ignored outputs to reduce the number of calculations performed. To do so, use the isargout function to query whether the output argument is wanted. For example:

function [out1, out2] = long_function (x, y, z)
  if (isargout (1))
    ## Long calculation
    ...
    out1 = result;
  endif
  ...
endfunction

[Built-in Function] isargout (k)

Within a function, return a logical value indicating whether the argument k will be assigned to a variable on output. If the result is false, the argument has been ignored during the function call through the use of the tilde (~) special output argument. Functions can use isargout to avoid performing unnecessary calculations for outputs which are unwanted. If k is outside the range 1:max (nargout), the function returns false. k can also be an array, in which case the function works element-by-element and a logical array is returned. At the top level, isargout returns an error.

See also: [nargout], page 172, [nargin], page 169, [varargin], page 174, [varargout], page 174, [nthargout], page 171.

11.6 Variable-length Return Lists

It is possible to return a variable number of output arguments from a function using a syntax that’s similar to the one used with the special varargin parameter name. To let a function return a variable number of output arguments the special output parameter name varargout is used. As with varargin, varargout is a cell array that will contain the requested output arguments.

As an example the following function sets the first output argument to 1, the second to 2, and so on.

function varargout = one_to_n ()
  for i = 1:nargout
    varargout{i} = i;
  endfor
endfunction

When called this function returns values like this

[a, b, c] = one_to_n ()
     ⇒ a =  1

     ⇒ b =  2
     ⇒ c =  3

If varargin (varargout) does not appear as the last element of the input (output) parameter list, then it is not special, and is handled the same as any other parameter name.

[Function File] [r1, r2, ..., rn] = deal (a)
[Function File] [r1, r2, ..., rn] = deal (a1, a2, . . ., an)

Copy the input parameters into the corresponding output parameters. If only one input parameter is supplied, its value is copied to each of the outputs. For example,

     [a, b, c] = deal (x, y, z);

is equivalent to

     a = x; b = y; c = z;

and

          [a, b, c] = deal (x);

is equivalent to

          a = b = c = x;

11.7 Returning from a Function

The body of a user-defined function can contain a return statement. This statement returns control to the rest of the Octave program. It looks like this:

     return

Unlike the return statement in C, Octave’s return statement cannot be used to return a value from a function. Instead, you must assign values to the list of return variables that are part of the function statement. The return statement simply makes it easier to exit a function from a deeply nested loop or conditional statement.

Here is an example of a function that checks to see if any elements of a vector are nonzero.

     function retval = any_nonzero (v)
       retval = 0;
       for i = 1:length (v)
         if (v (i) != 0)
           retval = 1;
           return;
         endif
       endfor
       printf ("no nonzero elements found\n");
     endfunction

Note that this function could not have been written using the break statement to exit the loop once a nonzero value is found without adding extra logic to avoid printing the message if the vector does contain a nonzero element.

[Keyword] return

When Octave encounters the keyword return inside a function or script, it returns control to the caller immediately. At the top level, the return statement is ignored. A return statement is assumed at the end of every function definition.

11.8 Default Arguments

Since Octave supports variable number of input arguments, it is very useful to assign default values to some input arguments. When an input argument is declared in the argument list it is possible to assign a default value to the argument like this

     function name (arg1 = val1, ...)
       body
     endfunction

If no value is assigned to arg1 by the user, it will have the value val1.

As an example, the following function implements a variant of the classic “Hello, World” program.

     function hello (who = "World")
       printf ("Hello, %s!\n", who);
     endfunction

When called without an input argument the function prints the following

     hello ();
          ⊣ Hello, World!

and when it’s called with an input argument it prints the following

     hello ("Beautiful World of Free Software");
          ⊣ Hello, Beautiful World of Free Software!

Sometimes it is useful to explicitly tell Octave to use the default value of an input argument. This can be done writing a ‘:’ as the value of the input argument when calling the function.

     hello (:);
          ⊣ Hello, World!

11.9 Function Files

Except for simple one-shot programs, it is not practical to have to define all the functions you need each time you need them. Instead, you will normally want to save them in a file so that you can easily edit them, and save them for use at a later time.

Octave does not require you to load function definitions from files before using them. You simply need to put the function definitions in a place where Octave can find them.

When Octave encounters an identifier that is undefined, it first looks for variables or functions that are already compiled and currently listed in its symbol table. If it fails to find a definition there, it searches a list of directories (the path) for files ending in ‘.m’ that have the same base name as the undefined identifier.^{1} Once Octave finds a file with a name that matches, the contents of the file are read. If it defines a single function, it is compiled and executed. See Section 11.10 [Script Files], page 189, for more information about how you can define more than one function in a single file.

When Octave defines a function from a function file, it saves the full name of the file it read and the time stamp on the file. If the time stamp on the file changes, Octave may reload the file. When Octave is running interactively, time stamp checking normally happens at most once each time Octave prints the prompt. Searching for new function definitions also occurs if the current working directory changes.

Checking the time stamp allows you to edit the definition of a function while Octave is running, and automatically use the new function definition without having to restart your Octave session.

To avoid degrading performance unnecessarily by checking the time stamps on functions that are not likely to change, Octave assumes that function files in the directory tree ‘octave-home/share/octave/version/m’ will not change, so it doesn’t have to check their time stamps every time the functions defined in those files are used. This is normally a very good assumption and provides a significant improvement in performance for the function files that are distributed with Octave.

1 The ‘.m’ suffix was chosen for compatibility with matlab.

If you know that your own function files will not change while you are running Octave, you can improve performance by calling ignore_function_time_stamp ("all"), so that Octave will ignore the time stamps for all function files. Passing "system" to this function resets the default behavior.

[Command] edit name

[Command] edit field value
[Command] value = edit get field

Edit the named function, or change editor settings. If edit is called with the name of a file or function as its argument it will be opened in the text editor defined by EDITOR.

• If the function name is available in a file on your path and that file is modifiable, then it will be edited in place. If it is a system function, then it will first be copied to the directory HOME (see below) and then edited. If no file is found, then the m-file variant, ending with ".m", will be considered. If still no file is found, then variants with a leading "@" and then with both a leading "@" and trailing ".m" will be considered.

• If name is the name of a function defined in the interpreter but not in an m-file, then an m-file will be created in HOME to contain that function along with its current definition.

• If name.cc is specified, then it will search for name.cc in the path and try to modify it, otherwise it will create a new ‘.cc’ file in the current directory. If name happens to be an m-file or interpreter defined function, then the text of that function will be inserted into the .cc file as a comment.

• If ‘name.ext’ is on your path then it will be edited, otherwise the editor will be started with ‘name.ext’ in the current directory as the filename. If ‘name.ext’ is not modifiable, it will be copied to HOME before editing. Warning: You may need to clear name before the new definition is available. If you are editing a .cc file, you will need to execute mkoctfile `name.cc' before the definition will be available.

If edit is called with field and value variables, the value of the control field field will be set to value. If an output argument is requested and the first input argument is get then edit will return the value of the control field field. If the control field does not exist, edit will return a structure containing all fields and values. Thus, edit get all returns a complete control structure. The following control fields are used:

‘home’ This is the location of user local m-files. Be sure it is in your path. The default is ‘~/octave’.

‘author’ This is the name to put after the "## Author:" field of new functions. By default it guesses from the gecos field of the password database.

‘email’ This is the e-mail address to list after the name in the author field. By default it guesses <$LOGNAME@$HOSTNAME>, and if $HOSTNAME is not defined it uses uname -n. You probably want to override this. Be sure to use the format <user@host>.

‘license’

‘gpl’ GNU General Public License (default).

‘bsd’ BSD-style license without advertising clause.

‘pd’ Public domain.

‘"text"’ Your own default copyright and license.

Unless you specify ‘pd’, edit will prepend the copyright statement with "Copyright (C) yyyy Function Author".

‘mode’ This value determines whether the editor should be started in async mode (editor is started in the background and Octave continues) or sync mode (Octave waits until the editor exits). Set it to "sync" to start the editor in sync mode. The default is "async" (see [system], page 721).

‘editinplace’

Determines whether files should be edited in place, without regard to whether they are modifiable or not. The default is false.

[Built-in Function] mfilename () [Built-in Function] mfilename ("fullpath")

[Built-in Function] mfilename ("fullpathext")

Return the name of the currently executing file. When called from outside an m-file return the empty string. Given the argument "fullpath", include the directory part of the file name, but not the extension. Given the argument "fullpathext", include the directory part of the file name and the extension.

[Built-in Function] val = ignore_function_time_stamp ()
[Built-in Function] old_val = ignore_function_time_stamp (new_val)

Query or set the internal variable that controls whether Octave checks the time stamp on files each time it looks up functions defined in function files. If the internal variable is set to "system", Octave will not automatically recompile function files in subdirectories of ‘octave-home/lib/version’ if they have changed since they were last compiled, but will recompile other function files in the search path if they change. If set to "all", Octave will not recompile any function files unless their definitions are removed with clear. If set to "none", Octave will always check time stamps on files to determine whether functions defined in function files need to recompiled.

11.9.1 Manipulating the Load Path

When a function is called, Octave searches a list of directories for a file that contains the function declaration. This list of directories is known as the load path. By default the load path contains a list of directories distributed with Octave plus the current working directory. To see your current load path call the path function without any input or output arguments.

It is possible to add or remove directories to or from the load path using addpath and rmpath. As an example, the following code adds ‘~/Octave’ to the load path.

addpath ("~/Octave")

After this the directory ‘~/Octave’ will be searched for functions.

[Built-in Function] addpath (dir1, …) [Built-in Function] addpath (dir1, …, option)

Add named directories to the function search path. If option is "-begin" or 0 (the default), prepend the directory name to the current path. If option is "-end" or 1, append the directory name to the current path. Directories added to the path must exist. In addition to accepting individual directory arguments, lists of directory names separated by pathsep are also accepted. For example:

     addpath ("dir1:/dir2:~/dir3")

See also: [path], page 181, [rmpath], page 181, [genpath], page 181, [pathdef], page 182, [savepath], page 181, [pathsep], page 182.

[Built-in Function] genpath (dir) [Built-in Function] genpath (dir, skip, …)

Return a path constructed from dir and all its subdirectories. If additional string parameters are given, the resulting path will exclude directories with those names.

[Built-in Function] rmpath (dir1, …)

Remove dir1, … from the current function search path. In addition to accepting individual directory arguments, lists of directory names separated by pathsep are also accepted. For example:

     rmpath ("dir1:/dir2:~/dir3")

See also: [path], page 181, [addpath], page 180, [genpath], page 181, [pathdef], page 182, [savepath], page 181, [pathsep], page 182.

[Function File] savepath () [Function File] savepath (file) [Function File] status = savepath (…)

Save the unique portion of the current function search path that is not set during Octave’s initialization process to file. If file is omitted, ‘~/.octaverc’ is used. If successful, savepath returns 0.

See also: [path], page 181, [addpath], page 180, [rmpath], page 181, [genpath], page 181, [pathdef], page 182.

[Built-in Function] path (…)

Modify or display Octave’s load path. If nargin and nargout are zero, display the elements of Octave’s load path in an easy to read format. If nargin is zero and nargout is greater than zero, return the current load path. If nargin is greater than zero, concatenate the arguments, separating them with pathsep. Set the internal search path to the result and return it. No checks are made for duplicate elements.

See also: [addpath], page 180, [rmpath], page 181, [genpath], page 181, [pathdef], page 182, [savepath], page 181, [pathsep], page 182.

[Function File] val = pathdef ()

Return the default path for Octave. The path information is extracted from one of three sources. The possible sources, in order of preference, are:

 1. ‘~/.octaverc’
 2. ‘<octave-home>/.../<version>/m/startup/octaverc’

3. Octave’s path prior to changes by any octaverc.

See also: [path], page 181, [addpath], page 180, [rmpath], page 181, [genpath], page 181, [savepath], page 181.

[Built-in Function] val = pathsep ()

[Built-in Function] old_val = pathsep (new_val)

Query or set the character used to separate directories in a path.

See also: [filesep], page 714.

[Built-in Function] rehash ()

Reinitialize Octave’s load path directory cache.

[Built-in Function] file_in_loadpath (file)

[Built-in Function] file_in_loadpath (file, "all")

Return the absolute name of file if it can be found in the list of directories specified by path. If no file is found, return an empty character string. If the first argument is a cell array of strings, search each directory of the loadpath for element of the cell array and return the first that matches. If the second optional argument "all" is supplied, return a cell array containing the list of all files that have the same name in the path. If no files are found, return an empty cell array.

See also: [file in path], page 714, [find dir in path], page 182, [path], page 181.

[Built-in Function] restoredefaultpath (…)

Restore Octave’s path to its initial state at startup.

See also: [path], page 181, [addpath], page 180, [rmpath], page 181, [genpath], page 181, [pathdef], page 182, [savepath], page 181, [pathsep], page 182.

[Built-in Function] command_line_path (…)

Return the command line path variable.

See also: [path], page 181, [addpath], page 180, [rmpath], page 181, [genpath], page 181, [pathdef], page 182, [savepath], page 181, [pathsep], page 182.

[Built-in Function] find_dir_in_path (dir) [Built-in Function] find_dir_in_path (dir, "all")

Return the full name of the path element matching dir. The match is performed at the end of each path element. For example, if dir is "foo/bar", it matches

the path element "/some/dir/foo/bar", but not "/some/dir/foo/bar/baz"
"/some/dir/allfoo/bar".

The second argument is optional. If it is supplied, return a cell array containing all name matches rather than just the first.

See also: [file in path], page 714, [file in loadpath], page 182, [path], page 181.

11.9.2 Subfunctions

A function file may contain secondary functions called subfunctions. These secondary functions are only visible to the other functions in the same function file. For example, a file ‘f.m’ containing

     function f ()
       printf ("in f, calling g\n");
       g ()
     endfunction
     function g ()
       printf ("in g, calling h\n");
       h ()
     endfunction
     function h ()
       printf ("in h\n")
     endfunction

defines a main function f and two subfunctions. The subfunctions g and h may only be called from the main function f or from the other subfunctions, but not from outside the file ‘f.m’.

11.9.3 Private Functions

In many cases one function needs to access one or more helper functions. If the helper function is limited to the scope of a single function, then subfunctions as discussed above might be used. However, if a single helper function is used by more than one function, then this is no longer possible. In this case the helper functions might be placed in a subdirectory, called "private", of the directory in which the functions needing access to this helper function are found.

As a simple example, consider a function func1, that calls a helper function func2 to do much of the work. For example:

     function y = func1 (x)
       y = func2 (x);
     endfunction

Then if the path to func1 is <directory>/func1.m, and if func2 is found in the directory <directory>/private/func2.m, then func2 is only available for use of the functions, like

func1, that are found in <directory>.

11.9.4 Nested Functions

Nested functions are similar to subfunctions in that only the main function is visible outside the file. However, they also allow for child functions to access the local variables in their parent function. This shared access mimics using a global variable to share information — but a global variable which is not visible to the rest of Octave. As a programming strategy, sharing data this way can create code which is difficult to maintain. It is recommended to use subfunctions in place of nested functions when possible.

As a simple example, consider a parent function foo, that calls a nested child function bar, with a shared variable x.

function y = foo ()
x = 10;
bar ();
y = x;
function bar ()
  x = 20;
endfunction
endfunction
foo ()
⇒ 20

Notice that there is no special syntax for sharing x. This can lead to problems with accidental variable sharing between a parent function and its child. While normally variables are inherited, child function parameters and return values are local to the child function.

Now consider the function foobar that uses variables x and y. foobar calls a nested function foo which takes x as a parameter and returns y. foo then calls bat which does some computation.

function z = foobar ()
x = 0;
y = 0;
z = foo (5);
z += x + y;
function y = foo (x)
  y = x + bat ();
  function z = bat ()
    z = x;
  endfunction
endfunction
endfunction
foobar ()
  ⇒ 10

It is important to note that the x and y in foobar remain zero, as in foo they are a return value and parameter respectively. The x in bat refers to the x in foo.

Variable inheritance leads to a problem for eval and scripts. If a new variable is created in a parent function, it is not clear what should happen in nested child functions. For example, consider a parent function foo with a nested child function bar:

function y = foo (to_eval)
bar ();
eval (to_eval);
function bar ()
  eval ("x = 100;");
  eval ("y = x;");
endfunction
endfunction
foo ("x = 5;")
  ⇒ error: can not add variable "x" to a static workspace
foo ("y = 10;")
  ⇒ 10
foo ("")
  ⇒ 100

The parent function foo is unable to create a new variable x, but the child function bar was successful. Furthermore, even in an eval statement y in bar is the same y as in its parent function foo. The use of eval in conjunction with nested functions is best avoided.

As with subfunctions, only the first nested function in a file may be called from the outside. Inside a function the rules are more complicated. In general a nested function may call:

0. Globally visible functions

1. Any function that the nested function’s parent can call

2. Sibling functions (functions that have the same parents)

3. Direct children

As a complex example consider a parent function ex_top with two child functions, ex_a and ex_b. In addition, ex_a has two more child functions, ex_aa and ex_ab. For example:

function ex_top ()
## Can call: ex_top, ex_a, and ex_b
## Can NOT call: ex_aa and ex_ab
function ex_a ()
  ## Call call everything
  function ex_aa ()
    ## Can call everything
  endfunction
  function ex_ab ()
    ## Can call everything
  endfunction
endfunction

       function ex_b ()
         ## Can call: ex_top, ex_a, and ex_b
         ## Can NOT call: ex_aa and ex_ab
       endfunction
     endfunction

11.9.5 Overloading and Autoloading

Functions can be overloaded to work with different input arguments. For example, the operator ’+’ has been overloaded in Octave to work with single, double, uint8, int32, and many other arguments. The preferred way to overload functions is through classes and object oriented programming (see Section 34.4.1 [Function Overloading], page 683). Occasionally, however, one needs to undo user overloading and call the default function associated with a specific type. The builtin function exists for this purpose.

[Built-in Function] [...] = builtin (f, …)

Call the base function f even if f is overloaded to another function for the given type signature.

This is normally useful when doing object-oriented programming and there is a requirement to call one of Octave’s base functions rather than the overloaded one of a new class.

A trivial example which redefines the sin function to be the cos function shows how

     builtin works.
          sin (0)
            ⇒ 0
          function y = sin (x), y = cos (x); endfunction
          sin (0)
            ⇒ 1
          builtin ("sin", 0)
            ⇒ 0

A single dynamically linked file might define several functions. However, as Octave searches for functions based on the functions filename, Octave needs a manner in which to find each of the functions in the dynamically linked file. On operating systems that support symbolic links, it is possible to create a symbolic link to the original file for each of the functions which it contains.

However, there is at least one well known operating system that doesn’t support symbolic links. Making copies of the original file for each of the functions is undesirable as it increases the amount of disk space used by Octave. Instead Octave supplies the autoload function, that permits the user to define in which file a certain function will be found.

[Built-in Function] autoload_map = autoload ()

[Built-in Function] autoload (function, file) [Built-in Function] autoload (…, "remove")

Define function to autoload from file.

The second argument, file, should be an absolute file name or a file name in the same directory as the function or script from which the autoload command was run. file should not depend on the Octave load path.

Normally, calls to autoload appear in PKG ADD script files that are evaluated when a directory is added to Octave’s load path. To avoid having to hardcode directory names in file, if file is in the same directory as the PKG ADD script then

     autoload ("foo", "bar.oct");

will load the function foo from the file bar.oct. The above usage when bar.oct is not in the same directory, or usages such as

     autoload ("foo", file_in_loadpath ("bar.oct"))

are strongly discouraged, as their behavior may be unpredictable.

With no arguments, return a structure containing the current autoload map.

If a third argument "remove" is given, the function is cleared and not loaded anymore during the current Octave session.

See also: [PKG ADD], page 754.

11.9.6 Function Locking

It is sometime desirable to lock a function into memory with the mlock function. This is typically used for dynamically linked functions in Oct-files or mex-files that contain some initialization, and it is desirable that calling clear does not remove this initialization.

As an example,

function my_function ()
  mlock ();
  ...

prevents my_function from being removed from memory after it is called, even if clear is called. It is possible to determine if a function is locked into memory with the mislocked, and to unlock a function with munlock, which the following illustrates.

my_function ();
mislocked ("my_function")
⇒ ans = 1
munlock ("my_function");
mislocked ("my_function")
⇒ ans = 0

A common use of mlock is to prevent persistent variables from being removed from memory, as the following example shows:

function count_calls ()
  mlock ();
  persistent calls = 0;
  printf ("'count_calls' has been called %d times\n",
          ++calls);
endfunction
count_calls ();
⊣ 'count_calls' has been called 1 times
clear count_calls
count_calls ();
⊣ 'count_calls' has been called 2 times

mlock might equally be used to prevent changes to a function from having effect in Octave, though a similar effect can be had with the ignore_function_time_stamp function.

[Built-in Function] mlock ()

Lock the current function into memory so that it can’t be cleared.

See also: [munlock], page 188, [mislocked], page 188, [persistent], page 124.

[Built-in Function] munlock () [Built-in Function] munlock (fcn)

Unlock the named function fcn. If no function is named then unlock the current function.

See also: [mlock], page 188, [mislocked], page 188, [persistent], page 124.

[Built-in Function] mislocked () [Built-in Function] mislocked (fcn)

Return true if the named function fcn is locked. If no function is named then return true if the current function is locked.

See also: [mlock], page 188, [munlock], page 188, [persistent], page 124.

11.9.7 Function Precedence

Given the numerous different ways that Octave can define a function, it is possible and even likely that multiple versions of a function, might be defined within a particular scope. The precedence of which function will be used within a particular scope is given by

1. Subfunction A subfunction with the required function name in the given scope.

2. Private function A function defined within a private directory of the directory which contains the current function.

3. Class constructor A function that constuctors a user class as defined in chapter Chapter 34 [Object Oriented Programming], page 673.

4. Class method An overloaded function of a class as in chapter Chapter 34 [Object Oriented Programming], page 673.

5. Command-line Function A function that has been defined on the command-line.

6. Autoload function A function that is marked as autoloaded with See [autoload], page 186.

7. A Function on the Path A function that can be found on the users load-path. There can also be Oct-file, mex-file or m-file versions of this function and the precedence between these versions are in that order.

8. Built-in function A function that is a part of core Octave such as numel, size, etc.

11.10 Script Files

A script file is a file containing (almost) any sequence of Octave commands. It is read and evaluated just as if you had typed each command at the Octave prompt, and provides a convenient way to perform a sequence of commands that do not logically belong inside a function.

Unlike a function file, a script file must not begin with the keyword function. If it does, Octave will assume that it is a function file, and that it defines a single function that should be evaluated as soon as it is defined.

A script file also differs from a function file in that the variables named in a script file are not local variables, but are in the same scope as the other variables that are visible on the command line.

Even though a script file may not begin with the function keyword, it is possible to define more than one function in a single script file and load (but not execute) all of them at once. To do this, the first token in the file (ignoring comments and other white space) must be something other than function. If you have no other statements to evaluate, you can use a statement that has no effect, like this:

     # Prevent Octave from thinking that this
     # is a function file:
     1;
     # Define function one:
     function one ()
       ...

To have Octave read and compile these functions into an internal form, you need to make sure that the file is in Octave’s load path (accessible through the path function), then simply type the base name of the file that contains the commands. (Octave uses the same rules to search for script files as it does to search for function files.)

If the first token in a file (ignoring comments) is function, Octave will compile the function and try to execute it, printing a message warning about any non-whitespace characters that appear after the function definition.

Note that Octave does not try to look up the definition of any identifier until it needs to evaluate it. This means that Octave will compile the following statements if they appear in a script file, or are typed at the command line,

     # not a function file:
     1;
     function foo ()
       do_something ();
     endfunction
     function do_something ()
       do_something_else ();
     endfunction

even though the function do_something is not defined before it is referenced in the function foo. This is not an error because Octave does not need to resolve all symbols that are referenced by a function until the function is actually evaluated.

Since Octave doesn’t look for definitions until they are needed, the following code will always print ‘bar = 3’ whether it is typed directly on the command line, read from a script file, or is part of a function body, even if there is a function or script file called ‘bar.m’ in Octave’s path.

     eval ("bar = 3");
     bar

Code like this appearing within a function body could fool Octave if definitions were resolved as the function was being compiled. It would be virtually impossible to make Octave clever enough to evaluate this code in a consistent fashion. The parser would have to be able to perform the call to eval at compile time, and that would be impossible unless all the references in the string to be evaluated could also be resolved, and requiring that would be too restrictive (the string might come from user input, or depend on things that are not known until the function is evaluated).

Although Octave normally executes commands from script files that have the name ‘file.m’, you can use the function source to execute commands from any file.

[Built-in Function] source (file)

Parse and execute the contents of file.

This is equivalent to executing commands from a script file, but without requiring the file to be named ‘file.m’.

See also: [run], page 152.

11.11 Function Handles, Anonymous Functions, Inline Functions

It can be very convenient store a function in a variable so that it can be passed to a different function. For example, a function that performs numerical minimization needs access to the function that should be minimized.

11.11.1 Function Handles

A function handle is a pointer to another function and is defined with the syntax

     @function-name

For example,

f = @sin;

creates a function handle called f that refers to the function sin.

Function handles are used to call other functions indirectly, or to pass a function as an argument to another function like quad or fsolve. For example:

f = @sin;
quad (f, 0, pi)
    ⇒ 2

You may use feval to call a function using function handle, or simply write the name of the function handle followed by an argument list. If there are no arguments, you must use an empty argument list ‘()’. For example:

f = @sin;
feval (f, pi/4)
    ⇒ 0.70711
f (pi/4)
    ⇒ 0.70711

[Built-in Function] is_function_handle (x)

Return true if x is a function handle.

See also: [isa], page 39, [typeinfo], page 39, [class], page 39, [functions], page 191.

[Built-in Function] s = functions (fcn_handle)

Return a structure containing information about the function handle fcn handle.

The structure s always contains these 3 fields:

function The function name. For an anonymous function (no name) this will be the actual function definition.

type Type of the function.

anonymous The function is anonymous.

private The function is private.

overloaded The function overloads an existing function.

simple The function is a built-in or m-file function.

subfunction The function is a subfunction within an m-file.

file The m-file that will be called to perform the function. This field is empty for anonymous and built-in functions.

In addition, some function types may return more information in additional fields.

Warning: functions is provided for debugging purposes only. It’s behavior may change in the future and programs should not depend on a particular output.

[Built-in Function] func2str (fcn_handle)

Return a string containing the name of the function referenced by the function handle fcn handle.

See also: [str2func], page 192, [functions], page 191.

[Built-in Function] str2func (fcn_name) [Built-in Function] str2func (fcn_name, "global")

Return a function handle constructed from the string fcn name. If the optional "global" argument is passed, locally visible functions are ignored in the lookup.

See also: [func2str], page 192, [inline], page 192.

11.11.2 Anonymous Functions

Anonymous functions are defined using the syntax

@(argument-list) expression

Any variables that are not found in the argument list are inherited from the enclosing scope. Anonymous functions are useful for creating simple unnamed functions from expressions or for wrapping calls to other functions to adapt them for use by functions like quad. For example,

f = @(x) x.^2;
quad (f, 0, 10)
    ⇒ 333.33

creates a simple unnamed function from the expression x.^2 and passes it to quad,

quad (@(x) sin (x), 0, pi)
    ⇒ 2

wraps another function, and

a = 1;
b = 2;
quad (@(x) betainc (x, a, b), 0, 0.4)
    ⇒ 0.13867

adapts a function with several parameters to the form required by quad. In this example, the values of a and b that are passed to betainc are inherited from the current environment.

11.11.3 Inline Functions

An inline function is created from a string containing the function body using the inline function. The following code defines the function f(x) = x^{2} + 2.

f = inline ("x^2 + 2");

After this it is possible to evaluate f at any x by writing f(x).

[Built-in Function] inline (str) [Built-in Function] inline (str, arg1, …) [Built-in Function] inline (str, n)

Create an inline function from the character string str. If called with a single argument, the arguments of the generated function are extracted from the function itself. The generated function arguments will then be in alphabetical order. It should be noted that i, and j are ignored as arguments due to the ambiguity between their use as a variable or their use as an inbuilt constant. All arguments followed by a parenthesis are considered to be functions. If no arguments are found, a function taking a single argument named x will be created. If the second and subsequent arguments are character strings, they are the names of the arguments of the function. If the second argument is an integer n, the arguments are "x", "P1", …, "PN". Programming Note: The use of inline is discouraged and it may be removed from a future version of Octave. The preferred way to create functions from strings is through the use of anonymous functions (see Section 11.11.2 [Anonymous Functions], page 192) or str2func.

See also: [argnames], page 193, [formula], page 193, [vectorize], page 458, [str2func], page 192.

[Built-in Function] argnames (fun)

Return a cell array of character strings containing the names of the arguments of the inline function fun.

See also: [inline], page 192, [formula], page 193, [vectorize], page 458.

[Built-in Function] formula (fun)

Return a character string representing the inline function fun. Note that char (fun) is equivalent to formula (fun).

See also: [argnames], page 193, [inline], page 192, [vectorize], page 458.

[Function File] symvar (s)

Identify the argument names in the function defined by a string. Common constant names such as pi, NaN, Inf, eps, i or j are ignored. The arguments that are found are returned in a cell array of strings. If no variables are found then the returned cell array is empty.

11.12 Commands

Commands are a special class of functions that only accept string input arguments. A command can be called as an ordinary function, but it can also be called without the parentheses. For example,

my_command hello world

is equivalent to

my_command ("hello", "world")

The general form of a command call is

cmdname arg1 arg2 ...

which translates directly to

cmdname ("arg1", "arg2", ...)

Any regular function can be used as a command if it accepts string input arguments. For example:

     toupper lower_case_arg
        ⇒ ans = LOWER_CASE_ARG

One difficulty of commands occurs when one of the string input arguments is stored in a variable. Because Octave can’t tell the difference between a variable name and an ordinary string, it is not possible to pass a variable as input to a command. In such a situation a command must be called as a function. For example:

     strvar = "hello world";
     toupper strvar
        ⇒ ans = STRVAR
     toupper (strvar)
        ⇒ ans = HELLO WORLD

11.13 Organization of Functions Distributed with Octave

Many of Octave’s standard functions are distributed as function files. They are loosely organized by topic, in subdirectories of ‘octave-home/lib/octave/version/m’, to make it easier to find them.

The following is a list of all the function file subdirectories, and the types of functions you will find there.

‘audio’ Functions for playing and recording sounds.

‘deprecated’

Out-of-date functions which will eventually be removed from Octave.

‘elfun’ Elementary functions, principally trigonometric.

‘@ftp’ Class functions for the FTP object.

‘general’ Miscellaneous matrix manipulations, like flipud, rot90, and triu, as well as other basic functions, like ismatrix, nargchk, etc.

‘geometry’

Functions related to Delaunay triangulation.

‘help’ Functions for Octave’s built-in help system.

‘image’ Image processing tools. These functions require the X Window System.

‘io’ Input-output functions.

‘linear-algebra’

Functions for linear algebra.

‘miscellaneous’

Functions that don’t really belong anywhere else.

‘optimization’

Functions related to minimization, optimization, and root finding.

‘path’ Functions to manage the directory path Octave uses to find functions.

‘pkg’ Package manager for installing external packages of functions in Octave.

‘plot’ Functions for displaying and printing two- and three-dimensional graphs.

‘polynomial’

Functions for manipulating polynomials.

‘prefs’ Functions implementing user-defined preferences. ‘set’ Functions for creating and manipulating sets of unique values. ‘signal’ Functions for signal processing applications. ‘sparse’ Functions for handling sparse matrices. ‘specfun’ Special functions such as bessel or factor.

‘special-matrix’

Functions that create special matrix forms such as Hilbert or Vandermonde matrices.

‘startup’ Octave’s system-wide startup file.

‘statistics’

Statistical functions.

‘strings’ Miscellaneous string-handling functions. ‘testfun’ Functions for performing unit tests on other functions. ‘time’ Functions related to time and date processing.

12 Errors and Warnings

Octave includes several functions for printing error and warning messages. When you write functions that need to take special action when they encounter abnormal conditions, you should print the error messages using the functions described in this chapter.

Since many of Octave’s functions use these functions, it is also useful to understand them, so that errors and warnings can be handled.

12.1 Handling Errors

An error is something that occurs when a program is in a state where it doesn’t make sense to continue. An example is when a function is called with too few input arguments. In this situation the function should abort with an error message informing the user of the lacking input arguments.

Since an error can occur during the evaluation of a program, it is very convenient to be able to detect that an error occurred, so that the error can be fixed. This is possible with the try statement described in Section 10.9 [The try Statement], page 164.

12.1.1 Raising Errors

The most common use of errors is for checking input arguments to functions. The following example calls the error function if the function f is called without any input arguments.

     function f (arg1)
       if (nargin == 0)
         error ("not enough input arguments");
       endif
     endfunction

When the error function is called, it prints the given message and returns to the Octave prompt. This means that no code following a call to error will be executed.

[Built-in Function] error (template, …) [Built-in Function] error (id, template, …)

Format the optional arguments under the control of the template string template using the same rules as the printf family of functions (see Section 14.2.4 [Formatted Output], page 243) and print the resulting message on the stderr stream. The message is prefixed by the character string ‘error: ’. Calling error also sets Octave’s internal error state such that control will return to the top level without evaluating any more commands. This is useful for aborting from functions or scripts. If the error message does not end with a new line character, Octave will print a traceback of all the function calls leading to the error. For example, given the following function definitions:

          function f () g (); end
          function g () h (); end
          function h () nargin == 1 || error ("nargin != 1"); end

calling the function f will result in a list of messages that can help you to quickly locate the exact location of the error:

f ()
error: nargin != 1
error: called from:
error:   error at line -1, column -1
error:   h at line 1, column 27
error:   g at line 1, column 15
error:   f at line 1, column 15

If the error message ends in a new line character, Octave will print the message but will not display any traceback messages as it returns control to the top level. For example, modifying the error message in the previous example to end in a new line causes Octave to only print a single message:

function h () nargin == 1 || error ("nargin != 1\n"); end
f ()
error: nargin != 1

A null string ("") input to error will be ignored and the code will continue running as if the statement were a NOP. This is for compatibility with matlab. It also makes it possible to write code such as

err_msg = "";
if (CONDITION 1)
  err_msg = "CONDITION 1 found";
elseif (CONDITION2)
  err_msg = "CONDITION 2 found";
...
endif
error (err_msg);

which will only stop execution if an error has been found.

Implementation Note: For compatibility with matlab, escape sequences (e.g., "\n" => newline) are processed in template regardless of whether template has been defined within single quotes as long as there are two or more input arguments. Use a second backslash to stop interpolation of the escape sequence (e.g., "\\n") or use the

regexptranslate function.

See also: [warning], page 204, [lasterror], page 201.

Since it is common to use errors when there is something wrong with the input to a function, Octave supports functions to simplify such code. When the print_usage function is called, it reads the help text of the function calling print_usage, and presents a useful error. If the help text is written in Texinfo it is possible to present an error message that only contains the function prototypes as described by the @deftypefn parts of the help text. When the help text isn’t written in Texinfo, the error message contains the entire help message.

Consider the following function.

## -*- texinfo -*-
## @deftypefn {Function File} f (@var{arg1})
## Function help text goes here...
## @end deftypefn
function f (arg1)
  if (nargin == 0)
    print_usage ();
  endif
endfunction

When it is called with no input arguments it produces the following error.

f ()
⊣ error: Invalid call to f.  Correct usage is:

⊣

⊣ -- Function File: f (ARG1)

⊣ ⊣

⊣ Additional help for built-in functions and operators is
⊣ available in the online version of the manual.  Use the command
⊣ `doc <topic>' to search the manual index.

⊣

⊣ Help and information about Octave is also available on the WWW
⊣ at http://www.octave.org and via the help@octave.org
⊣ mailing list.

[Function File] print_usage () [Function File] print_usage (name)

Print the usage message for a function. When called with no input arguments the print_usage function displays the usage message of the currently executing function.

See also: [help], page 20.

[Built-in Function] usage (msg)

Print the message msg, prefixed by the string ‘usage: ’, and set Octave’s internal error state such that control will return to the top level without evaluating any more commands. This is useful for aborting from functions.

After usage is evaluated, Octave will print a traceback of all the function calls leading to the usage message.

You should use this function for reporting problems errors that result from an improper call to a function, such as calling a function with an incorrect number of arguments, or with arguments of the wrong type. For example, most functions distributed with Octave begin with code like this

     if (nargin != 2)
       usage ("foo (a, b)");
     endif

to check for the proper number of arguments.

[Function File] beep ()

Produce a beep from the speaker (or visual bell).

See also: [puts], page 242, [fputs], page 242, [printf], page 243, [fprintf], page 243.

[Built-in Function] val = beep_on_error ()

[Built-in Function] old_val = beep_on_error (new_val)

[Built-in Function] beep_on_error (new_val, "local")

Query or set the internal variable that controls whether Octave will try to ring the terminal bell before printing an error message.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

12.1.2 Catching Errors

When an error occurs, it can be detected and handled using the try statement as described in Section 10.9 [The try Statement], page 164. As an example, the following piece of code counts the number of errors that occurs during a for loop.

number_of_errors = 0;
for n = 1:100
  try
    ...
  catch
    number_of_errors++;
  end_try_catch
endfor

The above example treats all errors the same. In many situations it can however be necessary to discriminate between errors, and take different actions depending on the error. The lasterror function returns a structure containing information about the last error that occurred. As an example, the code above could be changed to count the number of errors related to the ‘*’ operator.

number_of_errors = 0;
for n = 1:100
  try
    ...
  catch
    msg = lasterror.message;
    if (strfind (msg, "operator *"))
      number_of_errors++;
    endif
  end_try_catch
endfor

Alternatively, the output of the lasterror function can be found in a variable indicated immediately after the catch keyword, as in the example below showing how to redirect an error as a warning:

try
  ...
catch err
  warning(err.identifier, err.message);
  ...
end_try_catch

[Built-in Function] lasterr = lasterror () [Built-in Function] lasterror (err) [Built-in Function] lasterror ("reset")

Query or set the last error message structure. When called without arguments, return a structure containing the last error message and other information related to this error. The elements of the structure are:

message The text of the last error message

identifier

The message identifier of this error message

stack A structure containing information on where the message occurred. This may be an empty structure if the information cannot be obtained. The fields of the structure are:

file The name of the file where the error occurred

name The name of function in which the error occurred

line The line number at which the error occurred

column An optional field with the column number at which the error occurred

The last error structure may be set by passing a scalar structure, err, as input. Any fields of err that match those above are set while any unspecified fields are initialized with default values. If lasterror is called with the argument "reset", all fields are set to their default values.

See also: [lasterr], page 201, [error], page 197, [lastwarn], page 205.

[Built-in Function] [msg, msgid] = lasterr () [Built-in Function] lasterr (msg) [Built-in Function] lasterr (msg, msgid)

Query or set the last error message. When called without input arguments, return the last error message and message identifier. With one argument, set the last error message to msg. With two arguments, also set the last message identifier.

See also: [lasterror], page 201, [error], page 197, [lastwarn], page 205.

It is also possible to assign an identification string to an error. If an error has such an ID the user can catch this error as will be shown in the next example. To assign an ID to an error, simply call error with two string arguments, where the first is the identification string, and the second is the actual error. Note that error IDs are in the format "NAMESPACE:ERROR-NAME". The namespace "Octave" is used for Octave’s own errors. Any other string is available as a namespace for user’s own errors.

The next example counts indexing errors. The errors are caught using the field identifier of the structure returned by the function lasterror.

     number_of_errors = 0;
     for n = 1:100
       try
         ...
       catch
         id = lasterror.identifier;
         if (strcmp (id, "Octave:invalid-indexing"))
           number_of_errors++;
         endif
       end_try_catch
     endfor

The functions distributed with Octave can issue one of the following errors.

Octave:invalid-context

Indicates the error was generated by an operation that cannot be executed in the scope from which it was called. For example, the function print_usage () when called from the Octave prompt raises this error.

Octave:invalid-input-arg

Indicates that a function was called with invalid input arguments.

Octave:invalid-fun-call

Indicates that a function was called in an incorrect way, e.g., wrong number of input arguments.

Octave:invalid-indexing

Indicates that a data-type was indexed incorrectly, e.g., real-value index for arrays, non-existent field of a structure.

Octave:bad-alloc

Indicates that memory couldn’t be allocated.

Octave:undefined-function

Indicates a call to a function that is not defined. The function may exist but Octave is unable to find it in the search path.

When an error has been handled it is possible to raise it again. This can be useful when an error needs to be detected, but the program should still abort. This is possible using the rethrow function. The previous example can now be changed to count the number of errors related to the ‘*’ operator, but still abort if another kind of error occurs.

number_of_errors = 0;
for n = 1:100
  try
    ...
  catch
    msg = lasterror.message;
    if (strfind (msg, "operator *"))
      number_of_errors++;
    else
      rethrow (lasterror);
    endif
  end_try_catch
endfor

[Built-in Function] rethrow (err)

Reissue a previous error as defined by err. err is a structure that must contain at least the "message" and "identifier" fields. err can also contain a field "stack" that gives information on the assumed location of the error. Typically err is returned

from lasterror.

See also: [lasterror], page 201, [lasterr], page 201, [error], page 197.

[Built-in Function] err = errno () [Built-in Function] err = errno (val) [Built-in Function] err = errno (name)

Return the current value of the system-dependent variable errno, set its value to val and return the previous value, or return the named error code given name as a character string, or -1 if name is not found.

[Built-in Function] errno_list ()

Return a structure containing the system-dependent errno values.

12.1.3 Recovering From Errors

Octave provides several ways of recovering from errors. There are try/catch blocks,

unwind_protect/unwind_protect_cleanup blocks, and finally the onCleanup command.

The onCleanup command associates an ordinary Octave variable (the trigger) with an arbitrary function (the action). Whenever the Octave variable ceases to exist—whether due to a function return, an error, or simply because the variable has been removed with clear—then the assigned function is executed.

The function can do anything necessary for cleanup such as closing open file handles, printing an error message, or restoring global variables to their initial values. The last example is a very convenient idiom for Octave code. For example:

function rand42
  old_state = rand ("state");
  restore_state = onCleanup (@() rand ("state", old_state);
  rand ("state", 42);
  ...
endfunction  # rand generator state restored by onCleanup

[Built-in Function] obj = onCleanup (function)

Create a special object that executes a given function upon destruction. If the object is copied to multiple variables (or cell or struct array elements) or returned from a function, function will be executed after clearing the last copy of the object. Note that if multiple local onCleanup variables are created, the order in which they are called is unspecified. For similar functionality See Section 10.8 [The unwind protect Statement], page 164.

12.2 Handling Warnings

Like an error, a warning is issued when something unexpected happens. Unlike an error, a warning doesn’t abort the currently running program. A simple example of a warning is when a number is divided by zero. In this case Octave will issue a warning and assign the value Inf to the result.

     a = 1/0
          ⊣ warning: division by zero
          ⇒ a = Inf

12.2.1 Issuing Warnings

It is possible to issue warnings from any code using the warning function. In its most simple form, the warning function takes a string describing the warning as its input argument. As an example, the following code controls if the variable ‘a’ is non-negative, and if not issues a warning and sets ‘a’ to zero.

     a = -1;
     if (a < 0)
       warning ("'a' must be non-negative.  Setting 'a' to zero.");
       a = 0;
     endif
          ⊣ 'a' must be non-negative.  Setting 'a' to zero.

Since warnings aren’t fatal to a running program, it is not possible to catch a warning using the try statement or something similar. It is however possible to access the last warning as a string using the lastwarn function.

It is also possible to assign an identification string to a warning. If a warning has such an ID the user can enable and disable this warning as will be described in the next section. To assign an ID to a warning, simply call warning with two string arguments, where the first is the identification string, and the second is the actual warning. Note that warning IDs are in the format "NAMESPACE:WARNING-NAME". The namespace "Octave" is used for Octave’s own warnings. Any other string is available as a namespace for user’s own warnings.

[Built-in Function] warning (template, …) [Built-in Function] warning (id, template, …) [Built-in Function] warning ("on", id) [Built-in Function] warning ("off", id) [Built-in Function] warning ("query", id) [Built-in Function] warning ("error", id)

[Built-in Function] warning (state, id, "local")

Format the optional arguments under the control of the template string template using the same rules as the printf family of functions (see Section 14.2.4 [Formatted Output], page 243) and print the resulting message on the stderr stream. The message is prefixed by the character string ‘warning: ’. You should use this function when you want to notify the user of an unusual condition, but only when it makes sense for your program to go on. The optional message identifier allows users to enable or disable warnings tagged by id. A message identifier is of the form "NAMESPACE:WARNING-NAME". Octave’s own warnings use the "Octave" namespace (see [XREFwarning ids], page 205). The special identifier "all" may be used to set the state of all warnings. If the first argument is "on" or "off", set the state of a particular warning using the identifier id. If the first argument is "query", query the state of this warning instead. If the identifier is omitted, a value of "all" is assumed. If you set the state of a warning to "error", the warning named by id is handled as if it were an error instead. So, for example, the following handles all warnings as errors:

     warning ("error");

If the state is "on", "off", or "error" and the third argument is "local", then the warning state will be set temporarily, until the end of the current function. Changes to warning states that are set locally affect the current function and all functions called from the current scope. The previous warning state is restored on return from the current function. The "local" option is ignored if used in the top-level workspace. Implementation Note: For compatibility with matlab, escape sequences (e.g., "\n" => newline) are processed in template regardless of whether template has been defined within single quotes as long as there are two or more input arguments. Use a second backslash to stop interpolation of the escape sequence (e.g., "\\n") or use the

regexptranslate function.

See also: [warning ids], page 205, [lastwarn], page 205, [error], page 197.

[Built-in Function] [msg, msgid] = lastwarn ()

[Built-in Function] lastwarn (msg) [Built-in Function] lastwarn (msg, msgid)

Query or set the last warning message. When called without input arguments, return the last warning message and message identifier. With one argument, set the last warning message to msg. With two arguments, also set the last message identifier.

See also: [warning], page 204, [lasterror], page 201, [lasterr], page 201.

The functions distributed with Octave can issue one of the following warnings.

Octave:abbreviated-property-match
     By default, the Octave:abbreviated-property-match warning is enabled.
Octave:array-to-scalar

If the Octave:array-to-scalar warning is enabled, Octave will warn when an implicit conversion from an array to a scalar value is attempted. By default,

     the Octave:array-to-scalar warning is disabled.

Octave:array-to-vector

If the Octave:array-to-vector warning is enabled, Octave will warn when an implicit conversion from an array to a vector value is attempted. By default,

the Octave:array-to-vector warning is disabled.
Octave:assign-as-truth-value

If the Octave:assign-as-truth-value warning is enabled, a warning is issued for statements like

     if (s = t) ...

since such statements are not common, and it is likely that the intent was to write

     if (s == t) ...

instead. There are times when it is useful to write code that contains assignments within the condition of a while or if statement. For example, statements like

     while (c = getc ())
       ...

are common in C programming. It is possible to avoid all warnings about such statements by disabling the Octave:assign-as-truth-value warning, but that may also let real errors like

     if (x = 1)  # intended to test (x == 1)!
       ...

slip by. In such cases, it is possible suppress errors for specific statements by writing them with an extra set of parentheses. For example, writing the previous example as

     while ((c = getc ()))
       ...

will prevent the warning from being printed for this statement, while allowing Octave to warn about other assignments used in conditional contexts. By default, the Octave:assign-as-truth-value warning is enabled.

Octave:associativity-change

If the Octave:associativity-change warning is enabled, Octave will warn about possible changes in the meaning of some code due to changes in associativity for some operators. Associativity changes have typically been made for matlab compatibility. By default, the Octave:associativity-change warning is enabled.

Octave:autoload-relative-file-name
If the Octave:autoload-relative-file-name is enabled, Octave will warn

when parsing autoload() function calls with relative paths to function files.

This usually happens when using autoload() calls in PKG ADD files, when the PKG ADD file is not in the same directory as the .oct file referred to by the

autoload() command. By default, the Octave:autoload-relative-file-name

warning is enabled.

Octave:broadcast

Warn when performing broadcasting operations. By default, this is enabled. See Section 19.2 [Broadcasting], page 459 in the chapter Vectorization and Faster Code Execution of the manual.

Octave:built-in-variable-assignment
By default, the Octave:built-in-variable-assignment warning is enabled.
Octave:deprecated-keyword

If the Octave:deprecated-keyword warning is enabled, a warning is issued when Octave encounters a keyword that is obsolete and scheduled for removal from Octave. By default, the Octave:deprecated-keyword warning is enabled.

Octave:divide-by-zero

If the Octave:divide-by-zero warning is enabled, a warning is issued when Octave encounters a division by zero. By default, the Octave:divide-by-zero warning is enabled.

Octave:fopen-file-in-path

By default, the Octave:fopen-file-in-path warning is enabled.

Octave:function-name-clash

If the Octave:function-name-clash warning is enabled, a warning is issued when Octave finds that the name of a function defined in a function file differs from the name of the file. (If the names disagree, the name declared inside the file is ignored.) By default, the Octave:function-name-clash warning is enabled.

Octave:future-time-stamp

If the Octave:future-time-stamp warning is enabled, Octave will print a warning if it finds a function file with a time stamp that is in the future. By default, the Octave:future-time-stamp warning is enabled.

Octave:glyph-render

By default, the Octave:glyph-render warning is enabled.

Octave:imag-to-real

If the Octave:imag-to-real warning is enabled, a warning is printed for implicit conversions of complex numbers to real numbers. By default, the

Octave:imag-to-real warning is disabled.
Octave:load-file-in-path

By default, the Octave:load-file-in-path warning is enabled.

Octave:logical-conversion

By default, the Octave:logical-conversion warning is enabled.

Octave:matlab-incompatible

Print warnings for Octave language features that may cause compatibility problems with matlab. By default, the Octave:matlab-incompatible warning is disabled. The –traditional or –braindead startup options for Octave may also be of use, see Section 2.1.1 [Command Line Options], page 15.

Octave:md5sum-file-in-path

By default, the Octave:md5sum-file-in-path warning is enabled.

Octave:missing-glyph

By default, the Octave:missing-glyph warning is enabled.

Octave:missing-semicolon

If the Octave:missing-semicolon warning is enabled, Octave will warn when statements in function definitions don’t end in semicolons. By default the

Octave:missing-semicolon warning is disabled.
Octave:mixed-string-concat

If the Octave:mixed-string-concat warning is enabled, print a warning when concatenating a mixture of double and single quoted strings. By default, the

Octave:mixed-string-concat warning is disabled.
Octave:neg-dim-as-zero

If the Octave:neg-dim-as-zero warning is enabled, print a warning for expressions like

     eye (-1)

By default, the Octave:neg-dim-as-zero warning is disabled.

Octave:nested-functions-coerced
By default, the Octave:nested-functions-coerced warning is enabled.
Octave:noninteger-range-as-index
By default, the Octave:noninteger-range-as-index warning is enabled.
Octave:num-to-str

If the Octave:num-to-str warning is enable, a warning is printed for implicit conversions of numbers to their ASCII character equivalents when strings are constructed using a mixture of strings and numbers in matrix notation. For example,

     [ "f", 111, 111 ] ⇒ "foo"

elicits a warning if the Octave:num-to-str warning is enabled. By default, the

Octave:num-to-str warning is enabled.
Octave:possible-matlab-short-circuit-operator
If the Octave:possible-matlab-short-circuit-operator warning is

enabled, Octave will warn about using the not short circuiting operators & and | inside if or while conditions. They normally never short circuit, but matlab always short circuits if any logical operators are used in a condition. You can turn on the option

     do_braindead_shortcircuit_evaluation (1)

if you would like to enable this short-circuit evaluation in Octave. Note that the && and || operators always short circuit in both Octave and matlab, so it’s only necessary to enable matlab-style short-circuiting if it’s too arduous to modify existing code that relies on this behavior. By default, the Octave:possible-

matlab-short-circuit-operator warning is enabled.
Octave:precedence-change

If the Octave:precedence-change warning is enabled, Octave will warn about possible changes in the meaning of some code due to changes in precedence for some operators. Precedence changes have typically been made for matlab compatibility. By default, the Octave:precedence-change warning is enabled.

Octave:recursive-path-search

By default, the Octave:recursive-path-search warning is enabled.

Octave:remove-init-dir

The path function changes the search path that Octave uses to find functions. It is possible to set the path to a value which excludes Octave’s own built-in functions. If the Octave:remove-init-dir warning is enabled then Octave will warn when the path function has been used in a way that may render Octave unworkable. By default, the Octave:remove-init-dir warning is enabled.

Octave:reload-forces-clear

If several functions have been loaded from the same file, Octave must clear all the functions before any one of them can be reloaded. If the Octave:reloadforces-clear warning is enabled, Octave will warn you when this happens, and print a list of the additional functions that it is forced to clear. By default,

the Octave:reload-forces-clear warning is enabled.
Octave:resize-on-range-error

If the Octave:resize-on-range-error warning is enabled, print a warning when a matrix is resized by an indexed assignment with indices outside the current bounds. By default, the ## Octave:resize-on-range-error warning is disabled.

Octave:separator-insert

Print warning if commas or semicolons might be inserted automatically in literal matrices. By default, the Octave:separator-insert warning is disabled.

Octave:shadowed-function

By default, the Octave:shadowed-function warning is enabled.

Octave:single-quote-string

Print warning if a single quote character is used to introduce a string constant. By default, the Octave:single-quote-string warning is disabled.

Octave:singular-matrix-div

By default, the Octave:singular-matrix-div warning is enabled.

Octave:sqrtm:SingularMatrix

By default, the Octave:sqrtm:SingularMatrix warning is enabled.

Octave:str-to-num

If the Octave:str-to-num warning is enabled, a warning is printed for implicit conversions of strings to their numeric ASCII equivalents. For example,

               "abc" + 0 ⇒ 97 98 99

elicits a warning if the Octave:str-to-num warning is enabled. By default, the Octave:str-to-num warning is disabled.

Octave:undefined-return-values

If the Octave:undefined-return-values warning is disabled, print a warning if a function does not define all the values in the return list which are expected.

          By default, the Octave:undefined-return-values warning is enabled.
Octave:variable-switch-label

If the Octave:variable-switch-label warning is enabled, Octave will print a warning if a switch label is not a constant or constant expression. By default,

          the Octave:variable-switch-label warning is disabled.

12.2.2 Enabling and Disabling Warnings

The warning function also allows you to control which warnings are actually printed to the screen. If the warning function is called with a string argument that is either "on" or "off" all warnings will be enabled or disabled.

It is also possible to enable and disable individual warnings through their string identifications. The following code will issue a warning

     warning ("example:non-negative-variable",
              "'a' must be non-negative.  Setting 'a' to zero.");

while the following won’t issue a warning

     warning ("off", "example:non-negative-variable");
     warning ("example:non-negative-variable",
              "'a' must be non-negative.  Setting 'a' to zero.");

13 Debugging

Octave includes a built-in debugger to aid in the development of scripts. This can be used to interrupt the execution of an Octave script at a certain point, or when certain conditions are met. Once execution has stopped, and debug mode is entered, the symbol table at the point where execution has stopped can be examined and modified to check for errors.

The normal command-line editing and history functions are available in debug mode.

13.1 Entering Debug Mode

There are two basic means of interrupting the execution of an Octave script. These are breakpoints (see Section 13.3 [Breakpoints], page 212), discussed in the next section, and interruption based on some condition.

Octave supports three means to stop execution based on the values set in the functions

debug_on_interrupt, debug_on_warning, and debug_on_error.
[Built-in Function] val = debug_on_interrupt ()
[Built-in Function] old_val = debug_on_interrupt (new_val)
[Built-in Function] debug_on_interrupt (new_val, "local")

Query or set the internal variable that controls whether Octave will try to enter debugging mode when it receives an interrupt signal (typically generated with C-c). If a second interrupt signal is received before reaching the debugging mode, a normal interrupt will occur. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [debug on error], page 211, [debug on warning], page 211.

[Built-in Function] val = debug_on_warning ()
[Built-in Function] old_val = debug_on_warning (new_val)

[Built-in Function] debug_on_warning (new_val, "local")

Query or set the internal variable that controls whether Octave will try to enter the debugger when a warning is encountered. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [debug on error], page 211, [debug on interrupt], page 211.

[Built-in Function] val = debug_on_error ()

[Built-in Function] old_val = debug_on_error (new_val)

[Built-in Function] debug_on_error (new_val, "local")

Query or set the internal variable that controls whether Octave will try to enter the debugger when an error is encountered. This will also inhibit printing of the normal traceback message (you will only see the top-level error message). When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [debug on warning], page 211, [debug on interrupt], page 211.

13.2 Leaving Debug Mode

Use either dbcont or return to leave the debug mode and continue the normal execution of the script.

[Command] dbcont

Leave command-line debugging mode and continue code execution normally.

See also: [dbstep], page 215, [dbquit], page 212.

To quit debug mode and return directly to the prompt without executing any additional code use dbquit.

[Command] dbquit

Quit debugging mode immediately without further code execution and return to the Octave prompt.

See also: [dbcont], page 212, [dbstep], page 215.

Finally, typing exit or quit at the debug prompt will result in Octave terminating normally.

13.3 Breakpoints

Breakpoints can be set in any m-file function by using the dbstop function.

[Built-in Function] rline = dbstop ("func")

[Built-in Function] rline = dbstop ("func", line)
[Built-in Function] rline = dbstop ("func", line1, line2, . . .)

Set a breakpoint in function func.

Arguments are

func Function name as a string variable. When already in debug mode this should be left out and only the line should be given.

line Line number where the breakpoint should be set. Multiple lines may be given as separate arguments or as a vector.

When called with a single argument func, the breakpoint is set at the first executable line in the named function.

The optional output rline is the real line number where the breakpoint was set. This can differ from specified line if the line is not executable. For example, if a breakpoint attempted on a blank line then Octave will set the real breakpoint at the next executable line.

See also: [dbclear], page 213, [dbstatus], page 213, [dbstep], page 215, [debug on error], page 211, [debug on warning], page 211, [debug on interrupt], page 211.

Breakpoints in class methods are also supported (e.g., dbstop ("@class/method")). However, breakpoints cannot be set in built-in functions (e.g., sin, etc.) or dynamically loaded functions (i.e., oct-files).

To set a breakpoint immediately upon entering a function use line number 1, or omit the line number entirely and just give the function name. When setting the breakpoint Octave will ignore the leading comment block, and the breakpoint will be set on the first executable statement in the function. For example:

dbstop ("asind", 1)
⇒ 29

Note that the return value of 29 means that the breakpoint was effectively set to line 29. The status of breakpoints in a function can be queried with dbstatus.

[Built-in Function] dbstatus () [Built-in Function] brk_list = dbstatus ()

[Built-in Function] brk_list = dbstatus ("func")

Report the location of active breakpoints. When called with no input or output arguments, print the list of all functions with breakpoints and the line numbers where those breakpoints are set. If a function name func is specified then only report breakpoints for the named function. The optional return argument brk list is a struct array with the following fields.

name The name of the function with a breakpoint.

file The name of the m-file where the function code is located.

line A line number, or vector of line numbers, with a breakpoint.

See also: [dbclear], page 213, [dbwhere], page 214.

Reusing the previous example, dbstatus ("asind") will return 29. The breakpoints listed can then be cleared with the dbclear function.

[Built-in Function] dbclear ("func") [Built-in Function] dbclear ("func", line, …) [Built-in Function] dbclear (line, …)

Delete a breakpoint in the function func. Arguments are

func Function name as a string variable. When already in debug mode this argument should be omitted and only the line number should be given.

line Line number from which to remove a breakpoint. Multiple lines may be given as separate arguments or as a vector.

When called without a line number specification all breakpoints in the named function are cleared. If the requested line is not a breakpoint no action is performed.

See also: [dbstop], page 212, [dbstatus], page 213, [dbwhere], page 214.

A breakpoint may also be set in a subfunction. For example, if a file contains the functions

function y = func1 (x)
  y = func2 (x);
endfunction
function y = func2 (x)
  y = x + 1;
endfunction

then a breakpoint can be set at the start of the subfunction directly with

dbstop (["func1", filemarker(), "func2"])
⇒ 5

Note that filemarker returns the character that marks subfunctions from the file containing them. Unless the default has been changed this character is ‘>’. Thus, a quicker and more normal way to set the breakpoint would be

dbstop func1>func2

Another simple way of setting a breakpoint in an Octave script is the use of the keyboard function.

[Built-in Function] keyboard () [Built-in Function] keyboard ("prompt")

This function is normally used for simple debugging. When the keyboard function is executed, Octave prints a prompt and waits for user input. The input strings are then evaluated and the results are printed. This makes it possible to examine the values of variables within a function, and to assign new values if necessary. To leave the prompt and return to normal execution type ‘return’ or ‘dbcont’. The keyboard function does not return an exit status. If keyboard is invoked without arguments, a default prompt of ‘debug> ’ is used.

See also: [dbstop], page 212, [dbcont], page 212, [dbquit], page 212.

The keyboard function is placed in a script at the point where the user desires that the execution be stopped. It automatically sets the running script into the debug mode.

13.4 Debug Mode

There are three additional support functions that allow the user to find out where in the execution of a script Octave entered the debug mode, and to print the code in the script surrounding the point where Octave entered debug mode.

[Command] dbwhere

In debugging mode, report the current file and line number where execution is stopped.

See also: [dbstatus], page 213, [dbcont], page 212, [dbstep], page 215, [dbup], page 216.

[Command] dbtype

[Command] dbtype lineno

[Command] dbtype startl:endl
[Command] dbtype startl:end
[Command] dbtype func
[Command] dbtype func lineno
[Command] dbtype func startl:endl
[Command] dbtype func startl:end

Display a script file with line numbers. When called with no arguments in debugging mode, display the script file currently being debugged. An optional range specification can be used to list only a portion of the file. The special keyword "end" is a valid line number specification for the last line of the file. When called with the name of a function, list that script file with line numbers.

See also: [dbwhere], page 214, [dbstatus], page 213, [dbstop], page 212.

[Command] dblist

[Command] dblist n

In debugging mode, list n lines of the function being debugged centered around the current line to be executed. If unspecified n defaults to 10 (+/- 5 lines)

See also: [dbwhere], page 214, [dbtype], page 214.

You may also use isdebugmode to determine whether the debugger is currently active.

[Built-in Function] isdebugmode ()

Return true if in debugging mode, otherwise false.

See also: [dbwhere], page 214, [dbstack], page 216, [dbstatus], page 213.

Debug mode also allows single line stepping through a function using the command

dbstep.

[Command] dbstep

[Command] dbstep n

[Command] dbstep in

[Command] dbstep out

[Command] dbnext …

In debugging mode, execute the next n lines of code. If n is omitted, execute the next single line of code. If the next line of code is itself defined in terms of an m-file remain in the existing function. Using dbstep in will cause execution of the next line to step into any m-files defined on the next line. Using dbstep out will cause execution to continue until the current function returns. dbnext is an alias for dbstep.

See also: [dbcont], page 212, [dbquit], page 212.

When in debug mode the RETURN key will execute the last entered command. This is useful, for example, after hitting a breakpoint and entering dbstep once. After that, one can advance line by line through the code with only a single key stroke.

13.5 Call Stack

The function being debugged may be the leaf node of a series of function calls. After examining values in the current subroutine it may turn out that the problem occurred in earlier pieces of code. Use dbup and dbdown to move up and down through the series of function calls to locate where variables first took on the wrong values. dbstack shows the entire series of function calls and at what level debugging is currently taking place.

[Command] dbstack

[Command] dbstack n

[Command] dbstack -completenames

[Built-in Function] [stack, idx] = dbstack (…)

Display or return current debugging function stack information. With optional argument n, omit the n innermost stack frames.

Although accepted, the argument -completenames is silently ignored. Octave always returns absolute file names. The arguments n and -completenames can be both specified in any order.

The optional return argument stack is a struct array with the following fields:

file The name of the m-file where the function code is located.

name The name of the function with a breakpoint.

line The line number of an active breakpoint.

column The column number of the line where the breakpoint begins.

scope Undocumented.

context Undocumented.

The return argument idx specifies which element of the stack struct array is currently active.

See also: [dbup], page 216, [dbdown], page 216, [dbwhere], page 214, [dbstatus], page 213.

[Built-in Function] dbup [Built-in Function] dbup (n)

In debugging mode, move up the execution stack n frames. If n is omitted, move up one frame.

See also: [dbstack], page 216, [dbdown], page 216.

[Built-in Function] dbdown [Built-in Function] dbdown (n)

In debugging mode, move down the execution stack n frames. If n is omitted, move down one frame.

See also: [dbstack], page 216, [dbup], page 216.

13.6 Profiling

Octave supports profiling of code execution on a per-function level. If profiling is enabled, each call to a function (supporting built-ins, operators, functions in oct- and mex-files, userdefined functions in Octave code and anonymous functions) is recorded while running Octave code. After that, this data can aid in analyzing the code behavior, and is in particular helpful for finding “hot spots” in the code which use up a lot of computation time and are the best targets to spend optimization efforts on.

The main command for profiling is profile, which can be used to start or stop the profiler and also to query collected data afterwards. The data is returned in an Octave data structure which can then be examined or further processed by other routines or tools.

[Command] profile on

[Command] profile off

[Command] profile resume

[Command] profile clear [Function File] S = profile ("status") [Function File] T = profile ("info")

Control the built-in profiler.

profile on

Start the profiler, clearing all previously collected data if there is any.

profile off

Stop profiling. The collected data can later be retrieved and examined

with calls like S = profile ("info").
profile clear

Clear all collected profiler data.

profile resume

Restart profiling without cleaning up the old data and instead all newly collected statistics are added to the already existing ones.

S = profile ("status")

Return a structure filled with certain information about the current status of the profiler. At the moment, the only field is ProfilerStatus which is either "on" or "off".

T = profile ("info")

Return the collected profiling statistics in the structure T. The flat profile is returned in the field FunctionTable which is an array of structures, each entry corresponding to a function which was called and for which profiling statistics are present. Furthermore, the field Hierarchical contains the hierarchical call-tree. Each node has an index into the FunctionTable identifying the function it corresponds to as well as data fields for number of calls and time spent at this level in the call-tree.

See also: [profshow], page 218, [profexplore], page 218.

An easy way to get an overview over the collected data is profshow. This function takes the profiler data returned by profile as input and prints a flat profile, for instance:

 Function Attr     Time (s)        Calls
----------------------------------------
   >myfib    R        2.195        13529
binary <=             0.061        13529
 binary -             0.050        13528
 binary +             0.026         6764

This shows that most of the run time was spent executing the function ‘myfib’, and some minor proportion evaluating the listed binary operators. Furthermore, it is shown how often the function was called and the profiler also records that it is recursive.

[Function File] profshow (data) [Function File] profshow (data, n)

Show flat profiler results. This command prints out profiler data as a flat profile. data is the structure returned by profile ("info"). If n is given, it specifies the number of functions to show in the profile; functions are sorted in descending order by total time spent in them. If there are more than n included in the profile, those will not be shown. n defaults to 20. The attribute column shows ‘R’ for recursive functions and nothing otherwise.

See also: [profexplore], page 218, [profile], page 217.

[Function File] profexplore () [Function File] profexplore (data)

Interactively explore hierarchical profiler output. Assuming data is the structure with profile data returned by profile ("info"), this command opens an interactive prompt that can be used to explore the call-tree. Type help to get a list of possible commands. If data is omitted, profile ("info") is called and used in its place.

See also: [profile], page 217, [profshow], page 218.

13.7 Profiler Example

Below, we will give a short example of a profiler session. See Section 13.6 [Profiling], page 217, for the documentation of the profiler functions in detail. Consider the code:

global N A;
N = 300;
A = rand (N, N);
function xt = timesteps (steps, x0, expM)
  global N;
  if (steps == 0)
    xt = NA (N, 0);
  else
    xt = NA (N, steps);

    x1 = expM * x0;
    xt(:, 1) = x1;
    xt(:, 2 : end) = timesteps (steps - 1, x1, expM);
  endif
endfunction
function foo ()
  global N A;
  initial = @(x) sin (x);
  x0 = (initial (linspace (0, 2 * pi, N)))';
  expA = expm (A);
  xt = timesteps (100, x0, expA);
endfunction
function fib = bar (N)
  if (N <= 2)
    fib = 1;
  else
    fib = bar (N - 1) + bar (N - 2);
  endif
endfunction

If we execute the two main functions, we get:

tic; foo; toc;
⇒ Elapsed time is 2.37338 seconds.
tic; bar (20); toc;
⇒ Elapsed time is 2.04952 seconds.

But this does not give much information about where this time is spent; for instance, whether the single call to expm is more expensive or the recursive time-stepping itself. To get a more detailed picture, we can use the profiler.

profile on;
foo;
profile off;
data = profile ("info");
profshow (data, 10);

This prints a table like:

        #  Function Attr     Time (s)        Calls
     ---------------------------------------------
        7      expm             1.034            1
        3  binary *             0.823          117
       41  binary \             0.188            1
       38  binary ^0.126            2
       43 timesteps    R        0.111          101
       44        NA             0.029          101
       39  binary +             0.024            8
       34      norm             0.011            1
       40  binary -             0.004          101
       33   balance             0.003            1

The entries are the individual functions which have been executed (only the 10 most important ones), together with some information for each of them. The entries like ‘binary *’ denote operators, while other entries are ordinary functions. They include both builtins like expm and our own routines (for instance timesteps). From this profile, we can immediately deduce that expm uses up the largest proportion of the processing time, even though it is only called once. The second expensive operation is the matrix-vector product in the routine timesteps.^{1}

Timing, however, is not the only information available from the profile. The attribute column shows us that timesteps calls itself recursively. This may not be that remarkable in this example (since it’s clear anyway), but could be helpful in a more complex setting. As to the question of why is there a ‘binary \’ in the output, we can easily shed some light on that too. Note that data is a structure array (Section 6.1.2 [Structure Arrays], page 101) which contains the field FunctionTable. This stores the raw data for the profile shown. The number in the first column of the table gives the index under which the shown function can be found there. Looking up data.FunctionTable(41) gives:

       scalar structure containing the fields:
         FunctionName = binary \
         TotalTime =  0.18765
         NumCalls =  1
         IsRecursive = 0
         Parents =  7
         Children = [](1x0)

Here we see the information from the table again, but have additional fields Parents and Children. Those are both arrays, which contain the indices of functions which have directly called the function in question (which is entry 7, expm, in this case) or been called by it (no functions). Hence, the backslash operator has been used internally by expm.

Now let’s take a look at bar. For this, we start a fresh profiling session (profile on does this; the old data is removed before the profiler is restarted):

1 We only know it is the binary multiplication operator, but fortunately this operator appears only at one

place in the code and thus we know which occurrence takes so much time. If there were multiple places,

we would have to use the hierarchical profile to find out the exact place which uses up the time which is

not covered in this example.

profile on;
bar (20);
profile off;
profshow (profile ("info"));

This gives:

#            Function Attr     Time (s)        Calls
-------------------------------------------------------
1                 bar    R        2.091        13529
2           binary <=             0.062        13529
3            binary -             0.042        13528
4            binary +             0.023         6764
5             profile             0.000            1
8               false             0.000            1
6              nargin             0.000            1
7           binary !=             0.000            1
9 __profiler_enable__0.000            1

Unsurprisingly, bar is also recursive. It has been called 13,529 times in the course of recursively calculating the Fibonacci number in a suboptimal way, and most of the time was spent in bar itself.

Finally, let’s say we want to profile the execution of both foo and bar together. Since we already have the run-time data collected for bar, we can restart the profiler without clearing the existing data and collect the missing statistics about foo. This is done by:

profile resume;
foo;
profile off;
profshow (profile ("info"), 10);

As you can see in the table below, now we have both profiles mixed together.

#  Function Attr     Time (s)        Calls
---------------------------------------------
1       bar    R        2.091        13529
16      expm             1.122            1
12  binary *             0.798          117
46  binary \             0.185            1
45  binary ^0.124            2
48 timesteps    R        0.115          101
2 binary <=             0.062        13529
3  binary -             0.045        13629
4  binary +             0.041         6772
49        NA             0.036          101

14 Input and Output

Octave supports several ways of reading and writing data to or from the prompt or a file. The simplest functions for data Input and Output (I/O) are easy to use, but only provide limited control of how data is processed. For more control, a set of functions modeled after the C standard library are also provided by Octave.

14.1 Basic Input and Output

14.1.1 Terminal Output

Since Octave normally prints the value of an expression as soon as it has been evaluated, the simplest of all I/O functions is a simple expression. For example, the following expression will display the value of ‘pi’

     pi ⊣ pi = 3.1416

This works well as long as it is acceptable to have the name of the variable (or ‘ans’) printed along with the value. To print the value of a variable without printing its name, use the function disp.

The format command offers some control over the way Octave prints values with disp and through the normal echoing mechanism.

[Built-in Function] disp (x)

Display the value of x. For example:

          disp ("The value of pi is:"), disp (pi)
               ⊣ the value of pi is:
               ⊣ 3.1416

Note that the output from disp always ends with a newline.

If an output value is requested, disp prints nothing and returns the formatted output in a string.

See also: [fdisp], page 234.

[Built-in Function] list_in_columns (arg, width, prefix)

Return a string containing the elements of arg listed in columns with an overall maximum width of width and optional prefix prefix. The argument arg must be a cell array of character strings or a character array. If width is not specified or is an empty matrix, or less than or equal to zero, the width of the terminal screen is used. Newline characters are used to break the lines in the output string. For example:

      list_in_columns ({"abc", "def", "ghijkl", "mnop", "qrs", "tuv"}, 20)
           ⇒ abc     mnop
              def     qrs
              ghijkl  tuv
      whos ans

⇒

           Variables in the current scope:
             Attr Name        Size                     Bytes  Class
             ==== ====        ====                     =====  =====
                  ans         1x37                        37  char
           Total is 37 elements using 37 bytes

See also: [terminal size], page 224.

[Built-in Function] terminal_size ()

Return a two-element row vector containing the current size of the terminal window in characters (rows and columns).

See also: [list in columns], page 223.

[Command] format

[Command] format options Reset or specify the format of the output produced by disp and Octave’s normal echoing mechanism. This command only affects the display of numbers but not how they are stored or computed. To change the internal representation from the default double use one of the conversion functions such as single, uint8, int64, etc.

By default, Octave displays 5 significant digits in a human readable form (option ‘short’ paired with ‘loose’ format for matrices). If format is invoked without any options, this default format is restored.

Valid formats for floating point numbers are listed in the following table.

short Fixed point format with 5 significant figures in a field that is a maximum of 10 characters wide. (default).

If Octave is unable to format a matrix so that columns line up on the decimal point and all numbers fit within the maximum field width then it switches to an exponential ‘e’ format.

long Fixed point format with 15 significant figures in a field that is a maximum of 20 characters wide.

As with the ‘short’ format, Octave will switch to an exponential ‘e’ format if it is unable to format a matrix properly using the current format.

short e

long e Exponential format. The number to be represented is split between a mantissa and an exponent (power of 10). The mantissa has 5 significant digits in the short format and 15 digits in the long format. For example, with the ‘short e’ format, pi is displayed as 3.1416e+00.

short E

long E Identical to ‘short e’ or ‘long e’ but displays an uppercase ‘E’ to indicate the exponent. For example, with the ‘long E’ format, pi is displayed as

3.14159265358979E+00.
short g

long g Optimally choose between fixed point and exponential format based on the magnitude of the number. For example, with the ‘short g’ format,

pi .^[2; 4; 8; 16; 32] is displayed as
     ans =
           9.8696
           97.409
           9488.5
       9.0032e+07
       8.1058e+15
short eng

long eng Identical to ‘short e’ or ‘long e’ but displays the value using an engineering format, where the exponent is divisible by 3. For example, with the ‘short eng’ format, 10 * pi is displayed as 31.4159e+00.

long G

short G Identical to ‘short g’ or ‘long g’ but displays an uppercase ‘E’ to indicate the exponent.

free

none Print output in free format, without trying to line up columns of matrices on the decimal point. This also causes complex numbers to be formatted as numeric pairs like this ‘(0.60419, 0.60709)’ instead of like this

‘0.60419 + 0.60709i’.

The following formats affect all numeric output (floating point and integer types).

+
+ chars
plus
plus chars

Print a ‘+’ symbol for nonzero matrix elements and a space for zero matrix elements. This format can be very useful for examining the structure of a large sparse matrix. The optional argument chars specifies a list of 3 characters to use for printing values greater than zero, less than zero and equal to zero. For example, with the ‘+ "+-."’ format, [1, 0, -1; -1, 0, 1] is displayed as

     ans = +.- -.+

bank Print in a fixed format with two digits to the right of the decimal point.

native-hex

Print the hexadecimal representation of numbers as they are stored in memory. For example, on a workstation which stores 8 byte real values in IEEE format with the least significant byte first, the value of pi when

printed in native-hex format is 400921fb54442d18.

hex The same as native-hex, but always print the most significant byte first.

native-bit

Print the bit representation of numbers as stored in memory. For example, the value of pi is

     01000000000010010010000111111011 01010100010001000010110100011000

(shown here in two 32 bit sections for typesetting purposes) when printed in native-bit format on a workstation which stores 8 byte real values in IEEE format with the least significant byte first.

bit The same as native-bit, but always print the most significant bits first.

rat Print a rational approximation, i.e., values are approximated as the ratio of small integers. For example, with the ‘rat’ format, pi is displayed as

355/113.

The following two options affect the display of all matrices.

compact Remove blank lines around column number labels and between matrices producing more compact output with more data per page.

loose Insert blank lines above and below column number labels and between matrices to produce a more readable output with less data per page. (default).

See also: [fixed point format], page 50, [output max field width], page 49, [output precision], page 50, [split long rows], page 50, [rats], page 428.

14.1.1.1 Paging Screen Output

When running interactively, Octave normally sends any output intended for your terminal that is more than one screen long to a paging program, such as less or more. This avoids the problem of having a large volume of output stream by before you can read it. With less (and some versions of more) you can also scan forward and backward, and search for specific items.

Normally, no output is displayed by the pager until just before Octave is ready to print the top level prompt, or read from the standard input (for example, by using the fscanf or scanf functions). This means that there may be some delay before any output appears on your screen if you have asked Octave to perform a significant amount of work with a single command statement. The function fflush may be used to force output to be sent to the pager (or any other stream) immediately.

You can select the program to run as the pager using the PAGER function, and you can turn paging off by using the function more.

[Command] more

[Command] more on

[Command] more off

Turn output pagination on or off. Without an argument, more toggles the current state. The current state can be determined via page_screen_output.

See also: [page screen output], page 227, [page output immediately], page 227, [PAGER], page 227, [PAGER FLAGS], page 227.

[Built-in Function] val = PAGER ()

[Built-in Function] old_val = PAGER (new_val)

[Built-in Function] PAGER (new_val, "local")

Query or set the internal variable that specifies the program to use to display terminal output on your system. The default value is normally "less", "more", or "pg", depending on what programs are installed on your system. See Appendix G [Installation], page 833. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [PAGER FLAGS], page 227, [page output immediately], page 227, [more], page 226, [page screen output], page 227.

[Built-in Function] val = PAGER_FLAGS ()

[Built-in Function] old_val = PAGER_FLAGS (new_val)

[Built-in Function] PAGER_FLAGS (new_val, "local")

Query or set the internal variable that specifies the options to pass to the pager. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [PAGER], page 227, [more], page 226, [page screen output], page 227, [page output immediately], page 227.

[Built-in Function] val = page_screen_output ()
[Built-in Function] old_val = page_screen_output (new_val)
[Built-in Function] page_screen_output (new_val, "local")

Query or set the internal variable that controls whether output intended for the terminal window that is longer than one page is sent through a pager. This allows you to view one screenful at a time. Some pagers (such as less—see Appendix G [Installation], page 833) are also capable of moving backward on the output. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [more], page 226, [page output immediately], page 227, [PAGER], page 227, [PAGER FLAGS], page 227.

[Built-in Function] val = page_output_immediately ()
[Built-in Function] old_val = page_output_immediately (new_val)

[Built-in Function] page_output_immediately (new_val, "local")

Query or set the internal variable that controls whether Octave sends output to the pager as soon as it is available. Otherwise, Octave buffers its output and waits until just before the prompt is printed to flush it to the pager.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [page screen output], page 227, [more], page 226, [PAGER], page 227, [PAGER FLAGS], page 227.

[Built-in Function] fflush (fid)

Flush output to fid. This is useful for ensuring that all pending output makes it to the screen before some other event occurs. For example, it is always a good idea to flush the standard output stream before calling input.

fflush returns 0 on success and an OS dependent error value (−1 on Unix) on error.

See also: [fopen], page 240, [fclose], page 242.

14.1.2 Terminal Input

Octave has three functions that make it easy to prompt users for input. The input and menu functions are normally used for managing an interactive dialog with a user, and the keyboard function is normally used for doing simple debugging.

[Built-in Function] ans = input (prompt) [Built-in Function] ans = input (prompt, "s")

Print prompt and wait for user input.

For example,

     input ("Pick a number, any number! ")

prints the prompt

     Pick a number, any number!

and waits for the user to enter a value. The string entered by the user is evaluated as an expression, so it may be a literal constant, a variable name, or any other valid expression.

Currently, input only returns one value, regardless of the number of values produced by the evaluation of the expression.

If you are only interested in getting a literal string value, you can call input with the character string "s" as the second argument. This tells Octave to return the string entered by the user directly, without evaluating it first.

Because there may be output waiting to be displayed by the pager, it is a good idea to always call fflush (stdout) before calling input. This will ensure that all pending output is written to the screen before your prompt.

See also: [yes or no], page 229, [kbhit], page 229, [pause], page 702, [menu], page 229, [listdlg], page 743.

[Function File] menu (title, opt1, …)

Print a title string followed by a series of options. Each option will be printed along with a number. The return value is the number of the option selected by the user. This function is useful for interactive programs. There is no limit to the number of options that may be passed in, but it may be confusing to present more than will fit easily on one screen.

See also: [input], page 228, [listdlg], page 743.

[Built-in Function] ans = yes_or_no ("prompt")

Ask the user a yes-or-no question.

Return logical true if the answer is yes or false if the answer is no. Takes one argument, prompt, which is the string to display when asking the question. prompt should end in a space; yes-or-no adds the string ‘(yes or no) ’ to it. The user must confirm the answer with RET and can edit it until it has been confirmed.

See also: [input], page 228.

For input, the normal command line history and editing functions are available at the prompt.

Octave also has a function that makes it possible to get a single character from the keyboard without requiring the user to type a carriage return.

[Built-in Function] kbhit () [Built-in Function] kbhit (1)

Read a single keystroke from the keyboard. If called with an argument, don’t wait for a keypress. For example,

     x = kbhit ();

will set x to the next character typed at the keyboard as soon as it is typed.

     x = kbhit (1);

is identical to the above example, but doesn’t wait for a keypress, returning the empty string if no key is available.

See also: [input], page 228, [pause], page 702.

14.1.3 Simple File I/O

The save and load commands allow data to be written to and read from disk files in various formats. The default format of files written by the save command can be controlled using

the functions save_default_options and save_precision.

As an example the following code creates a 3-by-3 matrix and saves it to the file

‘myfile.mat’.
A = [ 1:3; 4:6; 7:9 ];
save myfile.mat A

Once one or more variables have been saved to a file, they can be read into memory using the load command.

load myfile.mat
A

     ⊣ A =

⊣

     ⊣  1   2   3
     ⊣  4   5   6
     ⊣  7   8   9

[Command] save file

[Command] save options file

[Command] save options file v1 v2 …

[Command] save options file -struct STRUCT f1 f2 … Save the named variables v1, v2, …, in the file file. The special filename ‘-’ may be used to write output to the terminal. If no variable names are listed, Octave saves all the variables in the current scope. Otherwise, full variable names or pattern syntax can be used to specify the variables to save. If the ‘-struct’ modifier is used, fields f1 f2 … of the scalar structure STRUCT are saved as if they were variables with corresponding names. Valid options for the save command are listed in the following table. Options that modify the output format override the format specified

by save_default_options.

If save is invoked using the functional form

     save ("-option1", ..., "file", "v1", ...)

then the options, file, and variable name arguments (v1, …) must be specified as character strings.

-append Append to the destination instead of overwriting.

-ascii Save a single matrix in a text file without header or any other information.

-binary Save the data in Octave’s binary data format.

-float-binary

Save the data in Octave’s binary data format but only using single precision. Only use this format if you know that all the values to be saved can be represented in single precision.

-hdf5 Save the data in hdf5 format. (HDF5 is a free, portable binary format developed by the National Center for Supercomputing Applications at the University of Illinois.) This format is only available if Octave was built with a link to the hdf5 libraries.

-float-hdf5

Save the data in hdf5 format but only using single precision. Only use this format if you know that all the values to be saved can be represented in single precision.

-V7
-v7
-7
-mat7-binary

Save the data in matlab’s v7 binary data format.

-V6
-v6
-6
-mat
-mat-binary

Save the data in matlab’s v6 binary data format.

-V4
-v4
-4
-mat4-binary

Save the data in the binary format written by matlab version 4.

-text Save the data in Octave’s text data format. (default).

-zip

-z Use the gzip algorithm to compress the file. This works equally on files that are compressed with gzip outside of octave, and gzip can equally be used to convert the files for backward compatibility. This option is only available if Octave was built with a link to the zlib libraries.

The list of variables to save may use wildcard patterns containing the following special characters:

? Match any single character.

* Match zero or more characters.

[ list ] Match the list of characters specified by list. If the first character is ! or ^, match all characters except those specified by list. For example, the pattern [a-zA-Z] will match all lower and uppercase alphabetic characters. Wildcards may also be used in the field name specifications when using the ‘-struct’ modifier (but not in the struct name itself).

Except when using the matlab binary data file format or the ‘-ascii’ format, saving global variables also saves the global status of the variable. If the variable is restored at a later time using ‘load’, it will be restored as a global variable.

The command

     save -binary data a b*

saves the variable ‘a’ and all variables beginning with ‘b’ to the file ‘data’ in Octave’s binary format.

See also: [load], page 232, [save default options], page 231, [save header format string], page 232, [dlmread], page 235, [csvread], page 236, [fread], page 252.

There are three functions that modify the behavior of save.

[Built-in Function] val = save_default_options ()
[Built-in Function] old_val = save_default_options (new_val)

[Built-in Function] save_default_options (new_val, "local")

Query or set the internal variable that specifies the default options for the save command, and defines the default format. Typical values include "-ascii", "-text -zip". The default value is ‘-text’. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [save], page 230.

[Built-in Function] val = save_precision ()

[Built-in Function] old_val = save_precision (new_val)

[Built-in Function] save_precision (new_val, "local")

Query or set the internal variable that specifies the number of digits to keep when saving data in text format. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

[Built-in Function] val = save_header_format_string ()
[Built-in Function] old_val = save_header_format_string (new_val)
[Built-in Function] save_header_format_string (new_val, "local")

Query or set the internal variable that specifies the format string used for the comment line written at the beginning of text-format data files saved by Octave. The format string is passed to strftime and should begin with the character ‘#’ and contain no newline characters. If the value of save_header_format_string is the empty string, the header comment is omitted from text-format data files. The default value is

      "# Created by Octave VERSION, %a %b %d %H:%M:%S %Y %Z <USER@HOST>"

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [strftime], page 699, [save], page 230.

[Command] load file

[Command] load options file

[Command] load options file v1 v2 …

[Command] S = load ("options", "file", "v1", "v2", …)

[Command] load file options

[Command] load file options v1 v2 …

[Command] S = load ("file", "options", "v1", "v2", …)

Load the named variables v1, v2, …, from the file file. If no variables are specified then all variables found in the file will be loaded. As with save, the list of variables to extract can be full names or use a pattern syntax. The format of the file is automatically detected but may be overridden by supplying the appropriate option. If load is invoked using the functional form

     load ("-option1", ..., "file", "v1", ...)

then the options, file, and variable name arguments (v1, …) must be specified as character strings.

If a variable that is not marked as global is loaded from a file when a global symbol with the same name already exists, it is loaded in the global symbol table. Also, if a variable is marked as global in a file and a local symbol exists, the local symbol is moved to the global symbol table and given the value from the file.

If invoked with a single output argument, Octave returns data instead of inserting variables in the symbol table. If the data file contains only numbers (TAB- or space-delimited columns), a matrix of values is returned. Otherwise, load returns a structure with members corresponding to the names of the variables in the file.

The load command can read data stored in Octave’s text and binary formats, and matlab’s binary format. If compiled with zlib support, it can also load gzip-compressed files. It will automatically detect the type of file and do conversion from different floating point formats (currently only IEEE big and little endian, though other formats may be added in the future).

Valid options for load are listed in the following table.

-force This option is accepted for backward compatibility but is ignored. Octave now overwrites variables currently in memory with those of the same name found in the file.

-ascii Force Octave to assume the file contains columns of numbers in text format without any header or other information. Data in the file will be loaded as a single numeric matrix with the name of the variable derived from the name of the file.

-binary Force Octave to assume the file is in Octave’s binary format.

-hdf5 Force Octave to assume the file is in hdf5 format. (hdf5 is a free, portable binary format developed by the National Center for Supercomputing Applications at the University of Illinois.) Note that Octave can read hdf5 files not created by itself, but may skip some datasets in formats that it cannot support. This format is only available if Octave was built with a link to the hdf5 libraries.

-import This option is accepted for backward compatibility but is ignored. Octave can now support multi-dimensional HDF data and automatically modifies variable names if they are invalid Octave identifiers.

-mat
-mat-binary
-6
-v6
-7

-v7 Force Octave to assume the file is in matlab’s version 6 or 7 binary format.

-mat4-binary
-4
-v4

-V4 Force Octave to assume the file is in the binary format written by matlab version 4.

-text Force Octave to assume the file is in Octave’s text format.

See also: [save], page 230, [dlmwrite], page 234, [csvwrite], page 236, [fwrite], page 254.

[Function File] str = fileread (filename)

Read the contents of filename and return it as a string.

See also: [fread], page 252, [textread], page 236, [sscanf], page 249.

[Built-in Function] native_float_format ()

Return the native floating point format as a string

It is possible to write data to a file in a similar way to the disp function for writing data to the screen. The fdisp works just like disp except its first argument is a file pointer as created by fopen. As an example, the following code writes to data ‘myfile.txt’.

fid = fopen ("myfile.txt", "w");
fdisp (fid, "3/8 is ");
fdisp (fid, 3/8);
fclose (fid);

See Section 14.2.1 [Opening and Closing Files], page 240, for details on how to use fopen

and fclose.

[Built-in Function] fdisp (fid, x)

Display the value of x on the stream fid. For example:

     fdisp (stdout, "The value of pi is:"), fdisp (stdout, pi)
          ⊣ the value of pi is:
          ⊣ 3.1416

Note that the output from fdisp always ends with a newline.

See also: [disp], page 223.

Octave can also read and write matrices text files such as comma separated lists.

[Function File] dlmwrite (file, M)

[Function File] dlmwrite (file, M, delim, r, c)

[Function File] dlmwrite (file, M, key, val …) [Function File] dlmwrite (file, M, "-append", …) [Function File] dlmwrite (fid, …)

Write the matrix M to the named file using delimiters. file should be a file name or writable file ID given by fopen. The parameter delim specifies the delimiter to use to separate values on a row.

The value of r specifies the number of delimiter-only lines to add to the start of the file. The value of c specifies the number of delimiters to prepend to each line of data. If the argument "-append" is given, append to the end of file. In addition, the following keyword value pairs may appear at the end of the argument list:

"append" Either "on" or "off". See "-append" above.
"delimiter"

See delim above.

"newline"

The character(s) to use to separate each row. Three special cases exist for this option. "unix" is changed into "\n", "pc" is changed into "\r\n", and "mac" is changed into "\r". Other values for this option are kept as is.

"roffset"

See r above.

"coffset"

See c above.

"precision"

The precision to use when writing the file. It can either be a format string (as used by fprintf) or a number of significant digits.

     dlmwrite ("file.csv", reshape (1:16, 4, 4));
     dlmwrite ("file.tex", a, "delimiter", "&", "newline", "\\n")

See also: [dlmread], page 235, [csvread], page 236, [csvwrite], page 236.

[Built-in Function] data = dlmread (file) [Built-in Function] data = dlmread (file, sep)

[Built-in Function] data = dlmread (file, sep, r0, c0)
[Built-in Function] data = dlmread (file, sep, range)

[Built-in Function] data = dlmread (…, "emptyvalue", EMPTYVAL)

Read the matrix data from a text file. If not defined the separator between fields is determined from the file itself. Otherwise the separation character is defined by sep. Given two scalar arguments r0 and c0, these define the starting row and column of the data to be read. These values are indexed from zero, such that the first row corresponds to an index of zero. The range parameter may be a 4-element vector containing the upper left and lower right corner [R0,C0,R1,C1] where the lowest index value is zero. Alternatively, a spreadsheet style range such as "A2..Q15" or "T1:AA5" can be used. The lowest alphabetical index 'A' refers to the first column. The lowest row index is 1. file should be a file name or file id given by fopen. In the latter case, the file is read until end of file is reached. The "emptyvalue" option may be used to specify the value used to fill empty fields. The default is zero.

See also: [csvread], page 236, [textscan], page 236, [textread], page 236, [dlmwrite], page 234.

[Function File] csvwrite (filename, x)
[Function File] csvwrite (filename, x, dlm_opts)

Write the matrix x to the file filename in comma-separated-value format. This function is equivalent to

     dlmwrite (filename, x, ",", ...)

See also: [csvread], page 236, [dlmwrite], page 234, [dlmread], page 235.

[Function File] x = csvread (filename)
[Function File] x = csvread (filename, dlm_opts)

Read the comma-separated-value file filename into the matrix x. This function is equivalent to

     x = dlmread (filename, "," , ...)

See also: [csvwrite], page 236, [dlmread], page 235, [dlmwrite], page 234.

Formatted data from can be read from, or written to, text files as well.

[Function File] [a, ...] = textread (filename)
[Function File] [a, ...] = textread (filename, format)
[Function File] [a, ...] = textread (filename, format, n)
[Function File] [a, ...] = textread (filename, format, prop1, value1, . . .)
[Function File] [a, ...] = textread (filename, format, n, prop1, value1, . . .)

Read data from a text file. The file filename is read and parsed according to format. The function behaves like strread except it works by parsing a file instead of a string. See the documentation of strread for details. In addition to the options supported by strread, this function supports two more:

• "headerlines": The first value number of lines of filename are skipped.

• "endofline": Specify a single character or "\r\n". If no value is given, it will be inferred from the file. If set to "" (empty string) EOLs are ignored as delimiters.

The optional input n specifies the number of data lines to read; in this sense it differs slightly from the format repeat count in strread. If the format string is empty (not: omitted) and the file contains only numeric data (excluding headerlines), textread will return a rectangular matrix with the number of columns matching the number of numeric fields on the first data line of the file. Empty fields are returned as zero values.

See also: [strread], page 81, [load], page 232, [dlmread], page 235, [fscanf], page 248, [textscan], page 236.

[Function File] C = textscan (fid, format)
[Function File] C = textscan (fid, format, n)
[Function File] C = textscan (fid, format, param, value, . . .)
[Function File] C = textscan (fid, format, n, param, value, . . .)

[Function File] C = textscan (str, …)

[Function File] [C, position] = textscan (fid, . . .)

Read data from a text file or string.

The string str or file associated with fid is read from and parsed according to format. The function behaves like strread except it can also read from file instead of a string. See the documentation of strread for details.

In addition to the options supported by strread, this function supports a few more:

• "collectoutput": A value of 1 or true instructs textscan to concatenate consecutive columns of the same class in the output cell array. A value of 0 or false (default) leaves output in distinct columns.

• "endofline": Specify "\r", "\n" or "\r\n" (for CR, LF, or CRLF). If no value is given, it will be inferred from the file. If set to "" (empty string) EOLs are ignored as delimiters and added to whitespace.

• "headerlines": The first value number of lines of fid are skipped.

• "returnonerror": If set to numerical 1 or true (default), return normally when read errors have been encountered. If set to 0 or false, return an error and no data. As the string or file is read by columns rather than by rows, and because textscan is fairly forgiving as regards read errors, setting this option may have little or no actual effect.

When reading from a character string, optional input argument n specifies the number of times format should be used (i.e., to limit the amount of data read). When reading from file, n specifies the number of data lines to read; in this sense it differs slightly from the format repeat count in strread.

The output C is a cell array whose second dimension is determined by the number of format specifiers.

The second output, position, provides the position, in characters, from the beginning of the file.

If the format string is empty (not: omitted) and the file contains only numeric data (excluding headerlines), textscan will return data in a number of columns matching the number of numeric fields on the first data line of the file.

See also: [dlmread], page 235, [fscanf], page 248, [load], page 232, [strread], page 81, [textread], page 236.

The importdata function has the ability to work with a wide variety of data.

[Function File] A = importdata (fname)
[Function File] A = importdata (fname, delimiter)
[Function File] A = importdata (fname, delimiter, header_rows)
[Function File] [A, delimiter] = importdata (. . .)
[Function File] [A, delimiter, header_rows] = importdata (. . .)

Import data from the file fname.

Input parameters:

• fname The name of the file containing data.

• delimiter The character separating columns of data. Use \t for tab. (Only valid for ASCII files)

• header rows The number of header rows before the data begins. (Only valid for ASCII files)

Different file types are supported:

• ASCII table Import ASCII table using the specified number of header rows and the specified delimiter.

• Image file

• matlab file

• Spreadsheet files (depending on external software)

• WAV file

See also: [textscan], page 236, [dlmread], page 235, [csvread], page 236, [load], page 232.

14.1.3.1 Saving Data on Unexpected Exits

If Octave for some reason exits unexpectedly it will by default save the variables available in the workspace to a file in the current directory. By default this file is named ‘octave-workspace’ and can be loaded into memory with the load command. While the default behavior most often is reasonable it can be changed through the following functions.

[Built-in Function] val = crash_dumps_octave_core ()
[Built-in Function] old_val = crash_dumps_octave_core (new_val)
[Built-in Function] crash_dumps_octave_core (new_val, "local")

Query or set the internal variable that controls whether Octave tries to save all current variables to the file ‘octave-workspace’ if it crashes or receives a hangup, terminate or similar signal. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [octave core file limit], page 239, [octave core file name], page 239, [octave core file options], page 239.

[Built-in Function] val = sighup_dumps_octave_core ()
[Built-in Function] old_val = sighup_dumps_octave_core (new_val)
[Built-in Function] sighup_dumps_octave_core (new_val, "local")

Query or set the internal variable that controls whether Octave tries to save all current variables to the file ‘octave-workspace’ if it receives a hangup signal. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

[Built-in Function] val = sigterm_dumps_octave_core ()
[Built-in Function] old_val = sigterm_dumps_octave_core (new_val)

[Built-in Function] sigterm_dumps_octave_core (new_val, "local")

Query or set the internal variable that controls whether Octave tries to save all current variables to the file ‘octave-workspace’ if it receives a terminate signal. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

[Built-in Function] val = octave_core_file_options ()
[Built-in Function] old_val = octave_core_file_options (new_val)
[Built-in Function] octave_core_file_options (new_val, "local")

Query or set the internal variable that specifies the options used for saving the workspace data if Octave aborts. The value of octave_core_file_options should follow the same format as the options for the save function. The default value is Octave’s binary format. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [crash dumps octave core], page 238, [octave core file name], page 239, [octave core file limit], page 239.

[Built-in Function] val = octave_core_file_limit ()
[Built-in Function] old_val = octave_core_file_limit (new_val)
[Built-in Function] octave_core_file_limit (new_val, "local")

Query or set the internal variable that specifies the maximum amount of memory (in kilobytes) of the top-level workspace that Octave will attempt to save when writing data to the crash dump file (the name of the file is specified by octave core file name). If octave core file options flags specify a binary format, then octave core file limit will be approximately the maximum size of the file. If a text file format is used, then the file could be much larger than the limit. The default value is -1 (unlimited) When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [crash dumps octave core], page 238, [octave core file name], page 239, [octave core file options], page 239.

[Built-in Function] val = octave_core_file_name ()
[Built-in Function] old_val = octave_core_file_name (new_val)
[Built-in Function] octave_core_file_name (new_val, "local")

Query or set the internal variable that specifies the name of the file used for saving data from the top-level workspace if Octave aborts. The default value is "octave-

workspace"

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [crash dumps octave core], page 238, [octave core file name], page 239, [octave core file options], page 239.

14.2 C-Style I/O Functions

Octave’s C-style input and output functions provide most of the functionality of the C programming language’s standard I/O library. The argument lists for some of the input functions are slightly different, however, because Octave has no way of passing arguments by reference.

In the following, file refers to a file name and fid refers to an integer file number, as returned by fopen.

There are three files that are always available. Although these files can be accessed using their corresponding numeric file ids, you should always use the symbolic names given in the table below, since it will make your programs easier to understand.

[Built-in Function] stdin ()

Return the numeric value corresponding to the standard input stream. When Octave is used interactively, this is filtered through the command line editing functions.

See also: [stdout], page 240, [stderr], page 240.

[Built-in Function] stdout ()

Return the numeric value corresponding to the standard output stream. Data written to the standard output is normally filtered through the pager.

See also: [stdin], page 240, [stderr], page 240.

[Built-in Function] stderr ()

Return the numeric value corresponding to the standard error stream. Even if paging is turned on, the standard error is not sent to the pager. It is useful for error messages and prompts.

See also: [stdin], page 240, [stdout], page 240.

14.2.1 Opening and Closing Files

When reading data from a file it must be opened for reading first, and likewise when writing to a file. The fopen function returns a pointer to an open file that is ready to be read or written. Once all data has been read from or written to the opened file it should be closed. The fclose function does this. The following code illustrates the basic pattern for writing to a file, but a very similar pattern is used when reading a file.

filename = "myfile.txt";
fid = fopen (filename, "w");
# Do the actual I/O here...
fclose (fid);
[Built-in Function] [fid, msg] = fopen (name, mode, arch)

[Built-in Function] fid_list = fopen ("all")

[Built-in Function] [file, mode, arch] = fopen (fid)

The first form of the fopen function opens the named file with the specified mode (read-write, read-only, etc.) and architecture interpretation (IEEE big endian, IEEE little endian, etc.), and returns an integer value that may be used to refer to the file later. If an error occurs, fid is set to −1 and msg contains the corresponding system error message. The mode is a one or two character string that specifies whether the file is to be opened for reading, writing, or both. The second form of the fopen function returns a vector of file ids corresponding to all the currently open files, excluding the stdin, stdout, and stderr streams. The third form of the fopen function returns information about the open file given its file id. For example,

     myfile = fopen ("splat.dat", "r", "ieee-le");

opens the file ‘splat.dat’ for reading. If necessary, binary numeric values will be read assuming they are stored in IEEE format with the least significant bit first, and then converted to the native representation. Opening a file that is already open simply opens it again and returns a separate file id. It is not an error to open a file several times, though writing to the same file through several different file ids may produce unexpected results. The possible values ‘mode’ may have are

‘r’ Open a file for reading.

‘w’ Open a file for writing. The previous contents are discarded.

‘a’ Open or create a file for writing at the end of the file.

‘r+’ Open an existing file for reading and writing.

‘w+’ Open a file for reading or writing. The previous contents are discarded.

‘a+’ Open or create a file for reading or writing at the end of the file.

Append a "t" to the mode string to open the file in text mode or a "b" to open in binary mode. On Windows and Macintosh systems, text mode reading and writing automatically converts linefeeds to the appropriate line end character for the system (carriage-return linefeed on Windows, carriage-return on Macintosh). The default if no mode is specified is binary mode. Additionally, you may append a "z" to the mode string to open a gzipped file for reading or writing. For this to be successful, you must also open the file in binary mode. The parameter arch is a string specifying the default data format for the file. Valid values for arch are:

‘native’ The format of the current machine (this is the default).

‘ieee-be’ IEEE big endian format.

‘ieee-le’ IEEE little endian format.

however, conversions are currently only supported for ‘native’ ‘ieee-be’, and ‘ieee-le’ formats.

See also: [fclose], page 242, [fgets], page 243, [fgetl], page 242, [fscanf], page 248, [fread], page 252, [fputs], page 242, [fdisp], page 234, [fprintf], page 243, [fwrite], page 254, [fskipl], page 243, [fseek], page 257, [frewind], page 257, [ftell], page 256, [feof], page 256, [ferror], page 256, [fclear], page 256, [fflush], page 228, [freport], page 256.

[Built-in Function] fclose (fid) [Built-in Function] fclose ("all")

Close the specified file. If successful, fclose returns 0, otherwise, it returns -1. The second form of the fclose call closes all open files except stdout, stderr, and stdin.

See also: [fopen], page 240, [freport], page 256.

[Function File] is_valid_file_id (fid)

Return true if fid refers to an open file.

See also: [fopen], page 240.

14.2.2 Simple Output

Once a file has been opened for writing a string can be written to the file using the fputs function. The following example shows how to write the string ‘Free Software is needed

for Free Science’ to the file ‘free.txt’.
filename = "free.txt";
fid = fopen (filename, "w");
fputs (fid, "Free Software is needed for Free Science");
fclose (fid);

[Built-in Function] fputs (fid, string)

Write a string to a file with no formatting. Return a non-negative number on success and EOF on error.

See also: [fdisp], page 234, [fprintf], page 243, [fwrite], page 254, [fopen], page 240.

A function much similar to fputs is available for writing data to the screen. The puts function works just like fputs except it doesn’t take a file pointer as its input.

[Built-in Function] puts (string)

Write a string to the standard output with no formatting. Return a non-negative number on success and EOF on error.

See also: [fputs], page 242, [disp], page 223.

14.2.3 Line-Oriented Input

To read from a file it must be opened for reading using fopen. Then a line can be read from the file using fgetl as the following code illustrates

fid = fopen ("free.txt");
txt = fgetl (fid)
     ⊣ Free Software is needed for Free Science
fclose (fid);

This of course assumes that the file ‘free.txt’ exists and contains the line ‘Free Software

is needed for Free Science’.

[Built-in Function] str = fgetl (fid) [Built-in Function] str = fgetl (fid, len)

Read characters from a file, stopping after a newline, or EOF, or len characters have been read. The characters read, excluding the possible trailing newline, are returned as a string.

If len is omitted, fgetl reads until the next newline character. If there are no more characters to read, fgetl returns −1. To read a line and return the terminating newline see fgets.

See also: [fgets], page 243, [fscanf], page 248, [fread], page 252, [fopen], page 240.

[Built-in Function] str = fgets (fid) [Built-in Function] str = fgets (fid, len)

Read characters from a file, stopping after a newline, or EOF, or len characters have been read. The characters read, including the possible trailing newline, are returned as a string. If len is omitted, fgets reads until the next newline character. If there are no more characters to read, fgets returns −1. To read a line and discard the terminating newline see fgetl.

See also: [fputs], page 242, [fgetl], page 242, [fscanf], page 248, [fread], page 252, [fopen], page 240.

[Built-in Function] nlines = fskipl (fid)

[Built-in Function] nlines = fskipl (fid, count)

[Built-in Function] nlines = fskipl (fid, Inf)

Read and skip count lines from the file descriptor fid. fskipl discards characters until an end-of-line is encountered exactly count-times, or until the end-of-file marker is found. If count is omitted, it defaults to 1. count may also be Inf, in which case lines are skipped until the end of the file. This form is suitable for counting the number of lines in a file. Returns the number of lines skipped (end-of-line sequences encountered).

See also: [fgetl], page 242, [fgets], page 243, [fscanf], page 248, [fopen], page 240.

14.2.4 Formatted Output

This section describes how to call printf and related functions.

The following functions are available for formatted output. They are modeled after the C language functions of the same name, but they interpret the format template differently in order to improve the performance of printing vector and matrix values.

[Built-in Function] printf (template, …)

Print optional arguments under the control of the template string template to the stream stdout and return the number of characters printed. See the Formatted Output section of the GNU Octave manual for a complete description of the syntax of the template string.

See also: [fprintf], page 243, [sprintf], page 244, [scanf], page 249.

[Built-in Function] fprintf (fid, template, …)

This function is just like printf, except that the output is written to the stream fid instead of stdout. If fid is omitted, the output is written to stdout.

See also: [fputs], page 242, [fdisp], page 234, [fwrite], page 254, [fscanf], page 248, [printf], page 243, [sprintf], page 244, [fopen], page 240.

[Built-in Function] sprintf (template, …)

This is like printf, except that the output is returned as a string. Unlike the C library function, which requires you to provide a suitably sized string as an argument, Octave’s sprintf function returns the string, automatically sized to hold all of the items converted.

See also: [printf], page 243, [fprintf], page 243, [sscanf], page 249.

The printf function can be used to print any number of arguments. The template string argument you supply in a call provides information not only about the number of additional arguments, but also about their types and what style should be used for printing them.

Ordinary characters in the template string are simply written to the output stream as-is, while conversion specifications introduced by a ‘%’ character in the template cause subsequent arguments to be formatted and written to the output stream. For example,

     pct = 37;
     filename = "foo.txt";
     printf ("Processed %d%% of '%s'.\nPlease be patient.\n",
             pct, filename);

produces output like

     Processed 37% of 'foo.txt'.
     Please be patient.

This example shows the use of the ‘%d’ conversion to specify that a scalar argument should be printed in decimal notation, the ‘%s’ conversion to specify printing of a string argument, and the ‘%%’ conversion to print a literal ‘%’ character.

There are also conversions for printing an integer argument as an unsigned value in octal, decimal, or hexadecimal radix (‘%o’, ‘%u’, or ‘%x’, respectively); or as a character value (‘%c’).

Floating-point numbers can be printed in normal, fixed-point notation using the ‘%f’ conversion or in exponential notation using the ‘%e’ conversion. The ‘%g’ conversion uses either ‘%e’ or ‘%f’ format, depending on what is more appropriate for the magnitude of the particular number.

You can control formatting more precisely by writing modifiers between the ‘%’ and the character that indicates which conversion to apply. These slightly alter the ordinary behavior of the conversion. For example, most conversion specifications permit you to specify a minimum field width and a flag indicating whether you want the result left- or right-justified within the field.

The specific flags and modifiers that are permitted and their interpretation vary depending on the particular conversion. They’re all described in more detail in the following sections.

14.2.5 Output Conversion for Matrices

When given a matrix value, Octave’s formatted output functions cycle through the format template until all the values in the matrix have been printed. For example:

 printf ("%4.2f %10.2e %8.4g\n", hilb (3));

       ⊣ 1.00   5.00e-01   0.3333
       ⊣ 0.50   3.33e-01     0.25
       ⊣ 0.33   2.50e-01      0.2

If more than one value is to be printed in a single call, the output functions do not return to the beginning of the format template when moving on from one value to the next. This can lead to confusing output if the number of elements in the matrices are not exact multiples of the number of conversions in the format template. For example:

 printf ("%4.2f %10.2e %8.4g\n", [1, 2], [3, 4]);

       ⊣ 1.00   2.00e+00        3
       ⊣ 4.00

If this is not what you want, use a series of calls instead of just one.

14.2.6 Output Conversion Syntax

This section provides details about the precise syntax of conversion specifications that can appear in a printf template string.

Characters in the template string that are not part of a conversion specification are printed as-is to the output stream.

The conversion specifications in a printf template string have the general form:

 % flags width [ . precision ] type conversion

For example, in the conversion specifier ‘%-10.8ld’, the ‘-’ is a flag, ‘10’ specifies the field width, the precision is ‘8’, the letter ‘l’ is a type modifier, and ‘d’ specifies the conversion style. (This particular type specifier says to print a numeric argument in decimal notation, with a minimum of 8 digits left-justified in a field at least 10 characters wide.)

In more detail, output conversion specifications consist of an initial ‘%’ character followed in sequence by:

• Zero or more flag characters that modify the normal behavior of the conversion specification.

• An optional decimal integer specifying the minimum field width. If the normal conversion produces fewer characters than this, the field is padded with spaces to the specified width. This is a minimum value; if the normal conversion produces more characters than this, the field is not truncated. Normally, the output is right-justified within the field. You can also specify a field width of ‘*’. This means that the next argument in the argument list (before the actual value to be printed) is used as the field width. The value is rounded to the nearest integer. If the value is negative, this means to set the ‘-’ flag (see below) and to use the absolute value as the field width.

• An optional precision to specify the number of digits to be written for the numeric conversions. If the precision is specified, it consists of a period (‘.’) followed optionally by a decimal integer (which defaults to zero if omitted). You can also specify a precision of ‘*’. This means that the next argument in the argument list (before the actual value to be printed) is used as the precision. The value must be an integer, and is ignored if it is negative.

• An optional type modifier character. This character is ignored by Octave’s printf function, but is recognized to provide compatibility with the C language printf.

• A character that specifies the conversion to be applied.

The exact options that are permitted and how they are interpreted vary between the different conversion specifiers. See the descriptions of the individual conversions for information about the particular options that they use.

14.2.7 Table of Output Conversions

Here is a table summarizing what all the different conversions do:

‘%d’, ‘%i’ Print an integer as a signed decimal number. See Section 14.2.8 [Integer Conversions], page 247, for details. ‘%d’ and ‘%i’ are synonymous for output, but are different when used with scanf for input (see Section 14.2.13 [Table of Input Conversions], page 250).

‘%o’ Print an integer as an unsigned octal number. See Section 14.2.8 [Integer Conversions], page 247, for details.

‘%u’ Print an integer as an unsigned decimal number. See Section 14.2.8 [Integer Conversions], page 247, for details.

‘%x’, ‘%X’ Print an integer as an unsigned hexadecimal number. ‘%x’ uses lowercase letters and ‘%X’ uses uppercase. See Section 14.2.8 [Integer Conversions], page 247, for details.

‘%f’ Print a floating-point number in normal (fixed-point) notation. See Section 14.2.9 [Floating-Point Conversions], page 247, for details.

‘%e’, ‘%E’ Print a floating-point number in exponential notation. ‘%e’ uses lowercase letters and ‘%E’ uses uppercase. See Section 14.2.9 [Floating-Point Conversions], page 247, for details.

‘%g’, ‘%G’ Print a floating-point number in either normal (fixed-point) or exponential notation, whichever is more appropriate for its magnitude. ‘%g’ uses lowercase letters and ‘%G’ uses uppercase. See Section 14.2.9 [Floating-Point Conversions], page 247, for details.

‘%c’ Print a single character. See Section 14.2.10 [Other Output Conversions], page 248.

‘%s’ Print a string. See Section 14.2.10 [Other Output Conversions], page 248.

‘%%’ Print a literal ‘%’ character. See Section 14.2.10 [Other Output Conversions], page 248.

If the syntax of a conversion specification is invalid, unpredictable things will happen, so don’t do this. If there aren’t enough function arguments provided to supply values for all the conversion specifications in the template string, or if the arguments are not of the correct types, the results are unpredictable. If you supply more arguments than conversion specifications, the extra argument values are simply ignored; this is sometimes useful.

14.2.8 Integer Conversions

This section describes the options for the ‘%d’, ‘%i’, ‘%o’, ‘%u’, ‘%x’, and ‘%X’ conversion specifications. These conversions print integers in various formats.

The ‘%d’ and ‘%i’ conversion specifications both print an numeric argument as a signed decimal number; while ‘%o’, ‘%u’, and ‘%x’ print the argument as an unsigned octal, decimal, or hexadecimal number (respectively). The ‘%X’ conversion specification is just like ‘%x’ except that it uses the characters ‘ABCDEF’ as digits instead of ‘abcdef’.

The following flags are meaningful:

‘-’ Left-justify the result in the field (instead of the normal right-justification).

‘+’ For the signed ‘%d’ and ‘%i’ conversions, print a plus sign if the value is positive.

‘ ’ For the signed ‘%d’ and ‘%i’ conversions, if the result doesn’t start with a plus or minus sign, prefix it with a space character instead. Since the ‘+’ flag ensures that the result includes a sign, this flag is ignored if you supply both of them.

‘#’ For the ‘%o’ conversion, this forces the leading digit to be ‘0’, as if by increasing the precision. For ‘%x’ or ‘%X’, this prefixes a leading ‘0x’ or ‘0X’ (respectively) to the result. This doesn’t do anything useful for the ‘%d’, ‘%i’, or ‘%u’ conversions.

‘0’ Pad the field with zeros instead of spaces. The zeros are placed after any indication of sign or base. This flag is ignored if the ‘-’ flag is also specified, or if a precision is specified.

If a precision is supplied, it specifies the minimum number of digits to appear; leading zeros are produced if necessary. If you don’t specify a precision, the number is printed with as many digits as it needs. If you convert a value of zero with an explicit precision of zero, then no characters at all are produced.

14.2.9 Floating-Point Conversions

This section discusses the conversion specifications for floating-point numbers: the ‘%f’, ‘%e’, ‘%E’, ‘%g’, and ‘%G’ conversions.

The ‘%f’ conversion prints its argument in fixed-point notation, producing output of the form [-]ddd.ddd, where the number of digits following the decimal point is controlled by the precision you specify.

The ‘%e’ conversion prints its argument in exponential notation, producing output of the form [-]d.ddde[+|-]dd. Again, the number of digits following the decimal point is controlled by the precision. The exponent always contains at least two digits. The ‘%E’ conversion is similar but the exponent is marked with the letter ‘E’ instead of ‘e’.

The ‘%g’ and ‘%G’ conversions print the argument in the style of ‘%e’ or ‘%E’ (respectively) if the exponent would be less than -4 or greater than or equal to the precision; otherwise they use the ‘%f’ style. Trailing zeros are removed from the fractional portion of the result and a decimal-point character appears only if it is followed by a digit.

The following flags can be used to modify the behavior:

‘-’ Left-justify the result in the field. Normally the result is right-justified.

‘+’ Always include a plus or minus sign in the result.

‘ ’ If the result doesn’t start with a plus or minus sign, prefix it with a space instead. Since the ‘+’ flag ensures that the result includes a sign, this flag is ignored if you supply both of them.

‘#’ Specifies that the result should always include a decimal point, even if no digits follow it. For the ‘%g’ and ‘%G’ conversions, this also forces trailing zeros after the decimal point to be left in place where they would otherwise be removed.

‘0’ Pad the field with zeros instead of spaces; the zeros are placed after any sign. This flag is ignored if the ‘-’ flag is also specified.

The precision specifies how many digits follow the decimal-point character for the ‘%f’, ‘%e’, and ‘%E’ conversions. For these conversions, the default precision is 6. If the precision is explicitly 0, this suppresses the decimal point character entirely. For the ‘%g’ and ‘%G’ conversions, the precision specifies how many significant digits to print. Significant digits are the first digit before the decimal point, and all the digits after it. If the precision is 0 or not specified for ‘%g’ or ‘%G’, it is treated like a value of 1. If the value being printed cannot be expressed precisely in the specified number of digits, the value is rounded to the nearest number that fits.

14.2.10 Other Output Conversions

This section describes miscellaneous conversions for printf.

The ‘%c’ conversion prints a single character. The ‘-’ flag can be used to specify leftjustification in the field, but no other flags are defined, and no precision or type modifier can be given. For example:

     printf ("%c%c%c%c%c", "h", "e", "l", "l", "o");

prints ‘hello’.

The ‘%s’ conversion prints a string. The corresponding argument must be a string. A precision can be specified to indicate the maximum number of characters to write; otherwise characters in the string up to but not including the terminating null character are written to the output stream. The ‘-’ flag can be used to specify left-justification in the field, but no other flags or type modifiers are defined for this conversion. For example:

     printf ("%3s%-6s", "no", "where");

prints ‘ nowhere ’ (note the leading and trailing spaces).

14.2.11 Formatted Input

Octave provides the scanf, fscanf, and sscanf functions to read formatted input. There are two forms of each of these functions. One can be used to extract vectors of data from a file, and the other is more ‘C-like’.

[Built-in Function] [val, count, errmsg] = fscanf (fid, template, size)
[Built-in Function] [v1, v2, ..., count, errmsg] = fscanf (fid, template,
        "C")

In the first form, read from fid according to template, returning the result in the matrix val. The optional argument size specifies the amount of data to read and may be one of

Inf Read as much as possible, returning a column vector. nr Read up to nr elements, returning a column vector.

[nr, Inf] Read as much as possible, returning a matrix with nr rows. If the number of elements read is not an exact multiple of nr, the last column is padded with zeros.

[nr, nc] Read up to nr * nc elements, returning a matrix with nr rows. If the number of elements read is not an exact multiple of nr, the last column is padded with zeros.

If size is omitted, a value of Inf is assumed. A string is returned if template specifies only character conversions. The number of items successfully read is returned in count. If an error occurs, errmsg contains a system-dependent error message. In the second form, read from fid according to template, with each conversion specifier in template corresponding to a single scalar return value. This form is more “C-like”, and also compatible with previous versions of Octave. The number of successful conversions is returned in count See the Formatted Input section of the GNU Octave manual for a complete description of the syntax of the template string.

See also: [fgets], page 243, [fgetl], page 242, [fread], page 252, [scanf], page 249, [sscanf], page 249, [fopen], page 240.

[Built-in Function] [val, count, errmsg] = scanf (template, size)
[Built-in Function] [v1, v2, ..., count, errmsg]] = scanf (template, "C")

This is equivalent to calling fscanf with fid = stdin. It is currently not useful to call scanf in interactive programs.

See also: [fscanf], page 248, [sscanf], page 249, [printf], page 243.

[Built-in Function] [val, count, errmsg, pos] = sscanf (string, template,
   size)
[Built-in Function] [v1, v2, ..., count, errmsg] = sscanf (string,
   template, "C")

This is like fscanf, except that the characters are taken from the string string instead of from a stream. Reaching the end of the string is treated as an end-of-file condition. In addition to the values returned by fscanf, the index of the next character to be read is returned in pos.

See also: [fscanf], page 248, [scanf], page 249, [sprintf], page 244.

Calls to scanf are superficially similar to calls to printf in that arbitrary arguments are read under the control of a template string. While the syntax of the conversion specifications in the template is very similar to that for printf, the interpretation of the template is oriented more towards free-format input and simple pattern matching, rather than fixedfield formatting. For example, most scanf conversions skip over any amount of “white space” (including spaces, tabs, and newlines) in the input file, and there is no concept of precision for the numeric input conversions as there is for the corresponding output conversions. Ordinarily, non-whitespace characters in the template are expected to match characters in the input stream exactly.

When a matching failure occurs, scanf returns immediately, leaving the first nonmatching character as the next character to be read from the stream, and scanf returns all the items that were successfully converted.

The formatted input functions are not used as frequently as the formatted output functions. Partly, this is because it takes some care to use them properly. Another reason is that it is difficult to recover from a matching error.

14.2.12 Input Conversion Syntax

A scanf template string is a string that contains ordinary multibyte characters interspersed with conversion specifications that start with ‘%’.

Any whitespace character in the template causes any number of whitespace characters in the input stream to be read and discarded. The whitespace characters that are matched need not be exactly the same whitespace characters that appear in the template string. For example, write ‘ , ’ in the template to recognize a comma with optional whitespace before and after.

Other characters in the template string that are not part of conversion specifications must match characters in the input stream exactly; if this is not the case, a matching failure occurs.

The conversion specifications in a scanf template string have the general form:

     % flags width type conversion

In more detail, an input conversion specification consists of an initial ‘%’ character followed in sequence by:

• An optional flag character ‘*’, which says to ignore the text read for this specification. When scanf finds a conversion specification that uses this flag, it reads input as directed by the rest of the conversion specification, but it discards this input, does not return any value, and does not increment the count of successful assignments.

• An optional decimal integer that specifies the maximum field width. Reading of characters from the input stream stops either when this maximum is reached or when a non-matching character is found, whichever happens first. Most conversions discard initial whitespace characters, and these discarded characters don’t count towards the maximum field width. Conversions that do not discard initial whitespace are explicitly documented.

• An optional type modifier character. This character is ignored by Octave’s scanf function, but is recognized to provide compatibility with the C language scanf.

• A character that specifies the conversion to be applied.

The exact options that are permitted and how they are interpreted vary between the different conversion specifiers. See the descriptions of the individual conversions for information about the particular options that they allow.

14.2.13 Table of Input Conversions

Here is a table that summarizes the various conversion specifications:

‘%d’ Matches an optionally signed integer written in decimal. See Section 14.2.14 [Numeric Input Conversions], page 251.

‘%i’ Matches an optionally signed integer in any of the formats that the C language defines for specifying an integer constant. See Section 14.2.14 [Numeric Input Conversions], page 251.

‘%o’ Matches an unsigned integer written in octal radix. See Section 14.2.14 [Numeric Input Conversions], page 251.

‘%u’ Matches an unsigned integer written in decimal radix. See Section 14.2.14 [Numeric Input Conversions], page 251.

‘%x’, ‘%X’ Matches an unsigned integer written in hexadecimal radix. See Section 14.2.14 [Numeric Input Conversions], page 251.

‘%e’, ‘%f’, ‘%g’, ‘%E’, ‘%G’ Matches an optionally signed floating-point number. See Section 14.2.14 [Numeric Input Conversions], page 251.

‘%s’ Matches a string containing only non-whitespace characters. See Section 14.2.15 [String Input Conversions], page 252.

‘%c’ Matches a string of one or more characters; the number of characters read is controlled by the maximum field width given for the conversion. See Section 14.2.15 [String Input Conversions], page 252.

‘%%’ This matches a literal ‘%’ character in the input stream. No corresponding argument is used.

If the syntax of a conversion specification is invalid, the behavior is undefined. If there aren’t enough function arguments provided to supply addresses for all the conversion specifications in the template strings that perform assignments, or if the arguments are not of the correct types, the behavior is also undefined. On the other hand, extra arguments are simply ignored.

14.2.14 Numeric Input Conversions

This section describes the scanf conversions for reading numeric values.

The ‘%d’ conversion matches an optionally signed integer in decimal radix.

The ‘%i’ conversion matches an optionally signed integer in any of the formats that the C language defines for specifying an integer constant.

For example, any of the strings ‘10’, ‘0xa’, or ‘012’ could be read in as integers under the ‘%i’ conversion. Each of these specifies a number with decimal value 10.

The ‘%o’, ‘%u’, and ‘%x’ conversions match unsigned integers in octal, decimal, and hexadecimal radices, respectively.

The ‘%X’ conversion is identical to the ‘%x’ conversion. They both permit either uppercase or lowercase letters to be used as digits.

Unlike the C language scanf, Octave ignores the ‘h’, ‘l’, and ‘L’ modifiers.

14.2.15 String Input Conversions

This section describes the scanf input conversions for reading string and character values: ‘%s’ and ‘%c’.

The ‘%c’ conversion is the simplest: it matches a fixed number of characters, always. The maximum field with says how many characters to read; if you don’t specify the maximum, the default is 1. This conversion does not skip over initial whitespace characters. It reads precisely the next n characters, and fails if it cannot get that many.

The ‘%s’ conversion matches a string of non-whitespace characters. It skips and discards initial whitespace, but stops when it encounters more whitespace after having read something.

For example, reading the input:

      hello, world

with the conversion ‘%10c’ produces " hello, wo", but reading the same input with the conversion ‘%10s’ produces "hello,".

14.2.16 Binary I/O

Octave can read and write binary data using the functions fread and fwrite, which are patterned after the standard C functions with the same names. They are able to automatically swap the byte order of integer data and convert among the supported floating point formats as the data are read.

[Built-in Function] [val, count] = fread (fid, size, precision, skip, arch)

Read binary data of type precision from the specified file ID fid.

The optional argument size specifies the amount of data to read and may be one of

Inf Read as much as possible, returning a column vector.

nr Read up to nr elements, returning a column vector.

[nr, Inf] Read as much as possible, returning a matrix with nr rows. If the number of elements read is not an exact multiple of nr, the last column is padded with zeros.

[nr, nc] Read up to nr * nc elements, returning a matrix with nr rows. If the number of elements read is not an exact multiple of nr, the last column is padded with zeros.

If size is omitted, a value of Inf is assumed.

The optional argument precision is a string specifying the type of data to read and may be one of

     "schar"
     "signed char"

Signed character.

     "uchar"
     "unsigned char"

Unsigned character.

"int8"
"integer*1"

8-bit signed integer.

"int16"
"integer*2"

16-bit signed integer.

"int32"
"integer*4"

32-bit signed integer.

"int64"
"integer*8"

64-bit signed integer.

"uint8" 8-bit unsigned integer.

"uint16" 16-bit unsigned integer.

"uint32" 32-bit unsigned integer.

"uint64" 64-bit unsigned integer.

"single"
"float32"

"real*4" 32-bit floating point number.

"double"
"float64"

"real*8" 64-bit floating point number.

"char"

"char*1" Single character.

"short" Short integer (size is platform dependent).

"int" Integer (size is platform dependent).

"long" Long integer (size is platform dependent).

"ushort"
"unsigned short"

Unsigned short integer (size is platform dependent).

"uint"
"unsigned int"

Unsigned integer (size is platform dependent).

"ulong"
"unsigned long"

Unsigned long integer (size is platform dependent).

"float" Single precision floating point number (size is platform dependent).

The default precision is "uchar".

The precision argument may also specify an optional repeat count. For example, ‘32*single’ causes fread to read a block of 32 single precision floating point numbers. Reading in blocks is useful in combination with the skip argument.

The precision argument may also specify a type conversion. For example, ‘int16=>int32’ causes fread to read 16-bit integer values and return an array of 32-bit integer values. By default, fread returns a double precision array. The special form ‘*TYPE’ is shorthand for ‘TYPE=>TYPE’.

The conversion and repeat counts may be combined. For example, the specification ‘32*single=>single’ causes fread to read blocks of single precision floating point values and return an array of single precision values instead of the default array of double precision values.

The optional argument skip specifies the number of bytes to skip after each element (or block of elements) is read. If it is not specified, a value of 0 is assumed. If the final block read is not complete, the final skip is omitted. For example,

     fread (f, 10, "3*single=>single", 8)

will omit the final 8-byte skip because the last read will not be a complete block of 3 values.

The optional argument arch is a string specifying the data format for the file. Valid values are

"native" The format of the current machine.

"ieee-be"

IEEE big endian.

"ieee-le"

IEEE little endian.

The data read from the file is returned in val, and the number of values read is returned in count

See also: [fwrite], page 254, [fgets], page 243, [fgetl], page 242, [fscanf], page 248, [fopen], page 240.

[Built-in Function] count = fwrite (fid, data, precision, skip, arch)

Write data in binary form of type precision to the specified file ID fid, returning the number of values successfully written to the file.

The argument data is a matrix of values that are to be written to the file. The values are extracted in column-major order.

The remaining arguments precision, skip, and arch are optional, and are interpreted as described for fread.

The behavior of fwrite is undefined if the values in data are too large to fit in the specified precision.

See also: [fread], page 252, [fputs], page 242, [fprintf], page 243, [fopen], page 240.

14.2.17 Temporary Files

Sometimes one needs to write data to a file that is only temporary. This is most commonly used when an external program launched from within Octave needs to access data. When Octave exits all temporary files will be deleted, so this step need not be executed manually.

[Built-in Function] [fid, name, msg] = mkstemp (template, delete)

Return the file ID corresponding to a new temporary file with a unique name created from template. The last six characters of template must be XXXXXX and these are replaced with a string that makes the filename unique. The file is then created with mode read/write and permissions that are system dependent (on GNU/Linux systems, the permissions will be 0600 for versions of glibc 2.0.7 and later). The file is opened in binary mode and with the O_EXCL flag.

If the optional argument delete is supplied and is true, the file will be deleted automatically when Octave exits.

If successful, fid is a valid file ID, name is the name of the file, and msg is an empty string. Otherwise, fid is -1, name is empty, and msg contains a system-dependent error message.

See also: [tmpfile], page 255, [tmpnam], page 255, [P tmpdir], page 715.

[Built-in Function] [fid, msg] = tmpfile ()

Return the file ID corresponding to a new temporary file with a unique name. The file is opened in binary read/write ("w+b") mode. The file will be deleted automatically when it is closed or when Octave exits.

If successful, fid is a valid file ID and msg is an empty string. Otherwise, fid is -1 and msg contains a system-dependent error message.

See also: [tmpnam], page 255, [mkstemp], page 255, [P tmpdir], page 715.

[Built-in Function] tmpnam () [Built-in Function] tmpnam (dir) [Built-in Function] tmpnam (dir, prefix)

Return a unique temporary file name as a string.

If prefix is omitted, a value of "oct-" is used. If dir is also omitted, the default directory for temporary files is used. If dir is provided, it must exist, otherwise the default directory for temporary files is used. Since the named file is not opened, by tmpnam, it is possible (though relatively unlikely) that it will not be available by the time your program attempts to open it.

See also: [tmpfile], page 255, [mkstemp], page 255, [P tmpdir], page 715.

14.2.18 End of File and Errors

Once a file has been opened its status can be acquired. As an example the feof functions determines if the end of the file has been reached. This can be very useful when reading small parts of a file at a time. The following example shows how to read one line at a time from a file until the end has been reached.

filename = "myfile.txt";
fid = fopen (filename, "r");
while (! feof (fid) )
  text_line = fgetl (fid);
endwhile
fclose (fid);

Note that in some situations it is more efficient to read the entire contents of a file and then process it, than it is to read it line by line. This has the potential advantage of removing the loop in the above code.

[Built-in Function] feof (fid)

Return 1 if an end-of-file condition has been encountered for a given file and 0 otherwise. Note that it will only return 1 if the end of the file has already been encountered, not if the next read operation will result in an end-of-file condition.

See also: [fread], page 252, [fopen], page 240.

[Built-in Function] [err, msg] = ferror (fid) [Built-in Function] [err, msg] = ferror (fid, "clear")

Return 1 if an error condition has been encountered for the file ID fid and 0 otherwise. Note that it will only return 1 if an error has already been encountered, not if the next operation will result in an error condition.

The second argument is optional. If it is supplied, also clear the error condition.

See also: [fclear], page 256, [fopen], page 240.

[Built-in Function] fclear (fid)

Clear the stream state for the specified file.

See also: [fopen], page 240.

[Built-in Function] freport ()

Print a list of which files have been opened, and whether they are open for reading, writing, or both. For example:

     freport ()
          ⊣ number  mode  name

⊣

          ⊣  0     r  stdin
          ⊣  1     w  stdout
          ⊣  2     w  stderr
          ⊣  3     r  myfile

See also: [fopen], page 240, [fclose], page 242.

14.2.19 File Positioning

Three functions are available for setting and determining the position of the file pointer for a given file.

[Built-in Function] ftell (fid)

Return the position of the file pointer as the number of characters from the beginning of the file fid.

See also: [fseek], page 257, [feof], page 256, [fopen], page 240.

[Built-in Function] fseek (fid, offset) [Built-in Function] fseek (fid, offset, origin) [Built-in Function] status = fseek (…)

Set the file pointer to any location within the file fid. The pointer is positioned offset characters from the origin, which may be one of the predefined variables SEEK_CUR (current position), SEEK_SET (beginning), or SEEK_END (end of file) or strings "cof", "bof" or "eof". If origin is omitted, SEEK_SET is assumed. offset may be positive, negative, or zero but not all combinations of origin and offset can be realized. Return 0 on success and -1 on error.

See also: [fskipl], page 243, [frewind], page 257, [ftell], page 256, [fopen], page 240.

[Built-in Function] SEEK_SET () [Built-in Function] SEEK_CUR () [Built-in Function] SEEK_END ()

Return the numerical value to pass to fseek to perform one of the following actions:

SEEK_SET Position file relative to the beginning.

SEEK_CUR Position file relative to the current position.

SEEK_END Position file relative to the end.

See also: [fseek], page 257.

[Built-in Function] frewind (fid)

Move the file pointer to the beginning of the file fid, returning 0 for success, and -1 if an error was encountered. It is equivalent to fseek (fid, 0, SEEK_SET).

See also: [fseek], page 257, [ftell], page 256, [fopen], page 240.

The following example stores the current file position in the variable marker, moves the pointer to the beginning of the file, reads four characters, and then returns to the original position.

marker = ftell (myfile);
frewind (myfile);
fourch = fgets (myfile, 4);
fseek (myfile, marker, SEEK_SET);

15 Plotting

15.1 Introduction to Plotting

Earlier versions of Octave provided plotting through the use of gnuplot. This capability is still available. But, a newer plotting capability is provided by access to OpenGL. Which plotting system is used is controlled by the graphics_toolkit function. See Section 15.4.7 [Graphics Toolkits], page 375.

The function call graphics_toolkit ("fltk") selects the FLTK/OpenGL system, and graphics_toolkit ("gnuplot") selects the gnuplot system. The two systems may be used selectively through the use of the graphics_toolkit property of the graphics handle for each figure. This is explained in Section 15.3 [Graphics Data Structures], page 334. Caution: The FLTK toolkit uses single precision variables internally which limits the maximum value that can be displayed to approximately 10^{38}. If your data contains larger values you must use the gnuplot toolkit which supports values up to 10^{308}.

15.2 High-Level Plotting

Octave provides simple means to create many different types of two- and three-dimensional plots using high-level functions.

If you need more detailed control, see Section 15.3 [Graphics Data Structures], page 334 and Section 15.4 [Advanced Plotting], page 361.

15.2.1 Two-Dimensional Plots

The plot function allows you to create simple x-y plots with linear axes. For example,

     x = -10:0.1:10;
     plot (x, sin (x));

displays a sine wave shown in Figure 15.1. On most systems, this command will open a separate plot window to display the graph.

Figure 15.1: Simple Two-Dimensional Plot.

[Function File] plot (y) [Function File] plot (x, y) [Function File] plot (x, y, fmt) [Function File] plot (…, property, value, …) [Function File] plot (x1, y1, …, xn, yn) [Function File] plot (hax, …) [Function File] h = plot (…)

Produce 2-D plots. Many different combinations of arguments are possible. The simplest form is

     plot (y)

where the argument is taken as the set of y coordinates and the x coordinates are taken to be the range 1:numel (y). If more than one argument is given, they are interpreted as

     plot (y, property, value, ...)

or

     plot (x, y, property, value, ...)

or

     plot (x, y, fmt, ...)

and so on. Any number of argument sets may appear. The x and y values are interpreted as follows:

• If a single data argument is supplied, it is taken as the set of y coordinates and the x coordinates are taken to be the indices of the elements, starting with 1.

• If x and y are scalars, a single point is plotted.

• squeeze() is applied to arguments with more than two dimensions, but no more than two singleton dimensions.

• If both arguments are vectors, the elements of y are plotted versus the elements of x.

• If x is a vector and y is a matrix, then the columns (or rows) of y are plotted versus x. (using whichever combination matches, with columns tried first.)

• If the x is a matrix and y is a vector, y is plotted versus the columns (or rows) of x. (using whichever combination matches, with columns tried first.)

• If both arguments are matrices, the columns of y are plotted versus the columns of x. In this case, both matrices must have the same number of rows and columns and no attempt is made to transpose the arguments to make the number of rows match.

Multiple property-value pairs may be specified, but they must appear in pairs. These arguments are applied to the line objects drawn by plot. Useful properties

to modify are "linestyle", "linewidth", "color", "marker", "markersize",
"markeredgecolor", "markerfacecolor".

The fmt format argument can also be used to control the plot style. The format is composed of three parts: linestyle, markerstyle, color. When a markerstyle is specified, but no linestyle, only the markers are plotted. Similarly, if a linestyle is specified, but no markerstyle, then only lines are drawn. If both are specified then lines and markers will be plotted. If no fmt and no property/value pairs are given, then the default plot style is solid lines with no markers and the color determined by the "colororder" property of the current axes. Format arguments:

linestyle

‘-’  Use solid lines (default).
‘--’ Use dashed lines.
‘:’  Use dotted lines.
‘-.’ Use dash-dotted lines.

markerstyle

‘+’  crosshair
‘o’  circle
‘*’  star
‘.’  point
‘x’  cross
‘s’  square
‘d’  diamond
‘^’  upward-facing triangle
‘v’  downward-facing triangle
‘>’  right-facing triangle
‘<’  left-facing triangle
‘p’  pentagram
‘h’  hexagram

color

‘k’  blacK
‘r’  Red

‘g’  Green
‘b’  Blue
‘m’  Magenta
‘c’  Cyan
‘w’  White

";key;" Here "key" is the label to use for the plot legend.

The fmt argument may also be used to assign legend keys. To do so, include the desired label between semicolons after the formatting sequence described above, e.g., "+b;Key Title;". Note that the last semicolon is required and Octave will generate an error if it is left out. Here are some plot examples:

     plot (x, y, "or", x, y2, x, y3, "m", x, y4, "+")

This command will plot y with red circles, y2 with solid lines, y3 with solid magenta lines, and y4 with points displayed as ‘+’.

     plot (b, "*", "markersize", 10)

This command will plot the data in the variable b, with points displayed as ‘*’ and a marker size of 10.

     t = 0:0.1:6.3;
     plot (t, cos(t), "-;cos(t);", t, sin(t), "-b;sin(t);");

This will plot the cosine and sine functions and label them accordingly in the legend. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a vector of graphics handles to the created line objects. To save a plot, in one of several image formats such as PostScript or PNG, use the print command.

See also: [axis], page 285, [box], page 316, [grid], page 316, [hold], page 321, [legend], page 313, [title], page 313, [xlabel], page 315, [ylabel], page 315, [xlim], page 287, [ylim], page 287, [ezplot], page 288, [errorbar], page 276, [fplot], page 288, [line], page 337, [plot3], page 303, [polar], page 280, [loglog], page 264, [semilogx], page 263, [semilogy], page 263, [subplot], page 318.

The plotyy function may be used to create a plot with two independent y axes.

[Function File] plotyy (x1, y1, x2, y2) [Function File] plotyy (…, fun) [Function File] plotyy (…, fun1, fun2) [Function File] plotyy (hax, …) [Function File] [ax, h1, h2] = plotyy (…)

Plot two sets of data with independent y-axes. The arguments x1 and y1 define the arguments for the first plot and x1 and y2 for the second. By default the arguments are evaluated with feval (@plot, x, y). However the type of plot can be modified with the fun argument, in which case the plots are generated by feval (fun, x, y). fun can be a function handle, an inline function, or a string of a function name. The function to use for each of the plots can be independently defined with fun1 and fun2. If the first argument hax is an axes handle, then it defines the principal axis in which to plot the x1 and y1 data. The return value ax is a vector with the axis handles of the two y axes. h1 and h2 are handles to the objects generated by the plot commands.

     x = 0:0.1:2*pi;
     y1 = sin (x);
     y2 = exp (x - 1);
     ax = plotyy (x, y1, x - 1, y2, @plot, @semilogy);
     xlabel ("X");
     ylabel (ax(1), "Axis 1");
     ylabel (ax(2), "Axis 2");

See also: [plot], page 260.

The functions semilogx, semilogy, and loglog are similar to the plot function, but produce plots in which one or both of the axes use log scales.

[Function File] semilogx (y) [Function File] semilogx (x, y)

[Function File] semilogx (x, y, property, value, . . .)

[Function File] semilogx (x, y, fmt) [Function File] semilogx (hax, …) [Function File] h = semilogx (…)

Produce a 2-D plot using a logarithmic scale for the x-axis. See the documentation of plot for a description of the arguments that semilogx will accept. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created plot.

See also: [plot], page 260, [semilogy], page 263, [loglog], page 264.

[Function File] semilogy (y) [Function File] semilogy (x, y)

[Function File] semilogy (x, y, property, value, . . .)

[Function File] semilogy (x, y, fmt) [Function File] semilogy (h, …) [Function File] h = semilogy (…)

Produce a 2-D plot using a logarithmic scale for the y-axis. See the documentation of plot for a description of the arguments that semilogy will accept. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

The optional return value h is a graphics handle to the created plot.

See also: [plot], page 260, [semilogx], page 263, [loglog], page 264.

[Function File] loglog (y) [Function File] loglog (x, y) [Function File] loglog (x, y, prop, value, …) [Function File] loglog (x, y, fmt) [Function File] loglog (hax, …) [Function File] h = loglog (…)

Produce a 2-D plot using logarithmic scales for both axes. See the documentation of plot for a description of the arguments that loglog will accept. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created plot.

See also: [plot], page 260, [semilogx], page 263, [semilogy], page 263.

The functions bar, barh, stairs, and stem are useful for displaying discrete data. For example,

hist (randn (10000, 1), 30);

produces the histogram of 10,000 normally distributed random numbers shown in Figure 15.2.

Figure 15.2: Histogram.

[Function File] bar (y) [Function File] bar (x, y) [Function File] bar (…, w) [Function File] bar (…, style) [Function File] bar (…, prop, val, …)

[Function File] bar (hax, …) [Function File] h = bar (…, prop, val, …)

Produce a bar graph from two vectors of X-Y data. If only one argument is given, y, it is taken as a vector of Y values and the X coordinates are the range 1:numel (y). The optional input w controls the width of the bars. A value of 1.0 will cause each bar to exactly touch any adjacent bars. The default width is 0.8. If y is a matrix, then each column of y is taken to be a separate bar graph plotted on the same graph. By default the columns are plotted side-by-side. This behavior can be changed by the style argument which can take the following values:

"grouped" (default)

Side-by-side bars with a gap between bars and centered over the Xcoordinate.

"stacked"

Bars are stacked so that each X value has a single bar composed of multiple segments.

"hist" Side-by-side bars with no gap between bars and centered over the Xcoordinate.

"histc" Side-by-side bars with no gap between bars and left-aligned to the Xcoordinate.

Optional property/value pairs are passed directly to the underlying patch objects. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a vector of handles to the created "bar series" hggroups with one handle per column of the variable y. This series makes it possible to change a common element in one bar series object and have the change reflected in the other "bar series". For example,

     h = bar (rand (5, 10));
     set (h(1), "basevalue", 0.5);

changes the position on the base of all of the bar series. The following example modifies the face and edge colors using property/value pairs.

     bar (randn (1, 100), "facecolor", "r", "edgecolor", "b");

The color of the bars is taken from the figure’s colormap, such that

     bar (rand (10, 3));
     colormap (summer (64));

will change the colors used for the bars. The color of bars can also be set manually using the "facecolor" property as shown below.

     h = bar (rand (10, 3));
     set (h(1), "facecolor", "r")
     set (h(2), "facecolor", "g")
     set (h(3), "facecolor", "b")

See also: [barh], page 266, [hist], page 266, [pie], page 280, [plot], page 260, [patch], page 337.

[Function File] barh (y) [Function File] barh (x, y) [Function File] barh (…, w) [Function File] barh (…, style) [Function File] barh (…, prop, val, …) [Function File] barh (hax, …) [Function File] h = barh (…, prop, val, …)

Produce a horizontal bar graph from two vectors of X-Y data. If only one argument is given, it is taken as a vector of Y values and the X coordinates are the range 1:numel (y). The optional input w controls the width of the bars. A value of 1.0 will cause each bar to exactly touch any adjacent bars. The default width is 0.8. If y is a matrix, then each column of y is taken to be a separate bar graph plotted on the same graph. By default the columns are plotted side-by-side. This behavior can be changed by the style argument which can take the following values:

"grouped" (default)

Side-by-side bars with a gap between bars and centered over the Ycoordinate.

"stacked"

Bars are stacked so that each Y value has a single bar composed of multiple segments.

"hist" Side-by-side bars with no gap between bars and centered over the Ycoordinate.

"histc" Side-by-side bars with no gap between bars and left-aligned to the Ycoordinate.

Optional property/value pairs are passed directly to the underlying patch objects. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created bar series hggroup. For a description of the use of the bar series, see [bar], page 264.

See also: [bar], page 264, [hist], page 266, [pie], page 280, [plot], page 260, [patch], page 337.

[Function File] hist (y) [Function File] hist (y, x) [Function File] hist (y, nbins) [Function File] hist (y, x, norm) [Function File] hist (…, prop, val, …) [Function File] hist (hax, …) [Function File] [nn, xx] = hist (…)

Produce histogram counts or plots. With one vector input argument, y, plot a histogram of the values with 10 bins. The range of the histogram bins is determined by the range of the data. With one matrix input argument, y, plot a histogram where each bin contains a bar per input column.

Given a second vector argument, x, use that as the centers of the bins, with the width of the bins determined from the adjacent values in the vector.

If scalar, the second argument, nbins, defines the number of bins.

If a third argument is provided, the histogram is normalized such that the sum of the bars is equal to norm.

Extreme values are lumped into the first and last bins.

The histogram’s appearance may be modified by specifying property/value pairs. For example the face and edge color may be modified.

     hist (randn (1, 100), 25, "facecolor", "r", "edgecolor", "b");

The histogram’s colors also depend upon the current colormap.

     hist (rand (10, 3));
     colormap (summer ());

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

With two output arguments, produce the values nn (numbers of elements) and xx (bin centers) such that bar (xx, nn) will plot the histogram.

See also: [histc], page 567, [bar], page 264, [pie], page 280, [rose], page 273.

[Function File] stemleaf (x, caption)

[Function File] stemleaf (x, caption, stem_sz)

[Function File] plotstr = stemleaf (…)

Compute and display a stem and leaf plot of the vector x.

The input x should be a vector of integers. Any non-integer values will be converted to integer by x = fix (x). By default each element of x will be plotted with the last digit of the element as a leaf value and the remaining digits as the stem. For example, 123 will be plotted with the stem ‘12’ and the leaf ‘3’. The second argument, caption, should be a character array which provides a description of the data. It is included as a heading for the output.

The optional input stem sz sets the width of each stem. The stem width is determined by 10^(stem_sz + 1). The default stem width is 10.

The output of stemleaf is composed of two parts: a "Fenced Letter Display," followed by the stem-and-leaf plot itself. The Fenced Letter Display is described in Exploratory Data Analysis. Briefly, the entries are as shown:

             Fenced Letter Display
     #% nx|___________________nx = numel (x)
     M% mi|       md         |     mi median index, md median
     H% hi|hl              hu| hs  hi lower hinge index, hl,hu hinges,
     1    |x(1)         x(nx)|     hs h_spreadx(1), x(nx) first
                _______and last data value.
          ______|step |_______step 1.5*h_spread
         f|ifl            ifh|     inner fence, lower and higher
          |nfl            nfh|     no.\ of data points within fences
         F|ofl            ofh|     outer fence, lower and higher
          |nFl            nFh|     no.\ of data points outside outer
                                   fences

The stem-and-leaf plot shows on each line the stem value followed by the string made up of the leaf digits. If the stem sz is not 1 the successive leaf values are separated by ",". With no return argument, the plot is immediately displayed. If an output argument is provided, the plot is returned as an array of strings. The leaf digits are not sorted. If sorted leaf values are desired, use xs = sort (x) before calling stemleaf (xs). The stem and leaf plot and associated displays are described in: Ch. 3, Exploratory Data Analysis by J. W. Tukey, Addison-Wesley, 1977.

See also: [hist], page 266, [printd], page 268.

[Function File] printd (obj, filename)

[Function File] out_file = printd (…)

Convert any object acceptable to disp into the format selected by the suffix of filename. If the return argument out file is given, the name of the created file is returned. This function is intended to facilitate manipulation of the output of functions such

as stemleaf.

See also: [stemleaf], page 267.

[Function File] stairs (y) [Function File] stairs (x, y) [Function File] stairs (…, style) [Function File] stairs (…, prop, val, …) [Function File] stairs (hax, …) [Function File] h = stairs (…)

[Function File] [xstep, ystep] = stairs (. . .)

Produce a stairstep plot. The arguments x and y may be vectors or matrices. If only one argument is given, it is taken as a vector of Y values and the X coordinates are taken to be the indices of the elements. The style to use for the plot can be defined with a line style style of the same format as the plot command.

Multiple property/value pairs may be specified, but they must appear in pairs.

If the first argument hax is an axis handle, then plot into this axis, rather than the current axis handle returned by gca.

If one output argument is requested, return a graphics handle to the created plot. If two output arguments are specified, the data are generated but not plotted. For example,

     stairs (x, y);

and

     [xs, ys] = stairs (x, y);
     plot (xs, ys);

are equivalent.

See also: [bar], page 264, [hist], page 266, [plot], page 260, [stem], page 269.

[Function File] stem (y) [Function File] stem (x, y) [Function File] stem (…, linespec)

[Function File] stem (…, "filled")

[Function File] stem (…, prop, val, …) [Function File] stem (hax, …) [Function File] h = stem (…)

Plot a 2-D stem graph.

If only one argument is given, it is taken as the y-values and the x-coordinates are taken from the indices of the elements.

If y is a matrix, then each column of the matrix is plotted as a separate stem graph. In this case x can either be a vector, the same length as the number of rows in y, or it can be a matrix of the same size as y.

The default color is "b" (blue), the default line style is "-", and the default marker is "o". The line style can be altered by the linespec argument in the same manner as the plot command. If the "filled" argument is present the markers at the top of the stems will be filled in. For example,

     x = 1:10;
     y = 2*x;
     stem (x, y, "r");

plots 10 stems with heights from 2 to 20 in red;

Optional property/value pairs may be specified to control the appearance of the plot.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

The optional return value h is a handle to a "stem series" hggroup. The single hggroup handle has all of the graphical elements comprising the plot as its children; This allows the properties of multiple graphics objects to be changed by modifying just a single property of the "stem series" hggroup.

For example,

     x = [0:10]';
     y = [sin(x), cos(x)]
     h = stem (x, y);
     set (h(2), "color", "g");
     set (h(1), "basevalue", -1)

changes the color of the second "stem series" and moves the base line of the first. Stem Series Properties

linestyle The linestyle of the stem. (Default: "-")

linewidth The width of the stem. (Default: 0.5)

color The color of the stem, and if not separately specified, the marker. (Default: "b" [blue])

marker The marker symbol to use at the top of each stem. (Default: "o")

markeredgecolor The edge color of the marker. (Default: "color" property)

markerfacecolor The color to use for "filling" the marker. (Default: "none" [unfilled])

markersize The size of the marker. (Default: 6)

baseline The handle of the line object which implements the baseline. Use set with the returned handle to change graphic properties of the baseline.

basevalue The y-value where the baseline is drawn. (Default: 0)

See also: [stem3], page 270, [bar], page 264, [hist], page 266, [plot], page 260, [stairs], page 268.

[Function File] stem3 (x, y, z) [Function File] stem3 (…, linespec)

[Function File] stem3 (…, "filled")

[Function File] stem3 (…, prop, val, …) [Function File] stem3 (hax, …) [Function File] h = stem3 (…)

Plot a 3-D stem graph. Stems are drawn from the height z to the location in the x-y plane determined by x and y. The default color is "b" (blue), the default line style is "-", and the default marker is "o". The line style can be altered by the linespec argument in the same manner as the plot command. If the "filled" argument is present the markers at the top of the stems will be filled in. Optional property/value pairs may be specified to control the appearance of the plot. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

The optional return value h is a handle to the "stem series" hggroup containing the line and marker objects used for the plot. See [stem], page 269, for a description of the "stem series" object. Example:

     theta = 0:0.2:6;
     stem3 (cos (theta), sin (theta), theta);

plots 31 stems with heights from 0 to 6 lying on a circle. Implementation Note: Color definitions with RGB-triples are not valid.

See also: [stem], page 269, [bar], page 264, [hist], page 266, [plot], page 260.

[Function File] scatter (x, y) [Function File] scatter (x, y, s) [Function File] scatter (x, y, s, c) [Function File] scatter (…, style) [Function File] scatter (…, "filled") [Function File] scatter (…, prop, val, …) [Function File] scatter (hax, …) [Function File] h = scatter (…)

Draw a 2-D scatter plot. A marker is plotted at each point defined by the coordinates in the vectors x and y. The size of the markers is determined by s, which can be a scalar or a vector of the same length as x and y. If s is not given, or is an empty matrix, then a default value of 8 points is used. The color of the markers is determined by c, which can be a string defining a fixed color; a 3-element vector giving the red, green, and blue components of the color; a vector of the same length as x that gives a scaled index into the current colormap; or an Nx3 matrix defining the RGB color of each marker individually. The marker to use can be changed with the style argument, that is a string defining a marker in the same manner as the plot command. If no marker is specified it defaults to "o" or circles. If the argument "filled" is given then the markers are filled. Additional property/value pairs are passed directly to the underlying patch object. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created patch object. Example:

     x = randn (100, 1);
     y = randn (100, 1);
     scatter (x, y, [], sqrt (x.^2 + y.^2));

See also: [scatter3], page 306, [patch], page 337, [plot], page 260.

[Function File] plotmatrix (x, y) [Function File] plotmatrix (x) [Function File] plotmatrix (…, style) [Function File] plotmatrix (hax, …)

[Function File] [h, ax, bigax, p, pax] = plotmatrix (. . .)

Scatter plot of the columns of one matrix against another. Given the arguments x and y, that have a matching number of rows, plotmatrix plots a set of axes corresponding to

     plot (x(:, i), y(:, j))

Given a single argument x this is equivalent to

     plotmatrix (x, x)

except that the diagonal of the set of axes will be replaced with the histogram hist

(x(:, i)).

The marker to use can be changed with the style argument, that is a string defining a marker in the same manner as the plot command. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h provides handles to the individual graphics objects in the scatter plots, whereas ax returns the handles to the scatter plot axis objects. bigax is a hidden axis object that surrounds the other axes, such that the commands xlabel, title, etc., will be associated with this hidden axis. Finally, p returns the graphics objects associated with the histogram and pax the corresponding axes objects. Example:

     plotmatrix (randn (100, 3), "g+")

See also: [scatter], page 271, [plot], page 260.

[Function File] pareto (y) [Function File] pareto (y, x) [Function File] pareto (hax, …) [Function File] h = pareto (…)

Draw a Pareto chart. A Pareto chart is a bar graph that arranges information in such a way that priorities for process improvement can be established; It organizes and displays information to show the relative importance of data. The chart is similar to the histogram or bar chart, except that the bars are arranged in decreasing magnitude from left to right along the x-axis. The fundamental idea (Pareto principle) behind the use of Pareto diagrams is that the majority of an effect is due to a small subset of the causes. For quality improvement, the first few contributing causes (leftmost bars as presented on the diagram) to a problem usually account for the majority of the result. Thus, targeting these "major causes" for elimination results in the most cost-effective improvement scheme. Typically only the magnitude data y is present in which case x is taken to be the range 1 : length (y). If x is given it may be a string array, a cell array of strings, or a numerical vector. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a 2-element vector with a graphics handle for the created bar plot and a second handle for the created line plot.

An example of the use of pareto is

     Cheese = {"Cheddar", "Swiss", "Camembert", ...
               "Munster", "Stilton", "Blue"};
     Sold = [105, 30, 70, 10, 15, 20];
     pareto (Sold, Cheese);

See also: [bar], page 264, [barh], page 266, [hist], page 266, [pie], page 280, [plot], page 260.

[Function File] rose (th) [Function File] rose (th, nbins) [Function File] rose (th, bins) [Function File] rose (hax, …) [Function File] h = rose (…) [Function File] [thout rout] = rose (…)

Plot an angular histogram. With one vector argument, th, plot the histogram with 20 angular bins. If th is a matrix then each column of th produces a separate histogram. If nbins is given and is a scalar, then the histogram is produced with nbin bins. If bins is a vector, then the center of each bin is defined by the values of bins and the number of bins is given by the number of elements in bins. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a vector of graphics handles to the line objects representing each histogram. If two output arguments are requested then no plot is made and the polar vectors necessary to plot the histogram are returned instead.

     [th, r] = rose ([2*randn(1e5,1), pi + 2*randn(1e5,1)]);
     polar (th, r);

See also: [hist], page 266, [polar], page 280.

The contour, contourf and contourc functions produce two-dimensional contour plots from three-dimensional data.

[Function File] contour (z) [Function File] contour (z, vn) [Function File] contour (x, y, z) [Function File] contour (x, y, z, vn) [Function File] contour (…, style) [Function File] contour (hax, …) [Function File] [c, h] = contour (…)

Create a 2-D contour plot. Plot level curves (contour lines) of the matrix z, using the contour matrix c computed by contourc from the same arguments; see the latter for their interpretation. The appearance of contour lines can be defined with a line style style in the same manner as plot. Only line style and color are used; Any markers defined by style are ignored.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional output c are the contour levels in contourc format. The optional return value h is a graphics handle to the hggroup comprising the contour lines. Example:

     x = 0:2;
     y = x;
     z = x' * y;
     contour (x, y, z, 2:3)

See also: [ezcontour], page 289, [contourc], page 274, [contourf], page 274, [contour3], page 275, [clabel], page 315, [meshc], page 293, [surfc], page 295, [caxis], page 287, [colormap], page 660, [plot], page 260.

[Function File] contourf (z) [Function File] contourf (z, vn) [Function File] contourf (x, y, z) [Function File] contourf (x, y, z, vn) [Function File] contourf (…, style) [Function File] contourf (hax, …) [Function File] [c, h] = contourf (…)

Create a 2-D contour plot with filled intervals. Plot level curves (contour lines) of the matrix z and fill the region between lines with colors from the current colormap. The level curves are taken from the contour matrix c computed by contourc for the same arguments; see the latter for their interpretation. The appearance of contour lines can be defined with a line style style in the same manner as plot. Only line style and color are used; Any markers defined by style are ignored. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional output c are the contour levels in contourc format. The optional return value h is a graphics handle to the hggroup comprising the contour lines. The following example plots filled contours of the peaks function.

     [x, y, z] = peaks (50);
     contourf (x, y, z, -7:9)

See also: [ezcontourf], page 290, [contour], page 273, [contourc], page 274, [contour3], page 275, [clabel], page 315, [meshc], page 293, [surfc], page 295, [caxis], page 287, [colormap], page 660, [plot], page 260.

[Function File] [c, lev] = contourc (z)
[Function File] [c, lev] = contourc (z, vn)
[Function File] [c, lev] = contourc (x, y, z)

[Function File] [c, lev] = contourc (x, y, z, vn)

Compute contour lines (isolines of constant Z value). The matrix z contains height values above the rectangular grid determined by x and y. If only a single input z is provided then x is taken to be 1:rows (z) and y is taken

to be 1:columns (z).

The optional input vn is either a scalar denoting the number of contour lines to compute or a vector containing the Z values where lines will be computed. When vn is a vector the number of contour lines is numel (vn). However, to compute a single contour line at a given value use vn = [val, val]. If vn is omitted it defaults to 10. The return value c is a 2xn matrix containing the contour lines in the following format

     c = [lev1, x1, x2, ..., levn, x1, x2, ...
          len1, y1, y2, ..., lenn, y1, y2, ...]

in which contour line n has a level (height) of levn and length of lenn. The optional return value lev is a vector with the Z values of of the contour levels. Example:

     x = 0:2;
     y = x;
     z = x' * y;
     contourc (x, y, z, 2:3)
        ⇒ 2.0000   2.0000   1.0000   3.0000   1.5000   2.0000
             2.0000   1.0000   2.0000   2.0000   2.0000   1.5000

See also: [contour], page 273, [contourf], page 274, [contour3], page 275, [clabel], page 315.

[Function File] contour3 (z) [Function File] contour3 (z, vn) [Function File] contour3 (x, y, z) [Function File] contour3 (x, y, z, vn) [Function File] contour3 (…, style) [Function File] contour3 (hax, …) [Function File] [c, h] = contour3 (…)

Create a 3-D contour plot. contour3 plots level curves (contour lines) of the matrix z at a Z level corresponding to each contour. This is in contrast to contour which plots all of the contour lines at the same Z level and produces a 2-D plot. The level curves are taken from the contour matrix c computed by contourc for the same arguments; see the latter for their interpretation. The appearance of contour lines can be defined with a line style style in the same manner as plot. Only line style and color are used; Any markers defined by style are ignored. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional output c are the contour levels in contourc format.

The optional return value h is a graphics handle to the hggroup comprising the contour lines. Example:

     contour3 (peaks (19));
     colormap cool;
     hold on;
     surf (peaks (19), "facecolor", "none", "edgecolor", "black");

See also: [contour], page 273, [contourc], page 274, [contourf], page 274, [clabel], page 315, [meshc], page 293, [surfc], page 295, [caxis], page 287, [colormap], page 660, [plot], page 260.

The errorbar, semilogxerr, semilogyerr, and loglogerr functions produce plots

with error bar markers. For example,

x = 0:0.1:10;
y = sin (x);
yp =  0.1 .* randn (size (x));
ym = -0.1 .* randn (size (x));
errorbar (x, sin (x), ym, yp);

produces the figure shown in Figure 15.3.

Figure 15.3: Errorbar plot.

[Function File] errorbar (y, ey) [Function File] errorbar (y, …, fmt) [Function File] errorbar (x, y, ey) [Function File] errorbar (x, y, err, fmt)

[Function File] errorbar (x, y, lerr, uerr, fmt)

[Function File] errorbar (x, y, ex, ey, fmt) [Function File] errorbar (x, y, lx, ux, ly, uy, fmt) [Function File] errorbar (x1, y1, …, fmt, xn, yn, …)

[Function File] errorbar (hax, …) [Function File] h = errorbar (…)

Create a 2-D plot with errorbars. Many different combinations of arguments are possible. The simplest form is

     errorbar (y, ey)

where the first argument is taken as the set of y coordinates, the second argument ey are the errors around the y values, and the x coordinates are taken to be the indices of the elements (1:numel (y)). The general form of the function is

     errorbar (x, y, err1, ..., fmt, ...)

After the x and y arguments there can be 1, 2, or 4 parameters specifying the error values depending on the nature of the error values and the plot format fmt.

err (scalar)

When the error is a scalar all points share the same error value. The errorbars are symmetric and are drawn from data-err to data+err. The fmt argument determines whether err is in the x-direction, y-direction (default), or both.

err (vector or matrix)

Each data point has a particular error value. The errorbars are symmetric and are drawn from data(n)-err(n) to data(n)+err(n).

lerr, uerr (scalar)

The errors have a single low-side value and a single upper-side value. The errorbars are not symmetric and are drawn from data-lerr to data+uerr.

lerr, uerr (vector or matrix)

Each data point has a low-side error and an upper-side error. The errorbars are not symmetric and are drawn from data(n)-lerr(n) to data(n)+uerr(n).

Any number of data sets (x1,y1, x2,y2, …) may appear as long as they are separated by a format string fmt. If y is a matrix, x and the error parameters must also be matrices having the same dimensions. The columns of y are plotted versus the corresponding columns of x and errorbars are taken from the corresponding columns of the error parameters. If fmt is missing, the yerrorbars ("~") plot style is assumed. If the fmt argument is supplied then it is interpreted, as in normal plots, to specify the line style, marker, and color. In addition, fmt may include an errorbar style which must precede the ordinary format codes. The following errorbar styles are supported:

‘~’ Set yerrorbars plot style (default).

‘>’ Set xerrorbars plot style.

‘~>’ Set xyerrorbars plot style.

‘#~’ Set yboxes plot style.

‘#’ Set xboxes plot style.

‘#~>’ Set xyboxes plot style.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a handle to the hggroup object representing the data plot and errorbars. Note: For compatibility with matlab a line is drawn through all data points. However, most scientific errorbar plots are a scatter plot of points with errorbars. To accomplish this, add a marker style to the fmt argument such as ".". Alternatively, remove the line by modifying the returned graphic handle with set (h, "linestyle",

"none").

Examples:

     errorbar (x, y, ex, ">.r")

produces an xerrorbar plot of y versus x with x errorbars drawn from x-ex to x+ex. The marker "." is used so no connecting line is drawn and the errorbars appear in red.

     errorbar (x, y1, ey, "~",
               x, y2, ly, uy)

produces yerrorbar plots with y1 and y2 versus x. Errorbars for y1 are drawn from y1-ey to y1+ey, errorbars for y2 from y2-ly to y2+uy.

     errorbar (x, y, lx, ux,
               ly, uy, "~>")

produces an xyerrorbar plot of y versus x in which x errorbars are drawn from x-lx to x+ux and y errorbars from y-ly to y+uy.

See also: [semilogxerr], page 278, [semilogyerr], page 279, [loglogerr], page 279, [plot], page 260.

[Function File] semilogxerr (y, ey) [Function File] semilogxerr (y, …, fmt) [Function File] semilogxerr (x, y, ey)

[Function File] semilogxerr (x, y, err, fmt)
[Function File] semilogxerr (x, y, lerr, uerr, fmt)
[Function File] semilogxerr (x, y, ex, ey, fmt)
[Function File] semilogxerr (x, y, lx, ux, ly, uy, fmt)

[Function File] semilogxerr (x1, y1, …, fmt, xn, yn, …) [Function File] semilogxerr (hax, …) [Function File] h = semilogxerr (…)

Produce 2-D plots using a logarithmic scale for the x-axis and errorbars at each data point. Many different combinations of arguments are possible. The most common form is

     semilogxerr (x, y, ey, fmt)

which produces a semi-logarithmic plot of y versus x with errors in the y-scale defined by ey and the plot format defined by fmt. See [errorbar], page 276, for available formats and additional information.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

See also: [errorbar], page 276, [semilogyerr], page 279, [loglogerr], page 279.

[Function File] semilogyerr (y, ey) [Function File] semilogyerr (y, …, fmt) [Function File] semilogyerr (x, y, ey)

[Function File] semilogyerr (x, y, err, fmt)
[Function File] semilogyerr (x, y, lerr, uerr, fmt)
[Function File] semilogyerr (x, y, ex, ey, fmt)
[Function File] semilogyerr (x, y, lx, ux, ly, uy, fmt)

[Function File] semilogyerr (x1, y1, …, fmt, xn, yn, …) [Function File] semilogyerr (hax, …) [Function File] h = semilogyerr (…)

Produce 2-D plots using a logarithmic scale for the y-axis and errorbars at each data point. Many different combinations of arguments are possible. The most common form is

     semilogyerr (x, y, ey, fmt)

which produces a semi-logarithmic plot of y versus x with errors in the y-scale defined by ey and the plot format defined by fmt. See [errorbar], page 276, for available formats and additional information. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

See also: [errorbar], page 276, [semilogxerr], page 278, [loglogerr], page 279.

[Function File] loglogerr (y, ey) [Function File] loglogerr (y, …, fmt) [Function File] loglogerr (x, y, ey) [Function File] loglogerr (x, y, err, fmt)

[Function File] loglogerr (x, y, lerr, uerr, fmt)

[Function File] loglogerr (x, y, ex, ey, fmt)

[Function File] loglogerr (x, y, lx, ux, ly, uy, fmt)

[Function File] loglogerr (x1, y1, …, fmt, xn, yn, …) [Function File] loglogerr (hax, …) [Function File] h = loglogerr (…)

Produce 2-D plots on a double logarithm axis with errorbars. Many different combinations of arguments are possible. The most common form is

     loglogerr (x, y, ey, fmt)

which produces a double logarithm plot of y versus x with errors in the y-scale defined by ey and the plot format defined by fmt. See [errorbar], page 276, for available formats and additional information. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

See also: [errorbar], page 276, [semilogxerr], page 278, [semilogyerr], page 279.

Finally, the polar function allows you to easily plot data in polar coordinates. However, the display coordinates remain rectangular and linear. For example,

polar (0:0.1:10*pi, 0:0.1:10*pi);

produces the spiral plot shown in Figure 15.4.

Figure 15.4: Polar plot.

[Function File] polar (theta, rho) [Function File] polar (theta, rho, fmt) [Function File] polar (cplx) [Function File] polar (cplx, fmt) [Function File] polar (hax, …) [Function File] h = polar (…)

Create a 2-D plot from polar coordinates theta and rho.

If a single complex input cplx is given then the real part is used for theta and the imaginary part is used for rho.

The optional argument fmt specifies the line format in the same way as plot.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

The optional return value h is a graphics handle to the created plot.

See also: [rose], page 273, [compass], page 283, [plot], page 260.

[Function File] pie (x) [Function File] pie (…, explode) [Function File] pie (…, labels) [Function File] pie (hax, …); [Function File] h = pie (…);

Plot a 2-D pie chart.

When called with a single vector argument, produce a pie chart of the elements in x. The size of the ith slice is the percentage that the element xi represents of the total

sum of x: pct = x(i) / sum (x).

The optional input explode is a vector of the same length as x that, if non-zero, "explodes" the slice from the pie chart. The optional input labels is a cell array of strings of the same length as x specifying the label for each slice. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a list of handles to the patch and text objects generating the plot. Note: If sum (x) ≤ 1 then the elements of x are interpreted as percentages directly and are not normalized by sum (x). Furthermore, if the sum is less than 1 then there will be a missing slice in the pie plot to represent the missing, unspecified percentage.

See also: [pie3], page 281, [bar], page 264, [hist], page 266, [rose], page 273.

[Function File] pie3 (x) [Function File] pie3 (…, explode) [Function File] pie3 (…, labels) [Function File] pie3 (hax, …); [Function File] h = pie3 (…);

Plot a 3-D pie chart. Called with a single vector argument, produces a 3-D pie chart of the elements in x. The size of the ith slice is the percentage that the element xi represents of the total

sum of x: pct = x(i) / sum (x).

The optional input explode is a vector of the same length as x that, if non-zero, "explodes" the slice from the pie chart. The optional input labels is a cell array of strings of the same length as x specifying the label for each slice. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a list of graphics handles to the patch, surface, and text objects generating the plot. Note: If sum (x) ≤ 1 then the elements of x are interpreted as percentages directly and are not normalized by sum (x). Furthermore, if the sum is less than 1 then there will be a missing slice in the pie plot to represent the missing, unspecified percentage.

See also: [pie], page 280, [bar], page 264, [hist], page 266, [rose], page 273.

[Function File] quiver (u, v) [Function File] quiver (x, y, u, v) [Function File] quiver (…, s) [Function File] quiver (…, style) [Function File] quiver (…, "filled") [Function File] quiver (hax, …)

[Function File] h = quiver (…)

Plot the (u, v) components of a vector field in an (x, y) meshgrid. If the grid is uniform, you can specify x and y as vectors. If x and y are undefined they are assumed to be (1:m, 1:n) where [m, n] = size

(u).

The variable s is a scalar defining a scaling factor to use for the arrows of the field relative to the mesh spacing. A value of 0 disables all scaling. The default value is 0.9. The style to use for the plot can be defined with a line style style of the same format as the plot command. If a marker is specified then markers at the grid points of the vectors are drawn rather than arrows. If the argument "filled" is given then the markers are filled. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to a quiver object. A quiver object regroups the components of the quiver plot (body, arrow, and marker), and allows them to be changed together. Example:

     [x, y] = meshgrid (1:2:20);
     h = quiver (x, y, sin (2*pi*x/10), sin (2*pi*y/10));
     set (h, "maxheadsize", 0.33);

See also: [quiver3], page 282, [compass], page 283, [feather], page 283, [plot], page 260.

[Function File] quiver3 (u, v, w) [Function File] quiver3 (x, y, z, u, v, w) [Function File] quiver3 (…, s) [Function File] quiver3 (…, style) [Function File] quiver3 (…, "filled") [Function File] quiver3 (hax, …) [Function File] h = quiver3 (…)

Plot the (u, v, w) components of a vector field in an (x, y, z) meshgrid. If the grid is uniform, you can specify x, y, and z as vectors. If x, y, and z are undefined they are assumed to be (1:m, 1:n, 1:p) where [m, n]

= size (u) and p = max (size (w)).

The variable s is a scalar defining a scaling factor to use for the arrows of the field relative to the mesh spacing. A value of 0 disables all scaling. The default value is 0.9. The style to use for the plot can be defined with a line style style of the same format as the plot command. If a marker is specified then markers at the grid points of the vectors are drawn rather than arrows. If the argument "filled" is given then the markers are filled. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

The optional return value h is a graphics handle to a quiver object. A quiver object regroups the components of the quiver plot (body, arrow, and marker), and allows them to be changed together.

     [x, y, z] = peaks (25);
     surf (x, y, z);
     hold on;
     [u, v, w] = surfnorm (x, y, z / 10);
     h = quiver3 (x, y, z, u, v, w);
     set (h, "maxheadsize", 0.33);

See also: [quiver], page 281, [compass], page 283, [feather], page 283, [plot], page 260.

[Function File] compass (u, v) [Function File] compass (z) [Function File] compass (…, style) [Function File] compass (hax, …) [Function File] h = compass (…)

Plot the (u, v) components of a vector field emanating from the origin of a polar plot. The arrow representing each vector has one end at the origin and the tip at [u(i), v(i)]. If a single complex argument z is given, then u = real (z) and v = imag (z). The style to use for the plot can be defined with a line style style of the same format as the plot command. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a vector of graphics handles to the line objects representing the drawn vectors.

     a = toeplitz ([1;randn(9,1)], [1,randn(1,9)]);
     compass (eig (a));

See also: [polar], page 280, [feather], page 283, [quiver], page 281, [rose], page 273, [plot], page 260.

[Function File] feather (u, v) [Function File] feather (z) [Function File] feather (…, style) [Function File] feather (hax, …) [Function File] h = feather (…)

Plot the (u, v) components of a vector field emanating from equidistant points on the x-axis. If a single complex argument z is given, then u = real (z) and v = imag (z). The style to use for the plot can be defined with a line style style of the same format as the plot command. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a vector of graphics handles to the line objects representing the drawn vectors.

     phi = [0 : 15 : 360] * pi/180;
     feather (sin (phi), cos (phi));

See also: [plot], page 260, [quiver], page 281, [compass], page 283.

[Function File] pcolor (x, y, c) [Function File] pcolor (c) [Function File] pcolor (hax, …) [Function File] h = pcolor (…)

Produce a 2-D density plot. A pcolor plot draws rectangles with colors from the matrix c over the two-dimensional region represented by the matrices x and y. x and y are the coordinates of the mesh’s vertices and are typically the output of meshgrid. If x and y are vectors, then a typical vertex is (x(j), y(i), c(i,j)). Thus, columns of c correspond to different x values and rows of c correspond to different y values. The values in c are scaled to span the range of the current colormap. Limits may be placed on the color axis by the command caxis, or by setting the clim property of the parent axis. The face color of each cell of the mesh is determined by interpolating the values of c for each of the cell’s vertices; Contrast this with imagesc which renders one cell for each element of c. shading modifies an attribute determining the manner by which the face color of each cell is interpolated from the values of c, and the visibility of the cells’ edges. By default the attribute is "faceted", which renders a single color for each cell’s face with the edge visible. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object.

See also: [caxis], page 287, [shading], page 306, [meshgrid], page 302, [contour], page 273, [imagesc], page 658.

[Function File] area (y) [Function File] area (x, y) [Function File] area (…, lvl) [Function File] area (…, prop, val, …) [Function File] area (hax, …) [Function File] h = area (…)

Area plot of the columns of y. This plot shows the contributions of each column value to the row sum. It is functionally similar to plot (x, cumsum (y, 2)), except that the area under the curve is shaded. If the x argument is omitted it defaults to 1:rows (y). A value lvl can be defined that determines where the base level of the shading under the curve should be defined. The default level is 0. Additional property/value pairs are passed directly to the underlying patch object.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the hggroup object comprising the area patch objects. The "BaseValue" property of the hggroup can be used to adjust the level where shading begins. Example: Verify identity sin^2 + cos^2 = 1

     t = linspace (0, 2*pi, 100)';
     y = [sin(t).^2, cos(t).^2)];
     area (t, y);
     legend ("sin^2", "cos^2", "location", "NorthEastOutside");

See also: [plot], page 260, [patch], page 337.

[Function File] comet (y) [Function File] comet (x, y) [Function File] comet (x, y, p) [Function File] comet (hax, …)

Produce a simple comet style animation along the trajectory provided by the input coordinate vectors (x, y). If x is not specified it defaults to the indices of y. The speed of the comet may be controlled by p, which represents the time each point is displayed before moving to the next one. The default for p is 0.1 seconds. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

See also: [comet3], page 285.

[Function File] comet3 (z) [Function File] comet3 (x, y, z) [Function File] comet3 (x, y, z, p) [Function File] comet3 (hax, …)

Produce a simple comet style animation along the trajectory provided by the input coordinate vectors (x, y, z). If only z is specified then x, y default to the indices of z. The speed of the comet may be controlled by p, which represents the time each point is displayed before moving to the next one. The default for p is 0.1 seconds. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

See also: [comet], page 285.

15.2.1.1 Axis Configuration

The axis function may be used to change the axis limits of an existing plot and various other axis properties, such as the aspect ratio and the appearance of tic marks.

[Function File] axis () [Function File] axis ([x lo x hi]) [Function File] axis ([x lo x hi y lo y hi])

[Function File] axis ([x lo x hi y lo y hi z lo z hi])

[Function File] axis (option)

[Function File] axis (…, option) [Function File] axis (hax, …) [Function File] limits = axis ()

Set axis limits and appearance. The argument limits should be a 2-, 4-, or 6-element vector. The first and second elements specify the lower and upper limits for the x-axis. The third and fourth specify the limits for the y-axis, and the fifth and sixth specify the limits for the z-axis. Without any arguments, axis turns autoscaling on. With one output argument, limits = axis returns the current axis limits. The vector argument specifying limits is optional, and additional string arguments may be used to specify various axis properties. For example,

     axis ([1, 2, 3, 4], "square");

forces a square aspect ratio, and

     axis ("tic", "labely");

turns tic marks on for all axes and tic mark labels on for the y-axis only. The following options control the aspect ratio of the axes. "square" Force a square aspect ratio. "equal" Force x distance to equal y-distance. "normal" Restore default aspect ratio. The following options control the way axis limits are interpreted. "auto" Set the specified axes to have nice limits around the data or all if no axes

are specified.

"manual" Fix the current axes limits. "tight" Fix axes to the limits of the data.

"image" Equivalent to "tight" and "equal".

The following options affect the appearance of tic marks. "on" Turn tic marks and labels on for all axes. "off" Turn tic marks off for all axes.

"tic[xyz]"

Turn tic marks on for all axes, or turn them on for the specified axes and off for the remainder.

"label[xyz]"

Turn tic labels on for all axes, or turn them on for the specified axes and off for the remainder.

"nolabel"

Turn tic labels off for all axes.

Note, if there are no tic marks for an axis, there can be no labels. The following options affect the direction of increasing values on the axes.

"ij" Reverse y-axis, so lower values are nearer the top.

"xy" Restore y-axis, so higher values are nearer the top.

If the first argument hax is an axes handle, then operate on this axes rather than the current axes returned by gca.

See also: [xlim], page 287, [ylim], page 287, [zlim], page 287, [daspect], page 307, [pbaspect], page 308, [box], page 316, [grid], page 316.

Similarly the axis limits of the colormap can be changed with the caxis function.

[Function File] caxis ([cmin cmax])

[Function File] caxis ("auto")

[Function File] caxis ("manual")

[Function File] caxis (hax, …) [Function File] limits = caxis ()

Query or set color axis limits for plots. The limits argument should be a 2-element vector specifying the lower and upper limits to assign to the first and last value in the colormap. Data values outside this range are clamped to the first and last colormap entries. If the "auto" option is given then automatic colormap limits are applied. The automatic algorithm sets cmin to the minimum data value and cmax to the maximum data value. If "manual" is specified then the "climmode" property is set to "manual" and the numeric values in the "clim" property are used for limits. If the first argument hax is an axes handle, then operate on this axis rather than the current axes returned by gca. Called without arguments the current color axis limits are returned.

See also: [colormap], page 660.

The xlim, ylim, and zlim functions may be used to get or set individual axis limits. Each has the same form.

[Function File] xlimits = xlim () [Function File] xmode = xlim ("mode") [Function File] xlim ([x_lo x_hi]) [Function File] xlim ("auto")

[Function File] xlim ("manual")

[Function File] xlim (hax, …)

Query or set the limits of the x-axis for the current plot. Called without arguments xlim returns the x-axis limits of the current plot. With the input query "mode", return the current x-limit calculation mode which is either

"auto" or "manual".

If passed a 2-element vector [x lo x hi], the limits of the x-axis are set to these values and the mode is set to "manual". The current plotting mode can be changed by using either "auto" or "manual" as the argument.

If the first argument hax is an axes handle, then operate on this axis rather than the current axes returned by gca.

See also: [ylim], page 287, [zlim], page 287, [axis], page 285, [set], page 342, [get], page 341, [gca], page 340.

15.2.1.2 Two-dimensional Function Plotting

Octave can plot a function from a function handle, inline function, or string defining the function without the user needing to explicitly create the data to be plotted. The function fplot also generates two-dimensional plots with linear axes using a function name and limits for the range of the x-coordinate instead of the x and y data. For example,

fplot (@sin, [-10, 10], 201);

produces a plot that is equivalent to the one above, but also includes a legend displaying the name of the plotted function.

[Function File] fplot (fn, limits) [Function File] fplot (…, tol) [Function File] fplot (…, n) [Function File] fplot (…, fmt) [Function File] [x, y] = fplot (…)

Plot a function fn within the range defined by limits. fn is a function handle, inline function, or string containing the name of the function to evaluate. The limits of the plot are of the form [xlo, xhi] or [xlo, xhi, ylo, yhi]. The next three arguments are all optional and any number of them may be given in any order. tol is the relative tolerance to use for the plot and defaults to 2e-3 (.2%). n is the minimum number of points to use. When n is specified, the maximum stepsize will be xhi - xlo / n. More than n points may still be used in order to meet the relative tolerance requirement. The fmt argument specifies the linestyle to be used by the plot command. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. With no output arguments the results are immediately plotted. With two output arguments the 2-D plot data is returned. The data can subsequently be plotted manually with plot (x, y). Example:

     fplot (@cos, [0, 2*pi])
     fplot ("[cos(x), sin(x)]", [0, 2*pi])

Note: fplot works best with continuous functions. Functions with discontinuities are unlikely to plot well. This restriction may be removed in the future.

See also: [ezplot], page 288, [plot], page 260.

Other functions that can create two-dimensional plots directly from a function include

ezplot, ezcontour, ezcontourf and ezpolar.

[Function File] ezplot (f) [Function File] ezplot (f2v) [Function File] ezplot (fx, fy) [Function File] ezplot (…, dom) [Function File] ezplot (…, n) [Function File] ezplot (hax, …) [Function File] h = ezplot (…)

Plot the 2-D curve defined by the function f. The function f may be a string, inline function, or function handle and can have either one or two variables. If f has one variable, then the function is plotted over the domain -2*pi < x < 2*pi with 500 points. If f2v is a function of two variables then the implicit function f(x,y) = 0 is calculated over the meshed domain -2*pi <= x | y <= 2*pi with 60 points in each dimension. For example:

     ezplot (@(x, y) x.^2 - y.^2 - 1)

If two functions are passed as inputs then the parametric function

     x = fx (t) y = fy (t)

is plotted over the domain -2*pi <= t <= 2*pi with 500 points. If dom is a two element vector, it represents the minimum and maximum values of both x and y, or t for a parametric plot. If dom is a four element vector, then the minimum and maximum values are [xmin xmax ymin ymax]. n is a scalar defining the number of points to use in plotting the function. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a vector of graphics handles to the created line objects.

See also: [plot], page 260, [ezplot3], page 308, [ezpolar], page 290, [ezcontour], page 289, [ezcontourf], page 290, [ezmesh], page 309, [ezmeshc], page 309, [ezsurf], page 310, [ezsurfc], page 311.

[Function File] ezcontour (f) [Function File] ezcontour (…, dom) [Function File] ezcontour (…, n) [Function File] ezcontour (hax, …) [Function File] h = ezcontour (…)

Plot the contour lines of a function. f is a string, inline function, or function handle with two arguments defining the function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi with 60 points in each dimension. If dom is a two element vector, it represents the minimum and maximum values of both x and y. If dom is a four element vector, then the minimum and maximum

values are [xmin xmax ymin ymax].

n is a scalar defining the number of points to use in each dimension.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created plot. Example:

     f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
     ezcontour (f, [-3, 3]);

See also: [contour], page 273, [ezcontourf], page 290, [ezplot], page 288, [ezmeshc], page 309, [ezsurfc], page 311.

[Function File] ezcontourf (f) [Function File] ezcontourf (…, dom) [Function File] ezcontourf (…, n) [Function File] ezcontourf (hax, …) [Function File] h = ezcontourf (…)

Plot the filled contour lines of a function. f is a string, inline function, or function handle with two arguments defining the function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi with 60 points in each dimension. If dom is a two element vector, it represents the minimum and maximum values of both x and y. If dom is a four element vector, then the minimum and maximum

values are [xmin xmax ymin ymax].

n is a scalar defining the number of points to use in each dimension. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created plot. Example:

     f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
     ezcontourf (f, [-3, 3]);

See also: [contourf], page 274, [ezcontour], page 289, [ezplot], page 288, [ezmeshc], page 309, [ezsurfc], page 311.

[Function File] ezpolar (f) [Function File] ezpolar (…, dom) [Function File] ezpolar (…, n) [Function File] ezpolar (hax, …) [Function File] h = ezpolar (…)

Plot a 2-D function in polar coordinates. The function f is a string, inline function, or function handle with a single argument. The expected form of the function is rho = f(theta). By default the plot is over the domain 0 <= theta <= 2*pi with 500 points. If dom is a two element vector, it represents the minimum and maximum values of theta. n is a scalar defining the number of points to use in plotting the function.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created plot. Example:

     ezpolar (@(t) sin (5/4 * t), [0, 8*pi]);

See also: [polar], page 280, [ezplot], page 288.

15.2.1.3 Two-dimensional Geometric Shapes

[Function File] rectangle () [Function File] rectangle (…, "Position", pos) [Function File] rectangle (…, "Curvature", curv) [Function File] rectangle (…, "EdgeColor", ec) [Function File] rectangle (…, "FaceColor", fc) [Function File] rectangle (hax, …) [Function File] h = rectangle (…)

Draw a rectangular patch defined by pos and curv. The variable pos(1:2) defines the lower left-hand corner of the patch and pos(3:4) defines its width and height. By default, the value of pos is [0, 0, 1, 1]. The variable curv defines the curvature of the sides of the rectangle and may be a scalar or two-element vector with values between 0 and 1. A value of 0 represents no curvature of the side, whereas a value of 1 means that the side is entirely curved into the arc of a circle. If curv is a two-element vector, then the first element is the curvature along the x-axis of the patch and the second along y-axis. If curv is a scalar, it represents the curvature of the shorter of the two sides of the rectangle and the curvature of the other side is defined by

     min (pos(1:2)) / max (pos(1:2)) * curv

Additional property/value pairs are passed to the underlying patch command. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created rectangle object.

See also: [patch], page 337, [line], page 337, [cylinder], page 312, [ellipsoid], page 312, [sphere], page 312.

15.2.2 Three-Dimensional Plots

The function mesh produces mesh surface plots. For example,

tx = ty = linspace (-8, 8, 41)';
[xx, yy] = meshgrid (tx, ty);
r = sqrt (xx .^2 + yy .^2) + eps;
tz = sin (r) ./ r;
mesh (tx, ty, tz);

produces the familiar “sombrero” plot shown in Figure 15.5. Note the use of the function meshgrid to create matrices of X and Y coordinates to use for plotting the Z data. The ndgrid function is similar to meshgrid, but works for N-dimensional matrices.

Figure 15.5: Mesh plot. The meshc function is similar to mesh, but also produces a plot of contours for the

surface.

The plot3 function displays arbitrary three-dimensional data, without requiring it to form a surface. For example,

  t = 0:0.1:10*pi;
  r = linspace (0, 1, numel (t));
  z = linspace (0, 1, numel (t));
  plot3 (r.*sin(t), r.*cos(t), z);

displays the spiral in three dimensions shown in Figure 15.6.

Figure 15.6: Three-dimensional spiral. Finally, the view function changes the viewpoint for three-dimensional plots.

[Function File] mesh (x, y, z) [Function File] mesh (z) [Function File] mesh (…, c) [Function File] mesh (…, prop, val, …) [Function File] mesh (hax, …) [Function File] h = mesh (…)

Plot a 3-D wireframe mesh. The wireframe mesh is plotted using rectangles. The vertices of the rectangles [x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y plane. z determines the height above the plane of each vertex. If only a single z matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows (z). Thus, columns of z correspond to different x values and rows of z correspond to different y values. The color of the mesh is computed by linearly scaling the z values to fit the range of the current colormap. Use caxis and/or change the colormap to control the appearance. Optionally, the color of the mesh can be specified independently of z by supplying a color matrix, c. Any property/value pairs are passed directly to the underlying surface object. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object.

See also: [ezmesh], page 309, [meshc], page 293, [meshz], page 294, [trimesh], page 628, [contour], page 273, [surf], page 295, [surface], page 339, [meshgrid], page 302, [hidden], page 294, [shading], page 306, [colormap], page 660, [caxis], page 287.

[Function File] meshc (x, y, z) [Function File] meshc (z) [Function File] meshc (…, c) [Function File] meshc (…, prop, val, …) [Function File] meshc (hax, …) [Function File] h = meshc (…)

Plot a 3-D wireframe mesh with underlying contour lines. The wireframe mesh is plotted using rectangles. The vertices of the rectangles [x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y plane. z determines the height above the plane of each vertex. If only a single z matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows (z). Thus, columns of z correspond to different x values and rows of z correspond to different y values. The color of the mesh is computed by linearly scaling the z values to fit the range of the current colormap. Use caxis and/or change the colormap to control the appearance. Optionally the color of the mesh can be specified independently of z by supplying a color matrix, c. Any property/value pairs are passed directly to the underlying surface object. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

The optional return value h is a 2-element vector with a graphics handle to the created surface object and to the created contour plot.

See also: [ezmeshc], page 309, [mesh], page 292, [meshz], page 294, [contour], page 273, [surfc], page 295, [surface], page 339, [meshgrid], page 302, [hidden], page 294, [shading], page 306, [colormap], page 660, [caxis], page 287.

[Function File] meshz (x, y, z) [Function File] meshz (z) [Function File] meshz (…, c) [Function File] meshz (…, prop, val, …) [Function File] meshz (hax, …) [Function File] h = meshz (…)

Plot a 3-D wireframe mesh with a surrounding curtain. The wireframe mesh is plotted using rectangles. The vertices of the rectangles [x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y plane. z determines the height above the plane of each vertex. If only a single z matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows (z). Thus, columns of z correspond to different x values and rows of z correspond to different y values. The color of the mesh is computed by linearly scaling the z values to fit the range of the current colormap. Use caxis and/or change the colormap to control the appearance. Optionally the color of the mesh can be specified independently of z by supplying a color matrix, c. Any property/value pairs are passed directly to the underlying surface object. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object.

See also: [mesh], page 292, [meshc], page 293, [contour], page 273, [surf], page 295, [surface], page 339, [waterfall], page 307, [meshgrid], page 302, [hidden], page 294, [shading], page 306, [colormap], page 660, [caxis], page 287.

[Command] hidden

[Command] hidden "on"

[Command] hidden "off" [Function File] mode = hidden (…)

Control mesh hidden line removal. When called with no argument the hidden line removal state is toggled. When called with one of the modes "on" or "off" the state is set accordingly. The optional output argument mode is the current state. Hidden Line Removal determines what graphic objects behind a mesh plot are visible. The default is for the mesh to be opaque and lines behind the mesh are not visible. If hidden line removal is turned off then objects behind the mesh can be seen through the faces (openings) of the mesh, although the mesh grid lines are still opaque.

See also: [mesh], page 292, [meshc], page 293, [meshz], page 294, [ezmesh], page 309, [ezmeshc], page 309, [trimesh], page 628, [waterfall], page 307.

[Function File] surf (x, y, z) [Function File] surf (z) [Function File] surf (…, c) [Function File] surf (…, prop, val, …) [Function File] surf (hax, …) [Function File] h = surf (…)

Plot a 3-D surface mesh. The surface mesh is plotted using shaded rectangles. The vertices of the rectangles [x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y plane. z determines the height above the plane of each vertex. If only a single z matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows (z). Thus, columns of z correspond to different x values and rows of z correspond to different y values. The color of the surface is computed by linearly scaling the z values to fit the range of the current colormap. Use caxis and/or change the colormap to control the appearance. Optionally, the color of the surface can be specified independently of z by supplying a color matrix, c. Any property/value pairs are passed directly to the underlying surface object. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object. Note: The exact appearance of the surface can be controlled with the shading command or by using set to control surface object properties.

See also: [ezsurf], page 310, [surfc], page 295, [surfl], page 296, [surfnorm], page 297, [trisurf], page 628, [contour], page 273, [mesh], page 292, [surface], page 339, [meshgrid], page 302, [hidden], page 294, [shading], page 306, [colormap], page 660, [caxis], page 287.

[Function File] surfc (x, y, z) [Function File] surfc (z) [Function File] surfc (…, c) [Function File] surfc (…, prop, val, …) [Function File] surfc (hax, …) [Function File] h = surfc (…)

Plot a 3-D surface mesh with underlying contour lines. The surface mesh is plotted using shaded rectangles. The vertices of the rectangles [x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y plane. z determines the height above the plane of each vertex. If only a single z matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows (z). Thus, columns of z correspond to different x values and rows of z correspond to different y values. The color of the surface is computed by linearly scaling the z values to fit the range of the current colormap. Use caxis and/or change the colormap to control the appearance.

Optionally, the color of the surface can be specified independently of z by supplying a color matrix, c. Any property/value pairs are passed directly to the underlying surface object. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object. Note: The exact appearance of the surface can be controlled with the shading command or by using set to control surface object properties.

See also: [ezsurfc], page 311, [surf], page 295, [surfl], page 296, [surfnorm], page 297, [trisurf], page 628, [contour], page 273, [mesh], page 292, [surface], page 339, [meshgrid], page 302, [hidden], page 294, [shading], page 306, [colormap], page 660, [caxis], page 287.

[Function File] surfl (z) [Function File] surfl (x, y, z) [Function File] surfl (…, lsrc) [Function File] surfl (x, y, z, lsrc, P)

[Function File] surfl (…, "cdata") [Function File] surfl (…, "light")

[Function File] surfl (hax, …) [Function File] h = surfl (…)

Plot a 3-D surface using shading based on various lighting models. The surface mesh is plotted using shaded rectangles. The vertices of the rectangles [x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y plane. z determines the height above the plane of each vertex. If only a single z matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows (z). Thus, columns of z correspond to different x values and rows of z correspond to different y values. The default lighting mode "cdata", changes the cdata property of the surface object to give the impression of a lighted surface. Warning: The alternative mode "light" mode which creates a light object to illuminate the surface is not implemented (yet). The light source location can be specified using lsrc. It can be given as a 2-element vector [azimuth, elevation] in degrees, or as a 3-element vector [lx, ly, lz]. The default value is rotated 45 degrees counterclockwise to the current view. The material properties of the surface can specified using a 4-element vector P = [AM D SP exp] which defaults to p = [0.55 0.6 0.4 10].

"AM" strength of ambient light "D" strength of diffuse reflection "SP" strength of specular reflection "EXP" specular exponent If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object. Example:

     colormap (bone (64));
     surfl (peaks);
     shading interp;

See also: [diffuse], page 302, [specular], page 302, [surf], page 295, [shading], page 306, [colormap], page 660, [caxis], page 287.

[Function File] surfnorm (x, y, z) [Function File] surfnorm (z)

[Function File] [nx, ny, nz] = surfnorm (. . .)

[Function File] surfnorm (h, …)

Find the vectors normal to a meshgridded surface. The meshed gridded surface is defined by x, y, and z. If x and y are not defined, then it is assumed that they are given by

     [x, y] = meshgrid (1:rows (z),
                        1:columns (z));

If no return arguments are requested, a surface plot with the normal vectors to the surface is plotted. Otherwise the components of the normal vectors at the mesh gridded points are returned in nx, ny, and nz.

The normal vectors are calculated by taking the cross product of the diagonals of each of the quadrilaterals in the meshgrid to find the normal vectors of the centers of these quadrilaterals. The four nearest normal vectors to the meshgrid points are then averaged to obtain the normal to the surface at the meshgridded points.

An example of the use of surfnorm is

     surfnorm (peaks (25));

See also: [surf], page 295, [quiver3], page 282.

[Function File] [fv] = isosurface (val, iso)
[Function File] [fv] = isosurface (x, y, z, val, iso)

[Function File] [fv] = isosurface (…, "noshare", "verbose") [Function File] [fvc] = isosurface (…, col)

[Function File] [f, v] = isosurface (x, y, z, val, iso)
[Function File] [f, v, c] = isosurface (x, y, z, val, iso, col)
[Function File] isosurface (x, y, z, val, iso, col, opt)

If called with one output argument and the first input argument val is a threedimensional array that contains the data of an isosurface geometry and the second input argument iso keeps the isovalue as a scalar value then return a structure array fv that contains the fields Faces and Vertices at computed points [x, y, z] = meshgrid (1:l, 1:m, 1:n). The output argument fv can directly be taken as an input argument for the patch function.

If called with further input arguments x, y and z which are three–dimensional arrays with the same size than val then the volume data is taken at those given points.

The string input argument "noshare" is only for compatibility and has no effect. If given the string input argument "verbose" then print messages to the command line interface about the current progress.

298 GNU Octave

If called with the input argument col which is a three-dimensional array of the same size than val then take those values for the interpolation of coloring the isosurface geometry. Add the field FaceVertexCData to the structure array fv. If called with two or three output arguments then return the information about the faces f, vertices v and color data c as separate arrays instead of a single structure array. If called with no output argument then directly process the isosurface geometry with the patch command. For example,

[x, y, z] = meshgrid (1:5, 1:5, 1:5);
val = rand (5, 5, 5);
isosurface (x, y, z, val, .5);

will directly draw a random isosurface geometry in a graphics window. Another example for an isosurface geometry with different additional coloring

N = 15;    # Increase number of vertices in each direction
iso = .4;  # Change isovalue to .1 to display a sphere
lin = linspace (0, 2, N);
[x, y, z] = meshgrid (lin, lin, lin);
c = abs ((x-.5).^2 + (y-.5).^2 + (z-.5).^2);
figure (); # Open another figure window
subplot (2,2,1); view (-38, 20);
[f, v] = isosurface (x, y, z, c, iso);
p = patch ("Faces", f, "Vertices", v, "EdgeColor", "none");
set (gca, "PlotBoxAspectRatioMode", "manual", ...
          "PlotBoxAspectRatio", [1 1 1]);
# set (p, "FaceColor", "green", "FaceLighting", "phong");
# light ("Position", [1 1 5]); # Available with the JHandles package
subplot (2,2,2); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "EdgeColor", "blue");
set (gca, "PlotBoxAspectRatioMode", "manual", ...
          "PlotBoxAspectRatio", [1 1 1]);
# set (p, "FaceColor", "none", "FaceLighting", "phong");
# light ("Position", [1 1 5]);
subplot (2,2,3); view (-38, 20);
[f, v, c] = isosurface (x, y, z, c, iso, y);
p = patch ("Faces", f, "Vertices", v, "FaceVertexCData", c, ...
           "FaceColor", "interp", "EdgeColor", "none");
set (gca, "PlotBoxAspectRatioMode", "manual", ...
          "PlotBoxAspectRatio", [1 1 1]);
# set (p, "FaceLighting", "phong");
# light ("Position", [1 1 5]);
subplot (2,2,4); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "FaceVertexCData", c, ...
           "FaceColor", "interp", "EdgeColor", "blue");
set (gca, "PlotBoxAspectRatioMode", "manual", ...
          "PlotBoxAspectRatio", [1 1 1]);
# set (p, "FaceLighting", "phong");
# light ("Position", [1 1 5]);

See also: [isonormals], page 299, [isocolors], page 300.

Chapter 15: Plotting 299

[Function File] [n] = isonormals (val, v)
[Function File] [n] = isonormals (val, p)
[Function File] [n] = isonormals (x, y, z, val, v)
[Function File] [n] = isonormals (x, y, z, val, p)

[Function File] [n] = isonormals (…, "negate") [Function File] isonormals (…, p)

If called with one output argument and the first input argument val is a threedimensional array that contains the data for an isosurface geometry and the second input argument v keeps the vertices of an isosurface then return the normals n in form of a matrix with the same size than v at computed points [x, y, z] = meshgrid (1:l, 1:m, 1:n). The output argument n can be taken to manually set VertexNormals of a patch.

If called with further input arguments x, y and z which are three–dimensional arrays with the same size than val then the volume data is taken at those given points. Instead of the vertices data v a patch handle p can be passed to this function.

If given the string input argument "negate" as last input argument then compute the reverse vector normals of an isosurface geometry.

If no output argument is given then directly redraw the patch that is given by the patch handle p.

For example:

function [] = isofinish (p)
  set (gca, "PlotBoxAspectRatioMode", "manual", ...
            "PlotBoxAspectRatio", [1 1 1]);
  set (p, "VertexNormals", -get (p,"VertexNormals")); # Revert normals
  set (p, "FaceColor", "interp");
  ## set (p, "FaceLighting", "phong");
  ## light ("Position", [1 1 5]); # Available with JHandles
endfunction
N = 15;    # Increase number of vertices in each direction
iso = .4;  # Change isovalue to .1 to display a sphere
lin = linspace (0, 2, N);
[x, y, z] = meshgrid (lin, lin, lin);
c = abs ((x-.5).^2 + (y-.5).^2 + (z-.5).^2);
figure (); # Open another figure window
subplot (2,2,1); view (-38, 20);
[f, v, cdat] = isosurface (x, y, z, c, iso, y);
p = patch ("Faces", f, "Vertices", v, "FaceVertexCData", cdat, ...
           "FaceColor", "interp", "EdgeColor", "none");
isofinish (p); ## Call user function isofinish
subplot (2,2,2); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "FaceVertexCData", cdat, ...
           "FaceColor", "interp", "EdgeColor", "none");
isonormals (x, y, z, c, p); # Directly modify patch
isofinish (p);
subplot (2,2,3); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "FaceVertexCData", cdat, ...
           "FaceColor", "interp", "EdgeColor", "none");
n = isonormals (x, y, z, c, v); # Compute normals of isosurface

      set (p, "VertexNormals", n);    # Manually set vertex normals
      isofinish (p);
      subplot (2,2,4); view (-38, 20);
      p = patch ("Faces", f, "Vertices", v, "FaceVertexCData", cdat, ...
                 "FaceColor", "interp", "EdgeColor", "none");
      isonormals (x, y, z, c, v, "negate"); # Use reverse directly
      isofinish (p);

See also: [isosurface], page 297, [isocolors], page 300.

[Function File] [cd] = isocolors (c, v) [Function File] [cd] = isocolors (x, y, z, c, v) [Function File] [cd] = isocolors (x, y, z, r, g, b, v) [Function File] [cd] = isocolors (r, g, b, v) [Function File] [cd] = isocolors (…, p) [Function File] isocolors (…)

If called with one output argument and the first input argument c is a three-dimensional array that contains color values and the second input argument v keeps the vertices of a geometry then return a matrix cd with color data information for the geometry at computed points [x, y, z] = meshgrid (1:l, 1:m, 1:n). The output argument cd can be taken to manually set FaceVertexCData of a patch. If called with further input arguments x, y and z which are three–dimensional arrays of the same size than c then the color data is taken at those given points. Instead of the color data c this function can also be called with RGB values r, g, b. If input argumnets x, y, z are not given then again meshgrid computed values are taken. Optionally, the patch handle p can be given as the last input argument to all variations of function calls instead of the vertices data v. Finally, if no output argument is given then directly change the colors of a patch that is given by the patch handle p. For example:

     function [] = isofinish (p)
       set (gca, "PlotBoxAspectRatioMode", "manual", ...
                 "PlotBoxAspectRatio", [1 1 1]);
       set (p, "FaceColor", "interp");
       ## set (p, "FaceLighting", "flat");
       ## light ("Position", [1 1 5]); ## Available with JHandles
     endfunction
     N = 15;    # Increase number of vertices in each direction
     iso = .4;  # Change isovalue to .1 to display a sphere
     lin = linspace (0, 2, N);
     [x, y, z] = meshgrid (lin, lin, lin);
     c = abs ((x-.5).^2 + (y-.5).^2 + (z-.5).^2);
     figure (); # Open another figure window
     subplot (2,2,1); view (-38, 20);
     [f, v] = isosurface (x, y, z, c, iso);
     p = patch ("Faces", f, "Vertices", v, "EdgeColor", "none");

cdat = rand (size (c));       # Compute random patch color data
isocolors (x, y, z, cdat, p); # Directly set colors of patch
isofinish (p);                # Call user function isofinish
subplot (2,2,2); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "EdgeColor", "none");
[r, g, b] = meshgrid (lin, 2-lin, 2-lin);
cdat = isocolors (x, y, z, c, v); # Compute color data vertices
set (p, "FaceVertexCData", cdat); # Set color data manually
isofinish (p);
subplot (2,2,3); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "EdgeColor", "none");
cdat = isocolors (r, g, b, c, p); # Compute color data patch
set (p, "FaceVertexCData", cdat); # Set color data manually
isofinish (p);
subplot (2,2,4); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "EdgeColor", "none");
r = g = b = repmat ([1:N] / N, [N, 1, N]); # Black to white
cdat = isocolors (x, y, z, r, g, b, v);
set (p, "FaceVertexCData", cdat);
isofinish (p);

See also: [isosurface], page 297, [isonormals], page 299.

[Function File] shrinkfaces (p, sf)

[Function File] nfv = shrinkfaces (p, sf)
[Function File] nfv = shrinkfaces (fv, sf)
[Function File] nfv = shrinkfaces (f, v, sf)
[Function File] [nf, nv] = shrinkfaces (. . .)

Reduce the faces area for a given patch, structure or explicit faces and points matrices by a scale factor sf. The structure fv must contain the fields "faces" and "vertices". If the factor sf is omitted then a default of 0.3 is used.

Given a patch handle as the first input argument and no output parameters, perform the shrinking of the patch faces in place and redraw the patch.

If called with one output argument, return a structure with fields "faces", "vertices", and "facevertexcdata" containing the data after shrinking which can then directly be used as an input argument for the patch function.

Performing the shrinking on faces which are not convex can lead to undesired results.

For example,

     [phi r] = meshgrid (linspace (0, 1.5*pi, 16), linspace (1, 2, 4));
     tri = delaunay (phi(:), r(:));
     v = [r(:).*sin(phi(:)) r(:).*cos(phi(:))];
     clf ()
     p = patch ("Faces", tri, "Vertices", v, "FaceColor", "none");
     fv = shrinkfaces (p);
     patch (fv)
     axis equal
     grid on

draws a triangulated 3/4 circle and the corresponding shrunken version.

See also: [patch], page 337.

[Function File] diffuse (sx, sy, sz, lv)

Calculate diffuse reflection strength of a surface defined by the normal vector elements sx, sy, sz. The light source location vector lv can be given as 2-element vector [azimuth, elevation] in degrees or as 3-element vector [lx, ly, lz].

See also: [specular], page 302, [surfl], page 296.

[Function File] specular (sx, sy, sz, lv, vv) [Function File] specular (sx, sy, sz, lv, vv, se)

Calculate specular reflection strength of a surface defined by the normal vector elements sx, sy, sz using Phong’s approximation. The light source location and viewer location vectors can be specified using parameter lv and vv respectively. The location vectors can given as 2-element vectors [azimuth, elevation] in degrees or as 3-element vectors [x, y, z]. An optional sixth argument describes the specular exponent (spread) se.

See also: [diffuse], page 302, [surfl], page 296.

[Function File] [xx, yy] = meshgrid (x, y)
[Function File] [xx, yy, zz] = meshgrid (x, y, z)
[Function File] [xx, yy] = meshgrid (x)
[Function File] [xx, yy, zz] = meshgrid (x)

Given vectors of x and y coordinates, return matrices xx and yy corresponding to a full 2-D grid. The rows of xx are copies of x, and the columns of yy are copies of y. If y is omitted, then it is assumed to be the same as x. If the optional z input is given, or zz is requested, then the output will be a full 3-D grid. meshgrid is most frequently used to produce input for a 2-D or 3-D function that will be plotted. The following example creates a surface plot of the “sombrero” function.

     f = @(x,y) sin (sqrt (x.^2 + y.^2)) ./ sqrt (x.^2 + y.^2);
     range = linspace (-8, 8, 41);
     [X, Y] = meshgrid (range, range);
     Z = f (X, Y);
     surf (X, Y, Z);

Programming Note: meshgrid is restricted to 2-D or 3-D grid generation. The ndgrid function will generate 1-D through N-D grids. However, the functions are not completely equivalent. If x is a vector of length M and y is a vector of length N, then meshgrid will produce an output grid which is NxM. ndgrid will produce an output which is MxN (transpose) for the same input. Some core functions expect meshgrid input and others expect ndgrid input. Check the documentation for the function in question to determine the proper input format.

See also: [ndgrid], page 303, [mesh], page 292, [contour], page 273, [surf], page 295.

[Function File] [y1, y2, ..., yn] = ndgrid (x1, x2, . . ., xn)
[Function File] [y1, y2, ..., yn] = ndgrid (x)

Given n vectors x1, …, xn, ndgrid returns n arrays of dimension n. The elements of the i-th output argument contains the elements of the vector xi repeated over all dimensions different from the i-th dimension. Calling ndgrid with only one input argument x is equivalent to calling ndgrid with all n input arguments equal to x: [y1, y2, …, yn] = ndgrid (x, …, x) Programming Note: ndgrid is very similar to the function meshgrid except that the first two dimensions are transposed in comparison to meshgrid. Some core functions expect meshgrid input and others expect ndgrid input. Check the documentation for the function in question to determine the proper input format.

See also: [meshgrid], page 302.

[Function File] plot3 (x, y, z) [Function File] plot3 (x, y, z, prop, value, …) [Function File] plot3 (x, y, z, fmt) [Function File] plot3 (x, cplx) [Function File] plot3 (cplx) [Function File] plot3 (hax, …) [Function File] h = plot3 (…)

Produce 3-D plots. Many different combinations of arguments are possible. The simplest form is

     plot3 (x, y, z)

in which the arguments are taken to be the vertices of the points to be plotted in three dimensions. If all arguments are vectors of the same length, then a single continuous line is drawn. If all arguments are matrices, then each column of is treated as a separate line. No attempt is made to transpose the arguments to make the number of rows match. If only two arguments are given, as

     plot3 (x, cplx)

the real and imaginary parts of the second argument are used as the y and z coordinates, respectively. If only one argument is given, as

     plot3 (cplx)

the real and imaginary parts of the argument are used as the y and z values, and they are plotted versus their index.

Arguments may also be given in groups of three as

     plot3 (x1, y1, z1, x2, y2, z2, ...)

in which each set of three arguments is treated as a separate line or set of lines in three dimensions. To plot multiple one- or two-argument groups, separate each group with an empty format string, as

     plot3 (x1, c1, "", c2, "", ...)

Multiple property-value pairs may be specified which will affect the line objects drawn by plot3. If the fmt argument is supplied it will format the line objects in the same manner as plot. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created plot. Example:

     z = [0:0.05:5];
     plot3 (cos (2*pi*z), sin (2*pi*z), z, ";helix;");
     plot3 (z, exp (2i*pi*z), ";complex sinusoid;");

See also: [ezplot3], page 308, [plot], page 260.

[Function File] view (azimuth, elevation)
[Function File] view ([azimuth elevation])

[Function File] view ([x y z]) [Function File] view (2) [Function File] view (3) [Function File] view (hax, …)

[Function File] [azimuth, elevation] = view ()

Query or set the viewpoint for the current axes. The parameters azimuth and elevation can be given as two arguments or as 2-element vector. The viewpoint can also be specified with Cartesian coordinates x, y, and z. The call view (2) sets the viewpoint to azimuth = 0 and elevation = 90, which is the default for 2-D graphs. The call view (3) sets the viewpoint to azimuth = -37.5 and elevation = 30, which is the default for 3-D graphs. If the first argument hax is an axes handle, then operate on this axis rather than the current axes returned by gca. If no inputs are given, return the current azimuth and elevation.

[Function File] slice (x, y, z, v, sx, sy, sz) [Function File] slice (x, y, z, v, xi, yi, zi) [Function File] slice (v, sx, sy, sz) [Function File] slice (v, xi, yi, zi) [Function File] slice (…, method) [Function File] slice (hax, …)

[Function File] h = slice (…)

Plot slices of 3-D data/scalar fields. Each element of the 3-dimensional array v represents a scalar value at a location given by the parameters x, y, and z. The parameters x, x, and z are either 3-dimensional arrays of the same size as the array v in the "meshgrid" format or vectors. The parameters xi, etc. respect a similar format to x, etc., and they represent the points at which the array vi is interpolated using interp3. The vectors sx, sy, and sz contain points of orthogonal slices of the respective axes. If x, y, z are omitted, they are assumed to be x = 1:size (v, 2), y = 1:size (v, 1)

and z = 1:size (v, 3).

method is one of:

"nearest"

Return the nearest neighbor.

"linear" Linear interpolation from nearest neighbors.

"cubic" Cubic interpolation from four nearest neighbors (not implemented yet).

"spline" Cubic spline interpolation—smooth first and second derivatives throughout the curve.

The default method is "linear". If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object. Examples:

     [x, y, z] = meshgrid (linspace (-8, 8, 32));
     v = sin (sqrt (x.^2 + y.^2 + z.^2)) ./ (sqrt (x.^2 + y.^2 + z.^2));
     slice (x, y, z, v, [], 0, []);
     [xi, yi] = meshgrid (linspace (-7, 7));
     zi = xi + yi;
     slice (x, y, z, v, xi, yi, zi);

See also: [interp3], page 622, [surface], page 339, [pcolor], page 284.

[Function File] ribbon (y) [Function File] ribbon (x, y) [Function File] ribbon (x, y, width) [Function File] ribbon (hax, …) [Function File] h = ribbon (…)

Plot a ribbon plot for the columns of y vs. x. The optional parameter width specifies the width of a single ribbon (default is 0.75). If x is omitted, a vector containing the row numbers is assumed (1:rows (Y)). If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a vector of graphics handles to the surface objects representing each ribbon.

See also: [surface], page 339, [waterfall], page 307.

[Function File] shading (type) [Function File] shading (hax, type)

Set the shading of patch or surface graphic objects.

Valid arguments for type are

"flat" Single colored patches with invisible edges.

"faceted"

Single colored patches with visible edges.

"interp" Color between patch vertices are interpolated and the patch edges are invisible.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

See also: [fill], page 338, [mesh], page 292, [patch], page 337, [pcolor], page 284, [surf], page 295, [surface], page 339, [hidden], page 294.

[Function File] scatter3 (x, y, z) [Function File] scatter3 (x, y, z, s) [Function File] scatter3 (x, y, z, s, c) [Function File] scatter3 (…, style) [Function File] scatter3 (…, "filled") [Function File] scatter3 (…, prop, val) [Function File] scatter3 (hax, …) [Function File] h = scatter3 (…)

Draw a 3-D scatter plot.

A marker is plotted at each point defined by the coordinates in the vectors x, y, and z.

The size of the markers is determined by s, which can be a scalar or a vector of the same length as x, y, and z. If s is not given, or is an empty matrix, then a default value of 8 points is used.

The color of the markers is determined by c, which can be a string defining a fixed color; a 3-element vector giving the red, green, and blue components of the color; a vector of the same length as x that gives a scaled index into the current colormap; or an Nx3 matrix defining the RGB color of each marker individually.

The marker to use can be changed with the style argument, that is a string defining a marker in the same manner as the plot command. If no marker is specified it defaults to "o" or circles. If the argument "filled" is given then the markers are filled.

Additional property/value pairs are passed directly to the underlying patch object.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

The optional return value h is a graphics handle to the hggroup object representing the points.

     [x, y, z] = peaks (20);
     scatter3 (x(:), y(:), z(:), [], z(:));

See also: [scatter], page 271, [patch], page 337, [plot], page 260.

[Function File] waterfall (x, y, z) [Function File] waterfall (z) [Function File] waterfall (…, c) [Function File] waterfall (…, prop, val, …) [Function File] waterfall (hax, …) [Function File] h = waterfall (…)

Plot a 3-D waterfall plot. A waterfall plot is similar to a meshz plot except only mesh lines for the rows of z (x-values) are shown. The wireframe mesh is plotted using rectangles. The vertices of the rectangles [x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y plane. z determines the height above the plane of each vertex. If only a single z matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows (z). Thus, columns of z correspond to different x values and rows of z correspond to different y values. The color of the mesh is computed by linearly scaling the z values to fit the range of the current colormap. Use caxis and/or change the colormap to control the appearance. Optionally the color of the mesh can be specified independently of z by supplying a color matrix, c. Any property/value pairs are passed directly to the underlying surface object. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object.

See also: [meshz], page 294, [mesh], page 292, [meshc], page 293, [contour], page 273, [surf], page 295, [surface], page 339, [ribbon], page 305, [meshgrid], page 302, [hidden], page 294, [shading], page 306, [colormap], page 660, [caxis], page 287.

15.2.2.1 Aspect Ratio

For three-dimensional plots the aspect ratio can be set for data with daspect and for the plot box with pbaspect. See Section 15.2.1.1 [Axis Configuration], page 285, for controlling the x-, y-, and z-limits for plotting.

[Function File] data_aspect_ratio = daspect ()
[Function File] daspect (data_aspect_ratio)

[Function File] daspect (mode)

[Function File] data_aspect_ratio_mode = daspect ("mode")

[Function File] daspect (hax, …)

Query or set the data aspect ratio of the current axes. The aspect ratio is a normalized 3-element vector representing the span of the x, y, and z-axis limits.

daspect (mode)

Set the data aspect ratio mode of the current axes. mode is either "auto" or

"manual".
daspect ("mode")

Return the data aspect ratio mode of the current axes.

daspect (hax, ...)

Operate on the axes in handle hax instead of the current axes.

See also: [axis], page 285, [pbaspect], page 308, [xlim], page 287, [ylim], page 287, [zlim], page 287.

[Function File] plot_box_aspect_ratio = pbaspect ( )
[Function File] pbaspect (plot_box_aspect_ratio)

[Function File] pbaspect (mode)

[Function File] plot_box_aspect_ratio_mode = pbaspect ("mode")

[Function File] pbaspect (hax, …)

Query or set the plot box aspect ratio of the current axes. The aspect ratio is a normalized 3-element vector representing the rendered lengths of the x, y, and z axes.

pbaspect(mode)

Set the plot box aspect ratio mode of the current axes. mode is either "auto" or

"manual".
pbaspect ("mode")

Return the plot box aspect ratio mode of the current axes.

pbaspect (hax, ...)

Operate on the axes in handle hax instead of the current axes.

See also: [axis], page 285, [daspect], page 307, [xlim], page 287, [ylim], page 287, [zlim], page 287.

15.2.2.2 Three-dimensional Function Plotting

[Function File] ezplot3 (fx, fy, fz) [Function File] ezplot3 (…, dom) [Function File] ezplot3 (…, n) [Function File] ezplot3 (hax, …) [Function File] h = ezplot3 (…)

Plot a parametrically defined curve in three dimensions. fx, fy, and fz are strings, inline functions, or function handles with one argument defining the function. By default the plot is over the domain 0 <= t <= 2*pi with 500 points. If dom is a two element vector, it represents the minimum and maximum values of t. n is a scalar defining the number of points to use in plotting the function. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created plot.

     fx = @(t) cos (t);
     fy = @(t) sin (t);
     fz = @(t) t;
     ezplot3 (fx, fy, fz, [0, 10*pi], 100);

See also: [plot3], page 303, [ezplot], page 288, [ezmesh], page 309, [ezsurf], page 310.

[Function File] ezmesh (f) [Function File] ezmesh (fx, fy, fz) [Function File] ezmesh (…, dom) [Function File] ezmesh (…, n) [Function File] ezmesh (…, "circ") [Function File] ezmesh (hax, …) [Function File] h = ezmesh (…)

Plot the mesh defined by a function. f is a string, inline function, or function handle with two arguments defining the function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi with 60 points in each dimension. If three functions are passed, then plot the parametrically defined function [fx (s,

t), fy (s, t), fz (s, t)].

If dom is a two element vector, it represents the minimum and maximum values of both x and y. If dom is a four element vector, then the minimum and maximum

values are [xmin xmax ymin ymax].

n is a scalar defining the number of points to use in each dimension. If the argument "circ" is given, then the function is plotted over a disk centered on the middle of the domain dom. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object. Example 1: 2-argument function

     f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
     ezmesh (f, [-3, 3]);

Example 2: parametrically defined function

     fx = @(s,t) cos (s) .* cos (t);
     fy = @(s,t) sin (s) .* cos (t);
     fz = @(s,t) sin (t);
     ezmesh (fx, fy, fz, [-pi, pi, -pi/2, pi/2], 20);

See also: [mesh], page 292, [ezmeshc], page 309, [ezplot], page 288, [ezsurf], page 310, [ezsurfc], page 311, [hidden], page 294.

[Function File] ezmeshc (f) [Function File] ezmeshc (fx, fy, fz) [Function File] ezmeshc (…, dom) [Function File] ezmeshc (…, n) [Function File] ezmeshc (…, "circ")

[Function File] ezmeshc (hax, …) [Function File] h = ezmeshc (…)

Plot the mesh and contour lines defined by a function. f is a string, inline function, or function handle with two arguments defining the function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi with 60 points in each dimension. If three functions are passed, then plot the parametrically defined function [fx (s,

t), fy (s, t), fz (s, t)].

If dom is a two element vector, it represents the minimum and maximum values of both x and y. If dom is a four element vector, then the minimum and maximum

values are [xmin xmax ymin ymax].

n is a scalar defining the number of points to use in each dimension. If the argument "circ" is given, then the function is plotted over a disk centered on the middle of the domain dom. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a 2-element vector with a graphics handle for the created mesh plot and a second handle for the created contour plot. Example: 2-argument function

     f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
     ezmeshc (f, [-3, 3]);

See also: [meshc], page 293, [ezmesh], page 309, [ezplot], page 288, [ezsurf], page 310, [ezsurfc], page 311, [hidden], page 294.

[Function File] ezsurf (f) [Function File] ezsurf (fx, fy, fz) [Function File] ezsurf (…, dom) [Function File] ezsurf (…, n) [Function File] ezsurf (…, "circ") [Function File] ezsurf (hax, …) [Function File] h = ezsurf (…)

Plot the surface defined by a function. f is a string, inline function, or function handle with two arguments defining the function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi with 60 points in each dimension. If three functions are passed, then plot the parametrically defined function [fx (s,

t), fy (s, t), fz (s, t)].

If dom is a two element vector, it represents the minimum and maximum values of both x and y. If dom is a four element vector, then the minimum and maximum

values are [xmin xmax ymin ymax].

n is a scalar defining the number of points to use in each dimension. If the argument "circ" is given, then the function is plotted over a disk centered on the middle of the domain dom.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created surface object. Example 1: 2-argument function

     f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
     ezsurf (f, [-3, 3]);

Example 2: parametrically defined function

     fx = @(s,t) cos (s) .* cos (t);
     fy = @(s,t) sin (s) .* cos (t);
     fz = @(s,t) sin (t);
     ezsurf (fx, fy, fz, [-pi, pi, -pi/2, pi/2], 20);

See also: [surf], page 295, [ezsurfc], page 311, [ezplot], page 288, [ezmesh], page 309, [ezmeshc], page 309, [shading], page 306.

[Function File] ezsurfc (f) [Function File] ezsurfc (fx, fy, fz) [Function File] ezsurfc (…, dom) [Function File] ezsurfc (…, n) [Function File] ezsurfc (…, "circ") [Function File] ezsurfc (hax, …) [Function File] h = ezsurfc (…)

Plot the surface and contour lines defined by a function. f is a string, inline function, or function handle with two arguments defining the function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi with 60 points in each dimension. If three functions are passed, then plot the parametrically defined function [fx (s,

t), fy (s, t), fz (s, t)].

If dom is a two element vector, it represents the minimum and maximum values of both x and y. If dom is a four element vector, then the minimum and maximum

values are [xmin xmax ymin ymax].

n is a scalar defining the number of points to use in each dimension. If the argument "circ" is given, then the function is plotted over a disk centered on the middle of the domain dom. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a 2-element vector with a graphics handle for the created surface plot and a second handle for the created contour plot. Example:

     f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
     ezsurfc (f, [-3, 3]);

See also: [surfc], page 295, [ezsurf], page 310, [ezplot], page 288, [ezmesh], page 309, [ezmeshc], page 309, [shading], page 306.

15.2.2.3 Three-dimensional Geometric Shapes

[Command] cylinder [Function File] cylinder (r) [Function File] cylinder (r, n) [Function File] cylinder (hax, …) [Function File] [x, y, z] = cylinder (…)

Plot a 3-D unit cylinder. The optional input r is a vector specifying the radius along the unit z-axis. The default is [1 1] indicating radius 1 at Z == 0 and at Z == 1. The optional input n determines the number of faces around the the circumference of the cylinder. The default value is 20. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. If outputs are requested cylinder returns three matrices in meshgrid format, such that surf (x, y, z) generates a unit cylinder. Example:

     [x, y, z] = cylinder (10:-1:0, 50);
     surf (x, y, z);
     title ("a cone");

See also: [ellipsoid], page 312, [rectangle], page 291, [sphere], page 312.

[Function File] sphere () [Function File] sphere (n) [Function File] sphere (hax, …) [Function File] [x, y, z] = sphere (…)

Plot a 3-D unit sphere. The optional input n determines the number of faces around the the circumference of the sphere. The default value is 20. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. If outputs are requested sphere returns three matrices in meshgrid format such that surf (x, y, z) generates a unit sphere. Example:

     [x, y, z] = sphere (40);
     surf (3*x, 3*y, 3*z);
     axis equal;
     title ("sphere of radius 3");

See also: [cylinder], page 312, [ellipsoid], page 312, [rectangle], page 291.

[Function File] ellipsoid (xc, yc, zc, xr, yr, zr, n)

[Function File] ellipsoid (…, n) [Function File] ellipsoid (hax, …) [Function File] [x, y, z] = ellipsoid (…)

Plot a 3-D ellipsoid.

The inputs xc, yc, zc specify the center of the ellipsoid. The inputs xr, yr, zr specify the semi-major axis lengths. The optional input n determines the number of faces around the the circumference of the cylinder. The default value is 20. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. If outputs are requested ellipsoid returns three matrices in meshgrid format, such that surf (x, y, z) generates the ellipsoid.

See also: [cylinder], page 312, [rectangle], page 291, [sphere], page 312.

15.2.3 Plot Annotations

You can add titles, axis labels, legends, and arbitrary text to an existing plot. For example:

x = -10:0.1:10;
plot (x, sin (x));
title ("sin(x) for x = -10:0.1:10");
xlabel ("x");
ylabel ("sin (x)");
text (pi, 0.7, "arbitrary text");
legend ("sin (x)");

The functions grid and box may also be used to add grid and border lines to the plot. By default, the grid is off and the border lines are on.

[Function File] title (string) [Function File] title (string, prop, val, …) [Function File] title (hax, …) [Function File] h = title (…)

Specify the string used as a title for the current axis. An optional list of property/value pairs can be used to change the appearance of the created title text object. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created text object.

See also: [xlabel], page 315, [ylabel], page 315, [zlabel], page 315, [text], page 315.

[Function File] legend (str1, str2, …) [Function File] legend (matstr) [Function File] legend (cellstr)

[Function File] legend (…, "location", pos) [Function File] legend (…, "orientation", orient)

[Function File] legend (hax, …) [Function File] legend (hobjs, …) [Function File] legend (hax, hobjs, …) [Function File] legend ("option")

[Function File] [hleg, hleg_obj, hplot, labels] = legend (. . .)

Display a legend for the current axes using the specified strings as labels.

Legend entries may be specified as individual character string arguments, a character array, or a cell array of character strings. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. If the handles, hobjs, are not specified then the legend’s strings will be associated with the axes’ descendants. legend works on line graphs, bar graphs, etc. A plot must exist before legend is called. The optional parameter pos specifies the location of the legend as follows:

pos  location of the legend
north  center top
south  center bottom
east  right center
west  left center
northeast right top (default)
northwest left top
southeast right bottom
southwest left bottom
outside  can be appended to any location string

The optional parameter orient determines if the key elements are placed vertically or horizontally. The allowed values are "vertical" (default) or "horizontal". The following customizations are available using option: "show" Show legend on the plot "hide" Hide legend on the plot

"toggle" Toggles between "hide" and "show"

"boxon" Show a box around legend (default) "boxoff" Hide the box around legend "right" Place label text to the right of the keys (default) "left" Place label text to the left of the keys "off" Delete the legend object The optional output values are hleg The graphics handle of the legend object. hleg obj Graphics handles to the text and line objects which make up the legend. hplot Graphics handles to the plot objects which were used in making the leg-

end.

labels A cell array of strings of the labels in the legend. The legend label text is either provided in the call to legend or is taken from the DisplayName property of graphics objects. If no labels or DisplayNames are available, then the label text is simply "data1", "data2", …, "dataN". Implementation Note: A legend is implemented as an additional axes object of the current figure with the "tag" set to "legend". Properties of the legend object may be manipulated directly by using set.

[Function File] text (x, y, string) [Function File] text (x, y, z, string) [Function File] text (…, prop, val, …) [Function File] h = text (…)

Create a text object with text string at position x, y, (z) on the current axes.

Multiple locations can be specified if x, y, (z) are vectors. Multiple strings can be specified with a character matrix or a cell array of strings.

Optional property/value pairs may be used to control the appearance of the text.

The optional return value h is a vector of graphics handles to the created text objects.

See also: [gtext], page 332, [title], page 313, [xlabel], page 315, [ylabel], page 315, [zlabel], page 315.

See Section 15.3.3.5 [Text Properties], page 353 for the properties that you can set.

[Function File] xlabel (string)

[Function File] xlabel (string, property, val, . . .)

[Function File] xlabel (hax, …) [Function File] h = xlabel (…)

Specify the string used to label the x-axis of the current axis.

An optional list of property/value pairs can be used to change the properties of the created text label.

If the first argument hax is an axes handle, then operate on this axis rather than the current axes returned by gca.

The optional return value h is a graphics handle to the created text object.

See also: [ylabel], page 315, [zlabel], page 315, [datetick], page 707, [title], page 313, [text], page 315.

[Function File] clabel (c, h) [Function File] clabel (c, h, v) [Function File] clabel (c, h, "manual") [Function File] clabel (c) [Function File] clabel (…, prop, val, …) [Function File] h = clabel (…)

Add labels to the contours of a contour plot.

The contour levels are specified by the contour matrix c which is returned by contour, contourc, contourf, and contour3. Contour labels are rotated to match the local line orientation and centered on the line. The position of labels along the contour line is chosen randomly.

If the argument h is a handle to a contour group object, then label this plot rather than the one in the current axes returned by gca.

By default, all contours are labeled. However, the contours to label can be specified by the vector v. If the "manual" argument is given then the contours to label can be selected with the mouse.

Additional property/value pairs that are valid properties of text objects can be given and are passed to the underlying text objects. Moreover, the contour group property "LabelSpacing" is available which determines the spacing between labels on a contour to be specified. The default is 144 points, or 2 inches. The optional return value h is a vector of graphics handles to the text objects representing each label. The "userdata" property of the text objects contains the numerical value of the contour label. An example of the use of clabel is

     [c, h] = contour (peaks (), -4 : 6);
     clabel (c, h, -4:2:6, "fontsize", 12);

See also: [contour], page 273, [contourf], page 274, [contour3], page 275, [meshc], page 293, [surfc], page 295, [text], page 315.

[Command] box on

[Command] box off

[Command] box [Function File] box (hax, …)

Control display of the axis border. The argument may be either "on" or "off". If it is omitted, the current box state is toggled. If the first argument hax is an axes handle, then operate on this axis rather than the current axes returned by gca.

See also: [axis], page 285, [grid], page 316.

[Command] grid

[Command] grid on

[Command] grid off

[Command] grid minor

[Command] grid minor on

[Command] grid minor off

[Function File] grid (hax, …)

Control the display of plot grid lines. The function state input may be either "on" or "off". If it is omitted, the current grid state is toggled. When the first argument is "minor" all subsequent commands modify the minor grid rather than the major grid. If the first argument hax is an axes handle, then operate on this axis rather than the current axes returned by gca. To control the grid lines for an individual axis use the set function. For example:

     set (gca, "ygrid", "on");

See also: [axis], page 285, [box], page 316.

[Command] colorbar [Function File] colorbar (loc)

[Function File] colorbar (delete_option)

[Function File] colorbar (hcb, …) [Function File] colorbar (hax, …) [Function File] colorbar (…, "peer", hax, …) [Function File] colorbar (…, "location", loc, …) [Function File] colorbar (…, prop, val, …) [Function File] h = colorbar (…)

Add a colorbar to the current axes.

A colorbar displays the current colormap along with numerical rulings so that the color scale can be interpreted.

The optional input loc determines the location of the colorbar. Valid values for loc are

"EastOutside"

Place the colorbar outside the plot to the right. This is the default.

"East" Place the colorbar inside the plot to the right.

"WestOutside"

Place the colorbar outside the plot to the left.

"West" Place the colorbar inside the plot to the left.

"NorthOutside"

Place the colorbar above the plot.

"North" Place the colorbar at the top of the plot.

"SouthOutside"

Place the colorbar under the plot.

"South" Place the colorbar at the bottom of the plot.

To remove a colorbar from a plot use any one of the following keywords for the

delete option: "delete", "hide", "off".

If the argument "peer" is given, then the following argument is treated as the axes handle in which to add the colorbar. Alternatively, If the first argument hax is an axes handle, then the colorbar is added to this axis, rather than the current axes returned by gca.

If the first argument hcb is a handle to a colorbar object, then operate on this colorbar directly.

Additional property/value pairs are passed directly to the underlying axes object.

The optional return value h is a graphics handle to the created colorbar object.

Implementation Note: A colorbar is created as an additional axes to the current figure with the "tag" property set to "colorbar". The created axes object has the extra property "location" which controls the positioning of the colorbar.

See also: [colormap], page 660.

15.2.4 Multiple Plots on One Page

Octave can display more than one plot in a single figure. The simplest way to do this is to use the subplot function to divide the plot area into a series of subplot windows that are indexed by an integer. For example,

subplot (2, 1, 1)
fplot (@sin, [-10, 10]);
subplot (2, 1, 2)
fplot (@cos, [-10, 10]);

creates a figure with two separate axes, one displaying a sine wave and the other a cosine wave. The first call to subplot divides the figure into two plotting areas (two rows and one column) and makes the first plot area active. The grid of plot areas created by subplot is numbered in column-major order (top to bottom, left to right).

[Function File] subplot (rows, cols, index)

[Function File] subplot (rcn) [Function File] subplot (hax) [Function File] subplot (…, "align")

[Function File] subplot (…, "replace")

[Function File] subplot (…, "position", pos) [Function File] subplot (…, prop, val, …) [Function File] hax = subplot (…)

Set up a plot grid with rows by cols subwindows and set the current axes for plotting (gca) to the location given by index. If only one numeric argument is supplied, then it must be a three digit value specifying the number of rows in digit 1, the number of columns in digit 2, and the plot index in digit 3. The plot index runs row-wise; First, all columns in a row are numbered and then the next row is filled. For example, a plot with 2x3 grid will have plot indices running as follows:

1  2  3
4  5  6

index may also be a vector. In this case, the new axis will enclose the grid locations specified. The first demo illustrates this:

     demo ("subplot", 1)

The index of the subplot to make active may also be specified by its axes handle, hax, returned from a previous subplot command. If the option "align" is given then the plot boxes of the subwindows will align, but this may leave no room for axis tick marks or labels. If the option "replace" is given then the subplot axis will be reset, rather than just switching the current axis for plotting to the requested subplot.

The "position" property can be used to exactly position the subplot axes within the current figure. The option pos is a 4-element vector [x, y, width, height] that determines the location and size of the axes. The values in pos are normalized in the range [0,1]. Any property/value pairs are passed directly to the underlying axes object. If the output hax is requested, subplot returns the axis handle for the subplot. This is useful for modifying the properties of a subplot using set.

See also: [axes], page 336, [plot], page 260, [gca], page 340, [set], page 342.

15.2.5 Multiple Plot Windows

You can open multiple plot windows using the figure function. For example,

figure (1);
fplot (@sin, [-10, 10]);
figure (2);
fplot (@cos, [-10, 10]);

creates two figures, with the first displaying a sine wave and the second a cosine wave. Figure numbers must be positive integers.

[Command] figure

[Command] figure n [Function File] figure (n) [Function File] figure (…, "property", value, …) [Function File] h = figure (…)

Create a new figure window for plotting. If no arguments are specified, a new figure with the next available number is created. If called with an integer n, and no such numbered figure exists, then a new figure with the specified number is created. If the figure already exists then it is made visible and becomes the current figure for plotting. Multiple property-value pairs may be specified for the figure object, but they must appear in pairs. The optional return value h is a graphics handle to the created figure object.

See also: [axes], page 336, [gcf], page 340, [clf], page 322, [close], page 323.

15.2.6 Manipulation of Plot Windows

By default, Octave refreshes the plot window when a prompt is printed, or when waiting for input. The drawnow function is used to cause a plot window to be updated.

[Built-in Function] drawnow () [Built-in Function] drawnow ("expose")

[Built-in Function] drawnow (term, file, mono, debug_file)

Update figure windows and their children. The event queue is flushed and any callbacks generated are executed. With the optional argument "expose", only graphic objects are updated and no other events or callbacks are processed. The third calling form of drawnow is for debugging and is undocumented.

Only figures that are modified will be updated. The refresh function can also be used to force an update of the current figure, even if it is not modified.

[Function File] refresh () [Function File] refresh (h)

Refresh a figure, forcing it to be redrawn. When called without an argument the current figure is redrawn. Otherwise, the figure with graphic handle h is redrawn.

See also: [drawnow], page 319.

Normally, high-level plot functions like plot or mesh call newplot to initialize the state of the current axes so that the next plot is drawn in a blank window with default property settings. To have two plots superimposed over one another, use the hold function. For example,

hold on;
x = -10:0.1:10;
plot (x, sin (x));
plot (x, cos (x));
hold off;

displays sine and cosine waves on the same axes. If the hold state is off, consecutive plotting commands like this will only display the last plot.

[Function File] newplot () [Function File] newplot (hfig) [Function File] newplot (hax) [Function File] hax = newplot (…)

Prepare graphics engine to produce a new plot. This function is called at the beginning of all high-level plotting functions. It is not normally required in user programs. newplot queries the "NextPlot" field of the current figure and axis to determine what to do.

Figure NextPlot  Action
"new"  Create a new figure and make it the current figure.
"add" (default)  Add new graphic objects to the current figure.

"replacechildren"

Delete child objects whose HandleVisibility is set to "on". Set NextPlot property to "add". This typically clears a figure, but leaves in place hidden objects such as menubars. This is equivalent

                     to clf.
"replace"

Delete all child objects of the figure and reset all figure properties to their defaults. However, the following four properties are not reset: Position, Units, PaperPosition, PaperUnits. This is equiva-

                     lent to clf reset.

Axis NextPlot  Action

"add"

Add new graphic objects to the current axes. This is equivalent

                     to hold on.
"replacechildren"

Delete child objects whose HandleVisibility is set to "on", but leave axis properties unmodified. This typically clears a plot, but preserves special settings such as log scaling for axes. This is equivalent to cla.

"replace" (default)

Delete all child objects of the axis and reset all axis properties to their defaults. However, the following properties are not reset: Position, Units. This is equivalent to cla reset.

If the optional input hfig or hax is given then prepare the specified figure or axes rather than the current figure and axes. The optional return value hax is a graphics handle to the created axes object (not figure). Caution: Calling newplot may change the current figure and current axis.

[Command] hold

[Command] hold on

[Command] hold off

[Command] hold all [Function File] hold (hax, …)

Toggle or set the "hold" state of the plotting engine which determines whether new graphic objects are added to the plot or replace the existing objects.

hold on Retain plot data and settings so that subsequent plot commands are displayed on a single graph.

hold all Retain plot line color, line style, data, and settings so that subsequent plot commands are displayed on a single graph with the next line color and style.

hold off Restore default graphics settings which clear the graph and reset axis properties before each new plot command. (default).

hold Toggle the current hold state.

When given the additional argument hax, the hold state is modified for this axis rather than the current axes returned by gca. To query the current hold state use the ishold function.

See also: [ishold], page 321, [cla], page 322, [clf], page 322, [newplot], page 320.

[Command] ishold [Function File] ishold (hax) [Function File] ishold (hfig)

Return true if the next plot will be added to the current plot, or false if the plot device will be cleared before drawing the next plot.

If the first argument is an axes handle hax or figure handle hfig then operate on this plot rather than the current one.

See also: [hold], page 321, [newplot], page 320.

To clear the current figure, call the clf function. To clear the current axis, call the cla function. To bring the current figure to the top of the window stack, call the shg function. To delete a graphics object, call delete on its index. To close the figure window, call the close function.

[Command] clf

[Command] clf reset

[Function File] clf (hfig) [Function File] clf (hfig, "reset") [Function File] h = clf (…)

Clear the current figure window. clf operates by deleting child graphics objects with visible handles (HandleVisibility

= "on").

If the optional argument "reset" is specified, delete all child objects including those with hidden handles and reset all figure properties to their defaults. However, the following properties are not reset: Position, Units, PaperPosition, PaperUnits. If the first argument hfig is a figure handle, then operate on this figure rather than the current figure returned by gcf. The optional return value h is the graphics handle of the figure window that was cleared.

See also: [cla], page 322, [close], page 323, [delete], page 323, [reset], page 361.

[Command] cla

[Command] cla reset

[Function File] cla (hax) [Function File] cla (hax, "reset")

Clear the current axes. cla operates by deleting child graphic objects with visible handles (HandleVisibility

= "on").

If the optional argument "reset" is specified, delete all child objects including those with hidden handles and reset all axis properties to their defaults. However, the following properties are not reset: Position, Units. If the first argument hax is an axes handle, then operate on this axis rather than the current axes returned by gca.

See also: [clf], page 322, [delete], page 323, [reset], page 361.

[Command] shg

Show the graph window. Currently, this is the same as executing drawnow.

See also: [drawnow], page 319, [figure], page 319.

[Function File] delete (file) [Function File] delete (handle)

Delete the named file or graphics handle. Deleting graphics objects is the proper way to remove features from a plot without clearing the entire figure.

See also: [clf], page 322, [cla], page 322, [unlink], page 708.

[Command] close

[Command] close (h)

[Command] close all

[Command] close all hidden Close figure window(s). When called with no arguments, close the current figure. This is equivalent to close (gcf). If the input h is a graphic handle, or vector of graphics handles, then close each figure in h. If the argument "all" is given then all figures with visible handles (HandleVisibility = "on") are closed. If the argument "all hidden" is given then all figures, including hidden ones, are closed. Implementation Note: close operates by calling the function specified by the "closerequestfcn" property for each figure. By default, the function closereq is used. It is possible that the function invoked will delay or abort removing the figure. To remove a figure without executing any callback functions use delete. When writing a callback function to close a window do not use close to avoid recursion.

See also: [closereq], page 323, [delete], page 323.

[Function File] closereq ()

Close the current figure and delete all graphics objects associated with it. By default, the "closerequestfcn" property of a new plot figure points to this function.

See also: [close], page 323, [delete], page 323.

15.2.7 Use of the interpreter Property

All text objects—such as titles, labels, legends, and text—include the property "interpreter", this property determines the manner in which special control sequences in the text are rendered. If the interpreter is set to "none", then no rendering occurs. Currently the "latex" interpreter is not implemented and is equivalent to "none".

The "tex" option implements a subset of TEX functionality when rendering text. This allows the insertion of special glyphs such as Greek characters or mathematical symbols. The special characters are inserted with a code following a backslash (\) character, as in the table Table 15.1.

In addition, the formatting of the text can be changed within the string by using the codes

\bf  Bold font

\it  Italic font
\sl  Oblique Font
\rm  Normal font

These may be used in conjunction with the { and } characters to limit the change in the font to part of the string. For example,

     xlabel ('{\bf H} = a {\bf V}')

where the character 'a' will not appear in a bold font. Note that to avoid having Octave interpret the backslash characters in the strings, the strings should be in single quotes.

It is also possible to change the fontname and size within the text

\fontname{fontname}  Specify the font to use
\fontsize{size}  Specify the size of the font to use

Finally, superscripting and subscripting can be controlled with the '^' and '_' characters. If the '^' or '_' is followed by a { character, then all of the block surrounded by the { } pair is super- or sub-scripted. Without the { } pair, only the character immediately following the '^' or '_' is super- or sub-scripted.

Chapter 15: Plotting 325

Greek Lowercase Letters

Code  Sym Code  Sym Code  Sym
\alpha  α  \beta  β  \gamma  γ
\delta  δ  \epsilon  ϵ  \zeta  ζ
\eta  η  \theta  θ  \vartheta  ϑ
\iota  ι  \kappa  κ  \lambda  λ
\mu  μ  \nu  ν  \xi  ξ
\o  o  \pi  π  \varpi  ϖ
\rho  ρ  \sigma  σ  \varsigma  ς
\tau  τ  \upsilon  υ  \phi  ϕ
\chi  χ  \psi  ψ  \omega  ω

Greek Uppercase Letters

Code  Sym Code  Sym Code  Sym
\Gamma  Γ  \Delta  Δ  \Theta  Θ
\Lambda  Λ  \Xi  Ξ  \Pi  Π
\Sigma  Σ  \Upsilon  Υ  \Phi  Φ
\Psi  Ψ  \Omega  Ω

Misc Symbols Type Ord

Code  Sym Code  Sym Code  Sym
\aleph  ℵ  \wp  ℘  \Re  ℜ
\Im  ℑ  \partial  ∂  \infty  ∞
\prime  ′  \nabla  ∇  \surd  √
\angle  ^{}\forall  ∀  \exists  ∃
\neg  ¬  \clubsuit  ♣  \diamondsuit  ♢
\heartsuit  ♡  \spadesuit  ♠

“Large” Operators

Code Sym Code Sym Code Sym
\int  ∫

Binary operators

Code  Sym Code  Sym Code  Sym
\pm  ±  \cdot  ·  \times  ×
\ast  ∗  \circ  ∘  \bullet  •
\div  ÷  \cap  ∩  \cup  ∪
\vee  ∨  \wedge  ∧  \oplus  ⊕
\otimes  ⊗  \oslash  ⊘

Table 15.1: Available special characters in TEX mode

Relations

Code  Sym Code  Sym Code  Sym
\leq  ≤  \subset  ⊂  \subseteq  ⊆
\in  ∈  \geq  ≥  \supset  ⊃
\supseteq  ⊇  \ni  ∋  \mid  |
\equiv  ≡  \sim  ∼  \approx  ≈
\cong  ≅  \propto  ∝  \perp  ⊥

Arrows

Code  Sym Code  Sym Code  Sym
\leftarrow  ←  \Leftarrow  ⇐  \rightarrow  →
\Rightarrow  ⇒  \leftrightarrow ↔  \uparrow  ↑
\downarrow  ↓

Openings and Closings

Code  Sym Code  Sym Code  Sym
\lfloor  ⌊  \langle  ⟨  \lceil  ⌈
\rfloor  ⌋  \rangle  ⟩  \rceil  ⌉

Alternate Names

Code Sym Code Sym Code Sym
\neq  ≠

Other (not in Appendix F Tables)

Code  Sym Code  Sym Code  Sym
\ldots  …  \0  ⊘  \copyright  \begin{matrix}c \\ ◯ \end{matrix}
\deg  ^{∘}

Table 15.1: Available special characters in TEX mode (cont.)

A complete example showing the capabilities of the extended text is

    x = 0:0.01:3;
    plot (x, erf (x));
    hold on;
    plot (x,x,"r");
    axis ([0, 3, 0, 1]);
    text (0.65, 0.6175, strcat ('\leftarrow x = {2/\surd\pi',
    ' {\fontsize{16}\int_{\fontsize{8}0}^{\fontsize{8}x}}',
    ' e^{-t^2} dt} = 0.6175'))

The result of which can be seen in Figure 15.7

Figure 15.7: Example of inclusion of text with the TEX interpreter

15.2.8 Printing and Saving Plots

The print command allows you to send plots to you printer and to save plots in a variety of formats. For example,

print -dpsc

prints the current figure to a color PostScript printer. And,

print -deps foo.eps

saves the current figure to an encapsulated PostScript file called ‘foo.eps’.

[Function File] print () [Function File] print (options)

[Function File] print (filename, options)
[Function File] print (h, filename, options)

Print a plot, or save it to a file. Both output formatted for printing (PDF and PostScript), and many bitmapped and vector image formats are supported. filename defines the name of the output file. If the file name has no suffix, one is inferred from the specified device and appended to the file name. If no filename is specified, the output is sent to the printer. h specifies the handle of the figure to print. If no handle is specified the current figure is used. For output to a printer, PostScript file, or PDF file, the paper size is specified by the figure’s papersize property. The location and size of the image on the page are specified by the figure’s paperposition property. The orientation of the page is specified by the figure’s paperorientation property. The width and height of images are specified by the figure’s paperpositon(3:4) property values.

The print command supports many options:

-fh Specify the handle, h, of the figure to be printed. The default is the current figure.

-Pprinter

Set the printer name to which the plot is sent if no filename is specified.

-Gghostscript_command

Specify the command for calling Ghostscript. For Unix and Windows the defaults are "gs" and "gswin32c", respectively.

-color

-mono Color or monochrome output.

-solid

-dashed Force all lines to be solid or dashed, respectively.

-portrait
-landscape

Specify the orientation of the plot for printed output. For non-printed output the aspect ratio of the output corresponds to the plot area defined by the "paperposition" property in the orientation specified. This option is equivalent to changing the figure’s "paperorientation" property.

-TextAlphaBits=n
-GraphicsAlphaBits=n

Octave is able to produce output for various printers, bitmaps, and vector formats by using Ghostscript. For bitmap and printer output anti-aliasing is applied using Ghostscript’s TextAlphaBits and GraphicsAlphaBits options. The default number of bits for each is 4. Allowed values for N are 1, 2, or 4.

-ddevice The available output format is specified by the option device, and is one of:

ps
ps2
psc

psc2 PostScript (level 1 and 2, mono and color). The FLTK graphics toolkit generates PostScript level 3.0.

eps
eps2
epsc

epsc2 Encapsulated PostScript (level 1 and 2, mono and color). The FLTK graphic toolkit generates PostScript level 3.0.

tex
epslatex
epslatexstandalone
pstex
pslatex

pdflatex Generate a L^{A}TEX (or TEX) file for labels and eps/ps/pdf for graphics. The file produced by epslatexstandalone can be processed directly by L^{A}TEX. The other formats are intended to be included in a L^{A}TEX (or TEX) document. The tex device is the same as the epslatex device. The pdflatex device is only available for the FLTK graphics toolkit.

tikz Generate a L^{A}TEX file using PGF/TikZ. For the FLTK toolkit the result is PGF.

ill

aifm Adobe Illustrator (Obsolete for Gnuplot versions > 4.2)

cdr

corel CorelDraw

dxf AutoCAD

emf

meta Microsoft Enhanced Metafile

fig XFig. For the Gnuplot graphics toolkit, the additional options ‘-textspecial’ or ‘-textnormal’ can be used to control whether the special flag should be set for the text in the figure. (default is ‘-textnormal’)

hpgl HP plotter language

mf Metafont

png Portable network graphics

jpg

jpeg JPEG image

gif GIF image (only available for the Gnuplot graphics toolkit)

pbm PBMplus

svg Scalable vector graphics

pdf Portable document format

If the device is omitted, it is inferred from the file extension, or if there is no filename it is sent to the printer as PostScript.

-dghostscript_device

Additional devices are supported by Ghostscript. Some examples are;

ljet2p HP LaserJet IIP

ljet3 HP LaserJet III deskjet HP DeskJet and DeskJet Plus

cdj550 HP DeskJet 550C

paintjet HP PointJet

pcx24b 24-bit color PCX file format

ppm Portable Pixel Map file format

pdfwrite Produces pdf output from eps

For a complete list, type system ("gs -h") to see what formats and devices are available. When Ghostscript output is sent to a printer the size is determined by the figure’s "papersize" property. When the output is sent to a file the size is determined by the plot box defined by the figure’s "paperposition" property.

-append Append PostScript or PDF output to a pre-existing file of the same type.

-rNUM Resolution of bitmaps in pixels per inch. For both metafiles and SVG the default is the screen resolution; for other formats it is 150 dpi. To specify screen resolution, use "-r0".

-loose

-tight Force a tight or loose bounding box for eps files. The default is loose.

-preview Add a preview to eps files. Supported formats are:

-interchange

Provide an interchange preview.

-metalfile

Provide a metafile preview.

-pict Provide pict preview.

-tiff Provide a tiff preview.

-Sxsize,ysize

Plot size in pixels for EMF, GIF, JPEG, PBM, PNG, and SVG. For PS, EPS, PDF, and other vector formats the plot size is in points. This option is equivalent to changing the size of the plot box associated with the "paperposition" property. When using the command form of the print function you must quote the xsize,ysize option. For example, by writing "-S640,480".

-Ffontname
-Ffontname:size

-F:size Use fontname and/or fontsize for all text. fontname is ignored for some devices: dxf, fig, hpgl, etc.

The filename and options can be given in any order. Example: Print to a file using the svg device.

     figure (1);
     clf ();
     surf (peaks);
     print -dsvg figure1.svg

Example: Print to an HP DeskJet 550C.

     clf ();
     surf (peaks);
     print -dcdj550

See also: [saveas], page 331, [orient], page 331, [figure], page 319.

[Function File] saveas (h, filename)

[Function File] saveas (h, filename, fmt)

Save graphic object h to the file filename in graphic format fmt. fmt should be one of the following formats:

ps PostScript

eps Encapsulated PostScript

jpg JPEG Image

png PNG Image

emf Enhanced Meta File

pdf Portable Document Format

All device formats specified in print may also be used. If fmt is omitted it is extracted from the extension of filename. The default format is "pdf".

     clf ();
     surf (peaks);
     saveas (1, "figure1.png");

See also: [print], page 327, [orient], page 331.

[Function File] orient (orientation)
[Function File] orient (hfig, orientation)
[Function File] orientation = orient ()
[Function File] orientation = orient (hfig)

Query or set the print orientation for figure hfig. Valid values for orientation are "portrait", "landscape", and "tall". The "landscape" option changes the orientation so the plot width is larger than the plot height. The "paperposition" is also modified so that the plot fills the page, while leaving a 0.25 inch border. The "tall" option sets the orientation to "portrait" and fills the page with the plot, while leaving a 0.25 inch border. The "portrait" option (default) changes the orientation so the plot height is larger than the plot width. It also restores the default "paperposition" property. When called with no arguments, return the current print orientation. If the argument hfig is omitted, then operate on the current figure returned by gcf.

See also: [print], page 327, [saveas], page 331.

15.2.9 Interacting with Plots

The user can select points on a plot with the ginput function or selection the position at which to place text on the plot with the gtext function using the mouse. Menus may also be created and populated with specific user commands via the uimenu function.

[Function File] [x, y, buttons] = ginput (n)
[Function File] [x, y, buttons] = ginput ()

Return the position and type of mouse button clicks and/or key strokes in the current figure window. If n is defined, then capture n events before returning. When n is not defined ginput will loop until the return key RET is pressed. The return values x, y are the coordinates where the mouse was clicked in the units of the current axes. The return value button is 1, 2, or 3 for the left, middle, or right button. If a key is pressed the ASCII value is returned in button.

See also: [gtext], page 332, [waitforbuttonpress], page 332.

[Function File] waitforbuttonpress ()
[Function File] b = waitforbuttonpress ()

Wait for mouse click or key press over the current figure window. The return value of b is 0 if a mouse button was pressed or 1 if a key was pressed.

See also: [waitfor], page 694, [ginput], page 332, [kbhit], page 229.

[Function File] gtext (s) [Function File] gtext ({s1, s2, …}) [Function File] gtext ({s1; s2; …}) [Function File] gtext (…, prop, val, …) [Function File] h = gtext (…)

Place text on the current figure using the mouse. The text is defined by the string s. If s is a cell string organized as a row vector then each string of the cell array is written to a separate line. If s is organized as a column vector then one string element of the cell array is placed for every mouse click. Optional property/value pairs are passed directly to the underlying text objects. The optional return value h is a graphics handle to the created text object(s).

See also: [ginput], page 332, [text], page 315.

[Function File] uimenu (property, value, …) [Function File] uimenu (h, property, value, …)

Create a uimenu object and return a handle to it. If h is omitted then a top-level menu for the current figure is created. If h is given then a submenu relative to h is created. uimenu objects have the following specific properties:

"accelerator"

A string containing the key combination together with CTRL to execute this menu entry (e.g., "x" for CTRL+x).

"callback"

Is the function called when this menu entry is executed. It can be either a function string (e.g., "myfun"), a function handle (e.g., @myfun) or a cell array containing the function handle and arguments for the callback function (e.g., {@myfun, arg1, arg2}).

"checked"

Can be set "on" or "off". Sets a mark at this menu entry.

"enable" Can be set "on" or "off". If disabled the menu entry cannot be selected and it is grayed out.

"foregroundcolor"

A color value setting the text color for this menu entry.

"label" A string containing the label for this menu entry. A "&"-symbol can be used to mark the "accelerator" character (e.g., "E&xit")

"position"

An scalar value containing the relative menu position. The entry with the lowest value is at the first position starting from left or top.

"separator"

Can be set "on" or "off". If enabled it draws a separator line above the current position. It is ignored for top level entries.

Examples:

     f = uimenu ("label", "&File", "accelerator", "f");
     e = uimenu ("label", "&Edit", "accelerator", "e");
     uimenu (f, "label", "Close", "accelerator", "q", ...
                "callback", "close (gcf)");
     uimenu (e, "label", "Toggle &Grid", "accelerator", "g", ...
                "callback", "grid (gca)");

See also: [figure], page 319.

15.2.10 Test Plotting Functions

The functions sombrero and peaks provide a way to check that plotting is working. Typing either sombrero or peaks at the Octave prompt should display a three-dimensional plot.

[Function File] sombrero () [Function File] sombrero (n) [Function File] z = sombrero (…) [Function File] [x, y, z] = sombrero (…)

Plot the familiar 3-D sombrero function. The function plotted is

z = \frac{sin(\sqrt{(x^{2} + y^{2})})}{\sqrt{(x^{2} + y^{2})}}

Called without a return argument, sombrero plots the surface of the above function over the meshgrid [-8,8] using surf.

If n is a scalar the plot is made with n grid lines. The default value for n is 41. When called with output arguments, return the data for the function evaluated over the meshgrid. This can subsequently be plotted with surf (x, y, z).

See also: [peaks], page 334, [meshgrid], page 302, [mesh], page 292, [surf], page 295.

[Function File] peaks () [Function File] peaks (n) [Function File] peaks (x, y) [Function File] z = peaks (…) [Function File] [x, y, z] = peaks (…)

Plot a function with lots of local maxima and minima. The function has the form

f(x, y) = 3(1 − x)^{2}e^{(−x^{2}−(y+1)^{2})} − 10 (\frac{x}{5} − x^{3} − y^{5})) − \frac{1}{3}e^{(−(x+1)^{2}−y^{2})}

Called without a return argument, peaks plots the surface of the above function using

surf.

If n is a scalar, peaks plots the value of the above function on an n-by-n mesh over the range [-3,3]. The default value for n is 49. If n is a vector, then it represents the grid values over which to calculate the function. If x and y are specified then the function value is calculated over the specified grid of vertices. When called with output arguments, return the data for the function evaluated over the meshgrid. This can subsequently be plotted with surf (x, y, z).

See also: [sombrero], page 333, [meshgrid], page 302, [mesh], page 292, [surf], page 295.

15.3 Graphics Data Structures

15.3.1 Introduction to Graphics Structures

The graphics functions use pointers, which are of class graphics handle, in order to address the data structures which control graphical displays. A graphics handle may point any one of a number of different object types. The objects are the graphics data structures. The

types of objects are: figure, axes, line, text, patch, surface, text and image.

Each of these objects has a function by the same name. and, each of these functions returns a graphics handle pointing to an object of corresponding type. In addition there are several functions which operate on properties of the graphics objects and which return handles: the functions plot and plot3 return a handle pointing to an object of type line, the function subplot returns a handle pointing to an object of type axes, the function fill returns a handle pointing to an object of type patch, the functions area, bar, barh,

contour, contourf, contour3, surf, mesh, surfc, meshc, errorbar, quiver, quiver3,

scatter, scatter3, stair, stem, stem3 each return a handle as documented in [Data Sources], page 367.

The graphics objects are arranged in a hierarchy:

1. The root is at 0. i.e., get (0) returns the properties of the root object.

2. Below the root are figure objects.

3. Below the figure objects are axes.

4. Below the axes objects are line, text, patch, surface, and image objects.

Graphics handles may be distinguished from function handles (see Section 11.11.1 [Function Handles], page 190) by means of the function ishandle. ishandle returns true if its argument is a handle of a graphics object. In addition, the figure object may be tested using isfigure. isfigure returns true only if its argument is a handle of a figure. The whos function can be used to show the object type of each currently defined graphics handle. (Note: this is not true today, but it is, I hope, considered an error in whos. It may be better to have whos just show graphics handle as the class, and provide a new function which, given a graphics handle, returns its object type. This could generalize the ishandle() functions and, in fact, replace them.)

The get and set commands are used to obtain and set the values of properties of graphics objects. In addition, the get command may be used to obtain property names.

For example, the property "type" of the graphics object pointed to by the graphics handle h may be displayed by:

     get (h, "type")

The properties and their current values are returned by get (h) where h is a handle of a graphics object. If only the names of the allowed properties are wanted they may be displayed by: get (h, "").

Thus, for example:

      h = figure ();
      get (h, "type")
      ans = figure
      get (h, "");
      error: get: ambiguous figure property name ; possible matches:
      __enhanced__hittest                resize
      __graphics_toolkit__integerhandle          resizefcn
      __guidata__interruptible          selected
      __modified__inverthardcopy         selectionhighlight
      __myhandle__keypressfcn            selectiontype
      __plot_stream__keyreleasefcn          tag
      alphamap               menubar                toolbar
      beingdeleted           mincolormap            type
      busyaction             name                   uicontextmenu
      buttondownfcn          nextplot               units
      children               numbertitle            userdata
      clipping               outerposition          visible
      closerequestfcn        paperorientation       windowbuttondownfcn
      color                  paperposition          windowbuttonmotionfcn
      colormap               paperpositionmode      windowbuttonupfcn
      createfcn              papersize              windowkeypressfcn
      currentaxes            papertype              windowkeyreleasefcn
      currentcharacter       paperunits             windowscrollwheelfcn
      currentobject          parent                 windowstyle
      currentpoint           pointer                wvisual
      deletefcn              pointershapecdata      wvisualmode

      dockcontrols           pointershapehotspot    xdisplay
      doublebuffer           position               xvisual
      filename               renderer               xvisualmode
      handlevisibility       renderermode

The root figure has index 0. Its properties may be displayed by: get (0, "").

The uses of get and set are further explained in [get], page 341, [set], page 342.

[Function File] res = isprop (h, "prop")

Return true if prop is a property of the object with handle h.

h may also be an array of handles in which case res will be a logical array indicating whether each handle has the property prop.

See also: [get], page 341, [set], page 342, [ismethod], page 675, [isobject], page 674.

15.3.2 Graphics Objects

The hierarchy of graphics objects was explained above. See Section 15.3.1 [Introduction to Graphics Structures], page 334. Here the specific objects are described, and the properties contained in these objects are discussed. Keep in mind that graphics objects are always referenced by handle.

root figure the top level of the hierarchy and the parent of all figure objects. The handle index of the root figure is 0.

figure A figure window.

axes A set of axes. This object is a child of a figure object and may be a parent of line, text, image, patch, or surface objects.

line A line in two or three dimensions.

text Text annotations.

image A bitmap image.

patch A filled polygon, currently limited to two dimensions.

surface A three-dimensional surface.

15.3.2.1 Creating Graphics Objects

You can create axes, line, patch, and surface objects directly using the axes, line, patch, fill, and surface functions. These objects become children of the current axes object.

[Function File] axes () [Function File] axes (property, value, …) [Function File] axes (hax) [Function File] h = axes (…)

Create an axes object and return a handle to it, or set the current axes to hax. Called without any arguments, or with property/value pairs, construct a new axes. For accepted properties and corresponding values, see [set], page 342. Called with a single axes handle argument hax, the function makes hax the current axis. It also restacks the axes in the corresponding figure so that hax is the first entry in the list of children. This causes hax to be displayed on top of any other axes objects (Z-order stacking).

See also: gca, set, get.

[Function File] line () [Function File] line (x, y) [Function File] line (x, y, property, value, …) [Function File] line (x, y, z) [Function File] line (x, y, z, property, value, …) [Function File] line (property, value, …) [Function File] line (hax, …) [Function File] h = line (…)

Create line object from x and y (and possibly z) and insert in the current axes. Multiple property-value pairs may be specified for the line object, but they must appear in pairs. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle (or vector of handles) to the line objects created.

See also: [image], page 657, [patch], page 337, [rectangle], page 291, [surface], page 339, [text], page 315.

[Function File] patch () [Function File] patch (x, y, c) [Function File] patch (x, y, z, c) [Function File] patch (fv) [Function File] patch ("Faces", faces, "Vertices", verts, …) [Function File] patch (…, prop, val, …) [Function File] patch (hax, …) [Function File] h = patch (…)

Create patch object in the current axes with vertices at locations (x, y) and of color c. If the vertices are matrices of size MxN then each polygon patch has M vertices and a total of N polygons will be created. If some polygons do not have M vertices use NaN to represent "no vertex". If the z input is present then 3-D patches will be created. The color argument c can take many forms. To create polygons which all share a single color use a string value (e.g., "r" for red), a scalar value which is scaled by caxis and indexed into the current colormap, or a 3-element RGB vector with the precise TrueColor. If c is a vector of length N then the ith polygon will have a color determined by scaling entry c(i) according to caxis and then indexing into the current colormap. More complicated coloring situations require directly manipulating patch property/value pairs. Instead of specifying polygons by matrices x and y, it is possible to present a unique list of vertices and then a list of polygon faces created from those vertices. In this case the "Vertices" matrix will be an Nx2 (2-D patch) or Nx3 (3-D path). The MxN "Faces" matrix describes M polygons having N vertices—each row describes a single polygon and each column entry is an index into the "Vertices" matrix to identify a vertex. The patch object can be created by directly passing the property/value pairs "Vertices"/verts, "Faces"/faces as inputs. A third input form is to create a structure fv with the fields "vertices", "faces",

and optionally "facevertexcdata".

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a graphics handle to the created patch object. Implementation Note: Patches are highly configurable objects. To truly customize them requires setting patch properties directly. Useful patch properties are: "cdata",

"edgecolor", "facecolor", "faces", "facevertexcdata".

See also: [fill], page 338, [get], page 341, [set], page 342.

[Function File] fill (x, y, c) [Function File] fill (x1, y1, c1, x2, y2, c2) [Function File] fill (…, prop, val) [Function File] fill (hax, …) [Function File] h = fill (…)

Create one or more filled 2-D polygons. The inputs x and y are the coordinates of the polygon vertices. If the inputs are matrices then the rows represent different vertices and each column produces a different polygon. fill will close any open polygons before plotting. The input c determines the color of the polygon. The simplest form is a single color specification such as a plot format or an RGB-triple. In this case the polygon(s) will have one unique color. If c is a vector or matrix then the color data is first scaled using caxis and then indexed into the current colormap. A row vector will color each polygon (a column from matrices x and y) with a single computed color. A matrix c of the same size as x and y will compute the color of each vertex and then interpolate the face color between the vertices. Multiple property/value pairs for the underlying patch object may be specified, but they must appear in pairs. If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca. The optional return value h is a vector of graphics handles to the created patch objects. Example: red square

     vertices = [0 0
                 1 0
                 1 1
                 0 1];
     fill (vertices(:,1), vertices(:,2), "r");
     axis ([-0.5 1.5, -0.5 1.5])
     axis equal

See also: [patch], page 337, [caxis], page 287, [colormap], page 660.

[Function File] surface (x, y, z, c) [Function File] surface (x, y, z) [Function File] surface (z, c) [Function File] surface (z) [Function File] surface (…, prop, val, …) [Function File] surface (hax, …) [Function File] h = surface (…)

Create a surface graphic object given matrices x and y from meshgrid and a matrix of values z corresponding to the x and y coordinates of the surface.

If x and y are vectors, then a typical vertex is (x(j), y(i), z(i,j)). Thus, columns of z correspond to different x values and rows of z correspond to different y values. If only a single input z is given then x is taken to be 1:rows (z) and y is 1:columns

(z).

Any property/value input pairs are assigned to the surface object.

If the first argument hax is an axes handle, then plot into this axis, rather than the current axes returned by gca.

The optional return value h is a graphics handle to the created surface object.

See also: [surf], page 295, [mesh], page 292, [patch], page 337, [line], page 337.

15.3.2.2 Handle Functions

To determine whether a variable is a graphics object index, or an index to an axes or figure,

use the functions ishandle, isaxes, and isfigure.

[Built-in Function] ishandle (h)

Return true if h is a graphics handle and false otherwise.

h may also be a matrix of handles in which case a logical array is returned that is true where the elements of h are graphics handles and false where they are not.

See also: [isaxes], page 339, [isfigure], page 340.

[Function File] ishghandle (h)

Return true if h is a graphics handle and false otherwise.

This function is equivalent to ishandle and is provided for compatibility with matlab.

See also: [ishandle], page 339.

[Function File] isaxes (h)

Return true if h is an axes graphics handle and false otherwise.

If h is a matrix then return a logical array which is true where the elements of h are axes graphics handles and false where they are not.

See also: [isaxes], page 339, [ishandle], page 339.

[Function File] isfigure (h)

Return true if h is a figure graphics handle and false otherwise. If h is a matrix then return a logical array which is true where the elements of h are figure graphics handles and false where they are not.

See also: [isaxes], page 339, [ishandle], page 339.

The function gcf returns an index to the current figure object, or creates one if none exists. Similarly, gca returns the current axes object, or creates one (and its parent figure object) if none exists.

[Function File] h = gcf ()

Return a handle to the current figure. The current figure is the default target for graphics output. If multiple figures exist, gcf returns the last created figure or the last figure that was clicked on with the mouse. If a current figure does not exist, create one and return its handle. The handle may then be used to examine or set properties of the figure. For example,

     fplot (@sin, [-10, 10]);
     fig = gcf ();
     set (fig, "numbertitle", "off", "name", "sin plot")

plots a sine wave, finds the handle of the current figure, and then renames the figure window to describe the contents. Note: To find the current figure without creating a new one if it does not exist, query the "CurrentFigure" property of the root graphics object.

     get (0, "currentfigure");

See also: [gca], page 340, [gco], page 341, [gcbf], page 363, [gcbo], page 363, [get], page 341, [set], page 342.

[Function File] h = gca ()

Return a handle to the current axis object. The current axis is the default target for graphics output. In the case of a figure with multiple axes, gca returns the last created axes or the last axes that was clicked on with the mouse. If no current axes object exists, create one and return its handle. The handle may then be used to examine or set properties of the axes. For example,

     ax = gca ();
     set (ax, "position", [0.5, 0.5, 0.5, 0.5]);

creates an empty axes object and then changes its location and size in the figure window. Note: To find the current axis without creating a new axes object if it does not exist, query the "CurrentAxes" property of a figure.

     get (gcf, "currentaxes");

See also: [gcf], page 340, [gco], page 341, [gcbf], page 363, [gcbo], page 363, [get], page 341, [set], page 342.

[Function File] h = gco () [Function File] h = gco (fig)

Return a handle to the current object of the current figure, or a handle to the current object of the figure with handle fig. The current object of a figure is the object that was last clicked on. It is stored in the "CurrentObject" property of the target figure. If the last mouse click did not occur on any child object of the figure, then the current object is the figure itself. If no mouse click occurred in the target figure, this function returns an empty matrix. Programming Note: The value returned by this function is not necessarily the same as the one returned by gcbo during callback execution. An executing callback can be interrupted by another callback and the current object may be changed.

See also: [gcbo], page 363, [gca], page 340, [gcf], page 340, [gcbf], page 363, [get], page 341, [set], page 342.

The get and set functions may be used to examine and set properties for graphics objects. For example,

get (0)
    ⇒ ans =
       {
         type = root
         currentfigure = [](0x0)
         children = [](0x0)
         visible = on
         ...
       }

returns a structure containing all the properties of the root figure. As with all functions in Octave, the structure is returned by value, so modifying it will not modify the internal root figure plot object. To do that, you must use the set function. Also, note that in this case, the currentfigure property is empty, which indicates that there is no current figure window.

The get function may also be used to find the value of a single property. For example,

get (gca (), "xlim")
    ⇒ [ 0 1 ]

returns the range of the x-axis for the current axes object in the current figure.

To set graphics object properties, use the set function. For example,

set (gca (), "xlim", [-10, 10]);

sets the range of the x-axis for the current axes object in the current figure to ‘[-10, 10]’. Additionally, calling set with a graphics object index as the only argument returns a structure containing the default values for all the properties for the given object type. For example,

set (gca ())

returns a structure containing the default property values for axes objects.

[Built-in Function] val = get (h) [Built-in Function] val = get (h, p)

Return the value of the named property p from the graphics handle h. If p is omitted, return the complete property list for h. If h is a vector, return a cell array including the property values or lists respectively.

See also: [set], page 342.

[Built-in Function] set (h, property, value, …) [Built-in Function] set (h, properties, values) [Built-in Function] set (h, pv)

Set named property values for the graphics handle (or vector of graphics handles) h. There are three ways how to give the property names and values:

• as a comma separated list of property, value pairs Here, each property is a string containing the property name, each value is a value of the appropriate type for the property.

• as a cell array of strings properties containing property names and a cell array values containing property values. In this case, the number of columns of values must match the number of elements in properties. The first column of values contains values for the first entry in properties, etc. The number of rows of values must be 1 or match the number of elements of h. In the first case, each handle in h will be assigned the same values. In the latter case, the first handle in h will be assigned the values from the first row of values and so on.

• as a structure array pv Here, the field names of pv represent the property names, and the field values give the property values. In contrast to the previous case, all elements of pv will be set in all handles in h independent of the dimensions of pv.

See also: [get], page 341.

[Function File] parent = ancestor (h, type)

[Function File] parent = ancestor (h, type, "toplevel")

Return the first ancestor of handle object h whose type matches type, where type is a character string. If type is a cell array of strings, return the first parent whose type matches any of the given type strings. If the handle object h itself is of type type, return h. If "toplevel" is given as a third argument, return the highest parent in the object hierarchy that matches the condition, instead of the first (nearest) one.

See also: [findobj], page 359, [findall], page 359, [allchild], page 342.

[Function File] h = allchild (handles)

Find all children, including hidden children, of a graphics object. This function is similar to get (h, "children"), but also returns hidden objects (HandleVisibility = "off"). If handles is a scalar, h will be a vector. Otherwise, h will be a cell matrix of the same size as handles and each cell will contain a vector of handles.

See also: [findall], page 359, [findobj], page 359, [get], page 341, [set], page 342.

[Function File] findfigs ()

Find all visible figures that are currently off the screen and move them onto the screen.

See also: [allchild], page 342, [figure], page 319, [get], page 341, [set], page 342.

Figures can be printed or saved in many graphics formats with print and saveas. Occasionally, however, it may be useful to save the original Octave handle graphic directly so that further modifications can be made such as modifying a title or legend.

This can be accomplished with the following functions by

fig_struct = hdl2struct (gcf);
save myplot.fig -struct fig_struct;
...
fig_struct = load ("myplot.fig");
struct2hdl (fig_struct);

[Function File] s = hdl2struct (h)

Return a structure, s, whose fields describe the properties of the object, and its children, associated with the handle, h.

The fields of the structure s are "type", "handle", "properties", "children", and
"special".

See also: [struct2hdl], page 343, [findobj], page 359.

[Function File] h = struct2hdl (s) [Function File] h = struct2hdl (s, p)

[Function File] h = struct2hdl (s, p, hilev)

Construct a graphics handle object h from the structure s.

The structure must contain the fields "handle", "type", "children", "properties",

and "special". If the handle of an existing figure or axes is specified, p, the new object will be created as a child of that object. If no parent handle is provided then a new figure and the necessary children will be constructed using the default values from the root figure. A third boolean argument hilev can be passed to specify whether the function should preserve listeners/callbacks, e.g., for legends or hggroups. The default is false.

See also: [hdl2struct], page 343, [findobj], page 359.

[Function File] hnew = copyobj (horig)
[Function File] hnew = copyobj (horig, hparent)

Construct a copy of the graphic object associated with handle horig and return a handle hnew to the new object. If a parent handle hparent (root, figure, axes, or hggroup) is specified, the copied object will be created as a child of hparent.

See also: [struct2hdl], page 343, [hdl2struct], page 343, [findobj], page 359.

15.3.3 Graphics Object Properties

In this Section the object properties are discussed in detail, starting with the root figure properties and continuing through the graphics object hierarchy.

15.3.3.1 Root Figure Properties

The root figure properties are:

__modified__
          — Values: "on", "off"
__myhandle__
beingdeleted
          — Values: "on", "off"
busyaction
buttondownfcn
callbackobject
children
clipping — Values: "on", "off"
createfcn
currentfigure
deletefcn
handlevisibility
          — Values: "on", "off"
hittest — Values: "on", "off"
interruptible
          — Values: "on", "off"
parent
screendepth
screenpixelsperinch
screensize
selected
selectionhighlight
screendepth
screenpixelsperinch
showhiddenhandles
          — Values: "on", "off"
tag
type
uicontextmenu
units
userdata
visible

15.3.3.2 Figure Properties

The figure properties are:

__graphics_toolkit__

— The graphics toolkit currently in use.

__enhanced__
__modified__
__myhandle__
__plot_stream__
alphamap
beingdeleted
          — Values: "on", "off"
busyaction
buttondownfcn

children Handle to children.

clipping — Values: "on", "off"
closerequestfcn

— Handle of function to call on close.

color

colormap An N-by-3 matrix containing the color map for the current axes.

paperorientation
createfcn
currentaxes

Handle to graphics object of current axes.

currentcharacter
currentobject
currentpoint

Holds the coordinates of the point over which the mouse pointer was when the mouse button was pressed. If a mouse callback function is defined, "currentpoint" holds the coordinates of the point over which the mouse pointer is when the function gets called.

deletefcn
dockcontrols
          — Values: "on", "off"
doublebuffer
          — Values: "on", "off"
filename
handlevisibility
          — Values: "on", "off"
hittest

integerhandle
interruptible
          — Values: "on", "off"
inverthardcopy
keypressfcn
          see "keypressfcn"
keyreleasefcn

With "keypressfcn", the keyboard callback functions. These callback functions get called when a key is pressed/released respectively. The functions are called with two input arguments. The first argument holds the handle of the calling figure. The second argument holds the event structure which has the following members:

          Character

The ASCII value of the key

Key lowercase value of the key

Modifier A cell array containing strings representing the modifiers pressed with the key. Possible values are "shift", "alt", and "control".

menubar
mincolormap
name

nextplot May be one of

          "new"
          "add"
          "replace"
          "replacechildren"
numbertitle
paperorientation

Indicates the orientation for printing. Either "landscape" or "portrait".

paperposition
paperpositionmode
papersize
papertype
paperunits
pointer
pointershapecdata
pointershapehotspot
position
renderer
renderermode
resize

resizefcn
selected
selectionhighlight
          — Values: "on", "off"
selectiontype
tag
toolbar
type
units
userdata

visible Either "on" or "off" to toggle display of the figure.

windowbuttondownfcn
          See "windowbuttonupfcn"
windowbuttonmotionfcn
          See "windowbuttonupfcn"
windowbuttonupfcn
          With "windowbuttondownfcn" and "windowbuttonmotionfcn", the mouse

callback functions. These callback functions get called when the mouse button is pressed, dragged, and released respectively. When these callback functions are called, the "currentpoint" property holds the current coordinates of the cursor.

windowscrollwheelfcn
windowstyle
wvisual
wvisualmode
xdisplay
xvisual
xvisualmode

15.3.3.3 Axes Properties

The axes properties are:

__modified__
__myhandle__
activepositionproperty
alim
alimmode
ambientlightcolor
beingdeleted

box Box surrounding axes. — Values: "on", "off"

busyaction
buttondownfcn
cameraposition
camerapositionmode
cameratarget
cameratargetmode
cameraupvector
cameraupvectormode
cameraviewangle
cameraviewanglemode
children

clim Two-element vector defining the limits for the c axis of an image. See pcolor property. Setting this property also forces the corresponding mode property to

          be set to "manual".
climmode Either "manual" or "auto".
clipping
color
colororder
createfcn
currentpoint

Holds the coordinates of the point over which the mouse pointer was when the mouse button was pressed. If a mouse callback function is defined, "currentpoint" holds the coordinates of the point over which the mouse pointer is when the function gets called.

dataaspectratio

A two-element vector specifying the relative height and width of the data displayed in the axes. Setting dataaspectratio to ‘1, 2]’ causes the length of one unit as displayed on the y-axis to be the same as the length of 2 units

          on the x-axis. Setting dataaspectratio also forces the dataaspectratiomode

property to be set to "manual".

dataaspectratiomode
          Either "manual" or "auto".
deletefcn
drawmode
fontangle
fontname
fontsize
fontunits
fontweight
gridlinestyle
handlevisibility
hittest

interpreter
interruptible
layer
linestyleorder
linewidth
minorgridlinestyle

nextplot May be one of

          "add"
          "replace"
          "replacechildren"
outerposition

A vector specifying the position of the plot, including titles, axes and legend. The four elements of the vector are the coordinates of the lower left corner and width and height of the plot, in units normalized to the width and height of the plot window. For example, [0.2, 0.3, 0.4, 0.5] sets the lower left corner of the axes at (0.2, 0.3) and the width and height to be 0.4 and 0.5 respectively. See also the position property.

parent
plotboxaspectratio
plotboxaspectratiomode

position A vector specifying the position of the plot, excluding titles, axes and legend. The four elements of the vector are the coordinates of the lower left corner and width and height of the plot, in units normalized to the width and height of the plot window. For example, [0.2, 0.3, 0.4, 0.5] sets the lower left corner of the axes at (0.2, 0.3) and the width and height to be 0.4 and 0.5 respectively. See also the outerposition property.

projection
selected
selectionhighlight
tag
tickdir
tickdirmode
ticklength
tightinset

title Index of text object for the axes title.

type
uicontextmenu
units
userdata

view A three element vector specifying the view point for three-dimensional plots.

visible Either "on" or "off" to toggle display of the axes.

x_normrendertransform
x_projectiontransform
x_rendertransform
x_viewporttransform
x_viewtransform
xaxislocation
          Either "top" or "bottom".
xcolor
xdir  Either "forward" or "reverse".

xgrid Either "on" or "off" to toggle display of grid lines. xlabel Indices to text objects for the axes labels. xlim Two-element vector defining the limits for the x-axis. Setting this property also

forces the corresponding mode property to be set to "manual".

xlimmode Either "manual" or "auto".
xminorgrid

Either "on" or "off" to toggle display of minor grid lines.

xminortick
xscale  Either "linear" or "log".

xtick Set position of tick marks. Setting this property also forces the corresponding

mode property to be set to "manual".

xticklabel

Setting this property also forces the corresponding mode property to be set to

          "manual".
xticklabelmode
          Either "manual" or "auto".
xtickmode
          Either "manual" or "auto".
yaxislocation
          Either "left" or "right"
ycolor
ydir  Either "forward" or "reverse".

ygrid Either "on" or "off" to toggle display of grid lines. ylabel Indices to text objects for the axes labels. ylim Two-element vectors defining the limits for the x, y, and z axes and the Setting

one of these properties also forces the corresponding mode property to be set

          to "manual".
ylimmode Either "manual" or "auto".
yminorgrid

Either "on" or "off" to toggle display of minor grid lines.

yminortick
yscale  Either "linear" or "log".

ytick Set position of tick marks. Setting this property also forces the corresponding mode property to be set to "manual".

yticklabel

Setting this property also forces the corresponding mode property to be set to

          "manual".
yticklabelmode
          Either "manual" or "auto".
ytickmode
          Either "manual" or "auto".
zcolor
zdir  Either "forward" or "reverse".

zgrid Either "on" or "off" to toggle display of grid lines.

zlabel Indices to text objects for the axes labels.

zlim Two-element vector defining the limits for z-axis. Setting this property also forces the corresponding mode property to be set to "manual".

zlimmode Either "manual" or "auto".
zminorgrid

Either "on" or "off" to toggle display of minor grid lines.

zminortick
zscale  Either "linear" or "log".

ztick Set position of tick marks. Setting this property also forces the corresponding mode property to be set to "manual".

zticklabel

Setting this property also forces the corresponding mode property to be set to

          "manual".
zticklabelmode
          Either "manual" or "auto".
ztickmode
          Either "manual" or "auto".

15.3.3.4 Line Properties

The line properties are:

__modified__
__myhandle__
beingdeleted
busyaction
buttondownfcn
children

clipping

color The RGB color of the line, or a color name. See Section 15.4.1 [Colors], page 361.

createfcn
deletefcn
displayname

The text of the legend entry corresponding to this line.

erasemode
handlevisibility
hittest
interpreter
interruptible

ldata The lower errorbar in the y direction to be plotted.

linestyle
linewidth

See Section 15.4.2 [Line Styles], page 361.

linewidth
marker
markeredgecolor
markerfacecolor
markersize

See Section 15.4.3 [Marker Styles], page 361.

parent
selected
selectionhighlight
tag
type

udata The upper errorbar in the y direction to be plotted.

uicontextmenu
userdata
visible

xdata The data to be plotted.

xdatasource

xldata The lower errorbar to be plotted.

xlim
xliminclude

xudata The upper errorbar to be plotted. ydata The data to be plotted.

ydatasource
ylim

yliminclude

zdata The data to be plotted.

zdatasource
zlim
zliminclude

15.3.3.5 Text Properties

The text properties are:

__modified__
__myhandle__
backgroundcolor
beingdeleted
busyaction
buttondownfcn
children
clipping

color The color of the text. See Section 15.4.1 [Colors], page 361.

createfcn
deletefcn
displayname

The text of the legend entry corresponding to this line.

edgecolor
editing
erasemode
fontangle

Flag whether the font is italic or normal. Valid values are "normal", "italic",

          and "oblique".

fontname The font used for the text.

fontsize The size of the font, in points to use.

fontunits
fontweight

Flag whether the font is bold, etc. Valid values are "normal", "bold", "demi",

          or "light".
handlevisibility
hittest
horizontalalignment
          May be "left", "center", or "right".
interpreter

Determines how the text is rendered. Valid values are "none", "tex", or

          "latex".

interruptible
linestyle
linewidth
margin
parent

position The coordinates of the text object.

rotation The angle of rotation for the displayed text, measured in degrees.

selected
selectionhighlight

string The character string contained by the text object.

tag
type
uicontextmenu
units  May be "normalized" or "graph".
userdata
verticalalignment
visible
xlim
xliminclude
ylim
yliminclude
zlim
zliminclude

15.3.3.6 Image Properties

The image properties are:

__modified__
__myhandle__
beingdeleted
busyaction
buttondownfcn

cdata The data for the image. Each pixel of the image corresponds to an element of cdata. The value of an element of cdata specifies the row-index into the colormap of the axes object containing the image. The color value found in the color map for the given index determines the color of the pixel.

cdatamapping
children
clim
climinclude
clipping

createfcn
deletefcn
handlevisibility
hittest
interruptible
parent
selected
selectionhighlight
tag
type
uicontextmenu
userdata
visible

xdata Two-element vector specifying the range of the x-coordinates for the image.

xlim
xliminclude

ydata Two-element vector specifying the range of the y-coordinates for the image.

ylim
yliminclude

15.3.3.7 Patch Properties

The patch properties are:

__modified__
__myhandle__
alim
aliminclude
alphadatamapping
ambientstrength
backfacelighting
beingdeleted
busyaction
buttondownfcn

cdata Data defining the patch object.

cdatamapping
children
clim
climinclude
clipping

createfcn
deletefcn
diffusestrength
displayname

The text of the legend entry corresponding to this line.

edgealpha
edgecolor

The color of the line defining the patch. See Section 15.4.1 [Colors], page 361.

edgelighting
erasemode
facealpha

A number in the range [0, 1] indicating the transparency of the patch.

facecolor

The fill color of the patch. See Section 15.4.1 [Colors], page 361.

facelighting
faces
facevertexalphadata
facevertexcdata
handlevisibility
hittest
interpreter
interruptible
linestyle

See Section 15.4.2 [Line Styles], page 361.

linewidth

See Section 15.4.2 [Line Styles], page 361.

marker See Section 15.4.3 [Marker Styles], page 361.

markeredgecolor

See Section 15.4.3 [Marker Styles], page 361.

markerfacecolor

See Section 15.4.3 [Marker Styles], page 361.

markersize

See Section 15.4.3 [Marker Styles], page 361.

normalmode
parent
selected
selectionhighlight
specularcolorreflectance
specularexponent
specularstrength
tag

type
uicontextmenu
userdata
vertexnormals
vertices
visible

xdata Data defining the patch object.

xlim
xliminclude

ydata Data defining the patch object.

ylim
yliminclude

zdata Data defining the patch object.

zlim
zliminclude

15.3.3.8 Surface Properties

The surface properties are:

__modified__
__myhandle__
alim
aliminclude
alphadata
alphadatamapping
ambientstrength
backfacelighting
beingdeleted
busyaction
buttondownfcn
cdata
cdatamapping
cdatasource
children
clim
climinclude
clipping
createfcn
deletefcn
diffusestrength
displayname

The text of the legend entry corresponding to this surface.

edgealpha
edgecolor
edgelighting
erasemode
facealpha
facecolor
facelighting
handlevisibility
hittest
interpreter
interruptible
linestyle
linewidth
marker
markeredgecolor
markerfacecolor
markersize
meshstyle
normalmode
parent
selected
selectionhighlight
specularcolorreflectance
specularexponent
specularstrength
tag
type
uicontextmenu
userdata
vertexnormals
visible

xdata The data determining the surface. The xdata and ydata elements are vectors and zdata must be a matrix.

xdatasource
xlim
xliminclude

ydata The data determining the surface. The xdata and ydata elements are vectors and zdata must be a matrix.

ydatasource
ylim
yliminclude

zdata The data determining the surface. The xdata and ydata elements are vectors and zdata must be a matrix.

zdatasource
zlim
zliminclude

15.3.4 Searching Properties

[Function File] h = findobj ()

[Function File] h = findobj (prop_name, prop_value, . . .)
[Function File] h = findobj (prop_name, prop_value, "-logical_op",
   prop_name, prop_value)

[Function File] h = findobj ("-property", prop_name)

[Function File] h = findobj ("-regexp", prop_name, pattern)

[Function File] h = findobj (hlist, …) [Function File] h = findobj (hlist, "flat", …) [Function File] h = findobj (hlist, "-depth", d, …)

Find graphics object with specified property values. The simplest form is

     findobj (prop_name, prop_value)

which returns the handles of all objects which have a property named prop name that has the value prop value. If multiple property/value pairs are specified then only objects meeting all of the conditions are returned. The search can be limited to a particular set of objects and their descendants, by passing a handle or set of handles hlist as the first argument. The depth of the object hierarchy to search can be limited with the "-depth" argument. An example of searching only three generations of children is:

     findobj (hlist, "-depth", 3, prop_name, prop_value)

Specifying a depth d of 0, limits the search to the set of objects passed in hlist. A depth d of 0 is equivalent to the "flat" argument. A specified logical operator may be applied to the pairs of prop name and prop value. The supported logical operators are: "-and", "-or", "-xor", "-not". Objects may also be matched by comparing a regular expression to the property values, where property values that match regexp (prop_value, pattern) are returned. Finally, objects may be matched by property name only by using the "-property" option. Implementation Note: The search only includes objects with visible handles (Handle-Visibility = "on"). See [findall], page 359, to search for all objects including hidden ones.

See also: [findall], page 359, [allchild], page 342, [get], page 341, [set], page 342.

[Function File] h = findall ()

[Function File] h = findall (prop_name, prop_value, . . .)
[Function File] h = findall (prop_name, prop_value, "-logical_op",
   prop_name, prop_value)

[Function File] h = findall ("-property", prop_name)

[Function File] h = findall ("-regexp", prop_name, pattern)

[Function File] h = findall (hlist, …) [Function File] h = findall (hlist, "flat", …) [Function File] h = findall (hlist, "-depth", d, …)

Find graphics object, including hidden ones, with specified property values.

The return value h is a list of handles to the found graphic objects.

findall performs the same search as findobj, but it includes hidden objects (HandleVisibility = "off"). For full documentation, see [findobj], page 359.

See also: [findobj], page 359, [allchild], page 342, [get], page 341, [set], page 342.

15.3.5 Managing Default Properties

Object properties have two classes of default values, factory defaults (the initial values) and user-defined defaults, which may override the factory defaults.

Although default values may be set for any object, they are set in parent objects and apply to child objects, of the specified object type. For example, setting the default color property of line objects to "green", for the root object, will result in all line objects inheriting the color "green" as the default value.

     set (0, "defaultlinecolor", "green");

sets the default line color for all objects. The rule for constructing the property name to set a default value is

     default + object-type + property-name

This rule can lead to some strange looking names, for example defaultlinelinewidth" specifies the default linewidth property for line objects.

The example above used the root figure object, 0, so the default property value will apply to all line objects. However, default values are hierarchical, so defaults set in a figure objects override those set in the root figure object. Likewise, defaults set in axes objects override those set in figure or root figure objects. For example,

     subplot (2, 1, 1);
     set (0, "defaultlinecolor", "red");
     set (1, "defaultlinecolor", "green");
     set (gca (), "defaultlinecolor", "blue");
     line (1:10, rand (1, 10));
     subplot (2, 1, 2);
     line (1:10, rand (1, 10));
     figure (2)
     line (1:10, rand (1, 10));

produces two figures. The line in first subplot window of the first figure is blue because it inherits its color from its parent axes object. The line in the second subplot window of the first figure is green because it inherits its color from its parent figure object. The line in the second figure window is red because it inherits its color from the global root figure parent object.

To remove a user-defined default setting, set the default property to the value "remove". For example,

     set (gca (), "defaultlinecolor", "remove");

removes the user-defined default line color setting from the current axes object. To quickly remove all user-defined defaults use the reset function.

[Built-in Function] reset (h, property)

Remove any defaults set for the handle h. The default figure properties of

     "position", "units", "windowstyle" and "paperunits" and the default axes

properties of "position" and "units" are not reset.

See also: [cla], page 322, [clf], page 322.

Getting the "default" property of an object returns a list of user-defined defaults set for the object. For example,

     get (gca (), "default");

returns a list of user-defined default values for the current axes object.

Factory default values are stored in the root figure object. The command

     get (0, "factory");

returns a list of factory defaults.

15.4 Advanced Plotting

15.4.1 Colors

Colors may be specified as RGB triplets with values ranging from zero to one, or by name.

Recognized color names include "blue", "black", "cyan", "green", "magenta", "red",
"white", and "yellow".

15.4.2 Line Styles

Line styles are specified by the following properties:

linestyle

May be one of

"-" Solid line. [default]

"--" Dashed line.

":" Dotted line.

"-." A dash-dot line.

"none" No line. Points will still be marked using the current Marker Style.

linewidth

A number specifying the width of the line. The default is 1. A value of 2 is twice as wide as the default, etc.

15.4.3 Marker Styles

Marker styles are specified by the following properties:

marker A character indicating a plot marker to be place at each data point, or "none", meaning no markers should be displayed.

markeredgecolor

The color of the edge around the marker, or "auto", meaning that the edge color is the same as the face color. See Section 15.4.1 [Colors], page 361.

markerfacecolor

The color of the marker, or "none" to indicate that the marker should not be filled. See Section 15.4.1 [Colors], page 361.

markersize

A number specifying the size of the marker. The default is 1. A value of 2 is twice as large as the default, etc.

The colstyle function will parse a plot-style specification and will return the color, line, and marker values that would result.

[Function File] [style, color, marker, msg] = colstyle (linespec)

Parse linespec and return the line style, color, and markers given. In the case of an error, the string msg will return the text of the error.

15.4.4 Callbacks

Callback functions can be associated with graphics objects and triggered after certain events occur. The basic structure of all callback function is

     function mycallback (src, data)
     ...
     endfunction

where src gives a handle to the source of the callback, and code gives some event specific data. This can then be associated with an object either at the objects creation or later with the set function. For example,

     plot (x, "DeleteFcn", @(s, e) disp ("Window Deleted"))

where at the moment that the plot is deleted, the message "Window Deleted" will be displayed.

Additional user arguments can be passed to callback functions, and will be passed after the 2 default arguments. For example:

     plot (x, "DeleteFcn", {@mycallback, "1"})
     ...
     function mycallback (src, data, a1)
       fprintf ("Closing plot %d\n", a1);
     endfunction

The basic callback functions that are available for all graphics objects are

• CreateFcn This is the callback that is called at the moment of the objects creation. It is not called if the object is altered in any way, and so it only makes sense to define this callback in the function call that defines the object. Callbacks that are added to CreateFcn later with the set function will never be executed.

• DeleteFcn This is the callback that is called at the moment an object is deleted.

• ButtonDownFcn This is the callback that is called if a mouse button is pressed while the pointer is over this object. Note, that the gnuplot interface does not respect this callback.

The object and figure that the event occurred in that resulted in the callback being called can be found with the gcbo and gcbf functions.

[Function File] h = gcbo () [Function File] [h, fig] = gcbo ()

Return a handle to the object whose callback is currently executing. If no callback is executing, this function returns the empty matrix. This handle is obtained from the root object property "CallbackObject". When called with a second output argument, return the handle of the figure containing the object whose callback is currently executing. If no callback is executing the second output is also set to the empty matrix.

See also: [gcbf], page 363, [gco], page 341, [gca], page 340, [gcf], page 340, [get], page 341, [set], page 342.

[Function File] fig = gcbf ()

Return a handle to the figure containing the object whose callback is currently executing. If no callback is executing, this function returns the empty matrix. The handle returned by this function is the same as the second output argument of gcbo.

See also: [gcbo], page 363, [gcf], page 340, [gco], page 341, [gca], page 340, [get], page 341, [set], page 342.

Callbacks can equally be added to properties with the addlistener function described below.

15.4.5 Application-defined Data

Octave has a provision for attaching application-defined data to a graphics handle. The data can be anything which is meaningful to the application, and will be completely ignored by Octave.

[Function File] setappdata (h, name, value)

Set the named application data to value for the object(s) with handle(s) h. If the application data with the specified name does not exist, it is created.

See also: [getappdata], page 363, [guidata], page 693, [get], page 341, [set], page 342, [getpref], page 695, [setpref], page 695.

[Function File] value = getappdata (h, name)
[Function File] appdata = getappdata (h)

Return the value for named application data for the object(s) with handle(s) h. getappdata(h) returns a structure, appdata, whose fields correspond to the appdata properties.

See also: [setappdata], page 363, [guidata], page 693, [get], page 341, [set], page 342, [getpref], page 695, [setpref], page 695.

[Function File] rmappdata (h, name)

Delete the named application data for the object(s) with handle(s) h.

[Function File] V = isappdata (h, name)

Return true if the named application data, name, exists for the object with handle h.

See also: [getappdata], page 363, [setappdata], page 363, [rmappdata], page 363.

15.4.6 Object Groups

A number of Octave high level plot functions return groups of other graphics objects or they return graphics objects that have their properties linked in such a way that changes to one of the properties results in changes in the others. A graphic object that groups other objects is an hggroup

[Function File] hggroup () [Function File] hggroup (hax) [Function File] hggroup (…, property, value, …) [Function File] h = hggroup (…)

Create handle graphics group object with axes parent hax. If no parent is specified, the group is created in the current axes. Multiple property/value pairs may be specified for the hggroup, but they must appear in pairs. The optional return value h is a graphics handle to the created hggroup object. Programming Note: An hggroup is a way to group base graphics objects such as line objects or patch objects into a single unit which can react appropriately. For example, the individual lines of a contour plot are collected into a single hggroup so that they can be made visible/invisible with a single command, set (hg_handle, "visible",

"off").

See also: [addproperty], page 364, [addlistener], page 365.

For example a simple use of a hggroup might be

x = 0:0.1:10;
hg = hggroup ();
plot (x, sin (x), "color", [1, 0, 0], "parent", hg);
hold on
plot (x, cos (x), "color", [0, 1, 0], "parent", hg);
set (hg, "visible", "off");

which groups the two plots into a single object and controls their visibility directly. The default properties of an hggroup are the same as the set of common properties for the other graphics objects. Additional properties can be added with the addproperty function.

[Built-in Function] addproperty (name, h, type) [Built-in Function] addproperty (name, h, type, arg, …)

Create a new property named name in graphics object h. type determines the type of the property to create. args usually contains the default value of the property, but additional arguments might be given, depending on the type of the property. The supported property types are:

string A string property. arg contains the default string value. any An un-typed property. This kind of property can hold any octave value.

args contains the default value.

radio A string property with a limited set of accepted values. The first argument must be a string with all accepted values separated by a vertical bar (’|’). The default value can be marked by enclosing it with a ’{’ ’}’ pair. The default value may also be given as an optional second string argument.

boolean A boolean property. This property type is equivalent to a radio property with "on|off" as accepted values. arg contains the default property value.

double A scalar double property. arg contains the default value.

handle A handle property. This kind of property holds the handle of a graphics object. arg contains the default handle value. When no default value is given, the property is initialized to the empty matrix.

data A data (matrix) property. arg contains the default data value. When no default value is given, the data is initialized to the empty matrix.

color A color property. arg contains the default color value. When no default color is given, the property is set to black. An optional second string argument may be given to specify an additional set of accepted string values (like a radio property).

type may also be the concatenation of a core object type and a valid property name for that object type. The property created then has the same characteristics as the referenced property (type, possible values, hidden state…). This allows to clone an existing property into the graphics object h. Examples:

     addproperty ("my_property", gcf, "string", "a string value");
     addproperty ("my_radio", gcf, "radio", "val_1|val_2|{val_3}");
     addproperty ("my_style", gcf, "linelinestyle", "--");

See also: [addlistener], page 365, [hggroup], page 364.

Once a property in added to an hggroup, it is not linked to any other property of either the children of the group, or any other graphics object. Add so to control the way in which this newly added property is used, the addlistener function is used to define a callback function that is executed when the property is altered.

[Built-in Function] addlistener (h, prop, fcn)

Register fcn as listener for the property prop of the graphics object h. Property listeners are executed (in order of registration) when the property is set. The new value is already available when the listeners are executed. prop must be a string naming a valid property in h. fcn can be a function handle, a string or a cell array whose first element is a function handle. If fcn is a function handle, the corresponding function should accept at least 2 arguments, that will be set to the object handle and the empty matrix respectively. If fcn is a string, it must be any valid octave expression. If fcn is a cell array, the first element must be a function handle with the same signature as described above. The next elements of the cell array are passed as additional arguments to the function. Example:

     function my_listener (h, dummy, p1)
       fprintf ("my_listener called with p1=%s\n", p1);
     endfunction
     addlistener (gcf, "position", {@my_listener, "my string"})

See also: [addproperty], page 364, [hggroup], page 364.

[Built-in Function] dellistener (h, prop, fcn)

Remove the registration of fcn as a listener for the property prop of the graphics object h. The function fcn must be the same variable (not just the same value), as was passed to the original call to addlistener. If fcn is not defined then all listener functions of prop are removed. Example:

     function my_listener (h, dummy, p1)
       fprintf ("my_listener called with p1=%s\n", p1);
     endfunction
     c = {@my_listener, "my string"};
     addlistener (gcf, "position", c);
     dellistener (gcf, "position", c);

An example of the use of these two functions might be

x = 0:0.1:10;
hg = hggroup ();
h = plot (x, sin (x), "color", [1, 0, 0], "parent", hg);
addproperty ("linestyle", hg, "linelinestyle", get (h, "linestyle"));
addlistener (hg, "linestyle", @update_props);
hold on
plot (x, cos (x), "color", [0, 1, 0], "parent", hg);
function update_props (h, d)
  set (get (h, "children"), "linestyle", get (h, "linestyle"));
endfunction

that adds a linestyle property to the hggroup and propagating any changes its value to the children of the group. The linkprop function can be used to simplify the above to be

x = 0:0.1:10;
hg = hggroup ();
h1 = plot (x, sin (x), "color", [1, 0, 0], "parent", hg);
addproperty ("linestyle", hg, "linelinestyle", get (h, "linestyle"));
hold on
h2 = plot (x, cos (x), "color", [0, 1, 0], "parent", hg);
hlink = linkprop ([hg, h1, h2], "color");

[Function File] hlink = linkprop (h, prop)
[Function File] hlink = linkprop (h, {prop1, prop2, . . .})

Link graphics object properties, such that a change in one is propagated to the others. prop can be a string for a single property, or a cell array of strings for multiple properties. h is an array of graphics handles which will have their properties linked. An example of the use of linkprop is

          x = 0:0.1:10;
          subplot (1,2,1);
          h1 = plot (x, sin (x));
          subplot (1,2,2);
          h2 = plot (x, cos (x));
          hlink = linkprop ([h1, h2], {"color","linestyle"});
          set (h1, "color", "green");
          set (h2, "linestyle", "--");

These capabilities are used in a number of basic graphics objects. The hggroup objects created by the functions of Octave contain one or more graphics object and are used to:

• group together multiple graphics objects,

• create linked properties between different graphics objects, and

• to hide the nominal user data, from the actual data of the objects.

For example the stem function creates a stem series where each hggroup of the stem series contains two line objects representing the body and head of the stem. The ydata property of the hggroup of the stem series represents the head of the stem, whereas the body of the stem is between the baseline and this value. For example

     h = stem (1:4)
     get (h, "xdata")
     ⇒ [  1   2   3   4]'
     get (get (h, "children")(1), "xdata")
     ⇒ [  1   1 NaN   2   2 NaN   3   3 NaN   4   4 NaN]'

shows the difference between the xdata of the hggroup of a stem series object and the underlying line.

The basic properties of such group objects is that they consist of one or more linked hggroup, and that changes in certain properties of these groups are propagated to other members of the group. Whereas, certain properties of the members of the group only apply to the current member.

In addition the members of the group can also be linked to other graphics objects through callback functions. For example the baseline of the bar or stem functions is a line object, whose length and position are automatically adjusted, based on changes to the corresponding hggroup elements.

15.4.6.1 Data Sources in Object Groups

All of the group objects contain data source parameters. There are string parameters that contain an expression that is evaluated to update the relevant data property of the group when the refreshdata function is called.

[Function File] refreshdata () [Function File] refreshdata (h)

[Function File] refreshdata (h, workspace)

Evaluate any ‘datasource’ properties of the current figure and update the plot if the corresponding data has changed.

If the first argument h is a list of graphic handles, then operate on these objects rather than the current figure returned by gcf.

The optional second argument workspace can take the following values:

"base" Evaluate the datasource properties in the base workspace. (default).

"caller" Evaluate the datasource properties in the workspace of the function that

               called refreshdata.

An example of the use of refreshdata is:

          x = 0:0.1:10;
          y = sin (x);
          plot (x, y, "ydatasource", "y");
          for i = 1 : 100
            pause (0.1);
            y = sin (x + 0.1*i);
            refreshdata ();
          endfor

15.4.6.2 Area Series

Area series objects are created by the area function. Each of the hggroup elements contains a single patch object. The properties of the area series are

basevalue

The value where the base of the area plot is drawn.

linewidth
linestyle

The line width and style of the edge of the patch objects making up the areas. See Section 15.4.2 [Line Styles], page 361.

edgecolor
facecolor

The line and fill color of the patch objects making up the areas. See Section 15.4.1 [Colors], page 361.

xdata

ydata The x and y coordinates of the original columns of the data passed to area prior to the cumulative summation used in the area function.

xdatasource
ydatasource

Data source variables.

15.4.6.3 Bar Series

Bar series objects are created by the bar or barh functions. Each hggroup element contains a single patch object. The properties of the bar series are

showbaseline
baseline
basevalue

The property showbaseline flags whether the baseline of the bar series is displayed (default is "on"). The handle of the graphics object representing the baseline is given by the baseline property and the y-value of the baseline by the basevalue property.

Changes to any of these properties are propagated to the other members of the bar series and to the baseline itself. Equally, changes in the properties of the base line itself are propagated to the members of the corresponding bar series.

barwidth
barlayout
horizontal

The property barwidth is the width of the bar corresponding to the width variable passed to bar or barh. Whether the bar series is "grouped" or "stacked" is determined by the barlayout property and whether the bars are horizontal or vertical by the horizontal property.

Changes to any of these property are propagated to the other members of the bar series.

linewidth
linestyle

The line width and style of the edge of the patch objects making up the bars. See Section 15.4.2 [Line Styles], page 361.

edgecolor
facecolor

The line and fill color of the patch objects making up the bars. See Section 15.4.1 [Colors], page 361.

xdata The nominal x positions of the bars. Changes in this property and propagated to the other members of the bar series.

ydata The y value of the bars in the hggroup.

xdatasource
ydatasource

Data source variables.

15.4.6.4 Contour Groups

Contour group objects are created by the contour, contourf and contour3 functions. The are equally one of the handles returned by the surfc and meshc functions. The properties of the contour group are

contourmatrix

A read only property that contains the data return by contourc used to create the contours of the plot.

fill A radio property that can have the values "on" or "off" that flags whether the contours to plot are to be filled.

zlevelmode

zlevel The radio property zlevelmode can have the values "none", "auto", or "manual". When its value is "none" there is no z component to the plotted contours. When its value is "auto" the z value of the plotted contours is at the same value as the contour itself. If the value is "manual", then the z value at which to plot the contour is determined by the zlevel property.

levellistmode
levellist
levelstepmode
levelstep

If levellistmode is "manual", then the levels at which to plot the contours is determined by levellist. If levellistmode is set to "auto", then the distance between contours is determined by levelstep. If both levellistmode and levelstepmode are set to "auto", then there are assumed to be 10 equal spaced contours.

textlistmode
textlist
textstepmode

textstep If textlistmode is "manual", then the labeled contours is determined by textlist. If textlistmode is set to "auto", then the distance between labeled

contours is determined by textstep. If both textlistmode and textstepmode

are set to "auto", then there are assumed to be 10 equal spaced labeled contours.

showtext Flag whether the contour labels are shown or not.

labelspacing

The distance between labels on a single contour in points.

linewidth
linestyle
linecolor

The properties of the contour lines. The properties linewidth and linestyle are similar to the corresponding properties for lines. The property linecolor is a color property (see Section 15.4.1 [Colors], page 361), that can also have the values of "none" or "auto". If linecolor is "none", then no contour line is drawn. If linecolor is "auto" then the line color is determined by the colormap.

xdata
ydata

zdata The original x, y, and z data of the contour lines.

xdatasource
ydatasource
zdatasource

Data source variables.

15.4.6.5 Error Bar Series

Error bar series are created by the errorbar function. Each hggroup element contains two line objects representing the data and the errorbars separately. The properties of the error bar series are

color The RGB color or color name of the line objects of the error bars. See Section 15.4.1 [Colors], page 361.

linewidth
linestyle

The line width and style of the line objects of the error bars. See Section 15.4.2 [Line Styles], page 361.

marker
markeredgecolor
markerfacecolor
markersize

The line and fill color of the markers on the error bars. See Section 15.4.1 [Colors], page 361.

xdata
ydata
ldata
udata
xldata

xudata The original x, y, l, u, xl, xu data of the error bars.

xdatasource
ydatasource
ldatasource
udatasource
xldatasource
xudatasource

Data source variables.

15.4.6.6 Line Series

Line series objects are created by the plot and plot3 functions and are of the type line. The properties of the line series with the ability to add data sources.

color The RGB color or color name of the line objects. See Section 15.4.1 [Colors], page 361.

linewidth
linestyle

The line width and style of the line objects. See Section 15.4.2 [Line Styles], page 361.

marker
markeredgecolor
markerfacecolor
markersize

The line and fill color of the markers. See Section 15.4.1 [Colors], page 361.

xdata
ydata

zdata The original x, y and z data.

xdatasource
ydatasource
zdatasource

Data source variables.

15.4.6.7 Quiver Group

Quiver series objects are created by the quiver or quiver3 functions. Each hggroup element of the series contains three line objects as children representing the body and head of the arrow, together with a marker as the point of origin of the arrows. The properties of the quiver series are

autoscale
autoscalefactor

Flag whether the length of the arrows is scaled or defined directly from the u, v and w data. If the arrow length is flagged as being scaled by the autoscale property, then the length of the autoscaled arrow is controlled by

          the autoscalefactor.
maxheadsize

This property controls the size of the head of the arrows in the quiver series. The default value is 0.2.

showarrowhead

Flag whether the arrow heads are displayed in the quiver plot.

color The RGB color or color name of the line objects of the quiver. See Section 15.4.1 [Colors], page 361.

linewidth
linestyle

The line width and style of the line objects of the quiver. See Section 15.4.2 [Line Styles], page 361.

marker
markerfacecolor
markersize

The line and fill color of the marker objects at the original of the arrows. See Section 15.4.1 [Colors], page 361.

xdata
ydata

zdata The origins of the values of the vector field.

udata
vdata

wdata The values of the vector field to plot.

xdatasource
ydatasource
zdatasource
udatasource
vdatasource
wdatasource

Data source variables.

15.4.6.8 Scatter Group

Scatter series objects are created by the scatter or scatter3 functions. A single hggroup element contains as many children as there are points in the scatter plot, with each child representing one of the points. The properties of the stem series are

linewidth

The line width of the line objects of the points. See Section 15.4.2 [Line Styles], page 361.

marker
markeredgecolor
markerfacecolor

The line and fill color of the markers of the points. See Section 15.4.1 [Colors], page 361.

xdata
ydata

zdata The original x, y and z data of the stems.

cdata The color data for the points of the plot. Each point can have a separate color, or a unique color can be specified.

sizedata The size data for the points of the plot. Each point can its own size or a unique size can be specified.

xdatasource
ydatasource
zdatasource
cdatasource
sizedatasource

Data source variables.

15.4.6.9 Stair Group

Stair series objects are created by the stair function. Each hggroup element of the series contains a single line object as a child representing the stair. The properties of the stair series are

color The RGB color or color name of the line objects of the stairs. See Section 15.4.1 [Colors], page 361.

linewidth
linestyle

The line width and style of the line objects of the stairs. See Section 15.4.2 [Line Styles], page 361.

marker
markeredgecolor
markerfacecolor
markersize

The line and fill color of the markers on the stairs. See Section 15.4.1 [Colors], page 361.

xdata

ydata The original x and y data of the stairs.

xdatasource
ydatasource

Data source variables.

15.4.6.10 Stem Series

Stem series objects are created by the stem or stem3 functions. Each hggroup element contains a single line object as a child representing the stems. The properties of the stem series are

showbaseline
baseline
basevalue

The property showbaseline flags whether the baseline of the stem series is displayed (default is "on"). The handle of the graphics object representing the baseline is given by the baseline property and the y-value (or z-value for stem3) of the baseline by the basevalue property.

Changes to any of these property are propagated to the other members of the stem series and to the baseline itself. Equally changes in the properties of the base line itself are propagated to the members of the corresponding stem series.

color The RGB color or color name of the line objects of the stems. See Section 15.4.1 [Colors], page 361.

linewidth
linestyle

The line width and style of the line objects of the stems. See Section 15.4.2 [Line Styles], page 361.

marker
markeredgecolor
markerfacecolor
markersize

The line and fill color of the markers on the stems. See Section 15.4.1 [Colors], page 361.

xdata
ydata

zdata The original x, y and z data of the stems.

xdatasource
ydatasource
zdatasource

Data source variables.

15.4.6.11 Surface Group

Surface group objects are created by the surf or mesh functions, but are equally one of the handles returned by the surfc or meshc functions. The surface group is of the type

surface.

The properties of the surface group are

edgecolor
facecolor

The RGB color or color name of the edges or faces of the surface. See Section 15.4.1 [Colors], page 361.

linewidth
linestyle

The line width and style of the lines on the surface. See Section 15.4.2 [Line Styles], page 361.

marker
markeredgecolor
markerfacecolor
markersize

The line and fill color of the markers on the surface. See Section 15.4.1 [Colors], page 361.

xdata
ydata
zdata

cdata The original x, y, z and c data.

xdatasource
ydatasource
zdatasource
cdatasource

Data source variables.

15.4.7 Graphics Toolkits

[Function File] name = graphics_toolkit ()
[Function File] name = graphics_toolkit (hlist)
[Function File] graphics_toolkit (name)
[Function File] graphics_toolkit (hlist, name)

Query or set the default graphics toolkit which is assigned to new figures.

With no inputs, return the current default graphics toolkit. If the input is a list of figure graphic handles, hlist, then return the name of the graphics toolkit in use for each figure. When called with a single input name set the default graphics toolkit to name. If the toolkit is not already loaded, it is initialized by calling the function __init_name__. If the first input is a list of figure handles, hlist, then the graphics toolkit is set to name for these figures only.

See also: [available graphics toolkits], page 376.

[Built-in Function] available_graphics_toolkits ()

Return a cell array of registered graphics toolkits.

See also: [graphics toolkit], page 375, [register graphics toolkit], page 376.

[Built-in Function] loaded_graphics_toolkits ()

Return a cell array of the currently loaded graphics toolkits.

See also: [available graphics toolkits], page 376.

[Built-in Function] register_graphics_toolkit (toolkit)

List toolkit as an available graphics toolkit.

See also: [available graphics toolkits], page 376.

15.4.7.1 Customizing Toolkit Behavior

The specific behavior of the backend toolkit may be modified using the following utility functions. Note: Not all functions apply to every graphics toolkit.

[Loadable Function] [prog, args] = gnuplot_binary ()
[Loadable Function] [old_prog, old_args] = gnuplot_binary (new_prog,

arg1, …)

Query or set the name of the program invoked by the plot command when the graphics toolkit is set to "gnuplot". Additional arguments to pass to the external plotting program may also be given. The default value is "gnuplot" with no additional arguments. See Appendix G [Installation], page 833.

See also: [graphics toolkit], page 375.

[Built-in Function] mode = gui_mode () [Built-in Function] gui_mode (mode)

Query or set the GUI mode for the current graphics toolkit. The mode argument can be one of the following strings:

"2d" Allows panning and zooming of current axes.

"3d" Allows rotating and zooming of current axes.

"none" Mouse inputs have no effect.

This function is currently implemented only for the FLTK graphics toolkit.

See also: [mouse wheel zoom], page 377.

[Loadable Function] val = mouse_wheel_zoom ()
[Loadable Function] old_val = mouse_wheel_zoom (new_val)

[Loadable Function] mouse_wheel_zoom (new_val, "local")

Query or set the mouse wheel zoom factor. The zoom factor is a number in the range (0,1) which is the percentage of the current axis limits that will be used when zooming. For example, if the current x-axis limits are [0, 50] and mouse_wheel_zoom is 0.4 (40%), then a zoom operation will change the limits by 20. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function. This function is currently implemented only for the FLTK graphics toolkit.

See also: [gui mode], page 376.

16 Matrix Manipulation

There are a number of functions available for checking to see if the elements of a matrix meet some condition, and for rearranging the elements of a matrix. For example, Octave can easily tell you if all the elements of a matrix are finite, or are less than some specified value. Octave can also rotate the elements, extract the upper- or lower-triangular parts, or sort the columns of a matrix.

16.1 Finding Elements and Checking Conditions

The functions any and all are useful for determining whether any or all of the elements of a matrix satisfy some condition. The find function is also useful in determining which elements of a matrix meet a specified condition.

[Built-in Function] any (x) [Built-in Function] any (x, dim)

For a vector argument, return true (logical 1) if any element of the vector is non-zero. For a matrix argument, return a row vector of logical ones and zeros with each element indicating whether any of the elements of the corresponding column of the matrix are non-zero. For example:

          any (eye (2, 4)) ⇒ [ 1, 1, 0, 0 ]

If the optional argument dim is supplied, work along dimension dim. For example:

          any (eye (2, 4), 2) ⇒ [ 1; 1 ]

See also: [all], page 379.

[Built-in Function] all (x) [Built-in Function] all (x, dim)

For a vector argument, return true (logical 1) if all elements of the vector are non-zero. For a matrix argument, return a row vector of logical ones and zeros with each element indicating whether all of the elements of the corresponding column of the matrix are non-zero. For example:

          all ([2, 3; 1, 0])) ⇒ [ 1, 0 ]

If the optional argument dim is supplied, work along dimension dim.

See also: [any], page 379.

Since the comparison operators (see Section 8.4 [Comparison Ops], page 140) return matrices of ones and zeros, it is easy to test a matrix for many things, not just whether the elements are nonzero. For example,

     all (all (rand (5) < 0.9))
          ⇒ 0

tests a random 5 by 5 matrix to see if all of its elements are less than 0.9.

Note that in conditional contexts (like the test clause of if and while statements) Octave treats the test as if you had typed all (all (condition)).

[Mapping Function] z = xor (x, y)

Return the exclusive or of the entries of x and y. For boolean expressions x and y, xor (x, y) is true if and only if one of x or y is true. Otherwise, for x and y both true or both false, xor returns false. The truth table for the xor operation is

x y z
0 0 0
1 0 1
0 1 1
1 1 0

See also: [and], page 142, [or], page 143, [not], page 143.

[Built-in Function] diff (x) [Built-in Function] diff (x, k) [Built-in Function] diff (x, k, dim)

If x is a vector of length n, diff (x) is the vector of first differences x_{2} − x_{1}, …, x_{n} − x_{n−1}. If x is a matrix, diff (x) is the matrix of column differences along the first nonsingleton dimension. The second argument is optional. If supplied, diff (x, k), where k is a non-negative integer, returns the k-th differences. It is possible that k is larger than the first nonsingleton dimension of the matrix. In this case, diff continues to take the differences along the next non-singleton dimension. The dimension along which to take the difference can be explicitly stated with the optional variable dim. In this case the k-th order differences are calculated along this dimension. In the case where k exceeds size (x, dim) an empty matrix is returned.

See also: [sort], page 387, [merge], page 144.

[Mapping Function] isinf (x)

Return a logical array which is true where the elements of x are are infinite and false where they are not. For example:

     isinf ([13, Inf, NA, NaN])
           ⇒ [ 0, 1, 0, 0 ]

See also: [isfinite], page 380, [isnan], page 380, [isna], page 42.

[Mapping Function] isnan (x)

Return a logical array which is true where the elements of x are NaN values and false where they are not. NA values are also considered NaN values. For example:

     isnan ([13, Inf, NA, NaN])
           ⇒ [ 0, 0, 1, 1 ]

See also: [isna], page 42, [isinf], page 380, [isfinite], page 380.

[Mapping Function] isfinite (x) [Mapping Function] finite (x)

Return a logical array which is true where the elements of x are finite values and false where they are not. For example:

     finite ([13, Inf, NA, NaN])
          ⇒ [ 1, 0, 0, 0 ]

See also: [isinf], page 380, [isnan], page 380, [isna], page 42.

[Function File] [err, y1, ...] = common_size (x1, . . .)

Determine if all input arguments are either scalar or of common size. If so, err is zero, and yi is a matrix of the common size with all entries equal to xi if this is a scalar or xi otherwise. If the inputs cannot be brought to a common size, err is 1, and yi is xi. For example:

     [errorcode, a, b] = common_size ([1 2; 3 4], 5)
          ⇒ errorcode = 0
          ⇒ a = [ 1, 2; 3, 4 ]
          ⇒ b = [ 5, 5; 5, 5 ]

This is useful for implementing functions where arguments can either be scalars or of common size.

[Built-in Function] idx = find (x) [Built-in Function] idx = find (x, n) [Built-in Function] idx = find (x, n, direction) [Built-in Function] [i, j] = find (…) [Built-in Function] [i, j, v] = find (…)

Return a vector of indices of nonzero elements of a matrix, as a row if x is a row vector or as a column otherwise. To obtain a single index for each matrix element, Octave pretends that the columns of a matrix form one long vector (like Fortran arrays are stored). For example:

     find (eye (2)) ⇒ [ 1; 4 ]

If two outputs are requested, find returns the row and column indices of nonzero elements of a matrix. For example:

     [i, j] = find (2 * eye (2))
         ⇒ i = [ 1; 2 ]

         ⇒ j = [ 1; 2 ]

If three outputs are requested, find also returns a vector containing the nonzero values. For example:

     [i, j, v] = find (3 * eye (2))
            ⇒ i = [ 1; 2 ]

            ⇒ j = [ 1; 2 ]
            ⇒ v = [ 3; 3 ]

If two inputs are given, n indicates the maximum number of elements to find from the beginning of the matrix or vector. If three inputs are given, direction should be one of "first" or "last", requesting only the first or last n indices, respectively. However, the indices are always returned in ascending order. Note that this function is particularly useful for sparse matrices, as it extracts the non-zero elements as vectors, which can then be used to create the original matrix. For example:

     sz = size (a);
     [i, j, v] = find (a);
     b = sparse (i, j, v, sz(1), sz(2));

See also: [nonzeros], page 494.

[Built-in Function] idx = lookup (table, y) [Built-in Function] idx = lookup (table, y, opt)

Lookup values in a sorted table. Usually used as a prelude to interpolation.

If table is increasing and idx = lookup (table, y), then table(idx(i)) <= y(i) <
table(idx(i+1)) for all y(i) within the table. If y(i) < table(1) then idx(i) is
0. If y(i) >= table(end) or isnan (y(i)) then idx(i) is n.

If the table is decreasing, then the tests are reversed. For non-strictly monotonic tables, empty intervals are always skipped. The result is undefined if table is not monotonic, or if table contains a NaN. The complexity of the lookup is O(M*log(N)) where N is the size of table and M is the size of y. In the special case when y is also sorted, the complexity is O(min(M*log(N),M+N)). table and y can also be cell arrays of strings (or y can be a single string). In this case, string lookup is performed using lexicographical comparison. If opts is specified, it must be a string with letters indicating additional options.

m         table(idx(i)) == val(i) if val(i) occurs in table; otherwise, idx(i)

is zero.

b idx(i) is a logical 1 or 0, indicating whether val(i) is contained in table or not.

l For numeric lookups the leftmost subinterval shall be extended to infinity (i.e., all indices at least 1)

r For numeric lookups the rightmost subinterval shall be extended to infinity (i.e., all indices at most n-1).

If you wish to check if a variable exists at all, instead of properties its elements may have, consult Section 7.3 [Status of Variables], page 126.

16.2 Rearranging Matrices

[Function File] fliplr (x)

Return a copy of x with the order of the columns reversed. In other words, x is flipped left-to-right about a vertical axis. For example:

     fliplr ([1, 2; 3, 4])
          ⇒ 2  1
              4  3

Note that fliplr only works with 2-D arrays. To flip N-D arrays use flipdim instead.

See also: [flipud], page 383, [flipdim], page 383, [rot90], page 383, [rotdim], page 383.

[Function File] flipud (x)

Return a copy of x with the order of the rows reversed. In other words, x is flipped upside-down about a horizontal axis. For example:

     flipud ([1, 2; 3, 4])
          ⇒ 3  4
              1  2

Note that flipud only works with 2-D arrays. To flip N-D arrays use flipdim instead.

See also: [fliplr], page 382, [flipdim], page 383, [rot90], page 383, [rotdim], page 383.

[Function File] flipdim (x) [Function File] flipdim (x, dim)

Return a copy of x flipped about the dimension dim. dim defaults to the first nonsingleton dimension. For example:

     flipdim ([1, 2; 3, 4], 2)
           ⇒ 2  1
               4  3

See also: [fliplr], page 382, [flipud], page 383, [rot90], page 383, [rotdim], page 383.

[Function File] rot90 (A) [Function File] rot90 (A, k)

Return a copy of A with the elements rotated counterclockwise in 90-degree increments. The second argument is optional, and specifies how many 90-degree rotations are to be applied (the default value is 1). Negative values of k rotate the matrix in a clockwise direction. For example,

     rot90 ([1, 2; 3, 4], -1)
         ⇒ 3  1
             4  2

rotates the given matrix clockwise by 90 degrees. The following are all equivalent statements:

     rot90 ([1, 2; 3, 4], -1)
     rot90 ([1, 2; 3, 4], 3)
     rot90 ([1, 2; 3, 4], 7)

Note that rot90 only works with 2-D arrays. To rotate N-D arrays use rotdim instead.

See also: [rotdim], page 383, [flipud], page 383, [fliplr], page 382, [flipdim], page 383.

[Function File] rotdim (x) [Function File] rotdim (x, n) [Function File] rotdim (x, n, plane)

Return a copy of x with the elements rotated counterclockwise in 90-degree increments. The second argument n is optional, and specifies how many 90-degree rotations are to be applied (the default value is 1). The third argument is also optional and defines the plane of the rotation. If present, plane is a two element vector containing two different valid dimensions of the matrix. When plane is not given the first two non-singleton dimensions are used. Negative values of n rotate the matrix in a clockwise direction. For example,

     rotdim ([1, 2; 3, 4], -1, [1, 2])
          ⇒ 3  1
              4  2

rotates the given matrix clockwise by 90 degrees. The following are all equivalent statements:

     rotdim ([1, 2; 3, 4], -1, [1, 2])
     rotdim ([1, 2; 3, 4], 3, [1, 2])
     rotdim ([1, 2; 3, 4], 7, [1, 2])

See also: [rot90], page 383, [flipud], page 383, [fliplr], page 382, [flipdim], page 383.

[Built-in Function] cat (dim, array1, array2, …, arrayN)

Return the concatenation of N-D array objects, array1, array2, …, arrayN along dimension dim.

     A = ones (2, 2);
     B = zeros (2, 2);
     cat (2, A, B)
       ⇒ 1 1 0 0
          1 1 0 0

Alternatively, we can concatenate A and B along the second dimension in the following way:

     [A, B]

dim can be larger than the dimensions of the N-D array objects and the result will thus have dim dimensions as the following example shows:

     cat (4, ones (2, 2), zeros (2, 2))
       ⇒ ans(:,:,1,1) =
            1 1
            1 1
          ans(:,:,1,2) =
            0 0
            0 0

See also: [horzcat], page 384, [vertcat], page 385.

[Built-in Function] horzcat (array1, array2, …, arrayN)

Return the horizontal concatenation of N-D array objects, array1, array2, …, arrayN along dimension 2.

Arrays may also be concatenated horizontally using the syntax for creating new matrices. For example:

     hcat = [ array1, array2, ... ]

See also: [cat], page 384, [vertcat], page 385.

[Built-in Function] vertcat (array1, array2, …, arrayN)

Return the vertical concatenation of N-D array objects, array1, array2, …, arrayN along dimension 1. Arrays may also be concatenated vertically using the syntax for creating new matrices. For example:

     vcat = [ array1; array2; ... ]

See also: [cat], page 384, [horzcat], page 384.

[Built-in Function] permute (A, perm)

Return the generalized transpose for an N-D array object A. The permutation vector perm must contain the elements 1:ndims (A) (in any order, but each element must appear only once).

See also: [ipermute], page 385.

[Built-in Function] ipermute (A, iperm)

The inverse of the permute function. The expression

     ipermute (permute (A, perm), perm)

returns the original array A.

See also: [permute], page 385.

[Built-in Function] reshape (A, m, n, …) [Built-in Function] reshape (A, [m n …]) [Built-in Function] reshape (A, …, [], …) [Built-in Function] reshape (A, size)

Return a matrix with the specified dimensions (m, n, …) whose elements are taken from the matrix A. The elements of the matrix are accessed in column-major order (like Fortran arrays are stored). The following code demonstrates reshaping a 1x4 row vector into a 2x2 square matrix.

     reshape ([1, 2, 3, 4], 2, 2)
           ⇒ 1  3
               2  4

Note that the total number of elements in the original matrix (prod (size (A))) must match the total number of elements in the new matrix (prod ([m n ...])). A single dimension of the return matrix may be left unspecified and Octave will determine its size automatically. An empty matrix ([]) is used to flag the unspecified dimension.

See also: [resize], page 385, [vec], page 389, [postpad], page 390, [cat], page 384, [squeeze], page 45.

[Built-in Function] resize (x, m) [Built-in Function] resize (x, m, n, …) [Built-in Function] resize (x, [m n …])

Resize x cutting off elements as necessary. In the result, element with certain indices is equal to the corresponding element of x if the indices are within the bounds of x; otherwise, the element is set to zero. In other words, the statement

     y = resize (x, dv)

is equivalent to the following code:

     y = zeros (dv, class (x));
     sz = min (dv, size (x));
     for i = 1:length (sz)
       idx{i} = 1:sz(i);
     endfor
     y(idx{:}) = x(idx{:});

but is performed more efficiently.

If only m is supplied, and it is a scalar, the dimension of the result is m-by-m. If m, n, … are all scalars, then the dimensions of the result are m-by-n-by-. . .. If given a vector as input, then the dimensions of the result are given by the elements of that vector.

An object can be resized to more dimensions than it has; in such case the missing dimensions are assumed to be 1. Resizing an object to fewer dimensions is not possible.

See also: [reshape], page 385, [postpad], page 390, [prepad], page 390, [cat], page 384.

[Function File] y = circshift (x, n)

Circularly shift the values of the array x. n must be a vector of integers no longer than the number of dimensions in x. The values of n can be either positive or negative, which determines the direction in which the values or x are shifted. If an element of n is zero, then the corresponding dimension of x will not be shifted. For example:

     x = [1, 2, 3; 4, 5, 6; 7, 8, 9];
     circshift (x, 1)
     ⇒ 7, 8, 9
         1, 2, 3
         4, 5, 6
     circshift (x, -2)
     ⇒ 7, 8, 9
         1, 2, 3
         4, 5, 6
     circshift (x, [0,1])
     ⇒ 3, 1, 2
         6, 4, 5
         9, 7, 8

See also: permute, ipermute, shiftdim.

[Function File] shift (x, b) [Function File] shift (x, b, dim)

If x is a vector, perform a circular shift of length b of the elements of x.

If x is a matrix, do the same for each column of x. If the optional dim argument is given, operate along this dimension.

[Function File] y = shiftdim (x, n)

[Function File] [y, ns] = shiftdim (x)

Shift the dimensions of x by n, where n must be an integer scalar. When n is positive, the dimensions of x are shifted to the left, with the leading dimensions circulated to the end. If n is negative, then the dimensions of x are shifted to the right, with n leading singleton dimensions added. Called with a single argument, shiftdim, removes the leading singleton dimensions, returning the number of dimensions removed in the second output argument ns. For example:

     x = ones (1, 2, 3);
     size (shiftdim (x, -1))
        ⇒ [1, 1, 2, 3]
     size (shiftdim (x, 1))
        ⇒ [2, 3]
     [b, ns] = shiftdim (x)
        ⇒ b = [1, 1, 1; 1, 1, 1]
        ⇒ ns = 1

See also: reshape, permute, ipermute, circshift, squeeze.

[Built-in Function] [s, i] = sort (x) [Built-in Function] [s, i] = sort (x, dim) [Built-in Function] [s, i] = sort (x, mode) [Built-in Function] [s, i] = sort (x, dim, mode)

Return a copy of x with the elements arranged in increasing order. For matrices, sort orders the elements within columns For example:

     sort ([1, 2; 2, 3; 3, 1])
        ⇒ 1  1
            2  2
            3  3

If the optional argument dim is given, then the matrix is sorted along the dimension defined by dim. The optional argument mode defines the order in which the values will be sorted. Valid values of mode are "ascend" or "descend". The sort function may also be used to produce a matrix containing the original row indices of the elements in the sorted matrix. For example:

     [s, i] = sort ([1, 2; 2, 3; 3, 1])
       ⇒ s = 1  1
              2  2
              3  3
       ⇒ i = 1  3
              2  1
              3  2

For equal elements, the indices are such that equal elements are listed in the order in which they appeared in the original list.

Sorting of complex entries is done first by magnitude (abs (z)) and for any ties by phase angle (angle (z)). For example:

     sort ([1+i; 1; 1-i])
         ⇒ 1 + 0i
            1 - 1i
            1 + 1i

NaN values are treated as being greater than any other value and are sorted to the end of the list. The sort function may also be used to sort strings and cell arrays of strings, in which case ASCII dictionary order (uppercase ’A’ precedes lowercase ’a’) of the strings is used. The algorithm used in sort is optimized for the sorting of partially ordered lists.

See also: [sortrows], page 388, [issorted], page 388.

[Function File] [s, i] = sortrows (A) [Function File] [s, i] = sortrows (A, c)

Sort the rows of the matrix A according to the order of the columns specified in c. If c is omitted, a lexicographical sort is used. By default ascending order is used however if elements of c are negative then the corresponding column is sorted in descending order.

See also: [sort], page 387.

[Built-in Function] issorted (a) [Built-in Function] issorted (a, mode) [Built-in Function] issorted (a, "rows", mode)

Return true if the array is sorted according to mode, which may be either

"ascending", "descending", or "either". By default, mode is "ascending".

NaNs are treated in the same manner as sort. If the optional argument "rows" is supplied, check whether the array is sorted by rows as output by the function sortrows (with no options). This function does not support sparse matrices.

See also: [sort], page 387, [sortrows], page 388.

[Built-in Function] nth_element (x, n) [Built-in Function] nth_element (x, n, dim)

Select the n-th smallest element of a vector, using the ordering defined by sort. In other words, the result is equivalent to sort(x)(n). n can also be a contiguous range, either ascending l:u or descending u:-1:l, in which case a range of elements is returned. If x is an array, nth_element operates along the dimension defined by dim, or the first non-singleton dimension if dim is not given. nth element encapsulates the C++ standard library algorithms nth element and partial sort. On average, the complexity of the operation is O(M*log(K)), where M = size (x, dim) and K = length (n). This function is intended for cases where the ratio K/M is small; otherwise, it may be better to use sort.

See also: [sort], page 387, [min], page 416, [max], page 416.

[Function File] tril (A) [Function File] tril (A, k) [Function File] tril (A, k, pack) [Function File] triu (A) [Function File] triu (A, k) [Function File] triu (A, k, pack)

Return a new matrix formed by extracting the lower (tril) or upper (triu) triangular part of the matrix A, and setting all other elements to zero. The second argument is optional, and specifies how many diagonals above or below the main diagonal should also be set to zero. The default value of k is zero, so that triu and tril normally include the main diagonal as part of the result. If the value of k is nonzero integer, the selection of elements starts at an offset of k diagonals above or below the main diagonal; above for positive k and below for negative k. The absolute value of k must not be greater than the number of sub-diagonals or super-diagonals. For example:

     tril (ones (3), -1)
          ⇒ 0  0  0
              1  0  0
              1  1  0

and

     tril (ones (3), 1)
          ⇒ 1  1  0
              1  1  1
              1  1  1

If the option "pack" is given as third argument, the extracted elements are not inserted into a matrix, but rather stacked column-wise one above other.

See also: [diag], page 390.

[Built-in Function] v = vec (x) [Built-in Function] v = vec (x, dim)

Return the vector obtained by stacking the columns of the matrix x one above the other. Without dim this is equivalent to x(:). If dim is supplied, the dimensions of v are set to dim with all elements along the last dimension. This is equivalent to

shiftdim (x(:), 1-dim).

See also: [vech], page 389, [resize], page 385, [cat], page 384.

[Function File] vech (x)

Return the vector obtained by eliminating all supradiagonal elements of the square matrix x and stacking the result one column above the other. This has uses in matrix calculus where the underlying matrix is symmetric and it would be pointless to keep values above the main diagonal.

See also: [vec], page 389.

[Function File] prepad (x, l) [Function File] prepad (x, l, c) [Function File] prepad (x, l, c, dim)

Prepend the scalar value c to the vector x until it is of length l. If c is not given, a value of 0 is used. If length (x) > l, elements from the beginning of x are removed until a vector of length l is obtained. If x is a matrix, elements are prepended or removed from each row. If the optional argument dim is given, operate along this dimension.

See also: [postpad], page 390, [cat], page 384, [resize], page 385.

[Function File] postpad (x, l) [Function File] postpad (x, l, c) [Function File] postpad (x, l, c, dim)

Append the scalar value c to the vector x until it is of length l. If c is not given, a value of 0 is used. If length (x) > l, elements from the end of x are removed until a vector of length l is obtained. If x is a matrix, elements are appended or removed from each row. If the optional argument dim is given, operate along this dimension.

See also: [prepad], page 390, [cat], page 384, [resize], page 385.

[Built-in Function] M = diag (v) [Built-in Function] M = diag (v, k) [Built-in Function] M = diag (v, m, n) [Built-in Function] v = diag (M) [Built-in Function] v = diag (M, k)

Return a diagonal matrix with vector v on diagonal k. The second argument is optional. If it is positive, the vector is placed on the k-th super-diagonal. If it is negative, it is placed on the -k-th sub-diagonal. The default value of k is 0, and the vector is placed on the main diagonal. For example:

     diag ([1, 2, 3], 1)
        ⇒ 0  1  0  0
            0  0  2  0
            0  0  0  3
            0  0  0  0

The 3-input form returns a diagonal matrix with vector v on the main diagonal and the resulting matrix being of size m rows x n columns. Given a matrix argument, instead of a vector, diag extracts the k-th diagonal of the matrix.

[Function File] blkdiag (A, B, C, …)

Build a block diagonal matrix from A, B, C, … All the arguments must be numeric and are two-dimensional matrices or scalars. If any argument is of type sparse, the output will also be sparse.

See also: [diag], page 390, [horzcat], page 384, [vertcat], page 385, [sparse], page 492.

16.3 Special Utility Matrices

[Built-in Function] eye (n) [Built-in Function] eye (m, n) [Built-in Function] eye ([m n]) [Built-in Function] eye (…, class)

Return an identity matrix. If invoked with a single scalar argument n, return a square NxN identity matrix. If supplied two scalar arguments (m, n), eye takes them to be the number of rows and columns. If given a vector with two elements, eye uses the values of the elements as the number of rows and columns, respectively. For example:

     eye (3) ⇒ 1  0  0 0  1  0 0  0  1

The following expressions all produce the same result:

     eye (2)

≡

     eye (2, 2)

≡

     eye (size ([1, 2; 3, 4])

The optional argument class, allows eye to return an array of the specified type, like

     val = zeros (n,m, "uint8")

Calling eye with no arguments is equivalent to calling it with an argument of 1. Any negative dimensions are treated as zero. These odd definitions are for compatibility with matlab.

See also: [speye], page 489, [ones], page 391, [zeros], page 392.

[Built-in Function] ones (n) [Built-in Function] ones (m, n) [Built-in Function] ones (m, n, k, …) [Built-in Function] ones ([m n …]) [Built-in Function] ones (…, class)

Return a matrix or N-dimensional array whose elements are all 1. If invoked with a single scalar integer argument n, return a square NxN matrix. If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with the given dimensions.

If you need to create a matrix whose values are all the same, you should use an expression like

     val_matrix = val * ones (m, n)

The optional argument class specifies the class of the return array and defaults to double. For example:

     val = ones (m,n, "uint8")

See also: [zeros], page 392.

[Built-in Function] zeros (n) [Built-in Function] zeros (m, n) [Built-in Function] zeros (m, n, k, …) [Built-in Function] zeros ([m n …]) [Built-in Function] zeros (…, class)

Return a matrix or N-dimensional array whose elements are all 0. If invoked with a single scalar integer argument, return a square NxN matrix. If invoked with two or more scalar integer arguments, or a vector of integer values, return an array with the given dimensions. The optional argument class specifies the class of the return array and defaults to double. For example:

     val = zeros (m,n, "uint8")

See also: [ones], page 391.

[Function File] repmat (A, m) [Function File] repmat (A, m, n) [Function File] repmat (A, [m n]) [Function File] repmat (A, [m n p …])

Form a block matrix of size m by n, with a copy of matrix A as each element. If n is not specified, form an m by m block matrix. For copying along more than two dimensions, specify the number of times to copy across each dimension m, n, p, …, in a vector in the second argument.

See also: [repelems], page 392.

[Built-in Function] repelems (x, r)

Construct a vector of repeated elements from x. r is a 2xN integer matrix specifying which elements to repeat and how often to repeat each element. Entries in the first row, r(1,j), select an element to repeat. The corresponding entry in the second row, r(2,j), specifies the repeat count. If x is a matrix then the columns of x are imagined to be stacked on top of each other for purposes of the selection index. A row vector is always returned. Conceptually the result is calculated as follows:

     y = [];
     for i = 1:columns (r)
       y = [y, x(r(1,i)*ones(1, r(2,i)))];
     endfor

See also: [repmat], page 392, [cat], page 384.

The functions linspace and logspace make it very easy to create vectors with evenly or logarithmically spaced elements. See Section 4.2 [Ranges], page 52.

[Built-in Function] linspace (base, limit) [Built-in Function] linspace (base, limit, n)

Return a row vector with n linearly spaced elements between base and limit. If the number of elements is greater than one, then the endpoints base and limit are always included in the range. If base is greater than limit, the elements are stored in decreasing order. If the number of points is not specified, a value of 100 is used. The linspace function always returns a row vector if both base and limit are scalars. If one, or both, of them are column vectors, linspace returns a matrix. For compatibility with matlab, return the second argument (limit) if fewer than two values are requested.

See also: [logspace], page 393.

[Function File] logspace (a, b) [Function File] logspace (a, b, n) [Function File] logspace (a, pi, n)

Return a row vector with n elements logarithmically spaced from 10^{a} to 10^{b}. If n is unspecified it defaults to 50. If b is equal to π, the points are between 10^{a} and π, not 10^{a} and 10^{π}, in order to be compatible with the corresponding matlab function. Also for compatibility with matlab, return the second argument b if fewer than two values are requested.

See also: [linspace], page 392.

[Built-in Function] rand (n) [Built-in Function] rand (m, n, …) [Built-in Function] rand ([m n …]) [Built-in Function] v = rand ("state")

[Built-in Function] rand ("state", v) [Built-in Function] rand ("state", "reset")

[Built-in Function] v = rand ("seed") [Built-in Function] rand ("seed", v)

[Built-in Function] rand ("seed", "reset") [Built-in Function] rand (…, "single") [Built-in Function] rand (…, "double")

Return a matrix with random elements uniformly distributed on the interval (0, 1). The arguments are handled the same as the arguments for eye. You can query the state of the random number generator using the form

     v = rand ("state")

This returns a column vector v of length 625. Later, you can restore the random number generator to the state v using the form

     rand ("state", v)

You may also initialize the state vector from an arbitrary vector of length ≤ 625 for v. This new state will be a hash based on the value of v, not v itself. By default, the generator is initialized from /dev/urandom if it is available, otherwise from CPU time, wall clock time, and the current fraction of a second. Note that this differs from matlab, which always initializes the state to the same state at startup. To obtain behavior comparable to matlab, initialize with a deterministic state vector in Octave’s startup files (see Section 2.1.2 [Startup Files], page 19).

To compute the pseudo-random sequence, rand uses the Mersenne Twister with a period of 2^{19937} − 1 (See M. Matsumoto and T. Nishimura, Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans. on Modeling and Computer Simulation Vol. 8, No. 1, pp. 3-30, January 1998,

http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html). Do not use for

cryptography without securely hashing several returned values together, otherwise the generator state can be learned after reading 624 consecutive values.

Older versions of Octave used a different random number generator. The new generator is used by default as it is significantly faster than the old generator, and produces random numbers with a significantly longer cycle time. However, in some circumstances it might be desirable to obtain the same random sequences as used by the old generators. To do this the keyword "seed" is used to specify that the old generators should be use, as in

     rand ("seed", val)

which sets the seed of the generator to val. The seed of the generator can be queried with

     s = rand ("seed")

However, it should be noted that querying the seed will not cause rand to use the old generators, only setting the seed will. To cause rand to once again use the new generators, the keyword "state" should be used to reset the state of the rand.

The state or seed of the generator can be reset to a new random value using the "reset" keyword.

The class of the value returned can be controlled by a trailing "double" or "single" argument. These are the only valid classes.

See also: [randn], page 395, [rande], page 395, [randg], page 396, [randp], page 396.

[Function File] randi (imax) [Function File] randi (imax, n) [Function File] randi (imax, m, n, …) [Function File] randi ([imin imax], …) [Function File] randi (…, "class")

Return random integers in the range 1:imax.

Additional arguments determine the shape of the return matrix. When no arguments are specified a single random integer is returned. If one argument n is specified then a square matrix (n x n) is returned. Two or more arguments will return a multidimensional matrix (m x n x …).

The integer range may optionally be described by a two element matrix with a lower and upper bound in which case the returned integers will be on the interval [imin, imax].

The optional argument class will return a matrix of the requested type. The default

is "double".

The following example returns 150 integers in the range 1-10.

     ri = randi (10, 150, 1)

Implementation Note: randi relies internally on rand which uses class "double" to represent numbers. This limits the maximum integer (imax) and range (imax - imin) to the value returned by the bitmax function. For IEEE floating point numbers this value is 2^{53} − 1.

See also: [rand], page 393.

[Built-in Function] randn (n) [Built-in Function] randn (m, n, …) [Built-in Function] randn ([m n …]) [Built-in Function] v = randn ("state") [Built-in Function] randn ("state", v)

[Built-in Function] randn ("state", "reset")

[Built-in Function] v = randn ("seed") [Built-in Function] randn ("seed", v)

[Built-in Function] randn ("seed", "reset") [Built-in Function] randn (…, "single") [Built-in Function] randn (…, "double")

Return a matrix with normally distributed random elements having zero mean and variance one. The arguments are handled the same as the arguments for rand. By default, randn uses the Marsaglia and Tsang “Ziggurat technique” to transform from a uniform to a normal distribution. The class of the value returned can be controlled by a trailing "double" or "single" argument. These are the only valid classes. Reference: G. Marsaglia and W.W. Tsang, Ziggurat Method for Generating Random Variables, J. Statistical Software, vol 5, 2000,

http://www.jstatsoft.org/v05/i08/)

See also: [rand], page 393, [rande], page 395, [randg], page 396, [randp], page 396.

[Built-in Function] rande (n) [Built-in Function] rande (m, n, …) [Built-in Function] rande ([m n …]) [Built-in Function] v = rande ("state") [Built-in Function] rande ("state", v)

[Built-in Function] rande ("state", "reset")

[Built-in Function] v = rande ("seed") [Built-in Function] rande ("seed", v)

[Built-in Function] rande ("seed", "reset") [Built-in Function] rande (…, "single") [Built-in Function] rande (…, "double")

Return a matrix with exponentially distributed random elements. The arguments are handled the same as the arguments for rand. By default, randn uses the Marsaglia and Tsang “Ziggurat technique” to transform from a uniform to an exponential distribution. The class of the value returned can be controlled by a trailing "double" or "single" argument. These are the only valid classes.

Reference: G. Marsaglia and W.W. Tsang, Ziggurat Method for Generating Random Variables, J. Statistical Software, vol 5, 2000,

http://www.jstatsoft.org/v05/i08/)

See also: [rand], page 393, [randn], page 395, [randg], page 396, [randp], page 396.

[Built-in Function] randp (l, n) [Built-in Function] randp (l, m, n, …) [Built-in Function] randp (l, [m n …]) [Built-in Function] v = randp ("state") [Built-in Function] randp ("state", v)

[Built-in Function] randp ("state", "reset")

[Built-in Function] v = randp ("seed") [Built-in Function] randp ("seed", v)

[Built-in Function] randp ("seed", "reset") [Built-in Function] randp (…, "single") [Built-in Function] randp (…, "double")

Return a matrix with Poisson distributed random elements with mean value parameter given by the first argument, l. The arguments are handled the same as the arguments for rand, except for the argument l. Five different algorithms are used depending on the range of l and whether or not l is a scalar or a matrix.

For scalar l ≤ 12, use direct method.

W.H. Press, et al., Numerical Recipes in C, Cambridge University Press, 1992.

For scalar l > 12, use rejection method.[1] W.H. Press, et al., Numerical Recipes in C, Cambridge University Press, 1992.

For matrix l ≤ 10, use inversion method.[2] E. Stadlober, et al., WinRand source code, available via FTP.

For matrix l > 10, use patchwork rejection method.

E. Stadlober, et al., WinRand source code, available via FTP, or H. Zechner, Efficient sampling from continuous and discrete unimodal distributions, Doctoral Dissertation, 156pp., Technical University Graz, Austria, 1994.

For l > 1e8, use normal approximation.

L. Montanet, et al., Review of Particle Properties, Physical Review D 50 p1284, 1994.

The class of the value returned can be controlled by a trailing "double" or "single" argument. These are the only valid classes.

See also: [rand], page 393, [randn], page 395, [rande], page 395, [randg], page 396.

[Built-in Function] randg (n) [Built-in Function] randg (m, n, …) [Built-in Function] randg ([m n …])

[Built-in Function] v = randg ("state") [Built-in Function] randg ("state", v)

[Built-in Function] randg ("state", "reset")

[Built-in Function] v = randg ("seed") [Built-in Function] randg ("seed", v)

[Built-in Function] randg ("seed", "reset") [Built-in Function] randg (…, "single") [Built-in Function] randg (…, "double")

Return a matrix with gamma (a,1) distributed random elements. The arguments are handled the same as the arguments for rand, except for the argument a. This can be used to generate many distributions:

gamma (a, b) for a > -1, b > 0
r = b * randg (a)
beta (a, b) for a > -1, b > -1
r1 = randg (a, 1)
r = r1 / (r1 + randg (b, 1))
Erlang (a, n)
r = a * randg (n)
chisq (df) for df > 0
r = 2 * randg (df / 2)

t (df) for 0 < df < inf (use randn if df is infinite)

r = randn () / sqrt (2 * randg (df / 2) / df)
F (n1, n2) for 0 < n1, 0 < n2
## r1 equals 1 if n1 is infinite
r1 = 2 * randg (n1 / 2) / n1
## r2 equals 1 if n2 is infinite
r2 = 2 * randg (n2 / 2) / n2
r = r1 / r2
negative binomial (n, p) for n > 0, 0 < p <= 1
r = randp ((1 - p) / p * randg (n))
non-central chisq (df, L), for df >= 0 and L > 0

(use chisq if L = 0)

r = randp (L / 2)
r(r > 0) = 2 * randg (r(r > 0))
r(df > 0) += 2 * randg (df(df > 0)/2)
Dirichlet (a1, ... ak)
r = (randg (a1), ..., randg (ak))
r = r / sum (r)

The class of the value returned can be controlled by a trailing "double" or "single" argument. These are the only valid classes.

See also: [rand], page 393, [randn], page 395, [rande], page 395, [randp], page 396.

The generators operate in the new or old style together, it is not possible to mix the two. Initializing any generator with "state" or "seed" causes the others to switch to the same style for future calls.

The state of each generator is independent and calls to different generators can be interleaved without affecting the final result. For example,

rand ("state", [11, 22, 33]);
randn ("state", [44, 55, 66]);
u = rand (100, 1);
n = randn (100, 1);

and

rand ("state", [11, 22, 33]);
randn ("state", [44, 55, 66]);
u = zeros (100, 1);
n = zeros (100, 1);
for i = 1:100
  u(i) = rand ();
  n(i) = randn ();
end

produce equivalent results. When the generators are initialized in the old style with "seed" only rand and randn are independent, because the old rande, randg and randp generators make calls to rand and randn.

The generators are initialized with random states at start-up, so that the sequences of random numbers are not the same each time you run Octave.^{1} If you really do need to reproduce a sequence of numbers exactly, you can set the state or seed to a specific value.

If invoked without arguments, rand and randn return a single element of a random sequence.

The original rand and randn functions use Fortran code from ranlib, a library of Fortran routines for random number generation, compiled by Barry W. Brown and James Lovato of the Department of Biomathematics at The University of Texas, M.D. Anderson Cancer Center, Houston, TX 77030.

[Built-in Function] randperm (n) [Built-in Function] randperm (n, m)

Return a row vector containing a random permutation of 1:n. If m is supplied, return m unique entries, sampled without replacement from 1:n. The complexity is O(n) in memory and O(m) in time, unless m < n/5, in which case O(m) memory is used as well. The randomization is performed using rand(). All permutations are equally likely.

See also: [perms], page 568.

1 The old versions of rand and randn obtain their initial seeds from the system clock.

16.4 Famous Matrices

The following functions return famous matrix forms.

[Function File] gallery (name) [Function File] gallery (name, args)

Create interesting matrices for testing.

[Function File] c = gallery ("cauchy", x) [Function File] c = gallery ("cauchy", x, y)

Create a Cauchy matrix.

[Function File] c = gallery ("chebspec", n) [Function File] c = gallery ("chebspec", n, k)

Create a Chebyshev spectral differentiation matrix.

[Function File] c = gallery ("chebvand", p) [Function File] c = gallery ("chebvand", m, p)

Create a Vandermonde-like matrix for the Chebyshev polynomials.

[Function File] a = gallery ("chow", n) [Function File] a = gallery ("chow", n, alpha) [Function File] a = gallery ("chow", n, alpha, delta)

Create a Chow matrix – a singular Toeplitz lower Hessenberg matrix.

[Function File] c = gallery ("circul", v)

Create a circulant matrix.

[Function File] a = gallery ("clement", n) [Function File] a = gallery ("clement", n, k)

Create a tridiagonal matrix with zero diagonal entries.

[Function File] c = gallery ("compar", a) [Function File] c = gallery ("compar", a, k)

Create a comparison matrix.

[Function File] a = gallery ("condex", n) [Function File] a = gallery ("condex", n, k) [Function File] a = gallery ("condex", n, k, theta)

Create a ‘counterexample’ matrix to a condition estimator.

[Function File] a = gallery ("cycol", [m n]) [Function File] a = gallery ("cycol", n) [Function File] a = gallery (…, k)

Create a matrix whose columns repeat cyclically.

[Function File] [c, d, e] = gallery ("dorr", n)

[Function File] [c, d, e] = gallery ("dorr", n, theta)

[Function File] a = gallery ("dorr", …)

Create a diagonally dominant, ill conditioned, tridiagonal matrix.

[Function File] a = gallery ("dramadah", n) [Function File] a = gallery ("dramadah", n, k)

Create a (0, 1) matrix whose inverse has large integer entries.

[Function File] a = gallery ("fiedler", c)

Create a symmetric Fiedler matrix.

[Function File] a = gallery ("forsythe", n) [Function File] a = gallery ("forsythe", n, alpha) [Function File] a = gallery ("forsythe", n, alpha, lambda)

Create a Forsythe matrix (a perturbed Jordan block).

[Function File] f = gallery ("frank", n) [Function File] f = gallery ("frank", n, k)

Create a Frank matrix (ill conditioned eigenvalues).

[Function File] c = gallery ("gcdmat", n)

Create a greatest common divisor matrix.

c is an n-by-n matrix whose values correspond to the greatest common divisor of its coordinate values, i.e., c(i,j) correspond gcd (i, j).

[Function File] a = gallery ("gearmat", n) [Function File] a = gallery ("gearmat", n, i) [Function File] a = gallery ("gearmat", n, i, j)

Create a Gear matrix.

[Function File] g = gallery ("grcar", n) [Function File] g = gallery ("grcar", n, k)

Create a Toeplitz matrix with sensitive eigenvalues.

[Function File] a = gallery ("hanowa", n) [Function File] a = gallery ("hanowa", n, d)

Create a matrix whose eigenvalues lie on a vertical line in the complex plane.

[Function File] v = gallery ("house", x) [Function File] [v, beta] = gallery ("house", x)

Create a householder matrix.

[Function File] a = gallery ("integerdata", imax, [M N …], j) [Function File] a = gallery ("integerdata", imax, M, N, …, j) [Function File] a = gallery ("integerdata", [imin, imax], [M N …], j) [Function File] a = gallery ("integerdata", [imin, imax], M, N, …, j) [Function File] a = gallery ("integerdata", …, "class")

Create a matrix with random integers in the range [1, imax]. If imin is given then the integers are in the range [imin, imax]. The second input is a matrix of dimensions describing the size of the output. The dimensions can also be input as comma-separated arguments. The input j is an integer index in the range [0, 2^32-1]. The values of the output matrix are always exactly the same (reproducibility) for a given size input and j index.

The final optional argument determines the class of the resulting matrix. Possible

     values for class: "uint8", "uint16", "uint32", "int8", "int16", int32", "single",
     "double". The default is "double".

[Function File] a = gallery ("invhess", x) [Function File] a = gallery ("invhess", x, y)

Create the inverse of an upper Hessenberg matrix.

[Function File] a = gallery ("invol", n)

Create an involutory matrix.

[Function File] a = gallery ("ipjfact", n) [Function File] a = gallery ("ipjfact", n, k)

Create an Hankel matrix with factorial elements.

[Function File] a = gallery ("jordbloc", n) [Function File] a = gallery ("jordbloc", n, lambda)

Create a Jordan block.

[Function File] u = gallery ("kahan", n) [Function File] u = gallery ("kahan", n, theta) [Function File] u = gallery ("kahan", n, theta, pert)

Create a Kahan matrix (upper trapezoidal).

[Function File] a = gallery ("kms", n) [Function File] a = gallery ("kms", n, rho)

Create a Kac-Murdock-Szego Toeplitz matrix.

[Function File] b = gallery ("krylov", a) [Function File] b = gallery ("krylov", a, x) [Function File] b = gallery ("krylov", a, x, j)

Create a Krylov matrix.

[Function File] a = gallery ("lauchli", n) [Function File] a = gallery ("lauchli", n, mu)

Create a Lauchli matrix (rectangular).

[Function File] a = gallery ("lehmer", n)

Create a Lehmer matrix (symmetric positive definite).

[Function File] t = gallery ("lesp", n)

Create a tridiagonal matrix with real, sensitive eigenvalues.

[Function File] a = gallery ("lotkin", n)

Create a Lotkin matrix.

[Function File] a = gallery ("minij", n)

Create a symmetric positive definite matrix MIN(i,j).

[Function File] a = gallery ("moler", n) [Function File] a = gallery ("moler", n, alpha)

Create a Moler matrix (symmetric positive definite).

[Function File] [a, t] = gallery ("neumann", n)

Create a singular matrix from the discrete Neumann problem (sparse).

[Function File] a = gallery ("normaldata", [M N …], j)

[Function File] a = gallery ("normaldata", M, N, …, j)

[Function File] a = gallery ("normaldata", …, "class")

Create a matrix with random samples from the standard normal distribution (mean = 0, std = 1). The first input is a matrix of dimensions describing the size of the output. The dimensions can also be input as comma-separated arguments. The input j is an integer index in the range [0, 2^32-1]. The values of the output matrix are always exactly the same (reproducibility) for a given size input and j index. The final optional argument determines the class of the resulting matrix. Possible values for class: "single", "double". The default is "double".

[Function File] q = gallery ("orthog", n) [Function File] q = gallery ("orthog", n, k)

Create orthogonal and nearly orthogonal matrices.

[Function File] a = gallery ("parter", n)

Create a Parter matrix (a Toeplitz matrix with singular values near pi).

[Function File] p = gallery ("pei", n) [Function File] p = gallery ("pei", n, alpha)

Create a Pei matrix.

[Function File] a = gallery ("Poisson", n)

Create a block tridiagonal matrix from Poisson’s equation (sparse).

[Function File] a = gallery ("prolate", n) [Function File] a = gallery ("prolate", n, w)

Create a prolate matrix (symmetric, ill-conditioned Toeplitz matrix).

[Function File] h = gallery ("randhess", x)

Create a random, orthogonal upper Hessenberg matrix.

[Function File] a = gallery ("rando", n) [Function File] a = gallery ("rando", n, k)

Create a random matrix with elements -1, 0 or 1.

[Function File] a = gallery ("randsvd", n) [Function File] a = gallery ("randsvd", n, kappa) [Function File] a = gallery ("randsvd", n, kappa, mode) [Function File] a = gallery ("randsvd", n, kappa, mode, kl) [Function File] a = gallery ("randsvd", n, kappa, mode, kl, ku)

Create a random matrix with pre-assigned singular values.

[Function File] a = gallery ("redheff", n)

Create a zero and ones matrix of Redheffer associated with the Riemann hypothesis.

[Function File] a = gallery ("riemann", n)

Create a matrix associated with the Riemann hypothesis.

[Function File] a = gallery ("ris", n)

Create a symmetric Hankel matrix.

[Function File] a = gallery ("smoke", n) [Function File] a = gallery ("smoke", n, k)

Create a complex matrix, with a ‘smoke ring’ pseudospectrum.

[Function File] t = gallery ("toeppd", n) [Function File] t = gallery ("toeppd", n, m) [Function File] t = gallery ("toeppd", n, m, w) [Function File] t = gallery ("toeppd", n, m, w, theta)

Create a symmetric positive definite Toeplitz matrix.

[Function File] p = gallery ("toeppen", n) [Function File] p = gallery ("toeppen", n, a) [Function File] p = gallery ("toeppen", n, a, b) [Function File] p = gallery ("toeppen", n, a, b, c) [Function File] p = gallery ("toeppen", n, a, b, c, d) [Function File] p = gallery ("toeppen", n, a, b, c, d, e)

Create a pentadiagonal Toeplitz matrix (sparse).

[Function File] a = gallery ("tridiag", x, y, z) [Function File] a = gallery ("tridiag", n) [Function File] a = gallery ("tridiag", n, c, d, e)

Create a tridiagonal matrix (sparse).

[Function File] t = gallery ("triw", n) [Function File] t = gallery ("triw", n, alpha) [Function File] t = gallery ("triw", n, alpha, k)

Create an upper triangular matrix discussed by Kahan, Golub and Wilkinson.

[Function File] a = gallery ("uniformdata", [M N …], j)

[Function File] a = gallery ("uniformdata", M, N, …, j)

[Function File] a = gallery ("uniformdata", …, "class")

Create a matrix with random samples from the standard uniform distribution (range [0,1]). The first input is a matrix of dimensions describing the size of the output. The dimensions can also be input as comma-separated arguments. The input j is an integer index in the range [0, 2^32-1]. The values of the output matrix are always exactly the same (reproducibility) for a given size input and j index. The final optional argument determines the class of the resulting matrix. Possible values for class: "single", "double". The default is "double".

[Function File] a = gallery ("wathen", nx, ny) [Function File] a = gallery ("wathen", nx, ny, k)

Create the Wathen matrix.

[Function File] [a, b] = gallery ("wilk", n)

Create various specific matrices devised/discussed by Wilkinson.

[Function File] hadamard (n)

Construct a Hadamard matrix (Hn) of size n-by-n. The size n must be of the form 2^{k} ∗p in which p is one of 1, 12, 20 or 28. The returned matrix is normalized, meaning

Hn(:,1) == 1 and Hn(1,:) == 1.

Some of the properties of Hadamard matrices are:

• kron (Hm, Hn) is a Hadamard matrix of size m-by-n.

  • Hn * Hn' = n * eye (n).

• The rows of Hn are orthogonal.

  • det (A) <= abs (det (Hn)) for all A with abs (A(i, j)) <= 1.

• Multiplying any row or column by -1 and the matrix will remain a Hadamard matrix.

See also: [compan], page 602, [hankel], page 404, [toeplitz], page 405.

[Function File] hankel (c) [Function File] hankel (c, r)

Return the Hankel matrix constructed from the first column c, and (optionally) the last row r. If the last element of c is not the same as the first element of r, the last element of c is used. If the second argument is omitted, it is assumed to be a vector of zeros with the same size as c.

A Hankel matrix formed from an m-vector c, and an n-vector r, has the elements

H(i, j) = \{ \begin{matrix}c_{i+j−1}, & i + j − 1 ≤ m; \\ r_{i+j−m}, & otherwise. \end{matrix}

See also: [hadamard], page 404, [toeplitz], page 405.

[Function File] hilb (n)

Return the Hilbert matrix of order n. The i, j element of a Hilbert matrix is defined as

H(i, j) = \frac{1}{(i + j − 1)}

Hilbert matrices are close to being singular which make them difficult to invert with numerical routines. Comparing the condition number of a random matrix 5x5 matrix with that of a Hilbert matrix of order 5 reveals just how difficult the problem is.

     cond (rand (5))
        ⇒ 14.392
     cond (hilb (5))
        ⇒ 4.7661e+05

See also: [invhilb], page 405.

[Function File] invhilb (n)

Return the inverse of the Hilbert matrix of order n. This can be computed exactly using

\begin{aligned}A_{ij} & = −1^{i+j}(i + j − 1) ( \begin{matrix}n + i − 1 \\ n − j \end{matrix} ) ( \begin{matrix}n + j − 1 \\ n − i \end{matrix} ) ( \begin{matrix}i + j − 2 \\ i − 2 \end{matrix} )^{2} \\ & = \frac{p(i)p(j)}{(i + j − 1)} \end{aligned}

where

p(k) = −1^{k} ( \begin{matrix}k + n − 1 \\ k − 1 \end{matrix} ) ( \begin{matrix}n \\ k \end{matrix} )

The validity of this formula can easily be checked by expanding the binomial coefficients in both formulas as factorials. It can be derived more directly via the theory of Cauchy matrices. See J. W. Demmel, Applied Numerical Linear Algebra, p. 92. Compare this with the numerical calculation of inverse (hilb (n)), which suffers from the ill-conditioning of the Hilbert matrix, and the finite precision of your computer’s floating point arithmetic.

See also: [hilb], page 404.

[Function File] magic (n)

Create an n-by-n magic square. A magic square is an arrangement of the integers 1:n^2 such that the row sums, column sums, and diagonal sums are all equal to the same value. Note: n must be greater than 2 for the magic square to exist.

[Function File] pascal (n) [Function File] pascal (n, t)

Return the Pascal matrix of order n if t = 0. t defaults to 0. Return the pseudo-lower triangular Cholesky factor of the Pascal matrix if t = 1 (The sign of some columns may be negative). This matrix is its own inverse, that is pascal (n, 1) ^2 == eye (n). If t = -1, return the true Cholesky factor with strictly positive values on the diagonal. If t = 2, return a transposed and permuted version of pascal (n, 1), which is the cube root of the identity matrix. That is, pascal (n, 2) ^3 == eye (n).

See also: [chol], page 442.

[Function File] rosser ()

Return the Rosser matrix. This is a difficult test case used to evaluate eigenvalue algorithms.

See also: [wilkinson], page 406, [eig], page 437.

[Function File] toeplitz (c) [Function File] toeplitz (c, r)

Return the Toeplitz matrix constructed from the first column c, and (optionally) the first row r. If the first element of r is not the same as the first element of c, the first element of c is used. If the second argument is omitted, the first row is taken to be the same as the first column.

A square Toeplitz matrix has the form:

[ \begin{matrix}c_{0} & r_{1} & r_{2} & ⋯ & r_{n} \\ c_{1} & c_{0} & r_{1} & ⋯ & r_{n−1} \\ c_{2} & c_{1} & c_{0} & ⋯ & r_{n−2} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ c_{n} & c_{n−1} & c_{n−2} & … & c_{0} \end{matrix} ]

See also: [hankel], page 404.

[Function File] vander (c) [Function File] vander (c, n)

Return the Vandermonde matrix whose next to last column is c. If n is specified, it determines the number of columns; otherwise, n is taken to be equal to the length of c. A Vandermonde matrix has the form:

[ \begin{matrix}c_{1}^{n−1} & ⋯ & c_{1}^{2} & c_{1} & 1 \\ c_{2}^{n−1} & ⋯ & c_{2}^{2} & c_{2} & 1 \\ ⋮ & ... & ⋮ & ⋮ & ⋮ \\ c_{n}^{n−1} & ⋯ & c_{n}^{2} & c_{n} & 1 \end{matrix} ]

See also: [polyfit], page 607.

[Function File] wilkinson (n)

Return the Wilkinson matrix of order n. Wilkinson matrices are symmetric and tridiagonal with pairs of nearly, but not exactly, equal eigenvalues. They are useful in testing the behavior and performance of eigenvalue solvers.

See also: [rosser], page 405, [eig], page 437.

17 Arithmetic

Unless otherwise noted, all of the functions described in this chapter will work for real and complex scalar, vector, or matrix arguments. Functions described as mapping functions apply the given operation individually to each element when given a matrix argument. For example:

sin ([1, 2; 3, 4])
     ⇒ 0.84147   0.90930
         0.14112  -0.75680

17.1 Exponents and Logarithms

[Mapping Function] exp (x)

Compute e^{x} for each element of x. To compute the matrix exponential, see Chapter 18 [Linear Algebra], page 435.

See also: [log], page 407.

[Mapping Function] expm1 (x)

Compute e^{x} − 1 accurately in the neighborhood of zero.

See also: [exp], page 407.

[Mapping Function] log (x)

Compute the natural logarithm, ln (x), for each element of x. To compute the matrix logarithm, see Chapter 18 [Linear Algebra], page 435.

See also: [exp], page 407, [log1p], page 407, [log2], page 407, [log10], page 407, [logspace], page 393.

[Function File] reallog (x)

Return the real-valued natural logarithm of each element of x. Report an error if any element results in a complex return value.

See also: [log], page 407, [realpow], page 408, [realsqrt], page 408.

[Mapping Function] log1p (x)

Compute ln (1 + x) accurately in the neighborhood of zero.

See also: [log], page 407, [exp], page 407, [expm1], page 407.

[Mapping Function] log10 (x)

Compute the base-10 logarithm of each element of x.

See also: [log], page 407, [log2], page 407, [logspace], page 393, [exp], page 407.

[Mapping Function] log2 (x) [Mapping Function] [f, e] = log2 (x)

Compute the base-2 logarithm of each element of x. If called with two output arguments, split x into binary mantissa and exponent so that \frac{1}{2} ≤ |f| < 1 and e is an integer. If x = 0, f = e = 0.

See also: [pow2], page 408, [log], page 407, [log10], page 407, [exp], page 407.

[Mapping Function] pow2 (x) [Mapping Function] pow2 (f, e)

With one argument, computes 2^{x} for each element of x.

With two arguments, returns f · 2^{e}.

See also: [log2], page 407, [nextpow2], page 408.

[Function File] nextpow2 (x)

If x is a scalar, return the first integer n such that 2^{n} ≥ |x|.

If x is a vector, return nextpow2 (length (x)).

See also: [pow2], page 408, [log2], page 407.

[Function File] realpow (x, y)

Compute the real-valued, element-by-element power operator. This is equivalent to x .^y, except that realpow reports an error if any return value is complex.

See also: [reallog], page 407, [realsqrt], page 408.

[Mapping Function] sqrt (x)

Compute the square root of each element of x. If x is negative, a complex result is returned. To compute the matrix square root, see Chapter 18 [Linear Algebra], page 435.

See also: [realsqrt], page 408, [nthroot], page 408.

[Function File] realsqrt (x)

Return the real-valued square root of each element of x. Report an error if any element results in a complex return value.

See also: [sqrt], page 408, [realpow], page 408, [reallog], page 407.

[Mapping Function] cbrt (x)

Compute the real cube root of each element of x. Unlike x^(1/3), the result will be negative if x is negative.

See also: [nthroot], page 408.

[Function File] nthroot (x, n)

Compute the n-th root of x, returning real results for real components of x. For example:

     nthroot (-1, 3)
     ⇒ -1
     (-1) ^(1 / 3)
     ⇒ 0.50000 - 0.86603i

x must have all real entries. n must be a scalar. If n is an even integer and X has negative entries, an error is produced.

See also: [realsqrt], page 408, [sqrt], page 408, [cbrt], page 408.

17.2 Complex Arithmetic

In the descriptions of the following functions, z is the complex number x + iy, where i is defined as \sqrt{−1}.

[Mapping Function] abs (z)

Compute the magnitude of z, defined as |z| = \sqrt{x^{2} + y^{2}}.

For example:

     abs (3 + 4i) ⇒ 5

[Mapping Function] arg (z) [Mapping Function] angle (z)

Compute the argument of z, defined as, θ = atan2(y, x), in radians.

For example:

     arg (3 + 4i) ⇒ 0.92730

[Mapping Function] conj (z)

Return the complex conjugate of z, defined as z̄ = x − iy.

See also: [real], page 409, [imag], page 409.

[Function File] cplxpair (z) [Function File] cplxpair (z, tol) [Function File] cplxpair (z, tol, dim)

Sort the numbers z into complex conjugate pairs ordered by increasing real part. Place the negative imaginary complex number first within each pair. Place all the real numbers (those with abs (imag (z) / z) < tol)) after the complex pairs.

If tol is unspecified the default value is 100*eps.

By default the complex pairs are sorted along the first non-singleton dimension of z. If dim is specified, then the complex pairs are sorted along this dimension.

Signal an error if some complex numbers could not be paired. Signal an error if all complex numbers are not exact conjugates (to within tol). Note that there is no defined order for pairs with identical real parts but differing imaginary parts.

      cplxpair (exp(2i*pi*[0:4]'/5)) == exp(2i*pi*[3; 2; 4; 1; 0]/5)

[Mapping Function] imag (z)

Return the imaginary part of z as a real number.

See also: [real], page 409, [conj], page 409.

[Mapping Function] real (z)

Return the real part of z.

See also: [imag], page 409, [conj], page 409.

17.3 Trigonometry

Octave provides the following trigonometric functions where angles are specified in radians. To convert from degrees to radians multiply by π/180 (e.g., sin (30 * pi/180) returns the sine of 30 degrees). As an alternative, Octave provides a number of trigonometric functions which work directly on an argument specified in degrees. These functions are named after the base trigonometric function with a ‘d’ suffix. For example, sin expects an angle in radians while sind expects an angle in degrees.

Octave uses the C library trigonometric functions. It is expected that these functions are defined by the ISO/IEC 9899 Standard. This Standard is available at:

http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1124.pdf. Section F.9.1 deals

with the trigonometric functions. The behavior of most of the functions is relatively straightforward. However, there are some exceptions to the standard behavior. Many of the exceptions involve the behavior for -0. The most complex case is atan2. Octave exactly implements the behavior given in the Standard. Including atan2(±0, −0) returns ±π.

It should be noted that matlab uses different definitions which apparently do not distinguish -0.

[Mapping Function] sin (x)

Compute the sine for each element of x in radians.

See also: [asin], page 410, [sind], page 412, [sinh], page 411.

[Mapping Function] cos (x)

Compute the cosine for each element of x in radians.

See also: [acos], page 411, [cosd], page 412, [cosh], page 411.

[Mapping Function] tan (z)

Compute the tangent for each element of x in radians.

See also: [atan], page 411, [tand], page 412, [tanh], page 411.

[Mapping Function] sec (x)

Compute the secant for each element of x in radians.

See also: [asec], page 411, [secd], page 413, [sech], page 411.

[Mapping Function] csc (x)

Compute the cosecant for each element of x in radians.

See also: [acsc], page 411, [cscd], page 413, [csch], page 411.

[Mapping Function] cot (x)

Compute the cotangent for each element of x in radians.

See also: [acot], page 411, [cotd], page 413, [coth], page 411.

[Mapping Function] asin (x)

Compute the inverse sine in radians for each element of x.

See also: [sin], page 410, [asind], page 413.

[Mapping Function] acos (x)

Compute the inverse cosine in radians for each element of x.

See also: [cos], page 410, [acosd], page 413.

[Mapping Function] atan (x)

Compute the inverse tangent in radians for each element of x.

See also: [tan], page 410, [atand], page 413.

[Mapping Function] asec (x)

Compute the inverse secant in radians for each element of x.

See also: [sec], page 410, [asecd], page 413.

[Mapping Function] acsc (x)

Compute the inverse cosecant in radians for each element of x.

See also: [csc], page 410, [acscd], page 413.

[Mapping Function] acot (x)

Compute the inverse cotangent in radians for each element of x.

See also: [cot], page 410, [acotd], page 413.

[Mapping Function] sinh (x)

Compute the hyperbolic sine for each element of x.

See also: [asinh], page 411, [cosh], page 411, [tanh], page 411.

[Mapping Function] cosh (x)

Compute the hyperbolic cosine for each element of x.

See also: [acosh], page 412, [sinh], page 411, [tanh], page 411.

[Mapping Function] tanh (x)

Compute hyperbolic tangent for each element of x.

See also: [atanh], page 412, [sinh], page 411, [cosh], page 411.

[Mapping Function] sech (x)

Compute the hyperbolic secant of each element of x.

See also: [asech], page 412.

[Mapping Function] csch (x)

Compute the hyperbolic cosecant of each element of x.

See also: [acsch], page 412.

[Mapping Function] coth (x)

Compute the hyperbolic cotangent of each element of x.

See also: [acoth], page 412.

[Mapping Function] asinh (x)

Compute the inverse hyperbolic sine for each element of x.

See also: [sinh], page 411.

[Mapping Function] acosh (x)

Compute the inverse hyperbolic cosine for each element of x.

See also: [cosh], page 411.

[Mapping Function] atanh (x)

Compute the inverse hyperbolic tangent for each element of x.

See also: [tanh], page 411.

[Mapping Function] asech (x)

Compute the inverse hyperbolic secant of each element of x.

See also: [sech], page 411.

[Mapping Function] acsch (x)

Compute the inverse hyperbolic cosecant of each element of x.

See also: [csch], page 411.

[Mapping Function] acoth (x)

Compute the inverse hyperbolic cotangent of each element of x.

See also: [coth], page 411.

[Mapping Function] atan2 (y, x)

Compute atan (y / x) for corresponding elements of y and x. Signal an error if y and x do not match in size and orientation.

See also: [tan], page 410, [tand], page 412, [tanh], page 411, [atanh], page 412.

Octave provides the following trigonometric functions where angles are specified in degrees. These functions produce true zeros at the appropriate intervals rather than the small round-off error that occurs when using radians. For example:

cosd (90)
     ⇒ 0
cos (pi/2)
     ⇒ 6.1230e-17

[Function File] sind (x)

Compute the sine for each element of x in degrees. Returns zero for elements where x/180 is an integer.

See also: [asind], page 413, [sin], page 410.

[Function File] cosd (x)

Compute the cosine for each element of x in degrees. Returns zero for elements where (x-90)/180 is an integer.

See also: [acosd], page 413, [cos], page 410.

[Function File] tand (x)

Compute the tangent for each element of x in degrees. Returns zero for elements where x/180 is an integer and Inf for elements where (x-90)/180 is an integer.

See also: [atand], page 413, [tan], page 410.

[Function File] secd (x)

Compute the secant for each element of x in degrees.

See also: [asecd], page 413, [sec], page 410.

[Function File] cscd (x)

Compute the cosecant for each element of x in degrees.

See also: [acscd], page 413, [csc], page 410.

[Function File] cotd (x)

Compute the cotangent for each element of x in degrees.

See also: [acotd], page 413, [cot], page 410.

[Function File] asind (x)

Compute the inverse sine in degrees for each element of x.

See also: [sind], page 412, [asin], page 410.

[Function File] acosd (x)

Compute the inverse cosine in degrees for each element of x.

See also: [cosd], page 412, [acos], page 411.

[Function File] atand (x)

Compute the inverse tangent in degrees for each element of x.

See also: [tand], page 412, [atan], page 411.

[Function File] atan2d (y, x)

Compute atan2 (y / x) in degrees for corresponding elements from y and x.

See also: [tand], page 412, [atan2], page 412.

[Function File] asecd (x)

Compute the inverse secant in degrees for each element of x.

See also: [secd], page 413, [asec], page 411.

[Function File] acscd (x)

Compute the inverse cosecant in degrees for each element of x.

See also: [cscd], page 413, [acsc], page 411.

[Function File] acotd (x)

Compute the inverse cotangent in degrees for each element of x.

See also: [cotd], page 413, [acot], page 411.

17.4 Sums and Products

[Built-in Function] sum (x) [Built-in Function] sum (x, dim)

[Built-in Function] sum (…, "native") [Built-in Function] sum (…, "double") [Built-in Function] sum (…, "extra")

Sum of elements along dimension dim. If dim is omitted, it defaults to the first non-singleton dimension.

If the optional argument "native" is given, then the sum is performed in the same type as the original argument, rather than in the default double type. For example:

     sum ([true, true])
        ⇒ 2
     sum ([true, true], "native")
        ⇒ true

On the contrary, if "double" is given, the sum is performed in double precision even for single precision inputs.

For double precision inputs, "extra" indicates that a more accurate algorithm than straightforward summation is to be used. For single precision inputs, "extra" is the same as "double". Otherwise, "extra" has no effect.

See also: [cumsum], page 414, [sumsq], page 415, [prod], page 414.

[Built-in Function] prod (x) [Built-in Function] prod (x, dim)

Product of elements along dimension dim. If dim is omitted, it defaults to the first non-singleton dimension.

See also: [cumprod], page 414, [sum], page 414.

[Built-in Function] cumsum (x) [Built-in Function] cumsum (x, dim)

[Built-in Function] cumsum (…, "native") [Built-in Function] cumsum (…, "double")

[Built-in Function] cumsum (…, "extra")

Cumulative sum of elements along dimension dim. If dim is omitted, it defaults to the first non-singleton dimension.

See sum for an explanation of the optional parameters "native", "double", and

"extra".

See also: [sum], page 414, [cumprod], page 414.

[Built-in Function] cumprod (x) [Built-in Function] cumprod (x, dim)

Cumulative product of elements along dimension dim. If dim is omitted, it defaults to the first non-singleton dimension.

See also: [prod], page 414, [cumsum], page 414.

[Built-in Function] sumsq (x) [Built-in Function] sumsq (x, dim)

Sum of squares of elements along dimension dim. If dim is omitted, it defaults to the first non-singleton dimension. This function is conceptually equivalent to computing

     sum (x .* conj (x), dim)

but it uses less memory and avoids calling conj if x is real.

See also: [sum], page 414, [prod], page 414.

17.5 Utility Functions

[Mapping Function] ceil (x)

Return the smallest integer not less than x. This is equivalent to rounding towards positive infinity. If x is complex, return ceil (real (x)) + ceil (imag (x)) * I.

     ceil ([-2.7, 2.7]) ⇒ -2    3

See also: [floor], page 415, [round], page 415, [fix], page 415.

[Mapping Function] fix (x)

Truncate fractional portion of x and return the integer portion. This is equivalent to rounding towards zero. If x is complex, return fix (real (x)) + fix (imag (x)) *

I.

     fix ([-2.7, 2.7]) ⇒ -2    2

See also: [ceil], page 415, [floor], page 415, [round], page 415.

[Mapping Function] floor (x)

Return the largest integer not greater than x. This is equivalent to rounding towards negative infinity. If x is complex, return floor (real (x)) + floor (imag (x)) *

I.

     floor ([-2.7, 2.7]) ⇒ -3    2

See also: [ceil], page 415, [round], page 415, [fix], page 415.

[Mapping Function] round (x)

Return the integer nearest to x. If x is complex, return round (real (x)) + round (imag (x)) * I. If there are two nearest integers, return the one further away from zero.

     round ([-2.7, 2.7]) ⇒ -3    3

See also: [ceil], page 415, [floor], page 415, [fix], page 415, [roundb], page 415.

[Mapping Function] roundb (x)

Return the integer nearest to x. If there are two nearest integers, return the even one (banker’s rounding). If x is complex, return roundb (real (x)) + roundb (imag

(x)) * I.

See also: [round], page 415.

[Built-in Function] max (x) [Built-in Function] max (x, y) [Built-in Function] max (x, [], dim) [Built-in Function] max (x, y, dim) [Built-in Function] [w, iw] = max (x)

For a vector argument, return the maximum value. For a matrix argument, return the maximum value from each column, as a row vector, or over the dimension dim if defined, in which case y should be set to the empty matrix (it’s ignored otherwise). For two matrices (or a matrix and scalar), return the pair-wise maximum. Thus,

     max (max (x))

returns the largest element of the matrix x, and

     max (2:5, pi) ⇒ 3.1416  3.1416  4.0000  5.0000

compares each element of the range 2:5 with pi, and returns a row vector of the maximum values. For complex arguments, the magnitude of the elements are used for comparison. If called with one input and two output arguments, max also returns the first index of the maximum value(s). Thus,

     [x, ix] = max ([1, 3, 5, 2, 5])
         ⇒ x = 5
             ix = 3

See also: [min], page 416, [cummax], page 417, [cummin], page 417.

[Built-in Function] min (x) [Built-in Function] min (x, y) [Built-in Function] min (x, [], dim) [Built-in Function] min (x, y, dim) [Built-in Function] [w, iw] = min (x)

For a vector argument, return the minimum value. For a matrix argument, return the minimum value from each column, as a row vector, or over the dimension dim if defined, in which case y should be set to the empty matrix (it’s ignored otherwise). For two matrices (or a matrix and scalar), return the pair-wise minimum. Thus,

     min (min (x))

returns the smallest element of x, and

     min (2:5, pi) ⇒ 2.0000  3.0000  3.1416  3.1416

compares each element of the range 2:5 with pi, and returns a row vector of the minimum values. For complex arguments, the magnitude of the elements are used for comparison. If called with one input and two output arguments, min also returns the first index of the minimum value(s). Thus,

     [x, ix] = min ([1, 3, 0, 2, 0])
         ⇒ x = 0
             ix = 3

See also: [max], page 416, [cummin], page 417, [cummax], page 417.

[Built-in Function] cummax (x) [Built-in Function] cummax (x, dim) [Built-in Function] [w, iw] = cummax (…)

Return the cumulative maximum values along dimension dim. If dim is unspecified it defaults to column-wise operation. For example:

     cummax ([1 3 2 6 4 5]) ⇒ 1  3  3  6  6  6

If called with two output arguments the index of the maximum value is also returned.

     [w, iw] = cummax ([1 3 2 6 4 5])

⇒

     w =  1  3  3  6  6  6
     iw = 1  2  2  4  4  4

See also: [cummin], page 417, [max], page 416, [min], page 416.

[Built-in Function] cummin (x) [Built-in Function] cummin (x, dim) [Built-in Function] [w, iw] = cummin (x)

Return the cumulative minimum values along dimension dim. If dim is unspecified it defaults to column-wise operation. For example:

     cummin ([5 4 6 2 3 1]) ⇒ 5  4  4  2  2  1

If called with two output arguments the index of the minimum value is also returned.

     [w, iw] = cummin ([5 4 6 2 3 1])

⇒

     w =  5  4  4  2  2  1
     iw = 1  2  2  4  4  6

See also: [cummax], page 417, [min], page 416, [max], page 416.

[Built-in Function] hypot (x, y) [Built-in Function] hypot (x, y, z, …)

Compute the element-by-element square root of the sum of the squares of x and y. This is equivalent to sqrt (x.^2 + y.^2), but calculated in a manner that avoids overflows for large values of x or y. hypot can also be called with more than 2 arguments; in this case, the arguments are accumulated from left to right:

     hypot (hypot (x, y), z)
     hypot (hypot (hypot (x, y), z), w), etc.

[Function File] dx = gradient (m)

[Function File] [dx, dy, dz, ...] = gradient (m)

[Function File] [...] = gradient (m, s)

[Function File] [...] = gradient (m, x, y, z, …)

[Function File] [...] = gradient (f, x0)
[Function File] [...] = gradient (f, x0, s)

[Function File] [...] = gradient (f, x0, x, y, …)

Calculate the gradient of sampled data or a function. If m is a vector, calculate the one-dimensional gradient of m. If m is a matrix the gradient is calculated for each dimension.

[dx, dy] = gradient (m) calculates the one dimensional gradient for x and y direction if m is a matrix. Additional return arguments can be use for multi-dimensional matrices.

A constant spacing between two points can be provided by the s parameter. If s is a scalar, it is assumed to be the spacing for all dimensions. Otherwise, separate values of the spacing can be supplied by the x, … arguments. Scalar values specify an equidistant spacing. Vector values for the x, … arguments specify the coordinate for that dimension. The length must match their respective dimension of m.

At boundary points a linear extrapolation is applied. Interior points are calculated with the first approximation of the numerical gradient

     y'(i) = 1/(x(i+1)-x(i-1)) * (y(i-1)-y(i+1)).

If the first argument f is a function handle, the gradient of the function at the points in x0 is approximated using central difference. For example, gradient (@cos, 0) approximates the gradient of the cosine function in the point x0 = 0. As with sampled data, the spacing values between the points from which the gradient is estimated can be set via the s or dx, dy, … arguments. By default a spacing of 1 is used.

See also: [diff], page 380, [del2], page 419.

[Built-in Function] dot (x, y, dim)

Compute the dot product of two vectors. If x and y are matrices, calculate the dot products along the first non-singleton dimension. If the optional argument dim is given, calculate the dot products along this dimension.

This is equivalent to sum (conj (X) .* Y, dim), but avoids forming a temporary array and is faster. When X and Y are column vectors, the result is equivalent to X'

* Y.

See also: [cross], page 418, [divergence], page 419.

[Function File] cross (x, y) [Function File] cross (x, y, dim)

Compute the vector cross product of two 3-dimensional vectors x and y.

     cross ([1,1,0], [0,1,1]) ⇒ [ 1; -1; 1 ]

If x and y are matrices, the cross product is applied along the first dimension with 3 elements. The optional argument dim forces the cross product to be calculated along the specified dimension.

See also: [dot], page 418, [curl], page 419, [divergence], page 419.

[Function File] div = divergence (x, y, z, fx, fy, fz)
[Function File] div = divergence (fx, fy, fz)
[Function File] div = divergence (x, y, fx, fy)
[Function File] div = divergence (fx, fy)

Calculate divergence of a vector field given by the arrays fx, fy, and fz or fx, fy respectively.

divF (x, y, z) = ∂_{x}F + ∂_{y}F + ∂_{z}F

The coordinates of the vector field can be given by the arguments x, y, z or x, y respectively.

See also: [curl], page 419, [gradient], page 417, [del2], page 419, [dot], page 418.

[Function File] [cx, cy, cz, v] = curl (x, y, z, fx, fy, fz)

[Function File] [cz, v] = curl (x, y, fx, fy) [Function File] [...] = curl (fx, fy, fz) [Function File] [...] = curl (fx, fy) [Function File] v = curl (…)

Calculate curl of vector field given by the arrays fx, fy, and fz or fx, fy respectively.

curlF (x, y, z) = (\frac{∂d}{∂y}F_{z} − \frac{∂d}{∂z}F_{y}, \frac{∂d}{∂z}F_{x} − \frac{∂d}{∂x}F_{z}, \frac{∂d}{∂x}F_{y} − \frac{∂d}{∂y}F_{x})

The coordinates of the vector field can be given by the arguments x, y, z or x, y respectively. v calculates the scalar component of the angular velocity vector in direction of the z-axis for two-dimensional input. For three-dimensional input the scalar rotation is calculated at each grid point in direction of the vector field at that point.

See also: [divergence], page 419, [gradient], page 417, [del2], page 419, [cross], page 418.

[Function File] d = del2 (M) [Function File] d = del2 (M, h) [Function File] d = del2 (M, dx, dy, …)

Calculate the discrete Laplace operator (∇^{2}). For a 2-dimensional matrix M this is defined as

d = \frac{1}{4} (\frac{d^{2}}{dx^{2}} M(x, y) + \frac{d^{2}}{dy^{2}} M(x, y))

For N-dimensional arrays the sum in parentheses is expanded to include second derivatives over the additional higher dimensions. The spacing between evaluation points may be defined by h, which is a scalar defining the equidistant spacing in all dimensions. Alternatively, the spacing in each dimension may be defined separately by dx, dy, etc. A scalar spacing argument defines equidistant spacing, whereas a vector argument can be used to specify variable spacing. The length of the spacing vectors must match the respective dimension of M. The default spacing value is 1. At least 3 data points are needed for each dimension. Boundary points are calculated from the linear extrapolation of interior points.

See also: [gradient], page 417, [diff], page 380.

[Function File] factorial (n)

Return the factorial of n where n is a positive integer. If n is a scalar, this is equivalent to prod (1:n). For vector or matrix arguments, return the factorial of each element in the array. For non-integers see the generalized factorial function gamma.

See also: [prod], page 414, [gamma], page 426.

[Function File] p = factor (q) [Function File] [p, n] = factor (q)

Return the prime factorization of q. That is, prod (p) == q and every element of p is a prime number. If q == 1, return 1. With two output arguments, return the unique primes p and their multiplicities. That

is, prod (p .^n) == q.

Implementation Note: The input q must not be greater than bitmax (9.0072e+15) in order to factor correctly.

See also: [gcd], page 420, [lcm], page 420, [isprime], page 63.

[Built-in Function] g = gcd (a1, a2, …) [Built-in Function] [g, v1, ...] = gcd (a1, a2, …)

Compute the greatest common divisor of a1, a2, . . .. If more than one argument is given all arguments must be the same size or scalar. In this case the greatest common divisor is calculated for each element individually. All elements must be ordinary or Gaussian (complex) integers. Note that for Gaussian integers, the gcd is not unique up to units (multiplication by 1, -1, i or -i), so an arbitrary greatest common divisor amongst four possible is returned. Example code:

     gcd ([15, 9], [20, 18]) ⇒ 5  9

Optional return arguments v1, etc., contain integer vectors such that, g = v_{1}a_{1} + v_{2}a_{2} + ⋯

See also: [lcm], page 420, [factor], page 420.

[Mapping Function] lcm (x, y) [Mapping Function] lcm (x, y, …)

Compute the least common multiple of x and y, or of the list of all arguments. All elements must be the same size or scalar.

See also: [factor], page 420, [gcd], page 420.

[Function File] chop (x, ndigits, base)

Truncate elements of x to a length of ndigits such that the resulting numbers are exactly divisible by base. If base is not specified it defaults to 10.

     chop (-pi, 5, 10)
        ⇒ -3.14200000000000
     chop (-pi, 5, 5)
        ⇒ -3.14150000000000

[Mapping Function] rem (x, y) [Mapping Function] fmod (x, y)

Return the remainder of the division x / y, computed using the expression

     x - y .* fix (x ./ y)

An error message is printed if the dimensions of the arguments do not agree, or if either of the arguments is complex.

See also: [mod], page 421.

[Mapping Function] mod (x, y)

Compute the modulo of x and y. Conceptually this is given by

     x - y .* floor (x ./ y)

and is written such that the correct modulus is returned for integer types. This function handles negative values correctly. That is, mod (-1, 3) is 2, not -1, as rem (-1, 3) returns. mod (x, 0) returns x. An error results if the dimensions of the arguments do not agree, or if either of the arguments is complex.

See also: [rem], page 421.

[Function File] primes (n)

Return all primes up to n. The algorithm used is the Sieve of Eratosthenes. Note that if you need a specific number of primes you can use the fact that the distance from one prime to the next is, on average, proportional to the logarithm of the prime. Integrating, one finds that there are about k primes less than k log(5k).

See also: [list primes], page 421, [isprime], page 63.

[Function File] list_primes () [Function File] list_primes (n)

List the first n primes. If n is unspecified, the first 25 primes are listed. The algorithm used is from page 218 of the TEXbook.

See also: [primes], page 421, [isprime], page 63.

[Mapping Function] sign (x)

Compute the signum function, which is defined as

sign(x) = \{ \begin{matrix}1, & x > 0; \\ 0, & x = 0; \\ −1, & x < 0. \end{matrix}

For complex arguments, sign returns x ./ abs (x). Note that sign (-0.0) is 0. Although IEEE 754 floating point allows zero to be signed, 0.0 and -0.0 compare equal. If you must test whether zero is signed, use the signbit function.

See also: [signbit], page 422.

[Mapping Function] signbit (x)

Return logical true if the value of x has its sign bit set. Otherwise return logical false. This behavior is consistent with the other logical functions. SeeSection 4.6 [Logical Values], page 59. The behavior differs from the C language function which returns non-zero if the sign bit is set.

This is not the same as x < 0.0, because IEEE 754 floating point allows zero to be signed. The comparison -0.0 < 0.0 is false, but signbit (-0.0) will return a nonzero value.

See also: [sign], page 421.

17.6 Special Functions

[Built-in Function] [a, ierr] = airy (k, z, opt)

Compute Airy functions of the first and second kind, and their derivatives.

      K   Function   Scale factor (if "opt" is supplied)
     ---  --------   ---------------------------------------
      0   Ai (Z)     exp ((2/3) * Z * sqrt (Z))
      1   dAi(Z)/dZ  exp ((2/3) * Z * sqrt (Z))
      2   Bi (Z)     exp (-abs (real ((2/3) * Z * sqrt (Z))))
      3   dBi(Z)/dZ  exp (-abs (real ((2/3) * Z * sqrt (Z))))

The function call airy (z) is equivalent to airy (0, z).

The result is the same size as z.

If requested, ierr contains the following status information and is the same size as the result.

0. Normal return.

1. Input error, return NaN.

2. Overflow, return Inf.

3. Loss of significance by argument reduction results in less than half of machine accuracy.

4. Complete loss of significance by argument reduction, return NaN.

5. Error—no computation, algorithm termination condition not met, return NaN.

[Built-in Function] [j, ierr] = besselj (alpha, x, opt)
[Built-in Function] [y, ierr] = bessely (alpha, x, opt)
[Built-in Function] [i, ierr] = besseli (alpha, x, opt)
[Built-in Function] [k, ierr] = besselk (alpha, x, opt)
[Built-in Function] [h, ierr] = besselh (alpha, k, x, opt)

Compute Bessel or Hankel functions of various kinds:

besselj Bessel functions of the first kind. If the argument opt is supplied, the result is multiplied by exp (-abs (imag (x))).

bessely Bessel functions of the second kind. If the argument opt is supplied, the result is multiplied by exp (-abs (imag (x))).

besseli

Modified Bessel functions of the first kind. If the argument opt is supplied, the result is multiplied by exp (-abs (real (x))).

besselk

Modified Bessel functions of the second kind. If the argument opt is supplied, the result is multiplied by exp (x).

besselh Compute Hankel functions of the first (k = 1) or second (k = 2) kind. If the argument opt is supplied, the result is multiplied by exp (-I*x) for k = 1 or exp (I*x) for k = 2.

If alpha is a scalar, the result is the same size as x. If x is a scalar, the result is the same size as alpha. If alpha is a row vector and x is a column vector, the result is a matrix with length (x) rows and length (alpha) columns. Otherwise, alpha and x must conform and the result will be the same size. The value of alpha must be real. The value of x may be complex. If requested, ierr contains the following status information and is the same size as the result.

0. Normal return.

1. Input error, return NaN.

2. Overflow, return Inf.

3. Loss of significance by argument reduction results in less than half of machine accuracy.

4. Complete loss of significance by argument reduction, return NaN.

5. Error—no computation, algorithm termination condition not met, return NaN.

[Mapping Function] beta (a, b)

For real inputs, return the Beta function,

B(a, b) = \frac{Γ(a)Γ(b)}{Γ(a + b)}.

See also: [betaln], page 424, [betainc], page 423.

[Mapping Function] betainc (x, a, b)

Return the regularized incomplete Beta function,

I(x, a, b) = \frac{1}{B(a, b)} ∫_{0}^{x} t^{(a−z)}(1 − t)^{(b−1)}dt.

If x has more than one component, both a and b must be scalars. If x is a scalar, a and b must be of compatible dimensions.

See also: [betaincinv], page 423, [beta], page 423, [betaln], page 424.

[Mapping Function] betaincinv (y, a, b)

Compute the inverse of the incomplete Beta function, i.e., x such that

     y == betainc (x, a, b)

See also: [betainc], page 423, [beta], page 423, [betaln], page 424.

[Mapping Function] betaln (a, b)

Return the natural logarithm of the Beta function,

betaln(a, b) = ln(B(a, b)) ≡ ln(\frac{Γ(a)Γ(b)}{Γ(a + b)}).

calculated in a way to reduce the occurrence of underflow.

See also: [beta], page 423, [betainc], page 423, [gammaln], page 428.

[Mapping Function] bincoeff (n, k)

Return the binomial coefficient of n and k, defined as

\binom{n}{k} = \frac{n(n − 1)(n − 2) ⋯ (n − k + 1)}{k!}

For example:

     bincoeff (5, 2)
        ⇒ 10

In most cases, the nchoosek function is faster for small scalar integer arguments. It also warns about loss of precision for big arguments.

See also: [nchoosek], page 568.

[Function File] commutation_matrix (m, n)

Return the commutation matrix K_{m,n} which is the unique mn×mn matrix such that K_{m,n} · vec(A) = vec(A^{T}) for all m × n matrices A.

If only one argument m is given, K_{m,m} is returned.

See Magnus and Neudecker (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics.

[Function File] duplication_matrix (n)

Return the duplication matrix D_{n} which is the unique n^{2} × n(n + 1)/2 matrix such that D_{n} ∗ vech(A) = vec(A) for all symmetric n × n matrices A.

See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics.

[Mapping Function] dawson (z)

Compute the Dawson (scaled imaginary error) function,

\frac{\sqrt{π}}{2} e^{−z^{2}}erfi(z) ≡ −i\frac{\sqrt{π}}{2} e^{−z^{2}}erf(iz)

See also: [erfc], page 425, [erf], page 425, [erfcx], page 426, [erfi], page 426, [erfinv], page 426, [erfcinv], page 426.

[Built-in Function] [sn, cn, dn, err] = ellipj (u, m)
[Built-in Function] [sn, cn, dn, err] = ellipj (u, m, tol)

Compute the Jacobi elliptic functions sn, cn, and dn of complex argument u and real parameter m. If m is a scalar, the results are the same size as u. If u is a scalar, the results are the same size as m. If u is a column vector and m is a row vector, the results are matrices with length (u) rows and length (m) columns. Otherwise, u and m must conform in size and the results will be the same size as the inputs. The value of u may be complex. The value of m must be 0 ≤ m ≤ 1. The optional input tol is currently ignored (matlab uses this to allow faster, less accurate approximation). If requested, err contains the following status information and is the same size as the result.

0. Normal return.

1. Error—no computation, algorithm termination condition not met, return NaN.

Reference: Milton Abramowitz and Irene A Stegun, Handbook of Mathematical Functions, Chapter 16 (Sections 16.4, 16.13, and 16.15), Dover, 1965.

See also: [ellipke], page 425.

[Function File] k = ellipke (m) [Function File] k = ellipke (m, tol) [Function File] [k, e] = ellipke (…)

Compute complete elliptic integrals of the first K(m) and second E(m) kind. m must be a scalar or real array with -Inf ≤ m ≤ 1. The optional input tol is currently ignored (matlab uses this to allow a faster, less accurate approximation). Called with only one output, elliptic integrals of the first kind are returned. Reference: Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, Chapter 17, Dover, 1965.

See also: [ellipj], page 425.

[Mapping Function] erf (z)

Compute the error function,

erf(z) = \frac{2}{\sqrt{π}} ∫_{0}^{z} e^{−t^{2}}dt

See also: [erfc], page 425, [erfcx], page 426, [erfi], page 426, [dawson], page 424, [erfinv], page 426, [erfcinv], page 426.

[Mapping Function] erfc (z)

Compute the complementary error function, 1 − erf(z).

See also: [erfcinv], page 426, [erfcx], page 426, [erfi], page 426, [dawson], page 424, [erf], page 425, [erfinv], page 426.

[Mapping Function] erfcx (z)

Compute the scaled complementary error function,

e^{z^{2}}erfc(z) ≡ e^{z^{2}}(1 − erf(z))

See also: [erfc], page 425, [erf], page 425, [erfi], page 426, [dawson], page 424, [erfinv], page 426, [erfcinv], page 426.

[Mapping Function] erfi (z)

Compute the imaginary error function,

−ierf(iz)

See also: [erfc], page 425, [erf], page 425, [erfcx], page 426, [dawson], page 424, [erfinv], page 426, [erfcinv], page 426.

[Mapping Function] erfinv (x)

Compute the inverse error function, i.e., y such that

     erf (y) == x

See also: [erf], page 425, [erfc], page 425, [erfcx], page 426, [erfi], page 426, [dawson], page 424, [erfcinv], page 426.

[Mapping Function] erfcinv (x)

Compute the inverse complementary error function, i.e., y such that

     erfc (y) == x

See also: [erfc], page 425, [erf], page 425, [erfcx], page 426, [erfi], page 426, [dawson], page 424, [erfinv], page 426.

[Function File] expint (x)

Compute the exponential integral:

E_{1}(x) = ∫_{x}^{∞} \frac{e^{−t}}{t} dt

Note: For compatibility, this functions uses the matlab definition of the exponential integral. Most other sources refer to this particular value as E_{1}(x), and the exponential integral is

Ei(x) = − ∫_{−x}^{∞} \frac{e^{−t}}{t} dt.

The two definitions are related, for positive real values of x, by E_{1}(−x) = −Ei(x)−iπ.

[Mapping Function] gamma (z)

Compute the Gamma function,

Γ(z) = ∫_{0}^{∞} t^{z−1}e^{−t}dt.

See also: [gammainc], page 427, [lgamma], page 428.

[Mapping Function] gammainc (x, a) [Mapping Function] gammainc (x, a, "lower") [Mapping Function] gammainc (x, a, "upper")

Compute the normalized incomplete gamma function,

γ(x, a) = \frac{1}{Γ(a)}∫_{0}^{x} t^{a−1}e^{−t}dt

with the limiting value of 1 as x approaches infinity. The standard notation is P (a, x), e.g., Abramowitz and Stegun (6.5.1). If a is scalar, then gammainc (x, a) is returned for each element of x and vice versa. If neither x nor a is scalar, the sizes of x and a must agree, and gammainc is applied element-by-element. By default the incomplete gamma function integrated from 0 to x is computed. If "upper" is given then the complementary function integrated from x to infinity is calculated. It should be noted that

     gammainc (x, a) ≡ 1 - gammainc (x, a, "upper")

See also: [gamma], page 426, [lgamma], page 428.

[Function File] l = legendre (n, x)

[Function File] l = legendre (n, x, normalization)

Compute the Legendre function of degree n and order m = 0 … N. The optional argument, normalization, may be one of "unnorm", "sch", or "norm". The default is "unnorm". The value of n must be a non-negative scalar integer. If the optional argument normalization is missing or is "unnorm", compute the Legendre function of degree n and order m and return all values for m = 0 … n. The return value has one dimension more than x. The Legendre Function of degree n and order m:

P_{n}^{m}(x) = (−1)^{m}(1 − x^{2})^{m/2} \frac{d^{m}}{dx^{m}} P_{n}(x)

with Legendre polynomial of degree n:

P (x) = \frac{1}{2^{n}n!}(\frac{d^{n}}{dx^{n}} (x^{2} − 1)^{n})

legendre (3, [-1.0, -0.9, -0.8]) returns the matrix:
      x  |   -1.0   |   -0.9   |   -0.8
     ------------------------------------
     m=0 | -1.00000 | -0.47250 | -0.08000
     m=1 |  0.00000 | -1.99420 | -1.98000
     m=2 |  0.00000 | -2.56500 | -4.32000
     m=3 |  0.00000 | -1.24229 | -3.24000

If the optional argument normalization is "sch", compute the Schmidt semi-normalized associated Legendre function. The Schmidt semi-normalized associated Legendre function is related to the unnormalized Legendre functions by the following: For Legendre functions of degree n and order 0:

SP_{n}^{0}(x) = P_{n}^{0}(x)

For Legendre functions of degree n and order m:

SP_{n}^{m}(x) = P_{n}^{m}(x)(−1)^{m}(\frac{2(n − m)!}{(n + m)!})^{0.5}

If the optional argument normalization is "norm", compute the fully normalized associated Legendre function. The fully normalized associated Legendre function is related to the unnormalized Legendre functions by the following: For Legendre functions of degree n and order m

NP_{n}^{m}(x) = P_{n}^{m}(x)(−1)^{m}(\frac{(n + 0.5)(n − m)!}{(n + m)!})^{0.5}

[Mapping Function] lgamma (x) [Mapping Function] gammaln (x)

Return the natural logarithm of the gamma function of x.

See also: [gamma], page 426, [gammainc], page 427.

17.7 Rational Approximations

[Function File] s = rat (x, tol) [Function File] [n, d] = rat (x, tol)

Find a rational approximation to x within the tolerance defined by tol using a continued fraction expansion. For example:

     rat (pi) = 3 + 1/(7 + 1/16) = 355/113
     rat (e) = 3 + 1/(-4 + 1/(2 + 1/(5 + 1/(-2 + 1/(-7)))))
             = 1457/536

Called with two arguments returns the numerator and denominator separately as two matrices.

See also: [rats], page 428.

[Built-in Function] rats (x, len)

Convert x into a rational approximation represented as a string. You can convert the string back into a matrix as follows:

     r = rats (hilb (4));
     x = str2num (r)

The optional second argument defines the maximum length of the string representing the elements of x. By default len is 9.

See also: [format], page 224, [rat], page 428.

17.8 Coordinate Transformations

[Function File] [theta, r] = cart2pol (x, y)
[Function File] [theta, r, z] = cart2pol (x, y, z)
[Function File] [theta, r] = cart2pol (C)
[Function File] [theta, r, z] = cart2pol (C)

[Function File] P = cart2pol (…)

Transform Cartesian to polar or cylindrical coordinates. theta describes the angle relative to the positive x-axis. r is the distance to the z-axis (0, 0, z). x, y (, and z) must be the same shape, or scalar. If called with a single matrix argument then each row of C represents the Cartesian coordinate (x, y (, z)). If only a single return argument is requested then return a matrix P where each row represents one polar/(cylindrical) coordinate (theta, phi (, z)).

See also: [pol2cart], page 429, [cart2sph], page 429, [sph2cart], page 429.

[Function File] [x, y] = pol2cart (theta, r)
[Function File] [x, y, z] = pol2cart (theta, r, z)

[Function File] [x, y] = pol2cart (P)

[Function File] [x, y, z] = pol2cart (P)

[Function File] C = pol2cart (…)

Transform polar or cylindrical to Cartesian coordinates. theta, r, (and z) must be the same shape, or scalar. theta describes the angle relative to the positive x-axis. r is the distance to the z-axis (0, 0, z). If called with a single matrix argument then each row of P represents the polar/(cylindrical) coordinate (theta, r (, z)). If only a single return argument is requested then return a matrix C where each row represents one Cartesian coordinate (x, y (, z)).

See also: [cart2pol], page 429, [sph2cart], page 429, [cart2sph], page 429.

[Function File] [theta, phi, r] = cart2sph (x, y, z)
[Function File] [theta, phi, r] = cart2sph (C)

[Function File] S = cart2sph (…)

Transform Cartesian to spherical coordinates. theta describes the angle relative to the positive x-axis. phi is the angle relative to the xy-plane. r is the distance to the origin (0, 0, 0). x, y, and z must be the same shape, or scalar. If called with a single matrix argument then each row of C represents the Cartesian coordinate (x, y, z). If only a single return argument is requested then return a matrix S where each row represents one spherical coordinate (theta, phi, r).

See also: [sph2cart], page 429, [cart2pol], page 429, [pol2cart], page 429.

[Function File] [x, y, z] = sph2cart (theta, phi, r)
[Function File] [x, y, z] = sph2cart (S)

[Function File] C = sph2cart (…)

Transform spherical to Cartesian coordinates. theta describes the angle relative to the positive x-axis. phi is the angle relative to the xy-plane. r is the distance to the origin (0, 0, 0). theta, phi, and r must be the same shape, or scalar. If called with a single matrix argument then each row of S represents the spherical coordinate (theta, phi, r). If only a single return argument is requested then return a matrix C where each row represents one Cartesian coordinate (x, y, z).

See also: [cart2sph], page 429, [pol2cart], page 429, [cart2pol], page 429.

17.9 Mathematical Constants

[Built-in Function] e [Built-in Function] e (n) [Built-in Function] e (n, m)

[Built-in Function] e (n, m, k, …)

[Built-in Function] e (…, class)

Return a scalar, matrix, or N-dimensional array whose elements are all equal to the base of natural logarithms. The constant e satisfies the equation log(e) = 1. When called with no arguments, return a scalar with the value e. When called with a single argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The optional argument class specifies the return type and may be either "double" or

"single".

See also: [log], page 407, [exp], page 407, [pi], page 430, [I], page 430.

[Built-in Function] pi [Built-in Function] pi (n) [Built-in Function] pi (n, m) [Built-in Function] pi (n, m, k, …) [Built-in Function] pi (…, class)

Return a scalar, matrix, or N-dimensional array whose elements are all equal to the ratio of the circumference of a circle to its diameter(π). Internally, pi is computed as

‘4.0 * atan (1.0)’.

When called with no arguments, return a scalar with the value of π. When called with a single argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The optional argument class specifies the return type and may be either

"double" or "single".

See also: [e], page 430, [I], page 430.

[Built-in Function] I [Built-in Function] I (n) [Built-in Function] I (n, m)

[Built-in Function] I (n, m, k, …)

[Built-in Function] I (…, class)

Return a scalar, matrix, or N-dimensional array whose elements are all equal to the pure imaginary unit, defined as \sqrt{−1}. I, and its equivalents i, j, and J, are functions so any of the names may be reused for other purposes (such as i for a counter variable). When called with no arguments, return a scalar with the value i. When called with a single argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The optional argument class specifies the return type and may be either "double" or

"single".

See also: [e], page 430, [pi], page 430, [log], page 407, [exp], page 407.

[Built-in Function] Inf [Built-in Function] Inf (n) [Built-in Function] Inf (n, m) [Built-in Function] Inf (n, m, k, …) [Built-in Function] Inf (…, class)

Return a scalar, matrix or N-dimensional array whose elements are all equal to the IEEE representation for positive infinity. Infinity is produced when results are too large to be represented using the the IEEE floating point format for numbers. Two common examples which produce infinity are division by zero and overflow.

     [ 1/0 e^800 ] ⇒ Inf   Inf

When called with no arguments, return a scalar with the value ‘Inf’. When called with a single argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The optional argument class specifies the return type and may be either

"double" or "single".

See also: [isinf], page 380, [NaN], page 431.

[Built-in Function] NaN [Built-in Function] NaN (n) [Built-in Function] NaN (n, m) [Built-in Function] NaN (n, m, k, …) [Built-in Function] NaN (…, class)

Return a scalar, matrix, or N-dimensional array whose elements are all equal to the IEEE symbol NaN (Not a Number). NaN is the result of operations which do not produce a well defined numerical result. Common operations which produce a NaN are arithmetic with infinity (∞ − ∞), zero divided by zero (0/0), and any operation involving another NaN value (5 + NaN). Note that NaN always compares not equal to NaN (NaN != NaN). This behavior is specified by the IEEE standard for floating point arithmetic. To find NaN values, use the isnan function.

When called with no arguments, return a scalar with the value ‘NaN’. When called with a single argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The optional argument class specifies the return type and may be either

"double" or "single".

See also: [isnan], page 380, [Inf], page 431.

[Built-in Function] eps [Built-in Function] eps (x) [Built-in Function] eps (n, m) [Built-in Function] eps (n, m, k, …) [Built-in Function] eps (…, class)

Return a scalar, matrix or N-dimensional array whose elements are all eps, the machine precision. More precisely, eps is the relative spacing between any two adjacent numbers in the machine’s floating point system. This number is obviously system dependent. On machines that support IEEE floating point arithmetic, eps is approximately 2.2204 × 10^{−16} for double precision and 1.1921 × 10^{−7} for single precision. When called with no arguments, return a scalar with the value eps (1.0). Given a single argument x, return the distance between x and the next largest value. When called with more than one argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The optional argument class specifies the return type and may be either "double" or

"single".

See also: [realmax], page 432, [realmin], page 432, [intmax], page 55, [bitmax], page 57.

[Built-in Function] realmax [Built-in Function] realmax (n) [Built-in Function] realmax (n, m) [Built-in Function] realmax (n, m, k, …) [Built-in Function] realmax (…, class)

Return a scalar, matrix or N-dimensional array whose elements are all equal to the largest floating point number that is representable. The actual value is system dependent. On machines that support IEEE floating point arithmetic, realmax is approximately 1.7977 × 10^{308} for double precision and 3.4028 × 10^{38} for single precision. When called with no arguments, return a scalar with the value realmax ("double"). When called with a single argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The optional argument class specifies the return type and may be either "double" or "single".

See also: [realmin], page 432, [intmax], page 55, [bitmax], page 57, [eps], page 432.

[Built-in Function] realmin [Built-in Function] realmin (n)

[Built-in Function] realmin (n, m) [Built-in Function] realmin (n, m, k, …) [Built-in Function] realmin (…, class)

Return a scalar, matrix or N-dimensional array whose elements are all equal to the smallest normalized floating point number that is representable. The actual value is system dependent. On machines that support IEEE floating point arithmetic, realmin is approximately 2.2251 × 10^{−308} for double precision and 1.1755 × 10^{−38} for single precision. When called with no arguments, return a scalar with the value realmin ("double"). When called with a single argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The optional argument class specifies the return type and may be either "double" or "single".

See also: [realmax], page 432, [intmin], page 55, [eps], page 432.

18 Linear Algebra

This chapter documents the linear algebra functions provided in Octave. Reference material for many of these functions may be found in Golub and Van Loan, Matrix Computations, 2nd Ed., Johns Hopkins, 1989, and in the lapack Users’ Guide, SIAM, 1992. The lapack Users’ Guide is available at: http://www.netlib.org/lapack/lug/

A common text for engineering courses is G. Strang, Linear Algebra and Its Applications, 4th Edition. It has become a widespread reference for linear algebra. An alternative is P. Lax Linear Algebra and Its Applications, and also is a good choice. It claims to be suitable for high school students with substantial mathematical interests as well as firstyear undergraduates.

18.1 Techniques Used for Linear Algebra

Octave includes a polymorphic solver that selects an appropriate matrix factorization depending on the properties of the matrix itself. Generally, the cost of determining the matrix type is small relative to the cost of factorizing the matrix itself. In any case the matrix type is cached once it is calculated so that it is not re-determined each time it is used in a linear equation.

The selection tree for how the linear equation is solved or a matrix inverse is formed is given by:

1. If the matrix is upper or lower triangular sparse use a forward or backward substitution using the lapack xTRTRS function, and goto 4.

2. If the matrix is square, Hermitian with a real positive diagonal, attempt Cholesky factorization using the lapack xPOTRF function.

3. If the Cholesky factorization failed or the matrix is not Hermitian with a real positive diagonal, and the matrix is square, factorize using the lapack xGETRF function.

4. If the matrix is not square, or any of the previous solvers flags a singular or near singular matrix, find a least squares solution using the lapack xGELSD function.

The user can force the type of the matrix with the matrix_type function. This overcomes the cost of discovering the type of the matrix. However, it should be noted that identifying the type of the matrix incorrectly will lead to unpredictable results, and so matrix_type should be used with care.

It should be noted that the test for whether a matrix is a candidate for Cholesky factorization, performed above, and by the matrix_type function, does not make certain that the matrix is Hermitian. However, the attempt to factorize the matrix will quickly detect a non-Hermitian matrix.

18.2 Basic Matrix Functions

[Built-in Function] AA = balance (A) [Built-in Function] AA = balance (A, opt) [Built-in Function] [DD, AA] = balance (A, opt) [Built-in Function] [D, P, AA] = balance (A, opt)

[Built-in Function] [CC, DD, AA, BB] = balance (A, B, opt)

Compute AA = DD \ A * DD in which AA is a matrix whose row and column norms are roughly equal in magnitude, and DD = P * D, in which P is a permutation matrix and D is a diagonal matrix of powers of two. This allows the equilibration to be computed without round-off. Results of eigenvalue calculation are typically improved by balancing first. If two output values are requested, balance returns the diagonal D and the permutation P separately as vectors. In this case, DD = eye(n)(:,P) * diag (D), where n is the matrix size. If four output values are requested, compute AA = CC*A*DD and BB = CC*B*DD, in which AA and BB have non-zero elements of approximately the same magnitude and CC and DD are permuted diagonal matrices as in DD for the algebraic eigenvalue problem. The eigenvalue balancing option opt may be one of:

"noperm", "S"

Scale only; do not permute.

"noscal", "P"

Permute only; do not scale.

Algebraic eigenvalue balancing uses standard lapack routines. Generalized eigenvalue problem balancing uses Ward’s algorithm (SIAM Journal on Scientific and Statistical Computing, 1981).

[Function File] cond (A) [Function File] cond (A, p)

Compute the p-norm condition number of a matrix. cond (A) is defined as ∥ A ∥_{p} ∗ ∥ A^{−1} ∥_{p} . By default, p = 2 is used which implies a (relatively slow) singular value decomposition. Other possible selections are p = 1, Inf, "fro" which are generally faster. See norm for a full discussion of possible p values. The condition number of a matrix quantifies the sensitivity of the matrix inversion operation when small changes are made to matrix elements. Ideally the condition number will be close to 1. When the number is large this indicates small changes (such as underflow or round-off error) will produce large changes in the resulting output. In such cases the solution results from numerical computing are not likely to be accurate.

See also: [condest], page 510, [rcond], page 441, [norm], page 439, [svd], page 450.

[Built-in Function] det (A) [Built-in Function] [d, rcond] = det (A)

Compute the determinant of A. Return an estimate of the reciprocal condition number if requested. Routines from lapack are used for full matrices and code from umfpack is used for sparse matrices.

The determinant should not be used to check a matrix for singularity. For that, use any of the condition number functions: cond, condest, rcond.

See also: [cond], page 436, [condest], page 510, [rcond], page 441.

[Built-in Function] lambda = eig (A) [Built-in Function] lambda = eig (A, B) [Built-in Function] [V, lambda] = eig (A) [Built-in Function] [V, lambda] = eig (A, B)

Compute the eigenvalues (and optionally the eigenvectors) of a matrix or a pair of matrices The algorithm used depends on whether there are one or two input matrices, if they are real or complex and if they are symmetric (Hermitian if complex) or nonsymmetric. The eigenvalues returned by eig are not ordered.

See also: [eigs], page 513, [svd], page 450.

[Built-in Function] g = givens (x, y) [Built-in Function] [c, s] = givens (x, y)

Return a 2 × 2 orthogonal matrix

G = [ \begin{matrix}c & s \\ −s′ & c \end{matrix} ]

such that

G [ \begin{matrix}x \\ y \end{matrix} ] = [ \begin{matrix}∗ \\ 0 \end{matrix} ]

with x and y scalars. For example:

     givens (1, 1)
        ⇒ 0.70711   0.70711
            -0.70711   0.70711

[Function File] [g, y] = planerot (x)

Given a two-element column vector, returns the 2 × 2 orthogonal matrix G such that

y = g * x and y(2) = 0.

See also: [givens], page 437.

[Built-in Function] x = inv (A) [Built-in Function] [x, rcond] = inv (A)

Compute the inverse of the square matrix A. Return an estimate of the reciprocal condition number if requested, otherwise warn of an ill-conditioned matrix if the reciprocal condition number is small. In general it is best to avoid calculating the inverse of a matrix directly. For example, it is both faster and more accurate to solve systems of equations (A*x = b) with y = A \ b, rather than y = inv (A) * b. If called with a sparse matrix, then in general x will be a full matrix requiring significantly more storage. Avoid forming the inverse of a sparse matrix if possible.

See also: [ldivide], page 139, [rdivide], page 140.

[Function File] x = linsolve (A, b) [Function File] x = linsolve (A, b, opts) [Function File] [x, R] = linsolve (…)

Solve the linear system A*x = b. With no options, this function is equivalent to the left division operator (x = A \ b) or the matrix-left-divide function (x = mldivide (A, b)). Octave ordinarily examines the properties of the matrix A and chooses a solver that best matches the matrix. By passing a structure opts to linsolve you can inform Octave directly about the matrix A. In this case Octave will skip the matrix examination and proceed directly to solving the linear system. Warning: If the matrix A does not have the properties listed in the opts structure then the result will not be accurate AND no warning will be given. When in doubt, let Octave examine the matrix and choose the appropriate solver as this step takes little time and the result is cached so that it is only done once per linear system. Possible opts fields (set value to true/false):

LT A is lower triangular

UT A is upper triangular

UHESS A is upper Hessenberg (currently makes no difference)

SYM A is symmetric or complex Hermitian (currently makes no difference)

POSDEF A is positive definite

RECT A is general rectangular (currently makes no difference)

TRANSA Solve A'*x = b by transpose (A) \ b

The optional second output R is the inverse condition number of A (zero if matrix is singular).

See also: [mldivide], page 139, [matrix type], page 438, [rcond], page 441.

[Built-in Function] type = matrix_type (A) [Built-in Function] type = matrix_type (A, "nocompute") [Built-in Function] A = matrix_type (A, type) [Built-in Function] A = matrix_type (A, "upper", perm) [Built-in Function] A = matrix_type (A, "lower", perm) [Built-in Function] A = matrix_type (A, "banded", nl, nu)

Identify the matrix type or mark a matrix as a particular type. This allows more rapid solutions of linear equations involving A to be performed. Called with a single argument, matrix_type returns the type of the matrix and caches it for future use. Called with more than one argument, matrix_type allows the type of the matrix to be defined. If the option "nocompute" is given, the function will not attempt to guess the type if it is still unknown. This is useful for debugging purposes. The possible matrix types depend on whether the matrix is full or sparse, and can be one of the following

"unknown"

Remove any previously cached matrix type, and mark type as unknown.

"full" Mark the matrix as full.

"positive definite"

Probable full positive definite matrix.

"diagonal"

Diagonal matrix. (Sparse matrices only)

"permuted diagonal"

Permuted Diagonal matrix. The permutation does not need to be specifically indicated, as the structure of the matrix explicitly gives this. (Sparse matrices only)

"upper" Upper triangular. If the optional third argument perm is given, the matrix is assumed to be a permuted upper triangular with the permutations defined by the vector perm.

"lower" Lower triangular. If the optional third argument perm is given, the matrix is assumed to be a permuted lower triangular with the permutations defined by the vector perm.

"banded"
"banded positive definite"

Banded matrix with the band size of nl below the diagonal and nu above it. If nl and nu are 1, then the matrix is tridiagonal and treated with specialized code. In addition the matrix can be marked as probably a positive definite. (Sparse matrices only)

"singular"

The matrix is assumed to be singular and will be treated with a minimum norm solution.

Note that the matrix type will be discovered automatically on the first attempt to solve a linear equation involving A. Therefore matrix_type is only useful to give Octave hints of the matrix type. Incorrectly defining the matrix type will result in incorrect results from solutions of linear equations; it is entirely the responsibility of the user to correctly identify the matrix type. Also, the test for positive definiteness is a low-cost test for a Hermitian matrix with a real positive diagonal. This does not guarantee that the matrix is positive definite, but only that it is a probable candidate. When such a matrix is factorized, a Cholesky factorization is first attempted, and if that fails the matrix is then treated with an LU factorization. Once the matrix has been factorized, matrix_type will return the correct classification of the matrix.

[Built-in Function] norm (A) [Built-in Function] norm (A, p) [Built-in Function] norm (A, p, opt)

Compute the p-norm of the matrix A. If the second argument is missing, p = 2 is assumed. If A is a matrix (or sparse matrix): p = 1 1-norm, the largest column sum of the absolute values of A.

p = 2 Largest singular value of A.

p = Inf or "inf"

Infinity norm, the largest row sum of the absolute values of A.

p = "fro"
          Frobenius norm of A, sqrt (sum (diag (A' * A))).

other p, p > 1

          maximum norm (A*x, p) such that norm (x, p) == 1

If A is a vector or a scalar:

p = Inf or "inf"
          max (abs (A)).
p = -Inf  min (abs (A)).
p = "fro"
          Frobenius norm of A, sqrt (sumsq (abs (A))).

p = 0 Hamming norm - the number of nonzero elements.

other p, p > 1

          p-norm of A, (sum (abs (A) .^p)) ^(1/p).

other p p < 1 the p-pseudonorm defined as above.

If opt is the value "rows", treat each row as a vector and compute its norm. The result is returned as a column vector. Similarly, if opt is "columns" or "cols" then compute the norms of each column and return a row vector.

See also: [cond], page 436, [svd], page 450.

[Function File] null (A) [Function File] null (A, tol)

Return an orthonormal basis of the null space of A. The dimension of the null space is taken as the number of singular values of A not greater than tol. If the argument tol is missing, it is computed as

     max (size (A)) * max (svd (A)) * eps

See also: [orth], page 440.

[Function File] orth (A) [Function File] orth (A, tol)

Return an orthonormal basis of the range space of A. The dimension of the range space is taken as the number of singular values of A greater than tol. If the argument tol is missing, it is computed as

     max (size (A)) * max (svd (A)) * eps

See also: [null], page 440.

[Built-in Function] [y, h] = mgorth (x, v)

Orthogonalize a given column vector x with respect to a set of orthonormal vectors comprising the columns of v using the modified Gram-Schmidt method. On exit, y is a unit vector such that:

       norm (y) = 1
       v' * y = 0
       x = [v, y]*h'

[Built-in Function] pinv (x) [Built-in Function] pinv (x, tol)

Return the pseudoinverse of x. Singular values less than tol are ignored.

If the second argument is omitted, it is taken to be

     tol = max (size (x)) * sigma_max (x) * eps,

where sigma_max (x) is the maximal singular value of x.

[Function File] rank (A) [Function File] rank (A, tol)

Compute the rank of matrix A, using the singular value decomposition.

The rank is taken to be the number of singular values of A that are greater than the specified tolerance tol. If the second argument is omitted, it is taken to be

     tol = max (size (A)) * sigma(1) * eps;

where eps is machine precision and sigma(1) is the largest singular value of A.

The rank of a matrix is the number of linearly independent rows or columns and determines how many particular solutions exist to a system of equations. Use null for finding the remaining homogenous solutions.

Example:

     x = [1 2 3 4 5 6 7 8 9]; rank (x) ⇒ 2

The number of linearly independent rows is only 2 because the final row is a linear combination of -1*row1 + 2*row2.

See also: [null], page 440, [sprank], page 511, [svd], page 450.

[Built-in Function] c = rcond (A)

Compute the 1-norm estimate of the reciprocal condition number as returned by lapack. If the matrix is well-conditioned then c will be near 1 and if the matrix is poorly conditioned it will be close to zero.

The matrix A must not be sparse. If the matrix is sparse then condest (A) or rcond (full (A)) should be used instead.

See also: [cond], page 436, [condest], page 510.

[Function File] trace (A)

Compute the trace of A, the sum of the elements along the main diagonal. The implementation is straightforward: sum (diag (A)).

See also: [eig], page 437.

[Function File] rref (A) [Function File] rref (A, tol) [Function File] [r, k] = rref (…)

Return the reduced row echelon form of A. tol defaults to eps * max (size (A)) *

norm (A, inf).

Called with two return arguments, k returns the vector of "bound variables", which are those columns on which elimination has been performed.

18.3 Matrix Factorizations

[Loadable Function] R = chol (A) [Loadable Function] [R, p] = chol (A) [Loadable Function] [R, p, Q] = chol (S) [Loadable Function] [R, p, Q] = chol (S, "vector") [Loadable Function] [L, ...] = chol (…, "lower") [Loadable Function] [L, ...] = chol (…, "upper")

Compute the Cholesky factor, R, of the symmetric positive definite matrix A, where R^{T} R = A. Called with one output argument chol fails if A or S is not positive definite. With two or more output arguments p flags whether the matrix was positive definite and chol does not fail. A zero value indicated that the matrix was positive definite and the R gives the factorization, and p will have a positive value otherwise. If called with 3 outputs then a sparsity preserving row/column permutation is applied to A prior to the factorization. That is R is the factorization of A(Q,Q) such that R^{T} R = Q^{T} AQ. The sparsity preserving permutation is generally returned as a matrix. However, given the flag "vector", Q will be returned as a vector such that R^{T} R = A(Q, Q). Called with either a sparse or full matrix and using the "lower" flag, chol returns the lower triangular factorization such that LL^{T} = A. For full matrices, if the "lower" flag is set only the lower triangular part of the matrix is used for the factorization, otherwise the upper triangular part is used. In general the lower triangular factorization is significantly faster for sparse matrices.

See also: [hess], page 444, [lu], page 444, [qr], page 446, [qz], page 448, [schur], page 450, [svd], page 450, [cholinv], page 442, [chol2inv], page 443, [cholupdate], page 443, [cholinsert], page 443, [choldelete], page 443, [cholshift], page 443.

[Loadable Function] cholinv (A)

Use the Cholesky factorization to compute the inverse of the symmetric positive definite matrix A.

See also: [chol], page 442, [chol2inv], page 443, [inv], page 437.

[Loadable Function] chol2inv (U)

Invert a symmetric, positive definite square matrix from its Cholesky decomposition, U. Note that U should be an upper-triangular matrix with positive diagonal elements. chol2inv (U) provides inv (U'*U) but it is much faster than using inv.

See also: [chol], page 442, [cholinv], page 442, [inv], page 437.

[Loadable Function] [R1, info] = cholupdate (R, u, op)

Update or downdate a Cholesky factorization. Given an upper triangular matrix R and a column vector u, attempt to determine another upper triangular matrix R1 such that

• R1’*R1 = R’*R + u*u’ if op is "+"

• R1’*R1 = R’*R - u*u’ if op is "-"

If op is "-", info is set to • 0 if the downdate was successful,

• 1 if R’*R - u*u’ is not positive definite,

• 2 if R is singular.

If info is not present, an error message is printed in cases 1 and 2.

See also: [chol], page 442, [cholinsert], page 443, [choldelete], page 443, [cholshift], page 443.

[Loadable Function] R1 = cholinsert (R, j, u)

[Loadable Function] [R1, info] = cholinsert (R, j, u)

Given a Cholesky factorization of a real symmetric or complex Hermitian positive definite matrix A = R’*R, R upper triangular, return the Cholesky factorization of A1, where A1(p,p) = A, A1(:,j) = A1(j,:)’ = u and p = [1:j-1,j+1:n+1]. u(j) should be positive. On return, info is set to

• 0 if the insertion was successful,

• 1 if A1 is not positive definite,

• 2 if R is singular.

If info is not present, an error message is printed in cases 1 and 2.

See also: [chol], page 442, [cholupdate], page 443, [choldelete], page 443, [cholshift], page 443.

[Loadable Function] R1 = choldelete (R, j)

Given a Cholesky factorization of a real symmetric or complex Hermitian positive definite matrix A = R’*R, R upper triangular, return the Cholesky factorization of A(p,p), where p = [1:j-1,j+1:n+1].

See also: [chol], page 442, [cholupdate], page 443, [cholinsert], page 443, [cholshift], page 443.

[Loadable Function] R1 = cholshift (R, i, j)

Given a Cholesky factorization of a real symmetric or complex Hermitian positive definite matrix A = R’*R, R upper triangular, return the Cholesky factorization of A(p,p), where p is the permutation

p = [1:i-1, shift(i:j, 1), j+1:n] if i < j

or

p = [1:j-1, shift(j:i,-1), i+1:n] if j < i.

See also: [chol], page 442, [cholupdate], page 443, [cholinsert], page 443, [choldelete], page 443.

[Built-in Function] H = hess (A) [Built-in Function] [P, H] = hess (A)

Compute the Hessenberg decomposition of the matrix A.

The Hessenberg decomposition is

A = P HP^{T}

where P is a square unitary matrix (P^{T} P = I), and H is upper Hessenberg (H_{i,j} = 0, ∀i ≥ j + 1).

The Hessenberg decomposition is usually used as the first step in an eigenvalue computation, but has other applications as well (see Golub, Nash, and Van Loan, IEEE Transactions on Automatic Control, 1979).

See also: [eig], page 437, [chol], page 442, [lu], page 444, [qr], page 446, [qz], page 448, [schur], page 450, [svd], page 450.

[Built-in Function] [L, U] = lu (A) [Built-in Function] [L, U, P] = lu (A) [Built-in Function] [L, U, P, Q] = lu (S) [Built-in Function] [L, U, P, Q, R] = lu (S) [Built-in Function] [...] = lu (S, thres) [Built-in Function] y = lu (…)

[Built-in Function] [...] = lu (…, "vector")

Compute the LU decomposition of A. If A is full subroutines from lapack are used and if A is sparse then umfpack is used. The result is returned in a permuted form, according to the optional return value P. For example, given the matrix a = [1, 2;

3, 4],
     [l, u, p] = lu (a)

returns

     l =
       1.00000  0.00000
       0.33333  1.00000
     u =
       3.00000  4.00000
       0.00000  0.66667
     p =
       0  1
       1  0

The matrix is not required to be square. When called with two or three output arguments and a spare input matrix, lu does not attempt to perform sparsity preserving column permutations. Called with a fourth output argument, the sparsity preserving column transformation Q is returned, such

that P * A * Q = L * U.

Called with a fifth output argument and a sparse input matrix, lu attempts to use a scaling factor R on the input matrix such that P * (R \ A) * Q = L * U. This typically leads to a sparser and more stable factorization. An additional input argument thres, that defines the pivoting threshold can be given. thres can be a scalar, in which case it defines the umfpack pivoting tolerance for both symmetric and unsymmetric cases. If thres is a 2-element vector, then the first element defines the pivoting tolerance for the unsymmetric umfpack pivoting strategy and the second for the symmetric strategy. By default, the values defined by spparms are used ([0.1, 0.001]). Given the string argument "vector", lu returns the values of P and Q as vector values, such that for full matrix, A (P,:) = L * U, and R(P,:) * A (:, Q) = L * U. With two output arguments, returns the permuted forms of the upper and lower triangular matrices, such that A = L * U. With one output argument y, then the matrix returned by the lapack routines is returned. If the input matrix is sparse then the matrix L is embedded into U to give a return value similar to the full case. For both full and sparse matrices, lu loses the permutation information.

See also: [luupdate], page 445, [chol], page 442, [hess], page 444, [qr], page 446, [qz], page 448, [schur], page 450, [svd], page 450.

[Built-in Function] [L, U] = luupdate (L, U, x, y) [Built-in Function] [L, U, P] = luupdate (L, U, P, x, y)

Given an LU factorization of a real or complex matrix A = L*U, L lower unit trapezoidal and U upper trapezoidal, return the LU factorization of A + x*y.’, where x and y are column vectors (rank-1 update) or matrices with equal number of columns (rank-k update). Optionally, row-pivoted updating can be used by supplying a row permutation (pivoting) matrix P; in that case, an updated permutation matrix is returned. Note that if L, U, P is a pivoted LU factorization as obtained by lu:

     [L, U, P] = lu (A);

then a factorization of A+x*y.' can be obtained either as

     [L1, U1] = lu (L, U, P*x, y)

or

     [L1, U1, P1] = lu (L, U, P, x, y)

The first form uses the unpivoted algorithm, which is faster, but less stable. The second form uses a slower pivoted algorithm, which is more stable. The matrix case is done as a sequence of rank-1 updates; thus, for large enough k, it will be both faster and more accurate to recompute the factorization from scratch. See also: [lu], page 444, [cholupdate], page 443, [qrupdate], page 447.

[Loadable Function] [Q, R, P] = qr (A) [Loadable Function] [Q, R, P] = qr (A, ’0’) [Loadable Function] [C, R] = qr (A, B) [Loadable Function] [C, R] = qr (A, B, ’0’)

Compute the QR factorization of A, using standard lapack subroutines. For example, given the matrix A = [1, 2; 3, 4],

     [Q, R] = qr (A)

returns

     Q =
       -0.31623  -0.94868
       -0.94868   0.31623
     R =
       -3.16228  -4.42719
        0.00000  -0.63246

The qr factorization has applications in the solution of least squares problems

min_{x} ∥Ax − b∥_{2}

for overdetermined systems of equations (i.e., A is a tall, thin matrix). The QR factorization is QR = A where Q is an orthogonal matrix and R is upper triangular. If given a second argument of '0', qr returns an economy-sized QR factorization, omitting zero rows of R and the corresponding columns of Q. If the matrix A is full, the permuted QR factorization [Q, R, P] = qr (A) forms the QR factorization such that the diagonal entries of R are decreasing in magnitude order. For example, given the matrix a = [1, 2; 3, 4],

     [Q, R, P] = qr (A)

returns

     Q =
       -0.44721  -0.89443
       -0.89443   0.44721
     R =
       -4.47214  -3.13050
        0.00000   0.44721
     P =
        0  1
        1  0

The permuted qr factorization [Q, R, P] = qr (A) factorization allows the construction of an orthogonal basis of span (A). If the matrix A is sparse, then compute the sparse QR factorization of A, using CSparse. As the matrix Q is in general a full matrix, this function returns the Q-less factorization R of A, such that R = chol (A' * A). If the final argument is the scalar 0 and the number of rows is larger than the number of columns, then an economy factorization is returned. That is R will have only size

(A,1) rows.

If an additional matrix B is supplied, then qr returns C, where C = Q' * B. This allows the least squares approximation of A \ B to be calculated as

     [C, R] = qr (A, B)
     x = R \ C

See also: [chol], page 442, [hess], page 444, [lu], page 444, [qz], page 448, [schur], page 450, [svd], page 450, [qrupdate], page 447, [qrinsert], page 447, [qrdelete], page 448, [qrshift], page 448.

[Loadable Function] [Q1, R1] = qrupdate (Q, R, u, v)

Given a QR factorization of a real or complex matrix A = Q*R, Q unitary and R upper trapezoidal, return the QR factorization of A + u*v’, where u and v are column vectors (rank-1 update) or matrices with equal number of columns (rank-k update). Notice that the latter case is done as a sequence of rank-1 updates; thus, for k large enough, it will be both faster and more accurate to recompute the factorization from scratch. The QR factorization supplied may be either full (Q is square) or economized (R is square).

See also: [qr], page 446, [qrinsert], page 447, [qrdelete], page 448, [qrshift], page 448.

[Loadable Function] [Q1, R1] = qrinsert (Q, R, j, x, orient)

Given a QR factorization of a real or complex matrix A = Q*R, Q unitary and R upper trapezoidal, return the QR factorization of [A(:,1:j-1) x A(:,j:n)], where u is a column vector to be inserted into A (if orient is "col"), or the QR factorization of [A(1:j-1,:);x;A(:,j:n)], where x is a row vector to be inserted into A (if orient is

"row").

The default value of orient is "col". If orient is "col", u may be a matrix and j an index vector resulting in the QR factorization of a matrix B such that B(:,j) gives u and B(:,j) = [] gives A. Notice that the latter case is done as a sequence of k insertions; thus, for k large enough, it will be both faster and more accurate to recompute the factorization from scratch.

If orient is "col", the QR factorization supplied may be either full (Q is square) or economized (R is square).

If orient is "row", full factorization is needed.

See also: [qr], page 446, [qrupdate], page 447, [qrdelete], page 448, [qrshift], page 448.

[Loadable Function] [Q1, R1] = qrdelete (Q, R, j, orient)

Given a QR factorization of a real or complex matrix A = Q*R, Q unitary and R upper trapezoidal, return the QR factorization of [A(:,1:j-1) A(:,j+1:n)], i.e., A with one column deleted (if orient is "col"), or the QR factorization of [A(1:j-1,:);A(j+1:n,:)], i.e., A with one row deleted (if orient is "row").

The default value of orient is "col".

If orient is "col", j may be an index vector resulting in the QR factorization of a matrix B such that A(:,j) = [] gives B. Notice that the latter case is done as a sequence of k deletions; thus, for k large enough, it will be both faster and more accurate to recompute the factorization from scratch.

If orient is "col", the QR factorization supplied may be either full (Q is square) or economized (R is square).

If orient is "row", full factorization is needed.

See also: [qr], page 446, [qrupdate], page 447, [qrinsert], page 447, [qrshift], page 448.

[Loadable Function] [Q1, R1] = qrshift (Q, R, i, j)

Given a QR factorization of a real or complex matrix A = Q*R, Q unitary and R upper trapezoidal, return the QR factorization of A(:,p), where p is the permutation

p = [1:i-1, shift(i:j, 1), j+1:n] if i < j

or

p = [1:j-1, shift(j:i,-1), i+1:n] if j < i.

See also: [qr], page 446, [qrupdate], page 447, [qrinsert], page 447, [qrdelete], page 448.

[Built-in Function] lambda = qz (A, B) [Built-in Function] lambda = qz (A, B, opt)

QZ decomposition of the generalized eigenvalue problem (Ax = sBx). There are three ways to call this function:

 1. lambda = qz (A, B)

Computes the generalized eigenvalues λ of (A − sB).

 2. [AA, BB, Q, Z, V, W, lambda] = qz (A, B)

Computes QZ decomposition, generalized eigenvectors, and generalized eigenvalues of (A − sB)

AV = BV diag(λ)
W^{T} A = diag(λ)W^{T} B
AA = Q^{T} AZ, BB = Q^{T} BZ

with Q and Z orthogonal (unitary)= I

 3. [AA,BB,Z{, lambda}] = qz (A, B, opt)

As in form [2], but allows ordering of generalized eigenpairs for (e.g.) solution of discrete time algebraic Riccati equations. Form 3 is not available for complex matrices, and does not compute the generalized eigenvectors V, W, nor the orthogonal matrix Q.

opt for ordering eigenvalues of the GEP pencil. The leading block of the revised pencil contains all eigenvalues that satisfy:

"N" = unordered (default)

"S" = small: leading block has all |lambda| ≤ 1

"B" = big: leading block has all |lambda| ≥ 1

"-" = negative real part: leading block has all eigenvalues in the open left half-plane

"+" = non-negative real part: leading block has all eigenvalues in the closed right half-plane

Note: qz performs permutation balancing, but not scaling (see [XREFbalance], page 435). The order of output arguments was selected for compatibility with matlab.

See also: [eig], page 437, [balance], page 435, [lu], page 444, [chol], page 442, [hess], page 444, [qr], page 446, [qzhess], page 449, [schur], page 450, [svd], page 450.

[Function File] [aa, bb, q, z] = qzhess (A, B)

Compute the Hessenberg-triangular decomposition of the matrix pencil (A, B), returning aa = q * A * z, bb = q * B * z, with q and z orthogonal. For example:

     [aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8])
          ⇒ aa = [ -3.02244, -4.41741;  0.92998,  0.69749 ]

          ⇒ bb = [ -8.60233, -9.99730;  0.00000, -0.23250 ]
          ⇒ q = [ -0.58124, -0.81373; -0.81373,  0.58124 ]
          ⇒ z = [ 1, 0; 0, 1 ]

The Hessenberg-triangular decomposition is the first step in Moler and Stewart’s QZ decomposition algorithm. Algorithm taken from Golub and Van Loan, Matrix Computations, 2nd edition.

See also: [lu], page 444, [chol], page 442, [hess], page 444, [qr], page 446, [qz], page 448, [schur], page 450, [svd], page 450.

[Built-in Function] S = schur (A) [Built-in Function] S = schur (A, "real") [Built-in Function] S = schur (A, "complex") [Built-in Function] S = schur (A, opt) [Built-in Function] [U, S] = schur (A, …)

Compute the Schur decomposition of A

S = U^{T} AU

where U is a unitary matrix (U^{T} U is identity) and S is upper triangular. The eigenvalues of A (and S) are the diagonal elements of S. If the matrix A is real, then the real Schur decomposition is computed, in which the matrix U is orthogonal and S is block upper triangular with blocks of size at most 2 × 2 along the diagonal. The diagonal elements of S (or the eigenvalues of the 2 × 2 blocks, when appropriate) are the eigenvalues of A and S. The default for real matrices is a real Schur decomposition. A complex decomposition may be forced by passing the flag "complex". The eigenvalues are optionally ordered along the diagonal according to the value of opt. opt = "a" indicates that all eigenvalues with negative real parts should be moved to the leading block of S (used in are), opt = "d" indicates that all eigenvalues with magnitude less than one should be moved to the leading block of S (used in dare), and opt = "u", the default, indicates that no ordering of eigenvalues should occur. The leading k columns of U always span the A-invariant subspace corresponding to the k leading eigenvalues of S. The Schur decomposition is used to compute eigenvalues of a square matrix, and has applications in the solution of algebraic Riccati equations in control (see are and

dare).

See also: [rsf2csf], page 450, [lu], page 444, [chol], page 442, [hess], page 444, [qr], page 446, [qz], page 448, [svd], page 450.

[Function File] [U, T] = rsf2csf (UR, TR)

Convert a real, upper quasi-triangular Schur form TR to a complex, upper triangular Schur form T. Note that the following relations hold: UR · T R · UR^{T} = UT U^{†} and U^{†}U is the identity matrix I. Note also that U and T are not unique.

See also: [schur], page 450.

[Function File] angle = subspace (A, B)

Determine the largest principal angle between two subspaces spanned by the columns of matrices A and B.

[Built-in Function] s = svd (A) [Built-in Function] [U, S, V] = svd (A) [Built-in Function] [U, S, V] = svd (A, econ)

Compute the singular value decomposition of A

A = USV^{†}

The function svd normally returns only the vector of singular values. When called with three return values, it computes U, S, and V. For example,

     svd (hilb (3))

returns

     ans = 1.4083189 0.1223271 0.0026873

and

     [u, s, v] = svd (hilb (3))

returns

     u =
       -0.82704   0.54745   0.12766
       -0.45986  -0.52829  -0.71375
       -0.32330  -0.64901   0.68867
     s =
       1.40832  0.00000  0.00000
       0.00000  0.12233  0.00000
       0.00000  0.00000  0.00269
     v =
       -0.82704   0.54745   0.12766
       -0.45986  -0.52829  -0.71375
       -0.32330  -0.64901   0.68867

If given a second argument, svd returns an economy-sized decomposition, eliminating the unnecessary rows or columns of U or V.

See also: [svd driver], page 451, [svds], page 515, [eig], page 437, [lu], page 444, [chol], page 442, [hess], page 444, [qr], page 446, [qz], page 448.

[Built-in Function] val = svd_driver ()

[Built-in Function] old_val = svd_driver (new_val)

[Built-in Function] svd_driver (new_val, "local")

Query or set the underlying lapack driver used by svd. Currently recognized values

are "gesvd" and "gesdd". The default is "gesvd".

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [svd], page 450.

[Function File] [housv, beta, zer] = housh (x, j, z)

Compute Householder reflection vector housv to reflect x to be the j-th column of identity, i.e.,

     (I - beta*housv*housv')x =  norm (x)*e(j) if x(j) < 0,
     (I - beta*housv*housv')x = -norm (x)*e(j) if x(j) >= 0

Inputs

x vector

j index into vector

z threshold for zero (usually should be the number 0)

Outputs (see Golub and Van Loan):

beta If beta = 0, then no reflection need be applied (zer set to 0)

housv householder vector

[Function File] [u, h, nu] = krylov (A, V, k, eps1, pflg)

Construct an orthogonal basis u of block Krylov subspace

     [v a*v a^2*v ... a^(k+1)*v]

Using Householder reflections to guard against loss of orthogonality. If V is a vector, then h contains the Hessenberg matrix such that a*u == u*h+rk*ek',

in which rk = a*u(:,k)-u*h(:,k), and ek' is the vector [0, 0, ..., 1] of length

k. Otherwise, h is meaningless. If V is a vector and k is greater than length (A) - 1, then h contains the Hessenberg matrix such that a*u == u*h. The value of nu is the dimension of the span of the Krylov subspace (based on eps1). If b is a vector and k is greater than m-1, then h contains the Hessenberg decomposition of A. The optional parameter eps1 is the threshold for zero. The default value is 1e-12. If the optional parameter pflg is nonzero, row pivoting is used to improve numerical behavior. The default value is 0. Reference: A. Hodel, P. Misra, Partial Pivoting in the Computation of Krylov Subspaces of Large Sparse Systems, Proceedings of the 42nd IEEE Conference on Decision and Control, December 2003.

18.4 Functions of a Matrix

[Function File] expm (A)

Return the exponential of a matrix, defined as the infinite Taylor series

exp(A) = I + A + \frac{A^{2}}{2!} + \frac{A^{3}}{3!} + ⋯

The Taylor series is not the way to compute the matrix exponential; see Moler and Van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review, 1978. This routine uses Ward’s diagonal Padé approximation method with three step preconditioning (SIAM Journal on Numerical Analysis, 1977). Diagonal Padé approximations are rational polynomials of matrices D_{q}(A)^{−1}N_{q}(A) whose Taylor series matches the first 2q + 1 terms of the Taylor series above; direct evaluation of the Taylor series (with the same preconditioning steps) may be desirable in lieu of the Padé approximation when D_{q}(A) is ill-conditioned.

See also: [logm], page 453, [sqrtm], page 453.

[Function File] s = logm (A) [Function File] s = logm (A, opt_iters) [Function File] [s, iters] = logm (…)

Compute the matrix logarithm of the square matrix A. The implementation utilizes a Padé approximant and the identity

     logm (A) = 2^k * logm (A^(1 / 2^k))

The optional argument opt iters is the maximum number of square roots to compute and defaults to 100. The optional output iters is the number of square roots actually computed.

See also: [expm], page 452, [sqrtm], page 453.

[Built-in Function] s = sqrtm (A)

[Built-in Function] [s, error_estimate] = sqrtm (A)

Compute the matrix square root of the square matrix A. Ref: N.J. Higham. A New sqrtm for matlab. Numerical Analysis Report No. 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999.

See also: [expm], page 452, [logm], page 453.

[Built-in Function] kron (A, B) [Built-in Function] kron (A1, A2, …)

Form the Kronecker product of two or more matrices, defined block by block as

     x = [ a(i,j)*b ]

For example:

     kron (1:4, ones (3, 1))
          ⇒ 1  2  3  4
              1  2  3  4
              1  2  3  4

If there are more than two input arguments A1, A2, …, An the Kronecker product is computed as

     kron (kron (A1, A2), ..., An)

Since the Kronecker product is associative, this is well-defined.

[Built-in Function] blkmm (A, B)

Compute products of matrix blocks. The blocks are given as 2-dimensional subarrays of the arrays A, B. The size of A must have the form [m,k,...] and size of B must be [k,n,...]. The result is then of size [m,n,...] and is computed as follows:

     for i = 1:prod (size (A)(3:end))
       C(:,:,i) = A(:,:,i) * B(:,:,i)
     endfor

[Built-in Function] x = syl (A, B, C)

Solve the Sylvester equation

AX + XB + C = 0

using standard lapack subroutines. For example:

     syl ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12]) ⇒ [ -0.50000, -0.66667; -0.66667, -0.50000 ]

18.5 Specialized Solvers

[Function File] x = bicg (A, b, rtol, maxit, M1, M2, x0)

[Function File] x = bicg (A, b, rtol, maxit, P)

[Function File] [x, flag, relres, iter, resvec] = bicg (A, b, . . .)

Solve A x = b using the Bi-conjugate gradient iterative method. − rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used. − maxit the maximum number of outer iterations, if not given or set to [] the default

    value min (20, numel (b)) is used.

− x0 the initial guess, if not given or set to [] the default value zeros (size (b)) is used.

A can be passed as a matrix or as a function handle or inline function f such that

f(x, "notransp") = A*x and f(x, "transp") = A'*x.

The preconditioner P is given as P = M1 * M2. Both M1 and M2 can be passed as a matrix or as a function handle or inline function g such that g(x, "notransp") = M1 \

x or g(x, "notransp") = M2 \ x and g(x, "transp") = M1' \ x or g(x, "transp")
= M2' \ x.

If called with more than one output parameter − flag indicates the exit status:

− 0: iteration converged to the within the chosen tolerance

− 1: the maximum number of iterations was reached before convergence − 3: the algorithm reached stagnation

(the value 2 is unused but skipped for compatibility). − relres is the final value of the relative residual. − iter is the number of iterations performed. − resvec is a vector containing the relative residual at each iteration.

See also: [bicgstab], page 454, [cgs], page 455, [gmres], page 456, [pcg], page 516.

[Function File] x = bicgstab (A, b, rtol, maxit, M1, M2, x0)
[Function File] x = bicgstab (A, b, rtol, maxit, P)
[Function File] [x, flag, relres, iter, resvec] = bicgstab (A, b, . . .)

Solve A x = b using the stabilizied Bi-conjugate gradient iterative method.

− rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used. − maxit the maximum number of outer iterations, if not given or set to [] the default

    value min (20, numel (b)) is used.

− x0 the initial guess, if not given or set to [] the default value zeros (size (b))

is used. A can be passed as a matrix or as a function handle or inline function f such that

f(x) = A*x.

The preconditioner P is given as P = M1 * M2. Both M1 and M2 can be passed as a matrix or as a function handle or inline function g such that g(x) = M1 \ x or g(x)

= M2 \ x.

If called with more than one output parameter − flag indicates the exit status:

− 0: iteration converged to the within the chosen tolerance

− 1: the maximum number of iterations was reached before convergence − 3: the algorithm reached stagnation (the value 2 is unused but skipped for compatibility).

− relres is the final value of the relative residual. − iter is the number of iterations performed. − resvec is a vector containing the relative residual at each iteration.

See also: [bicg], page 454, [cgs], page 455, [gmres], page 456, [pcg], page 516.

[Function File] x = cgs (A, b, rtol, maxit, M1, M2, x0)

[Function File] x = cgs (A, b, rtol, maxit, P)

[Function File] [x, flag, relres, iter, resvec] = cgs (A, b, . . .)

Solve A x = b, where A is a square matrix, using the Conjugate Gradients Squared method. − rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used. − maxit the maximum number of outer iterations, if not given or set to [] the default

    value min (20, numel (b)) is used.

− x0 the initial guess, if not given or set to [] the default value zeros (size (b))

is used. A can be passed as a matrix or as a function handle or inline function f such that

f(x) = A*x.

The preconditioner P is given as P = M1 * M2. Both M1 and M2 can be passed as a matrix or as a function handle or inline function g such that g(x) = M1 \ x or g(x)

= M2 \ x.

If called with more than one output parameter − flag indicates the exit status:

− 0: iteration converged to the within the chosen tolerance

− 1: the maximum number of iterations was reached before convergence − 3: the algorithm reached stagnation

(the value 2 is unused but skipped for compatibility). − relres is the final value of the relative residual. − iter is the number of iterations performed. − resvec is a vector containing the relative residual at each iteration.

See also: [pcg], page 516, [bicgstab], page 454, [bicg], page 454, [gmres], page 456.

[Function File] x = gmres (A, b, m, rtol, maxit, M1, M2, x0)

[Function File] x = gmres (A, b, m, rtol, maxit, P)

[Function File] [x, flag, relres, iter, resvec] = gmres (. . .)

Solve A x = b using the Preconditioned GMRES iterative method with restart, a.k.a. PGMRES(m). − rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used. − maxit is the maximum number of outer iterations, if not given or set to [] the

   default value min (10, numel (b) / restart) is used.

− x0 is the initial guess, if not given or set to [] the default value zeros (size (b))

is used. − m is the restart parameter, if not given or set to [] the default value numel (b)

is used.

Argument A can be passed as a matrix, function handle, or inline function f such

that f(x) = A*x.

The preconditioner P is given as P = M1 * M2. Both M1 and M2 can be passed as a matrix, function handle, or inline function g such that g(x) = M1\x or g(x) = M2\x. Besides the vector x, additional outputs are: − flag indicates the exit status:

0 : iteration converged to within the specified tolerance 1 : maximum number of iterations exceeded 2 : unused, but skipped for compatibility 3 : algorithm reached stagnation (no change between iterations)

− relres is the final value of the relative residual. − iter is a vector containing the number of outer iterations and total iterations

performed. − resvec is a vector containing the relative residual at each iteration.

See also: [bicg], page 454, [bicgstab], page 454, [cgs], page 455, [pcg], page 516.

19 Vectorization and Faster Code Execution

Vectorization is a programming technique that uses vector operations instead of element-by-element loop-based operations. Besides frequently producing more succinct Octave code, vectorization also allows for better optimization in the subsequent implementation. The optimizations may occur either in Octave’s own Fortran, C, or C++ internal implementation, or even at a lower level depending on the compiler and external numerical libraries used to build Octave. The ultimate goal is to make use of your hardware’s vector instructions if possible or to perform other optimizations in software.

Vectorization is not a concept unique to Octave, but it is particularly important because Octave is a matrix-oriented language. Vectorized Octave code will see a dramatic speed up (10X–100X) in most cases.

This chapter discusses vectorization and other techniques for writing faster code.

19.1 Basic Vectorization

To a very good first approximation, the goal in vectorization is to write code that avoids loops and uses whole-array operations. As a trivial example, consider

     for i = 1:n
       for j = 1:m
         c(i,j) = a(i,j) + b(i,j);
       endfor
     endfor

compared to the much simpler

     c = a + b;

This isn’t merely easier to write; it is also internally much easier to optimize. Octave delegates this operation to an underlying implementation which, among other optimizations, may use special vector hardware instructions or could conceivably even perform the additions in parallel. In general, if the code is vectorized, the underlying implementation has more freedom about the assumptions it can make in order to achieve faster execution.

This is especially important for loops with "cheap" bodies. Often it suffices to vectorize just the innermost loop to get acceptable performance. A general rule of thumb is that the "order" of the vectorized body should be greater or equal to the "order" of the enclosing loop.

As a less trivial example, instead of

     for i = 1:n-1
       a(i) = b(i+1) - b(i);
     endfor

write

     a = b(2:n) - b(1:n-1);

This shows an important general concept about using arrays for indexing instead of looping over an index variable. See Section 8.1 [Index Expressions], page 131. Also use boolean indexing generously. If a condition needs to be tested, this condition can also be written as a boolean index. For instance, instead of

     for i = 1:n
       if (a(i) > 5)
         a(i) -= 20
       endif
     endfor

write

     a(a>5) -= 20;

which exploits the fact that a > 5 produces a boolean index.

Use elementwise vector operators whenever possible to avoid looping (operators like .* and .^). See Section 8.3 [Arithmetic Ops], page 137. For simple inline functions, the vectorize function can do this automatically.

[Built-in Function] vectorize (fun)

Create a vectorized version of the inline function fun by replacing all occurrences of *, /, etc., with .*, ./, etc. This may be useful, for example, when using inline functions with numerical integration or optimization where a vector-valued function is expected.

          fcn = vectorize (inline ("x^2 - 1"))
             ⇒ fcn = f(x) = x.^2 - 1
          quadv (fcn, 0, 3)
             ⇒ 6

See also: [inline], page 192, [formula], page 193, [argnames], page 193.

Also exploit broadcasting in these elementwise operators both to avoid looping and unnecessary intermediate memory allocations. See Section 19.2 [Broadcasting], page 459.

Use built-in and library functions if possible. Built-in and compiled functions are very fast. Even with an m-file library function, chances are good that it is already optimized, or will be optimized more in a future release.

For instance, even better than

     a = b(2:n) - b(1:n-1);

is

     a = diff (b);

Most Octave functions are written with vector and array arguments in mind. If you find yourself writing a loop with a very simple operation, chances are that such a function already exists. The following functions occur frequently in vectorized code:

• Index manipulation • find

• sub2ind

• ind2sub

• sort

• unique

• lookup

• ifelse / merge • Repetition

• repmat

• repelems

• Vectorized arithmetic • sum

• prod

• cumsum

• cumprod

• sumsq

• diff

• dot

• cummax

• cummin

• Shape of higher dimensional arrays • reshape

• resize

• permute

• squeeze

• deal

19.2 Broadcasting

Broadcasting refers to how Octave binary operators and functions behave when their matrix or array operands or arguments differ in size. Since version 3.6.0, Octave now automatically broadcasts vectors, matrices, and arrays when using elementwise binary operators and functions. Broadly speaking, smaller arrays are “broadcast” across the larger one, until they have a compatible shape. The rule is that corresponding array dimensions must either 1. be equal, or

2. one of them must be 1.

In case all dimensions are equal, no broadcasting occurs and ordinary element-by-element arithmetic takes place. For arrays of higher dimensions, if the number of dimensions isn’t the same, then missing trailing dimensions are treated as 1. When one of the dimensions is 1, the array with that singleton dimension gets copied along that dimension until it matches the dimension of the other array. For example, consider

x = [1 2 3;
     4 5 6;
     7 8 9];
y = [10 20 30];
x + y

Without broadcasting, x + y would be an error because the dimensions do not agree. However, with broadcasting it is as if the following operation were performed:

     x = [1 2 3
          4 5 6
          7 8 9];
     y = [10 20 30
          10 20 30
          10 20 30];
     x + y
     ⇒  11   22   33
           14   25   36
           17   28   39

That is, the smaller array of size [1 3] gets copied along the singleton dimension (the number of rows) until it is [3 3]. No actual copying takes place, however. The internal implementation reuses elements along the necessary dimension in order to achieve the desired effect without copying in memory.

Both arrays can be broadcast across each other, for example, all pairwise differences of the elements of a vector with itself:

     y - y'
     ⇒  0   10   20
         -10    0   10
         -20  -10    0

Here the vectors of size [1 3] and [3 1] both get broadcast into matrices of size [3 3] before ordinary matrix subtraction takes place.

A special case of broadcasting that may be familiar is when all dimensions of the array being broadcast are 1, i.e., the array is a scalar. Thus for example, operations like x - 42 and max (x, 2) are basic examples of broadcasting.

For a higher-dimensional example, suppose img is an RGB image of size [m n 3] and we wish to multiply each color by a different scalar. The following code accomplishes this with broadcasting,

     img .*= permute ([0.8, 0.9, 1.2], [1, 3, 2]);

Note the usage of permute to match the dimensions of the [0.8, 0.9, 1.2] vector with

img.

For functions that are not written with broadcasting semantics, bsxfun can be useful for coercing them to broadcast.

[Built-in Function] bsxfun (f, A, B)

The binary singleton expansion function applier performs broadcasting, that is, applies a binary function f element-by-element to two array arguments A and B, and expands as necessary singleton dimensions in either input argument. f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be capable of accepting two column-vector arguments of equal length, or one column vector argument and a scalar. The dimensions of A and B must be equal or singleton. The singleton dimensions of the arrays will be expanded to the same dimensionality as the other array.

See also: [arrayfun], page 462, [cellfun], page 464.

Broadcasting is only applied if either of the two broadcasting conditions hold. As usual, however, broadcasting does not apply when two dimensions differ and neither is 1:

x = [1 2 3 4 5 6]; y = [10 20 30 40]; x + y

This will produce an error about nonconformant arguments.

Besides common arithmetic operations, several functions of two arguments also broadcast. The full list of functions and operators that broadcast is

plus      +  .+
minus     -  .-
times     .*
rdivide   ./
ldivide   .\
power     .^.**
lt        <
le        <=
eq        ==
gt        >
ge        >=
ne        !=  ~=
and       &
or        |
atan2
hypot
max
min
mod
rem
xor
+=  -=  .+=  .-=  .*=  ./=  .\=  .^=  .**=  &=  |=

Beware of resorting to broadcasting if a simpler operation will suffice. For matrices a and b, consider the following:

c = sum (permute (a, [1, 3, 2]) .* permute (b, [3, 2, 1]), 3);

This operation broadcasts the two matrices with permuted dimensions across each other during elementwise multiplication in order to obtain a larger 3-D array, and this array is then summed along the third dimension. A moment of thought will prove that this operation is simply the much faster ordinary matrix multiplication, c = a*b;.

A note on terminology: “broadcasting” is the term popularized by the Numpy numerical environment in the Python programming language. In other programming languages and environments, broadcasting may also be known as binary singleton expansion (BSX, in matlab, and the origin of the name of the bsxfun function), recycling (R programming language), single-instruction multiple data (SIMD), or replication.

19.2.1 Broadcasting and Legacy Code

The new broadcasting semantics almost never affect code that worked in previous versions of Octave. Consequently, all code inherited from matlab that worked in previous versions of Octave should still work without change in Octave. The only exception is code such as

     try
       c = a.*b;
     catch
       c = a.*a;
     end_try_catch

that may have relied on matrices of different size producing an error. Due to how broadcasting changes semantics with older versions of Octave, by default Octave warns if a broadcasting operation is performed. To disable this warning, refer to its ID (see [warning ids], page 205):

     warning ("off", "Octave:broadcast");

If you want to recover the old behavior and produce an error, turn this warning into an error:

     warning ("error", "Octave:broadcast");

For broadcasting on scalars that worked in previous versions of Octave, this warning will not be emitted.

19.3 Function Application

As a general rule, functions should already be written with matrix arguments in mind and should consider whole matrix operations in a vectorized manner. Sometimes, writing functions in this way appears difficult or impossible for various reasons. For those situations, Octave provides facilities for applying a function to each element of an array, cell, or struct.

[Function File] arrayfun (func, A) [Function File] x = arrayfun (func, A) [Function File] x = arrayfun (func, A, b, …)

[Function File] [x, y, ...] = arrayfun (func, A, . . .)

[Function File] arrayfun (…, "UniformOutput", val)

[Function File] arrayfun (…, "ErrorHandler", errfunc)

Execute a function on each element of an array. This is useful for functions that do not accept array arguments. If the function does accept array arguments it is better to call the function directly. The first input argument func can be a string, a function handle, an inline function, or an anonymous function. The input argument A can be a logic array, a numeric array, a string array, a structure array, or a cell array. By a call of the function arrayfun all elements of A are passed on to the named function func individually. The named function can also take more than two input arguments, with the input arguments given as third input argument b, fourth input argument c, … If given more than one array input argument then all input arguments must have the same sizes, for example:

     arrayfun (@atan2, [1, 0], [0, 1])
          ⇒ [ 1.5708   0.0000 ]

If the parameter val after a further string input argument "UniformOutput" is set true (the default), then the named function func must return a single element which then will be concatenated into the return value and is of type matrix. Otherwise, if that parameter is set to false, then the outputs are concatenated in a cell array. For example:

     arrayfun (@(x,y) x:y, "abc", "def", "UniformOutput", false)

⇒

        {
          [1,1] = abcd
          [1,2] = bcde
          [1,3] = cdef
        }

If more than one output arguments are given then the named function must return the number of return values that also are expected, for example:

     [A, B, C] = arrayfun (@find, [10; 0], "UniformOutput", false)

⇒

     A =
     {
        [1,1] =  1
        [2,1] = [](0x0)
     }
     B =
     {
        [1,1] =  1
        [2,1] = [](0x0)
     }
     C =
     {
        [1,1] =  10
        [2,1] = [](0x0)
     }

If the parameter errfunc after a further string input argument "ErrorHandler" is another string, a function handle, an inline function, or an anonymous function, then errfunc defines a function to call in the case that func generates an error. The definition of the function must be of the form

     function [...] = errfunc (s, ...)

where there is an additional input argument to errfunc relative to func, given by s. This is a structure with the elements "identifier", "message", and "index" giving, respectively, the error identifier, the error message, and the index of the array elements that caused the error. The size of the output argument of errfunc must have the same size as the output argument of func, otherwise a real error is thrown. For example:

     function y = ferr (s, x), y = "MyString"; endfunction
     arrayfun (@str2num, [1234],
               "UniformOutput", false, "ErrorHandler", @ferr)

⇒

        {
          [1,1] = MyString
        }

See also: [spfun], page 464, [cellfun], page 464, [structfun], page 466.

[Function File] y = spfun (f, S)

Compute f(S) for the non-zero values of S. This results in a sparse matrix with the same structure as S. The function f can be passed as a string, a function handle, or an inline function.

See also: [arrayfun], page 462, [cellfun], page 464, [structfun], page 466.

[Built-in Function] cellfun (name, C) [Built-in Function] cellfun ("size", C, k) [Built-in Function] cellfun ("isclass", C, class) [Built-in Function] cellfun (func, C) [Built-in Function] cellfun (func, C, D) [Built-in Function] [a, ...] = cellfun (…) [Built-in Function] cellfun (…, "ErrorHandler", errfunc)

[Built-in Function] cellfun (…, "UniformOutput", val)

Evaluate the function named name on the elements of the cell array C. Elements in C are passed on to the named function individually. The function name can be one of the functions

isempty Return 1 for empty elements.

islogical

Return 1 for logical elements.

isnumeric

Return 1 for numeric elements.

isreal Return 1 for real elements.

length Return a vector of the lengths of cell elements.

ndims Return the number of dimensions of each element.

numel
prodofsize

Return the number of elements contained within each cell element. The number is the product of the dimensions of the object at each cell element.

size Return the size along the k-th dimension.

isclass Return 1 for elements of class.

Additionally, cellfun accepts an arbitrary function func in the form of an inline function, function handle, or the name of a function (in a character string). The function can take one or more arguments, with the inputs arguments given by C, D, etc. Equally the function can return one or more output arguments. For example:

     cellfun ("atan2", {1, 0}, {0, 1})
          ⇒ [ 1.57080   0.00000 ]

The number of output arguments of cellfun matches the number of output arguments of the function. The outputs of the function will be collected into the output arguments of cellfun like this:

     function [a, b] = twoouts (x)
       a = x;
       b = x*x;
     endfunction
     [aa, bb] = cellfun (@twoouts, {1, 2, 3})

⇒

             aa =
                1 2 3
             bb =
                1 4 9

Note that per default the output argument(s) are arrays of the same size as the input arguments. Input arguments that are singleton (1x1) cells will be automatically expanded to the size of the other arguments. If the parameter "UniformOutput" is set to true (the default), then the function must return scalars which will be concatenated into the return array(s). If "UniformOutput" is false, the outputs are concatenated into a cell array (or cell arrays). For example:

     cellfun ("tolower", {"Foo", "Bar", "FooBar"},
              "UniformOutput", false)
     ⇒ {"foo", "bar", "foobar"}

Given the parameter "ErrorHandler", then errfunc defines a function to call in case func generates an error. The form of the function is

     function [...] = errfunc (s, ...)

where there is an additional input argument to errfunc relative to func, given by s. This is a structure with the elements "identifier", "message" and "index", giving respectively the error identifier, the error message, and the index into the input arguments of the element that caused the error. For example:

     function y = foo (s, x), y = NaN; endfunction
     cellfun ("factorial", {-1,2}, "ErrorHandler", @foo)
     ⇒ [NaN 2]

Use cellfun intelligently. The cellfun function is a useful tool for avoiding loops. It is often used with anonymous function handles; however, calling an anonymous function involves an overhead quite comparable to the overhead of an m-file function. Passing a handle to a built-in function is faster, because the interpreter is not involved in the internal loop. For example:

     a = {...}
     v = cellfun (@(x) det (x), a); # compute determinants
     v = cellfun (@det, a); # faster

See also: [arrayfun], page 462, [structfun], page 466, [spfun], page 464.

[Function File] structfun (func, S) [Function File] [A, ...] = structfun (…) [Function File] structfun (…, "ErrorHandler", errfunc)

[Function File] structfun (…, "UniformOutput", val)

Evaluate the function named name on the fields of the structure S. The fields of S are passed to the function func individually.

structfun accepts an arbitrary function func in the form of an inline function, function handle, or the name of a function (in a character string). In the case of a character string argument, the function must accept a single argument named x, and it must return a string value. If the function returns more than one argument, they are returned as separate output variables.

If the parameter "UniformOutput" is set to true (the default), then the function must return a single element which will be concatenated into the return value. If "UniformOutput" is false, the outputs are placed into a structure with the same fieldnames as the input structure.

     s.name1 = "John Smith";
     s.name2 = "Jill Jones";
     structfun (@(x) regexp (x, '(\w+)$', "matches"){1}, s,
                "UniformOutput", false)

⇒

        {
          name1 = Smith
          name2 = Jones
        }

Given the parameter "ErrorHandler", errfunc defines a function to call in case func generates an error. The form of the function is

     function [...] = errfunc (se, ...)

where there is an additional input argument to errfunc relative to func, given by se. This is a structure with the elements "identifier", "message" and "index", giving respectively the error identifier, the error message, and the index into the input arguments of the element that caused the error. For an example on how to use an error handler, see [cellfun], page 464.

See also: [cellfun], page 464, [arrayfun], page 462, [spfun], page 464.

19.4 Accumulation

Whenever it’s possible to categorize according to indices the elements of an array when performing a computation, accumulation functions can be useful.

[Function File] accumarray (subs, vals, sz, func, fillval, issparse)

[Function File] accumarray (subs, vals, …)

Create an array by accumulating the elements of a vector into the positions defined by their subscripts. The subscripts are defined by the rows of the matrix subs and the values by vals. Each row of subs corresponds to one of the values in vals. If vals is a scalar, it will be used for each of the row of subs. If subs is a cell array of vectors, all vectors must be of the same length, and the subscripts in the kth vector must correspond to the kth dimension of the result.

The size of the matrix will be determined by the subscripts themselves. However, if sz is defined it determines the matrix size. The length of sz must correspond to the number of columns in subs. An exception is if subs has only one column, in which case sz may be the dimensions of a vector and the subscripts of subs are taken as the indices into it.

The default action of accumarray is to sum the elements with the same subscripts. This behavior can be modified by defining the func function. This should be a function or function handle that accepts a column vector and returns a scalar. The result of the function should not depend on the order of the subscripts.

The elements of the returned array that have no subscripts associated with them are set to zero. Defining fillval to some other value allows these values to be defined. This behavior changes, however, for certain values of func. If func is min (respectively, max) then the result will be filled with the minimum (respectively, maximum) integer if vals is of integral type, logical false (respectively, logical true) if vals is of logical type, zero if fillval is zero and all values are non-positive (respectively, non-negative), and NaN otherwise.

By default accumarray returns a full matrix. If issparse is logically true, then a sparse matrix is returned instead.

The following accumarray example constructs a frequency table that in the first column counts how many occurrences each number in the second column has, taken from the vector x. Note the usage of unique for assigning to all repeated elements of x the same index (see [unique], page 597).

     x = [91, 92, 90, 92, 90, 89, 91, 89, 90, 100, 100, 100];
     [u, ~, j] = unique (x);
     [accumarray(j', 1), u']
       ⇒ 2    89
           3    90
           2    91
           2    92
           3   100

Another example, where the result is a multi-dimensional 3-D array and the default value (zero) appears in the output:

     accumarray ([1, 1, 1;
                  2, 1, 2;
                  2, 3, 2;
                  2, 1, 2;
                  2, 3, 2], 101:105)
     ⇒ ans(:,:,1) = [101, 0, 0; 0, 0, 0]

     ⇒ ans(:,:,2) = [0, 0, 0; 206, 0, 208]

The sparse option can be used as an alternative to the sparse constructor (see [sparse], page 492). Thus

     sparse (i, j, sv)

can be written with accumarray as

     accumarray ([i, j], sv', [], [], 0, true)

For repeated indices, sparse adds the corresponding value. To take the minimum instead, use min as an accumulator function:

     accumarray ([i, j], sv', [], @min, 0, true)

The complexity of accumarray in general for the non-sparse case is generally O(M+N), where N is the number of subscripts and M is the maximum subscript (linearized in multi-dimensional case). If func is one of @sum (default), @max, @min or @(x) {x}, an optimized code path is used. Note that for general reduction function the interpreter overhead can play a major part and it may be more efficient to do multiple accumarray calls and compute the results in a vectorized manner.

See also: [accumdim], page 468, [unique], page 597, [sparse], page 492.

[Function File] accumdim (subs, vals, dim, n, func, fillval)

Create an array by accumulating the slices of an array into the positions defined by their subscripts along a specified dimension. The subscripts are defined by the index vector subs. The dimension is specified by dim. If not given, it defaults to the first non-singleton dimension. The length of subs must be equal to size (vals, dim). The extent of the result matrix in the working dimension will be determined by the subscripts themselves. However, if n is defined it determines this extent. The default action of accumdim is to sum the subarrays with the same subscripts. This behavior can be modified by defining the func function. This should be a function or function handle that accepts an array and a dimension, and reduces the array along this dimension. As a special exception, the built-in min and max functions can be used directly, and accumdim accounts for the middle empty argument that is used in their calling. The slices of the returned array that have no subscripts associated with them are set to zero. Defining fillval to some other value allows these values to be defined. An example of the use of accumdim is:

     accumdim ([1, 2, 1, 2, 1], [ 7, -10,   4;
                                 -5, -12,   8;
                                -12,   2,   8;
                                -10,   9,  -3;
                                 -5,  -3, -13])
     ⇒ [-10,-11,-1;-15,-3,5]

See also: [accumarray], page 466.

19.5 JIT Compiler

Vectorization is the preferred technique for eliminating loops and speeding up code. Nevertheless, it is not always possible to replace every loop. In such situations it may be worth trying Octave’s experimental Just-In-Time (JIT) compiler.

A JIT compiler works by analyzing the body of a loop, translating the Octave statements into another language, compiling the new code segment into an executable, and then running the executable and collecting any results. The process is not simple and there is a significant amount of work to perform for each step. It can still make sense, however, if the number of loop iterations is large. Because Octave is an interpreted language every time through a loop Octave must parse the statements in the loop body before executing them. With a JIT compiler this is done just once when the body is translated to another language.

The JIT compiler is a very new feature in Octave and not all valid Octave statements can currently be accelerated. However, if no other technique is available it may be worth benchmarking the code with JIT enabled. The function jit_enable is used to turn compilation on or off. The function jit_startcnt sets the threshold for acceleration. Loops with iteration counts above jit_startcnt will be accelerated. The function debug_jit is not likely to be of use to anyone not working directly on the implementation of the JIT compiler.

[Built-in Function] val = jit_enable ()

[Built-in Function] old_val = jit_enable (new_val)

[Built-in Function] jit_enable (new_val, "local")

Query or set the internal variable that enables Octave’s JIT compiler.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [jit startcnt], page 469, [debug jit], page 469.

[Built-in Function] val = jit_startcnt ()

[Built-in Function] old_val = jit_startcnt (new_val)

[Built-in Function] jit_startcnt (new_val, "local")

Query or set the internal variable that determines whether JIT compilation will take place for a specific loop. Because compilation is a costly operation it does not make sense to employ JIT when the loop count is low. By default only loops with greater than 1000 iterations will be accelerated.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [jit enable], page 469, [debug jit], page 469.

[Built-in Function] val = debug_jit ()

[Built-in Function] old_val = debug_jit (new_val)

[Built-in Function] debug_jit (new_val, "local")

Query or set the internal variable that determines whether debugging/tracing is enabled for Octave’s JIT compiler. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [jit enable], page 469, [jit startcnt], page 469.

19.6 Miscellaneous Techniques

Here are some other ways of improving the execution speed of Octave programs.

• Avoid computing costly intermediate results multiple times. Octave currently does not eliminate common subexpressions. Also, certain internal computation results are cached for variables. For instance, if a matrix variable is used multiple times as an index, checking the indices (and internal conversion to integers) is only done once.

• Be aware of lazy copies (copy-on-write). When a copy of an object is created, the data is not immediately copied, but rather shared. The actual copying is postponed until the copied data needs to be modified. For example:

     a = zeros (1000); # create a 1000x1000 matrix
     b = a; # no copying done here
     b(1) = 1; # copying done here

Lazy copying applies to whole Octave objects such as matrices, cells, struct, and also individual cell or struct elements (not array elements). Additionally, index expressions also use lazy copying when Octave can determine that the indexed portion is contiguous in memory. For example:

     a = zeros (1000); # create a 1000x1000 matrix
     b = a(:,10:100);  # no copying done here
     b = a(10:100,:);  # copying done here

This applies to arrays (matrices), cell arrays, and structs indexed using ‘()’. Index expressions generating comma-separated lists can also benefit from shallow copying in some cases. In particular, when a is a struct array, expressions like {a.x}, {a(:,2).x} will use lazy copying, so that data can be shared between a struct array and a cell array. Most indexing expressions do not live longer than their parent objects. In rare cases, however, a lazily copied slice outlasts its parent, in which case it becomes orphaned, still occupying unnecessarily more memory than needed. To provide a remedy working in most real cases, Octave checks for orphaned lazy slices at certain situations, when a value is stored into a "permanent" location, such as a named variable or cell or struct element, and possibly economizes them. For example:

     a = zeros (1000); # create a 1000x1000 matrix
     b = a(:,10:100);  # lazy slice
     a = []; # the original "a" array is still allocated
     c{1} = b; # b is reallocated at this point

• Avoid deep recursion. Function calls to m-file functions carry a relatively significant overhead, so rewriting a recursion as a loop often helps. Also, note that the maximum level of recursion is limited.

• Avoid resizing matrices unnecessarily. When building a single result matrix from a series of calculations, set the size of the result matrix first, then insert values into it. Write

     result = zeros (big_n, big_m)
     for i = over:and_over
       ridx = ...
       cidx = ...
       result(ridx, cidx) = new_value ();
     endfor

instead of

     result = [];
     for i = ever:and_ever
       result = [ result, new_value() ];
     endfor

Sometimes the number of items can not be computed in advance, and stack-like operations are needed. When elements are being repeatedly inserted or removed from the end of an array, Octave detects it as stack usage and attempts to use a smarter memory management strategy by pre-allocating the array in bigger chunks. This strategy is also applied to cell and struct arrays.

     a = [];
     while (condition)
       ...
       a(end+1) = value; # "push" operation
       ...
       a(end) = []; # "pop" operation
       ...
     endwhile

• Avoid calling eval or feval excessively. Parsing input or looking up the name of a function in the symbol table are relatively expensive operations.

If you are using eval merely as an exception handling mechanism, and not because you need to execute some arbitrary text, use the try statement instead. See Section 10.9 [The try Statement], page 164.

• Use ignore_function_time_stamp when appropriate. If you are calling lots of functions, and none of them will need to change during your run, set the variable ignore_ function_time_stamp to "all". This will stop Octave from checking the time stamp of a function file to see if it has been updated while the program is being run.

19.7 Examples

The following are examples of vectorization questions asked by actual users of Octave and their solutions.

• For a vector A, the following loop

n = length (A);
B = zeros (n, 2);
for i = 1:length (A)
  ## this will be two columns, the first is the difference and
  ## the second the mean of the two elements used for the diff.
  B(i,:) = [A(i+1)-A(i), (A(i+1) + A(i))/2)];
endfor

can be turned into the following one-liner:

B = [diff(A)(:), 0.5*(A(1:end-1)+A(2:end))(:)]

Note the usage of colon indexing to flatten an intermediate result into a column vector. This is a common vectorization trick.

20 Nonlinear Equations

20.1 Solvers

Octave can solve sets of nonlinear equations of the form

f(x) = 0

using the function fsolve, which is based on the minpack subroutine hybrd. This is an iterative technique so a starting point must be provided. This also has the consequence that convergence is not guaranteed even if a solution exists.

[Function File] fsolve (fcn, x0, options)
[Function File] [x, fvec, info, output, fjac] = fsolve (fcn, . . .)

Solve a system of nonlinear equations defined by the function fcn. fcn should accept a vector (array) defining the unknown variables, and return a vector of left-hand sides of the equations. Right-hand sides are defined to be zeros. In other words, this function attempts to determine a vector x such that fcn (x) gives (approximately) all zeros. x0 determines a starting guess. The shape of x0 is preserved in all calls to fcn, but otherwise it is treated as a column vector. options is a structure specifying additional options. Currently, fsolve recognizes these options: "FunValCheck",

"OutputFcn", "TolX", "TolFun", "MaxIter", "MaxFunEvals", "Jacobian",
"Updating", "ComplexEqn" "TypicalX", "AutoScaling" and "FinDiffType".

If "Jacobian" is "on", it specifies that fcn, called with 2 output arguments, also returns the Jacobian matrix of right-hand sides at the requested point. "TolX" specifies the termination tolerance in the unknown variables, while "TolFun" is a tolerance for equations. Default is 1e-7 for both "TolX" and "TolFun". If "AutoScaling" is on, the variables will be automatically scaled according to the column norms of the (estimated) Jacobian. As a result, TolF becomes scalingindependent. By default, this option is off, because it may sometimes deliver unexpected (though mathematically correct) results. If "Updating" is "on", the function will attempt to use Broyden updates to update the Jacobian, in order to reduce the amount of Jacobian calculations. If your user function always calculates the Jacobian (regardless of number of output arguments), this option provides no advantage and should be set to false. "ComplexEqn" is "on", fsolve will attempt to solve complex equations in complex variables, assuming that the equations possess a complex derivative (i.e., are holomorphic). If this is not what you want, should unpack the real and imaginary parts of the system to get a real system. For description of the other options, see optimset. On return, fval contains the value of the function fcn evaluated at x, and info may be one of the following values:

1 Converged to a solution point. Relative residual error is less than specified by TolFun.

2 Last relative step size was less that TolX.

3 Last relative decrease in residual was less than TolF.

0 Iteration limit exceeded.

-3 The trust region radius became excessively small.

Note: If you only have a single nonlinear equation of one variable, using fzero is usually a much better idea. Note about user-supplied Jacobians: As an inherent property of the algorithm, Jacobian is always requested for a solution vector whose residual vector is already known, and it is the last accepted successful step. Often this will be one of the last two calls, but not always. If the savings by reusing intermediate results from residual calculation in Jacobian calculation are significant, the best strategy is to employ OutputFcn: After a vector is evaluated for residuals, if OutputFcn is called with that vector, then the intermediate results should be saved for future Jacobian evaluation, and should be kept until a Jacobian evaluation is requested or until OutputFcn is called with a different vector, in which case they should be dropped in favor of this most recent vector. A short example how this can be achieved follows:

     function [fvec, fjac] = user_func (x, optimvalues, state)
     persistent sav = [], sav0 = [];
     if (nargin == 1)
       ## evaluation call
       if (nargout == 1)
         sav0.x = x; # mark saved vector
         ## calculate fvec, save results to sav0.
       elseif (nargout == 2)
         ## calculate fjac using sav.
       endif
     else
       ## outputfcn call.
       if (all (x == sav0.x))
         sav = sav0;
       endif
       ## maybe output iteration status, etc.
     endif
     endfunction
     ## ...
     fsolve (@user_func, x0, optimset ("OutputFcn", @user_func, ...))

See also: [fzero], page 475, [optimset], page 558.

The following is a complete example. To solve the set of equations

\begin{aligned}−2x^{2} + 3xy + 4 sin(y) − 6 & = 0 \\ 3x^{2} − 2xy^{2} + 3 cos(x) + 4 & = 0 \end{aligned}

you first need to write a function to compute the value of the given function. For example:

function y = f (x)
y = zeros (2, 1);
y(1) = -2*x(1)^2 + 3*x(1)*x(2)   + 4*sin(x(2)) - 6;
y(2) =  3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4;
endfunction

Then, call fsolve with a specified initial condition to find the roots of the system of equations. For example, given the function f defined above,

[x, fval, info] = fsolve (@f, [1; 2])

results in the solution

x =
0.57983
2.54621
fval =
-5.7184e-10
 5.5460e-10
info = 1

A value of info = 1 indicates that the solution has converged.

When no Jacobian is supplied (as in the example above) it is approximated numerically. This requires more function evaluations, and hence is less efficient. In the example above we could compute the Jacobian analytically as

[ \begin{matrix}\frac{∂f_{1}}{∂x_{1}} & \frac{∂f_{1}}{∂x_{2}} \\ \frac{∂f_{2}}{∂x_{1}} & \frac{∂f_{2}}{∂x_{2}} \end{matrix} ] = [ \begin{matrix}3x_{2} − 4x_{1} & 4 cos(x_{2}) + 3x_{1} \\ −2x_{2}^{2} − 3 sin(x_{1}) + 6x_{1} & −4x_{1}x_{2} \end{matrix} ]

and compute it with the following Octave function

function [y, jac] = f (x)
y = zeros (2, 1);
y(1) = -2*x(1)^2 + 3*x(1)*x(2)   + 4*sin(x(2)) - 6;
y(2) =  3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4;
if (nargout == 2)
  jac = zeros (2, 2);
  jac(1,1) =  3*x(2) - 4*x(1);
  jac(1,2) =  4*cos(x(2)) + 3*x(1);
  jac(2,1) = -2*x(2)^2 - 3*sin(x(1)) + 6*x(1);
  jac(2,2) = -4*x(1)*x(2);
endif
endfunction

The Jacobian can then be used with the following call to fsolve:

[x, fval, info] = fsolve (@f, [1; 2], optimset ("jacobian", "on"));

which gives the same solution as before.

[Function File] fzero (fun, x0) [Function File] fzero (fun, x0, options)

[Function File] [x, fval, info, output] = fzero (. . .)

Find a zero of a univariate function.

fun is a function handle, inline function, or string containing the name of the function to evaluate. x0 should be a two-element vector specifying two points which bracket a zero. In other words, there must be a change in sign of the function between x0(1) and x0(2). More mathematically, the following must hold

     sign (fun(x0(1))) * sign (fun(x0(2))) <= 0

If x0 is a single scalar then several nearby and distant values are probed in an attempt to obtain a valid bracketing. If this is not successful, the function fails. options is a structure specifying additional options. Currently, fzero recognizes these op-

tions: "FunValCheck", "OutputFcn", "TolX", "MaxIter", "MaxFunEvals". For a

description of these options, see [optimset], page 558.

On exit, the function returns x, the approximate zero point and fval, the function value thereof. info is an exit flag that can have these values:

• 1 The algorithm converged to a solution.

• 0 Maximum number of iterations or function evaluations has been reached.

• -1 The algorithm has been terminated from user output function.

• -5 The algorithm may have converged to a singular point.

output is a structure containing runtime information about the fzero algorithm. Fields in the structure are:

• iterations Number of iterations through loop.

• nfev Number of function evaluations.

• bracketx A two-element vector with the final bracketing of the zero along the x-axis.

• brackety A two-element vector with the final bracketing of the zero along the y-axis.

See also: [optimset], page 558, [fsolve], page 473.

20.2 Minimizers

Often it is useful to find the minimum value of a function rather than just the zeroes where it crosses the x-axis. fminbnd is designed for the simpler, but very common, case of a univariate function where the interval to search is bounded. For unbounded minimization of a function with potentially many variables use fminunc or fminsearch. The two functions use different internal algorithms and some knowledge of the objective function is required. For functions which can be differentiated, fminunc is appropriate. For functions with discontinuities, or for which a gradient search would fail, use fminsearch. See Chapter 25 [Optimization], page 547, for minimization with the presence of constraint functions. Note that searches can be made for maxima by simply inverting the objective function (F_{max} = −F_{min}).

[Function File] [x, fval, info, output] = fminbnd (fun, a, b, options)

Find a minimum point of a univariate function. fun should be a function handle or name. a, b specify a starting interval. options is a structure specifying additional options. Currently, fminbnd recognizes these

options: "FunValCheck", "OutputFcn", "TolX", "MaxIter", "MaxFunEvals". For a

description of these options, see [optimset], page 558. On exit, the function returns x, the approximate minimum point and fval, the function value thereof. info is an exit flag that can have these values:

• 1 The algorithm converged to a solution.

• 0 Maximum number of iterations or function evaluations has been exhausted.

• -1 The algorithm has been terminated from user output function.

Notes: The search for a minimum is restricted to be in the interval bound by a and b. If you only have an initial point to begin searching from you will need to use an unconstrained minimization algorithm such as fminunc or fminsearch. fminbnd internally uses a Golden Section search strategy.

See also: [fzero], page 475, [fminunc], page 477, [fminsearch], page 478, [optimset], page 558.

[Function File] fminunc (fcn, x0)

[Function File] fminunc (fcn, x0, options)
[Function File] [x, fval, info, output, grad, hess] = fminunc (fcn, . . .)

Solve an unconstrained optimization problem defined by the function fcn. fcn should accept a vector (array) defining the unknown variables, and return the objective function value, optionally with gradient. fminunc attempts to determine a vector x such that fcn (x) is a local minimum. x0 determines a starting guess. The shape of x0 is preserved in all calls to fcn, but otherwise is treated as a column vector. options is a structure specifying additional options. Currently, fminunc rec-

ognizes these options: "FunValCheck", "OutputFcn", "TolX", "TolFun", "MaxIter",
"MaxFunEvals", "GradObj", "FinDiffType", "TypicalX", "AutoScaling".

If "GradObj" is "on", it specifies that fcn, when called with 2 output arguments, also returns the Jacobian matrix of partial first derivatives at the requested point. TolX specifies the termination tolerance for the unknown variables x, while TolFun is a tolerance for the objective function value fval. The default is 1e-7 for both options. For a description of the other options, see optimset. On return, x is the location of the minimum and fval contains the value of the objective function at x. info may be one of the following values:

1 Converged to a solution point. Relative gradient error is less than speci-

          fied by TolFun.

2 Last relative step size was less than TolX.

3 Last relative change in function value was less than TolFun.

0 Iteration limit exceeded—either maximum numer of algorithm iterations MaxIter or maximum number of function evaluations MaxFunEvals.

-1 Alogrithm terminated by OutputFcn.

-3 The trust region radius became excessively small.

Optionally, fminunc can return a structure with convergence statistics (output), the output gradient (grad) at the solution x, and approximate Hessian (hess) at the solution x. Notes: If have only a single nonlinear equation of one variable then using fminbnd is usually a much better idea. The algorithm used is a gradient search which depends on the objective function being differentiable. If the function has discontinuities it may be better to use a derivative-free algorithm such as fminsearch.

See also: [fminbnd], page 476, [fminsearch], page 478, [optimset], page 558.

[Function File] x = fminsearch (fun, x0)
[Function File] x = fminsearch (fun, x0, options)
[Function File] [x, fval] = fminsearch (. . .)

Find a value of x which minimizes the function fun. The search begins at the point x0 and iterates using the Nelder & Mead Simplex algorithm (a derivative-free method). This algorithm is better-suited to functions which have discontinuities or for which a gradient-based search such as fminunc fails. Options for the search are provided in the parameter options using the function

optimset. Currently, fminsearch accepts the options: "TolX", "MaxFunEvals",

"MaxIter", "Display". For a description of these options, see optimset. On exit, the function returns x, the minimum point, and fval, the function value thereof. Example usages:

     fminsearch (@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0])
     fminsearch (inline ("(x(1)-5).^2+(x(2)-8).^4", "x"), [0;0])

See also: [fminbnd], page 476, [fminunc], page 477, [optimset], page 558.

21 Diagonal and Permutation Matrices

21.1 Creating and Manipulating Diagonal/Permutation Matrices

A diagonal matrix is defined as a matrix that has zero entries outside the main diagonal; that is, D_{ij} = 0 if i ≠ j Most often, square diagonal matrices are considered; however, the definition can equally be applied to non-square matrices, in which case we usually speak of a rectangular diagonal matrix.

A permutation matrix is defined as a square matrix that has a single element equal to unity in each row and each column; all other elements are zero. That is, there exists a permutation (vector) p such that P_{ij} = 1 if j = p_{i} and P_{ij} = 0 otherwise.

Octave provides special treatment of real and complex rectangular diagonal matrices, as well as permutation matrices. They are stored as special objects, using efficient storage and algorithms, facilitating writing both readable and efficient matrix algebra expressions in the Octave language.

21.1.1 Creating Diagonal Matrices

The most common and easiest way to create a diagonal matrix is using the built-in function diag. The expression diag (v), with v a vector, will create a square diagonal matrix with elements on the main diagonal given by the elements of v, and size equal to the length of v. diag (v, m, n) can be used to construct a rectangular diagonal matrix. The result of these expressions will be a special diagonal matrix object, rather than a general matrix object.

Diagonal matrix with unit elements can be created using eye. Some other built-in functions can also return diagonal matrices. Examples include balance or inv.

Example:

       diag (1:4)

⇒

     Diagonal Matrix
        1   0   0   0
        0   2   0   0
        0   0   3   0
        0   0   0   4
       diag (1:3,5,3)

⇒

     Diagonal Matrix
        1   0   0
        0   2   0
        0   0   3
        0   0   0
        0   0   0

21.1.2 Creating Permutation Matrices

For creating permutation matrices, Octave does not introduce a new function, but rather overrides an existing syntax: permutation matrices can be conveniently created by indexing an identity matrix by permutation vectors. That is, if q is a permutation vector of length n, the expression

       P = eye (n) (:, q);

will create a permutation matrix - a special matrix object.

     eye (n) (q, :)

will also work (and create a row permutation matrix), as well as

     eye (n) (q1, q2).

For example:

       eye (4) ([1,3,2,4],:)

⇒

     Permutation Matrix
        1   0   0   0
        0   0   1   0
        0   1   0   0
        0   0   0   1
       eye (4) (:,[1,3,2,4])

⇒

     Permutation Matrix
        1   0   0   0
        0   0   1   0
        0   1   0   0
        0   0   0   1

Mathematically, an identity matrix is both diagonal and permutation matrix. In Octave, eye (n) returns a diagonal matrix, because a matrix can only have one class. You can convert this diagonal matrix to a permutation matrix by indexing it by an identity permutation, as shown below. This is a special property of the identity matrix; indexing other diagonal matrices generally produces a full matrix.

       eye (3)

⇒

     Diagonal Matrix
        1   0   0
        0   1   0
        0   0   1
       eye(3)(1:3,:)

⇒

     Permutation Matrix
        1   0   0
        0   1   0
        0   0   1

Some other built-in functions can also return permutation matrices. Examples include inv or lu.

21.1.3 Explicit and Implicit Conversions

The diagonal and permutation matrices are special objects in their own right. A number of operations and built-in functions are defined for these matrices to use special, more efficient code than would be used for a full matrix in the same place. Examples are given in further sections.

To facilitate smooth mixing with full matrices, backward compatibility, and compatibility with matlab, the diagonal and permutation matrices should allow any operation that works on full matrices, and will either treat it specially, or implicitly convert themselves to full matrices.

Instances include matrix indexing, except for extracting a single element or a leading submatrix, indexed assignment, or applying most mapper functions, such as exp.

An explicit conversion to a full matrix can be requested using the built-in function full. It should also be noted that the diagonal and permutation matrix objects will cache the result of the conversion after it is first requested (explicitly or implicitly), so that subsequent conversions will be very cheap.

21.2 Linear Algebra with Diagonal/Permutation Matrices

As has been already said, diagonal and permutation matrices make it possible to use efficient algorithms while preserving natural linear algebra syntax. This section describes in detail the operations that are treated specially when performed on these special matrix objects.

21.2.1 Expressions Involving Diagonal Matrices

Assume D is a diagonal matrix. If M is a full matrix, then D*M will scale the rows of M. That means, if S = D*M, then for each pair of indices i,j it holds

S_{ij} = D_{ii}M_{ij}

Similarly, M*D will do a column scaling.

The matrix D may also be rectangular, m-by-n where m != n. If m < n, then the expression D*M is equivalent to

     D(:,1:m) * M(1:m,:),

i.e., trailing n-m rows of M are ignored. If m > n, then D*M is equivalent to

     [D(1:n,n) * M; zeros(m-n, columns (M))],

i.e., null rows are appended to the result. The situation for right-multiplication M*D is analogous.

The expressions D \ M and M / D perform inverse scaling. They are equivalent to solving a diagonal (or rectangular diagonal) in a least-squares minimum-norm sense. In exact arithmetic, this is equivalent to multiplying by a pseudoinverse. The pseudoinverse of a rectangular diagonal matrix is again a rectangular diagonal matrix with swapped dimensions, where each nonzero diagonal element is replaced by its reciprocal. The matrix division algorithms do, in fact, use division rather than multiplication by reciprocals for better numerical accuracy; otherwise, they honor the above definition. Note that a diagonal matrix is never truncated due to ill-conditioning; otherwise, it would not be of much use for scaling. This is typically consistent with linear algebra needs. A full matrix that only happens to be diagonal (and is thus not a special object) is of course treated normally.

Multiplication and division by diagonal matrices work efficiently also when combined with sparse matrices, i.e., D*S, where D is a diagonal matrix and S is a sparse matrix scales the rows of the sparse matrix and returns a sparse matrix. The expressions S*D, D\S, S/D work analogically.

If D1 and D2 are both diagonal matrices, then the expressions

     D1 + D2 D1 - D2 D1 * D2 D1 / D2 D1 \ D2

again produce diagonal matrices, provided that normal dimension matching rules are obeyed. The relations used are same as described above.

Also, a diagonal matrix D can be multiplied or divided by a scalar, or raised to a scalar power if it is square, producing diagonal matrix result in all cases.

A diagonal matrix can also be transposed or conjugate-transposed, giving the expected result. Extracting a leading submatrix of a diagonal matrix, i.e., D(1:m,1:n), will produce a diagonal matrix, other indexing expressions will implicitly convert to full matrix.

Adding a diagonal matrix to a full matrix only operates on the diagonal elements. Thus,

     A = A + eps * eye (n)

is an efficient method of augmenting the diagonal of a matrix. Subtraction works analogically.

When involved in expressions with other element-by-element operators, .*, ./, .\ or .^, an implicit conversion to full matrix will take place. This is not always strictly necessary but chosen to facilitate better consistency with matlab.

21.2.2 Expressions Involving Permutation Matrices

If P is a permutation matrix and M a matrix, the expression P*M will permute the rows of M. Similarly, M*P will yield a column permutation. Matrix division P\M and M/P can be used to do inverse permutation.

The previously described syntax for creating permutation matrices can actually help an user to understand the connection between a permutation matrix and a permuting vector. Namely, the following holds, where I = eye (n) is an identity matrix:

       I(p,:) * M = (I*M) (p,:) = M(p,:)

Similarly,

       M * I(:,p) = (M*I) (:,p) = M(:,p)

The expressions I(p,:) and I(:,p) are permutation matrices.

A permutation matrix can be transposed (or conjugate-transposed, which is the same, because a permutation matrix is never complex), inverting the permutation, or equivalently, turning a row-permutation matrix into a column-permutation one. For permutation matrices, transpose is equivalent to inversion, thus P\M is equivalent to P'*M. Transpose of a permutation matrix (or inverse) is a constant-time operation, flipping only a flag internally, and thus the choice between the two above equivalent expressions for inverse permuting is completely up to the user’s taste.

Multiplication and division by permutation matrices works efficiently also when combined with sparse matrices, i.e., P*S, where P is a permutation matrix and S is a sparse matrix permutes the rows of the sparse matrix and returns a sparse matrix. The expressions S*P, P\S, S/P work analogically.

Two permutation matrices can be multiplied or divided (if their sizes match), performing a composition of permutations. Also a permutation matrix can be indexed by a permutation vector (or two vectors), giving again a permutation matrix. Any other operations do not generally yield a permutation matrix and will thus trigger the implicit conversion.

21.3 Functions That Are Aware of These Matrices

This section lists the built-in functions that are aware of diagonal and permutation matrices on input, or can return them as output. Passed to other functions, these matrices will in general trigger an implicit conversion. (Of course, user-defined dynamically linked functions may also work with diagonal or permutation matrices).

21.3.1 Diagonal Matrix Functions

inv and pinv can be applied to a diagonal matrix, yielding again a diagonal matrix. det will use an efficient straightforward calculation when given a diagonal matrix, as well as cond. The following mapper functions can be applied to a diagonal matrix without converting it to a full one: abs, real, imag, conj, sqrt. A diagonal matrix can also be returned from the balance and svd functions. The sparse function will convert a diagonal matrix efficiently to a sparse matrix.

21.3.2 Permutation Matrix Functions

inv and pinv will invert a permutation matrix, preserving its specialness. det can be applied to a permutation matrix, efficiently calculating the sign of the permutation (which is equal to the determinant).

A permutation matrix can also be returned from the built-in functions lu and qr, if a pivoted factorization is requested.

The sparse function will convert a permutation matrix efficiently to a sparse matrix. The find function will also work efficiently with a permutation matrix, making it possible to conveniently obtain the permutation indices.

21.4 Examples of Usage

The following can be used to solve a linear system A*x = b using the pivoted LU factorization:

       [L, U, P] = lu (A); ## now L*U = P*A
       x = U \ L \ P*b;

This is one way to normalize columns of a matrix X to unit norm:

       s = norm (X, "columns");
       X /= diag (s);

The same can also be accomplished with broadcasting (see Section 19.2 [Broadcasting], page 459):

       s = norm (X, "columns");
       X ./= s;

The following expression is a way to efficiently calculate the sign of a permutation, given by a permutation vector p. It will also work in earlier versions of Octave, but slowly.

       det (eye (length (p))(p, :))

Finally, here’s how to solve a linear system A*x = b with Tikhonov regularization (ridge regression) using SVD (a skeleton only):

       m = rows (A); n = columns (A);
       [U, S, V] = svd (A);
       ## determine the regularization factor alpha
       ## alpha = ...
       ## transform to orthogonal basis
       b = U'*b;
       ## Use the standard formula, replacing A with S.
       ## S is diagonal, so the following will be very fast and accurate.
       x = (S'*S + alpha^2 * eye (n)) \ (S' * b);
       ## transform to solution basis
       x = V*x;

21.5 Differences in Treatment of Zero Elements

Making diagonal and permutation matrices special matrix objects in their own right and the consequent usage of smarter algorithms for certain operations implies, as a side effect, small differences in treating zeros. The contents of this section apply also to sparse matrices, discussed in the following chapter. (see Chapter 22 [Sparse Matrices], page 487)

The IEEE floating point standard defines the result of the expressions 0*Inf and 0*NaN as NaN. This is widely agreed to be a good compromise. Numerical software dealing with structured and sparse matrices (including Octave) however, almost always makes a distinction between a "numerical zero" and an "assumed zero". A "numerical zero" is a zero value occurring in a place where any floating-point value could occur. It is normally stored somewhere in memory as an explicit value. An "assumed zero", on the contrary, is a zero matrix element implied by the matrix structure (diagonal, triangular) or a sparsity pattern; its value is usually not stored explicitly anywhere, but is implied by the underlying data structure.

The primary distinction is that an assumed zero, when multiplied by any number, or divided by any nonzero number, yields *always* a zero, even when, e.g., multiplied by Inf or divided by NaN. The reason for this behavior is that the numerical multiplication is not actually performed anywhere by the underlying algorithm; the result is just assumed to be zero. Equivalently, one can say that the part of the computation involving assumed zeros is performed symbolically, not numerically.

This behavior not only facilitates the most straightforward and efficient implementation of algorithms, but also preserves certain useful invariants, like:

• scalar * diagonal matrix is a diagonal matrix

• sparse matrix / scalar preserves the sparsity pattern

• permutation matrix * matrix is equivalent to permuting rows

all of these natural mathematical truths would be invalidated by treating assumed zeros as numerical ones.

Note that matlab does not strictly follow this principle and converts assumed zeros to numerical zeros in certain cases, while not doing so in other cases. As of today, there are no intentions to mimic such behavior in Octave.

Examples of effects of assumed zeros vs. numerical zeros:

     Inf * eye (3)

⇒

        Inf     0     0
          0   Inf     0
          0     0   Inf
     Inf * speye (3)

⇒

     Compressed Column Sparse (rows = 3, cols = 3, nnz = 3 [33%])
       (1, 1) -> Inf
       (2, 2) -> Inf
       (3, 3) -> Inf
     Inf * full (eye (3))

⇒

        Inf   NaN   NaN
        NaN   Inf   NaN
        NaN   NaN   Inf

diag (1:3) * [NaN; 1; 1]

⇒

NaN
  2
  3
sparse (1:3,1:3,1:3) * [NaN; 1; 1]

⇒

NaN
  2
  3
[1,0,0;0,2,0;0,0,3] * [NaN; 1; 1]

⇒

NaN
NaN
NaN

22 Sparse Matrices

22.1 Creation and Manipulation of Sparse Matrices

The size of mathematical problems that can be treated at any particular time is generally limited by the available computing resources. Both, the speed of the computer and its available memory place limitation on the problem size.

There are many classes of mathematical problems which give rise to matrices, where a large number of the elements are zero. In this case it makes sense to have a special matrix type to handle this class of problems where only the non-zero elements of the matrix are stored. Not only does this reduce the amount of memory to store the matrix, but it also means that operations on this type of matrix can take advantage of the a priori knowledge of the positions of the non-zero elements to accelerate their calculations.

A matrix type that stores only the non-zero elements is generally called sparse. It is the purpose of this document to discuss the basics of the storage and creation of sparse matrices and the fundamental operations on them.

22.1.1 Storage of Sparse Matrices

It is not strictly speaking necessary for the user to understand how sparse matrices are stored. However, such an understanding will help to get an understanding of the size of sparse matrices. Understanding the storage technique is also necessary for those users wishing to create their own oct-files.

There are many different means of storing sparse matrix data. What all of the methods have in common is that they attempt to reduce the complexity and storage given a priori knowledge of the particular class of problems that will be solved. A good summary of the available techniques for storing sparse matrix is given by Saad^{1}. With full matrices, knowledge of the point of an element of the matrix within the matrix is implied by its position in the computers memory. However, this is not the case for sparse matrices, and so the positions of the non-zero elements of the matrix must equally be stored.

An obvious way to do this is by storing the elements of the matrix as triplets, with two elements being their position in the array (rows and column) and the third being the data itself. This is conceptually easy to grasp, but requires more storage than is strictly needed.

The storage technique used within Octave is the compressed column format. It is similar to the Yale format.^{2} In this format the position of each element in a row and the data are stored as previously. However, if we assume that all elements in the same column are stored adjacent in the computers memory, then we only need to store information on the number of non-zero elements in each column, rather than their positions. Thus assuming that the matrix has more non-zero elements than there are columns in the matrix, we win in terms of the amount of memory used.

In fact, the column index contains one more element than the number of columns, with the first element always being zero. The advantage of this is a simplification in the code, in that there is no special case for the first or last columns. A short example, demonstrating this in C is.

1 Y. Saad "SPARSKIT: A basic toolkit for sparse matrix computation", 1994, http://www-users.cs.umn.edu/~saad/software/SPARSKIT/paper.ps 2 http://en.wikipedia.org/wiki/Sparse_matrix#Yale_format

       for (j = 0; j < nc; j++)
         for (i = cidx(j); i < cidx(j+1); i++)
            printf ("non-zero element (%i,%i) is %d\n",
                ridx(i), j, data(i));

A clear understanding might be had by considering an example of how the above applies to an example matrix. Consider the matrix

         1   2   0  0
         0   0   0  3
         0   0   0  4

The non-zero elements of this matrix are

        (1, 1) ⇒ 1

        (1, 2) ⇒ 2
        (2, 4) ⇒ 3
        (3, 4) ⇒ 4

This will be stored as three vectors cidx, ridx and data, representing the column indexing, row indexing and data respectively. The contents of these three vectors for the above matrix will be

       cidx = [0, 1, 2, 2, 4]
       ridx = [0, 0, 1, 2]
       data = [1, 2, 3, 4]

Note that this is the representation of these elements with the first row and column assumed to start at zero, while in Octave itself the row and column indexing starts at one. Thus the number of elements in the i-th column is given by cidx (i + 1) - cidx (i).

Although Octave uses a compressed column format, it should be noted that compressed row formats are equally possible. However, in the context of mixed operations between mixed sparse and dense matrices, it makes sense that the elements of the sparse matrices are in the same order as the dense matrices. Octave stores dense matrices in column major ordering, and so sparse matrices are equally stored in this manner.

A further constraint on the sparse matrix storage used by Octave is that all elements in the rows are stored in increasing order of their row index, which makes certain operations faster. However, it imposes the need to sort the elements on the creation of sparse matrices. Having disordered elements is potentially an advantage in that it makes operations such as concatenating two sparse matrices together easier and faster, however it adds complexity and speed problems elsewhere.

22.1.2 Creating Sparse Matrices

There are several means to create sparse matrix.

Returned from a function There are many functions that directly return sparse matrices. These include speye, sprand, diag, etc.

Constructed from matrices or vectors The function sparse allows a sparse matrix to be constructed from three vectors representing the row, column and data. Alternatively, the function spconvert uses a three column matrix format to allow easy importation of data from elsewhere.

Created and then filled The function sparse or spalloc can be used to create an empty matrix that is then filled by the user

From a user binary program The user can directly create the sparse matrix within an oct-file.

There are several basic functions to return specific sparse matrices. For example the sparse identity matrix, is a matrix that is often needed. It therefore has its own function to create it as speye (n) or speye (r, c), which creates an n-by-n or r-by-c sparse identity matrix.

Another typical sparse matrix that is often needed is a random distribution of random elements. The functions sprand and sprandn perform this for uniform and normal random distributions of elements. They have exactly the same calling convention, where sprand (r, c, d), creates an r-by-c sparse matrix with a density of filled elements of d.

Other functions of interest that directly create sparse matrices, are diag or its generalization spdiags, that can take the definition of the diagonals of the matrix and create the sparse matrix that corresponds to this. For example,

     s = diag (sparse (randn (1,n)), -1);

creates a sparse (n+1)-by-(n+1) sparse matrix with a single diagonal defined.

[Function File] [b, c] = spdiags (A) [Function File] b = spdiags (A, c) [Function File] b = spdiags (v, c, A) [Function File] b = spdiags (v, c, m, n)

A generalization of the function diag. Called with a single input argument, the nonzero diagonals c of A are extracted. With two arguments the diagonals to extract are given by the vector c. The other two forms of spdiags modify the input matrix by replacing the diagonals. They use the columns of v to replace the columns represented by the vector c. If the sparse matrix A is defined then the diagonals of this matrix are replaced. Otherwise a matrix of m by n is created with the diagonals given by v. Negative values of c represent diagonals below the main diagonal, and positive values of c diagonals above the main diagonal. For example:

          spdiags (reshape (1:12, 4, 3), [-1 0 1], 5, 4)
             ⇒ 5 10  0  0
                1  6 11  0
                0  2  7 12
                0  0  3  8
                0  0  0  4

[Function File] s = speye (m, n) [Function File] s = speye (m)

[Function File] s = speye (sz)

Return a sparse identity matrix of size mxn. The implementation is significantly more efficient than sparse (eye (m)) as the full matrix is not constructed. Called with a single argument a square matrix of size m-by-m is created. If called with a single vector argument sz, this argument is taken to be the size of the matrix to create.

See also: [sparse], page 492, [spdiags], page 489, [eye], page 391.

[Function File] r = spones (S)

Replace the non-zero entries of S with ones. This creates a sparse matrix with the same structure as S.

See also: [sparse], page 492, [sprand], page 490, [sprandn], page 490, [sprandsym], page 490, [spfun], page 464, [spy], page 496.

[Function File] sprand (m, n, d) [Function File] sprand (s)

Generate a random sparse matrix. The size of the matrix will be mxn, with a density of values given by d. d must be between 0 and 1 inclusive. Values will be uniformly distributed between 0 and 1. If called with a single matrix argument, a random sparse matrix is generated wherever the matrix S is non-zero.

See also: [sprandn], page 490, [sprandsym], page 490, [spones], page 490, [sparse], page 492.

[Function File] sprandn (m, n, d) [Function File] sprandn (s)

Generate a random sparse matrix. The size of the matrix will be mxn, with a density of values given by d. d must be between 0 and 1 inclusive. Values will be normally distributed with a mean of zero and a variance of 1. If called with a single matrix argument, a random sparse matrix is generated wherever the matrix S is non-zero.

See also: [sprand], page 490, [sprandsym], page 490, [spones], page 490, [sparse], page 492.

[Function File] sprandsym (n, d) [Function File] sprandsym (s)

Generate a symmetric random sparse matrix. The size of the matrix will be nxn, with a density of values given by d. d must be between 0 and 1 inclusive. Values will be normally distributed with a mean of zero and a variance of 1. If called with a single matrix argument, a random sparse matrix is generated wherever the matrix S is non-zero in its lower triangular part.

See also: [sprand], page 490, [sprandn], page 490, [spones], page 490, [sparse], page 492.

The recommended way for the user to create a sparse matrix, is to create two vectors containing the row and column index of the data and a third vector of the same size containing the data to be stored. For example,

       ri = ci = d = [];
       for j = 1:c
         ri = [ri; randperm(r,n)'];
         ci = [ci; j*ones(n,1)];
         d = [d; rand(n,1)];
       endfor
       s = sparse (ri, ci, d, r, c);

creates an r-by-c sparse matrix with a random distribution of n (<r) elements per column. The elements of the vectors do not need to be sorted in any particular order as Octave will sort them prior to storing the data. However, pre-sorting the data will make the creation of the sparse matrix faster.

The function spconvert takes a three or four column real matrix. The first two columns represent the row and column index respectively and the third and four columns, the real and imaginary parts of the sparse matrix. The matrix can contain zero elements and the elements can be sorted in any order. Adding zero elements is a convenient way to define the size of the sparse matrix. For example:

     s = spconvert ([1 2 3 4; 1 3 4 4; 1 2 3 0]')
     ⇒ Compressed Column Sparse (rows=4, cols=4, nnz=3)
           (1 , 1) -> 1
           (2 , 3) -> 2
           (3 , 4) -> 3

An example of creating and filling a matrix might be

     k = 5;
     nz = r * k;
     s = spalloc (r, c, nz)
     for j = 1:c
       idx = randperm (r);
       s (:, j) = [zeros(r - k, 1); ...
             rand(k, 1)] (idx);
     endfor

It should be noted, that due to the way that the Octave assignment functions are written that the assignment will reallocate the memory used by the sparse matrix at each iteration of the above loop. Therefore the spalloc function ignores the nz argument and does not pre-assign the memory for the matrix. Therefore, it is vitally important that code using to above structure should be vectorized as much as possible to minimize the number of assignments and reduce the number of memory allocations.

[Built-in Function] FM = full (SM)

Return a full storage matrix from a sparse, diagonal, permutation matrix, or a range.

See also: [sparse], page 492, [issparse], page 493.

[Built-in Function] s = spalloc (m, n, nz)

Create an m-by-n sparse matrix with pre-allocated space for at most nz nonzero elements. This is useful for building a matrix incrementally by a sequence of indexed assignments. Subsequent indexed assignments after spalloc will reuse the pre-allocated memory, provided they are of one of the simple forms

  • s(I:J) = x
  • s(:,I:J) = x
  • s(K:L,I:J) = x

and that the following conditions are met:

• the assignment does not decrease nnz (S).

• after the assignment, nnz (S) does not exceed nz.

• no index is out of bounds.

Partial movement of data may still occur, but in general the assignment will be more memory and time efficient under these circumstances. In particular, it is possible to efficiently build a pre-allocated sparse matrix from a contiguous block of columns. The amount of pre-allocated memory for a given matrix may be queried using the function nzmax.

See also: [nzmax], page 494, [sparse], page 492.

[Built-in Function] s = sparse (a) [Built-in Function] s = sparse (i, j, sv, m, n) [Built-in Function] s = sparse (i, j, sv) [Built-in Function] s = sparse (m, n) [Built-in Function] s = sparse (i, j, s, m, n, "unique") [Built-in Function] s = sparse (i, j, sv, m, n, nzmax)

Create a sparse matrix from a full matrix or row, column, value triplets. If a is a full matrix, convert it to a sparse matrix representation, removing all zero values in the process. Given the integer index vectors i and j, and a 1-by-nnz vector of real or complex values sv, construct the sparse matrix S(i(k),j(k)) = sv(k) with overall dimensions m and n. If any of sv, i or j are scalars, they are expanded to have a common size. If m or n are not specified their values are derived from the maximum index in the vectors i and j as given by m = max (i), n = max (j). Note: if multiple values are specified with the same i, j indices, the corresponding value in s will be the sum of the values at the repeated location. See accumarray for an example of how to produce different behavior, such as taking the minimum instead. If the option "unique" is given, and more than one value is specified at the same i, j indices, then the last specified value will be used. sparse (m, n) will create an empty mxn sparse matrix and is equivalent to sparse

([], [], [], m, n)

The argument nzmax is ignored but accepted for compatibility with matlab.

Example 1 (sum at repeated indices):

          i = [1 1 2]; j = [1 1 2]; sv = [3 4 5];
          sparse (i, j, sv, 3, 4)

⇒

          Compressed Column Sparse (rows = 3, cols = 4, nnz = 2 [17%])
            (1, 1) ->  7
            (2, 2) ->  5

Example 2 ("unique" option):

          i = [1 1 2]; j = [1 1 2]; sv = [3 4 5];
          sparse (i, j, sv, 3, 4, "unique")

⇒

          Compressed Column Sparse (rows = 3, cols = 4, nnz = 2 [17%])
            (1, 1) ->  4
            (2, 2) ->  5

See also: [full], page 491, [accumarray], page 466, [spalloc], page 492, [spdiags], page 489, [speye], page 489, [spones], page 490, [sprand], page 490, [sprandn], page 490, [sprandsym], page 490, [spconvert], page 493, [spfun], page 464.

[Function File] x = spconvert (m)

Convert a simple sparse matrix format easily generated by other programs into Octave’s internal sparse format.

The input m is either a 3 or 4 column real matrix, containing the row, column, real, and imaginary parts of the elements of the sparse matrix. An element with a zero real and imaginary part can be used to force a particular matrix size.

See also: [sparse], page 492.

The above problem of memory reallocation can be avoided in oct-files. However, the construction of a sparse matrix from an oct-file is more complex than can be discussed here. See Appendix A [External Code Interface], page 759, for a full description of the techniques involved.

22.1.3 Finding Information about Sparse Matrices

There are a number of functions that allow information concerning sparse matrices to be obtained. The most basic of these is issparse that identifies whether a particular Octave object is in fact a sparse matrix.

Another very basic function is nnz that returns the number of non-zero entries there are in a sparse matrix, while the function nzmax returns the amount of storage allocated to the sparse matrix. Note that Octave tends to crop unused memory at the first opportunity for sparse objects. There are some cases of user created sparse objects where the value returned by nzmax will not be the same as nnz, but in general they will give the same result. The function spstats returns some basic statistics on the columns of a sparse matrix including the number of elements, the mean and the variance of each column.

[Built-in Function] issparse (x)

Return true if x is a sparse matrix.

See also: [ismatrix], page 62.

[Built-in Function] n = nnz (a)

Return the number of non-zero elements in a.

See also: [nzmax], page 494, [nonzeros], page 494, [find], page 381.

[Function File] nonzeros (s)

Return a vector of the non-zero values of the sparse matrix s.

See also: [find], page 381, [nnz], page 494.

[Built-in Function] n = nzmax (SM)

Return the amount of storage allocated to the sparse matrix SM. Note that Octave tends to crop unused memory at the first opportunity for sparse objects. Thus, in general the value of nzmax will be the the same as nnz except for some cases of user-created sparse objects.

See also: [nnz], page 494, [spalloc], page 492, [sparse], page 492.

[Function File] [count, mean, var] = spstats (S)
[Function File] [count, mean, var] = spstats (S, j)

Return the stats for the non-zero elements of the sparse matrix S. count is the number of non-zeros in each column, mean is the mean of the non-zeros in each column, and var is the variance of the non-zeros in each column. Called with two input arguments, if S is the data and j is the bin number for the data, compute the stats for each bin. In this case, bins can contain data values of zero, whereas with spstats (S) the zeros may disappear.

When solving linear equations involving sparse matrices Octave determines the means to solve the equation based on the type of the matrix (see Section 22.2 [Sparse Linear Algebra], page 508). Octave probes the matrix type when the div (/) or ldiv (\) operator is first used with the matrix and then caches the type. However the matrix type function can be used to determine the type of the sparse matrix prior to use of the div or ldiv operators. For example,

a = tril (sprandn (1024, 1024, 0.02), -1) ...
    + speye (1024);
matrix_type (a);
ans = Lower

shows that Octave correctly determines the matrix type for lower triangular matrices. matrix type can also be used to force the type of a matrix to be a particular type. For example:

a = matrix_type (tril (sprandn (1024, ...
   1024, 0.02), -1) + speye (1024), "Lower");

This allows the cost of determining the matrix type to be avoided. However, incorrectly defining the matrix type will result in incorrect results from solutions of linear equations, and so it is entirely the responsibility of the user to correctly identify the matrix type

There are several graphical means of finding out information about sparse matrices. The first is the spy command, which displays the structure of the non-zero elements of the matrix. See Figure 22.1, for an example of the use of spy. More advanced graphical information can be obtained with the treeplot, etreeplot and gplot commands.

Figure 22.1: Structure of simple sparse matrix.

One use of sparse matrices is in graph theory, where the interconnections between nodes are represented as an adjacency matrix. That is, if the i-th node in a graph is connected to the j-th node. Then the ij-th node (and in the case of undirected graphs the ji-th node) of the sparse adjacency matrix is non-zero. If each node is then associated with a set of coordinates, then the gplot command can be used to graphically display the interconnections between nodes.

As a trivial example of the use of gplot consider the example,

     A = sparse ([2,6,1,3,2,4,3,5,4,6,1,5],
         [1,1,2,2,3,3,4,4,5,5,6,6],1,6,6);
     xy = [0,4,8,6,4,2;5,0,5,7,5,7]';
     gplot (A,xy)

which creates an adjacency matrix A where node 1 is connected to nodes 2 and 6, node 2 with nodes 1 and 3, etc. The coordinates of the nodes are given in the n-by-2 matrix xy. See Figure 22.2.

Figure 22.2: Simple use of the gplot command. The dependencies between the nodes of a Cholesky factorization can be calculated in

linear time without explicitly needing to calculate the Cholesky factorization by the etree command. This command returns the elimination tree of the matrix and can be displayed graphically by the command treeplot (etree (A)) if A is symmetric or treeplot (etree (A+A')) otherwise.

[Function File] spy (x) [Function File] spy (…, markersize) [Function File] spy (…, line_spec)

Plot the sparsity pattern of the sparse matrix x. If the argument markersize is given as a scalar value, it is used to determine the point size in the plot. If the string line spec is given it is passed to plot and determines the appearance of the plot.

See also: [plot], page 260, [gplot], page 497.

[Loadable Function] p = etree (S) [Loadable Function] p = etree (S, typ) [Loadable Function] [p, q] = etree (S, typ)

Return the elimination tree for the matrix S. By default S is assumed to be symmetric and the symmetric elimination tree is returned. The argument typ controls whether a symmetric or column elimination tree is returned. Valid values of typ are "sym" or "col", for symmetric or column elimination tree respectively. Called with a second argument, etree also returns the postorder permutations on the tree.

[Function File] etreeplot (A)

[Function File] etreeplot (A, node_style, edge_style)

Plot the elimination tree of the matrix A or A+A' if A in not symmetric. The optional parameters node style and edge style define the output style.

See also: [treeplot], page 497, [gplot], page 497.

[Function File] gplot (A, xy)

[Function File] gplot (A, xy, line_style)

[Function File] [x, y] = gplot (A, xy)

Plot a graph defined by A and xy in the graph theory sense. A is the adjacency matrix of the array to be plotted and xy is an n-by-2 matrix containing the coordinates of the nodes of the graph. The optional parameter line style defines the output style for the plot. Called with no output arguments the graph is plotted directly. Otherwise, return the coordinates of the plot in x and y.

See also: [treeplot], page 497, [etreeplot], page 496, [spy], page 496.

[Function File] treeplot (tree)

[Function File] treeplot (tree, node_style, edge_style)

Produce a graph of tree or forest. The first argument is vector of predecessors, optional parameters node style and edge style define the output style. The complexity of the algorithm is O(n) in terms of is time and memory requirements.

See also: [etreeplot], page 496, [gplot], page 497.

[Function File] treelayout (tree)

[Function File] treelayout (tree, permutation)

treelayout lays out a tree or a forest. The first argument tree is a vector of predecessors, optional parameter permutation is an optional postorder permutation. The complexity of the algorithm is O(n) in terms of time and memory requirements.

See also: [etreeplot], page 496, [gplot], page 497, [treeplot], page 497.

22.1.4 Basic Operators and Functions on Sparse Matrices

22.1.4.1 Sparse Functions

Many Octave functions have been overloaded to work with either sparse or full matrices. There is no difference in calling convention when using an overloaded function with a sparse matrix, however, there is also no access to potentially sparse-specific features. At any time the sparse matrix specific version of a function can be used by explicitly calling its function name.

The table below lists all of the sparse functions of Octave. Note that the names of the specific sparse forms of the functions are typically the same as the general versions with a sp prefix. In the table below, and in the rest of this article, the specific sparse versions of functions are used.

Generate sparse matrices:

spalloc, spdiags, speye, sprand, sprandn, sprandsym

Sparse matrix conversion:

full, sparse, spconvert

Manipulate sparse matrices issparse, nnz, nonzeros, nzmax, spfun, spones, spy Graph Theory:

etree, etreeplot, gplot, treeplot

Sparse matrix reordering:

amd, ccolamd, colamd, colperm, csymamd, dmperm, symamd, randperm, symrcm

Linear algebra:

condest, eigs, matrix type, normest, sprank, spaugment, svds

Iterative techniques:

luinc, pcg, pcr

Miscellaneous:

spparms, symbfact, spstats

In addition all of the standard Octave mapper functions (i.e., basic math functions that take a single argument) such as abs, etc. can accept sparse matrices. The reader is referred to the documentation supplied with these functions within Octave itself for further details.

22.1.4.2 Return Types of Operators and Functions

The two basic reasons to use sparse matrices are to reduce the memory usage and to not have to do calculations on zero elements. The two are closely related in that the computation time on a sparse matrix operator or function is roughly linear with the number of non-zero elements.

Therefore, there is a certain density of non-zero elements of a matrix where it no longer makes sense to store it as a sparse matrix, but rather as a full matrix. For this reason operators and functions that have a high probability of returning a full matrix will always return one. For example adding a scalar constant to a sparse matrix will almost always make it a full matrix, and so the example,

     speye (3) + 0
     ⇒ 1  0  0
       0  1  0
       0  0  1

returns a full matrix as can be seen.

Additionally, if sparse_auto_mutate is true, all sparse functions test the amount of memory occupied by the sparse matrix to see if the amount of storage used is larger than the amount used by the full equivalent. Therefore speye (2) * 1 will return a full matrix as the memory used is smaller for the full version than the sparse version.

As all of the mixed operators and functions between full and sparse matrices exist, in general this does not cause any problems. However, one area where it does cause a problem is where a sparse matrix is promoted to a full matrix, where subsequent operations would resparsify the matrix. Such cases are rare, but can be artificially created, for example

(fliplr (speye (3)) + speye (3)) - speye (3) gives a full matrix when it should give a

sparse one. In general, where such cases occur, they impose only a small memory penalty.

There is however one known case where this behavior of Octave’s sparse matrices will cause a problem. That is in the handling of the diag function. Whether diag returns a sparse or full matrix depending on the type of its input arguments. So

      a = diag (sparse ([1,2,3]), -1);

should return a sparse matrix. To ensure this actually happens, the sparse function, and other functions based on it like speye, always returns a sparse matrix, even if the memory used will be larger than its full representation.

[Built-in Function] val = sparse_auto_mutate ()
[Built-in Function] old_val = sparse_auto_mutate (new_val)
[Built-in Function] sparse_auto_mutate (new_val, "local")

Query or set the internal variable that controls whether Octave will automatically mutate sparse matrices to full matrices to save memory. For example:

          s = speye (3);
          sparse_auto_mutate (false);
          s(:, 1) = 1;
          typeinfo (s)
          ⇒ sparse matrix
          sparse_auto_mutate (true);
          s(1, :) = 1;
          typeinfo (s)
          ⇒ matrix

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

Note that the sparse_auto_mutate option is incompatible with matlab, and so it is off by default.

22.1.4.3 Mathematical Considerations

The attempt has been made to make sparse matrices behave in exactly the same manner as there full counterparts. However, there are certain differences and especially differences with other products sparse implementations.

First, the "./" and ".^" operators must be used with care. Consider what the examples

       s = speye (4);
       a1 = s .^2;
       a2 = s .^s;
       a3 = s .^-2;
       a4 = s ./ 2;
       a5 = 2 ./ s;
       a6 = s ./ s;

will give. The first example of s raised to the power of 2 causes no problems. However s raised element-wise to itself involves a large number of terms 0 .^0 which is 1. There s .^ s is a full matrix.

Likewise s .^-2 involves terms like 0 .^-2 which is infinity, and so s .^-2 is equally a full matrix.

For the "./" operator s ./ 2 has no problems, but 2 ./ s involves a large number of infinity terms as well and is equally a full matrix. The case of s ./ s involves terms like 0 ./ 0 which is a NaN and so this is equally a full matrix with the zero elements of s filled with NaN values.

The above behavior is consistent with full matrices, but is not consistent with sparse implementations in other products.

A particular problem of sparse matrices comes about due to the fact that as the zeros are not stored, the sign-bit of these zeros is equally not stored. In certain cases the sign-bit of zero is important. For example:

      a = 0 ./ [-1, 1; 1, -1];
      b = 1 ./ a
      ⇒ -Inf            Inf
          Inf           -Inf
      c = 1 ./ sparse (a)
      ⇒ Inf            Inf
          Inf            Inf

To correct this behavior would mean that zero elements with a negative sign-bit would need to be stored in the matrix to ensure that their sign-bit was respected. This is not done at this time, for reasons of efficiency, and so the user is warned that calculations where the sign-bit of zero is important must not be done using sparse matrices.

In general any function or operator used on a sparse matrix will result in a sparse matrix with the same or a larger number of non-zero elements than the original matrix. This is particularly true for the important case of sparse matrix factorizations. The usual way to address this is to reorder the matrix, such that its factorization is sparser than the factorization of the original matrix. That is the factorization of L * U = P * S * Q has sparser terms L and U than the equivalent factorization L * U = S.

Several functions are available to reorder depending on the type of the matrix to be factorized. If the matrix is symmetric positive-definite, then symamd or csymamd should be used. Otherwise amd, colamd or ccolamd should be used. For completeness the reordering functions colperm and randperm are also available.

See Figure 22.3, for an example of the structure of a simple positive definite matrix.

Figure 22.3: Structure of simple sparse matrix.

The standard Cholesky factorization of this matrix can be obtained by the same command that would be used for a full matrix. This can be visualized with the command r = chol (A); spy (r);. See Figure 22.4. The original matrix had 598 non-zero terms, while this Cholesky factorization has 10200, with only half of the symmetric matrix being stored. This is a significant level of fill in, and although not an issue for such a small test case, can represents a large overhead in working with other sparse matrices.

The appropriate sparsity preserving permutation of the original matrix is given by symamd and the factorization using this reordering can be visualized using the command q = symamd (A); r = chol (A(q,q)); spy (r). This gives 399 non-zero terms which is a significant improvement.

The Cholesky factorization itself can be used to determine the appropriate sparsity preserving reordering of the matrix during the factorization, In that case this might be obtained with three return arguments as [r, p, q] = chol (A); spy (r).

Figure 22.4: Structure of the unpermuted Cholesky factorization of the above matrix.

Figure 22.5: Structure of the permuted Cholesky factorization of the above matrix.

In the case of an asymmetric matrix, the appropriate sparsity preserving permutation is colamd and the factorization using this reordering can be visualized using the command q

= colamd (A); [l, u, p] = lu (A(:,q)); spy (l+u).

Finally, Octave implicitly reorders the matrix when using the div (/) and ldiv (\) operators, and so no the user does not need to explicitly reorder the matrix to maximize performance.

[Loadable Function] p = amd (S) [Loadable Function] p = amd (S, opts)

Return the approximate minimum degree permutation of a matrix. This permutation such that the Cholesky factorization of S (p, p) tends to be sparser than the Cholesky factorization of S itself. amd is typically faster than symamd but serves a similar purpose. The optional parameter opts is a structure that controls the behavior of amd. The fields of the structure are

opts.dense Determines what amd considers to be a dense row or column of the input matrix. Rows or columns with more than max(16, (dense * sqrt (n) entries, where n is the order of the matrix S, are ignored by amd during the calculation of the permutation The value of dense must be a positive scalar and its default value is 10.0

opts.aggressive If this value is a nonzero scalar, then amd performs aggressive absorption. The default is not to perform aggressive absorption.

The author of the code itself is Timothy A. Davis davis@cise.ufl.edu, University

of Florida (see http://www.cise.ufl.edu/research/sparse/amd).

See also: [symamd], page 506, [colamd], page 504.

[Loadable Function] p = ccolamd (S) [Loadable Function] p = ccolamd (S, knobs)

[Loadable Function] p = ccolamd (S, knobs, cmember)

[Loadable Function] [p, stats] = ccolamd (…)

Constrained column approximate minimum degree permutation. p = ccolamd (S) returns the column approximate minimum degree permutation vector for the sparse matrix S. For a non-symmetric matrix S, S(:, p) tends to have sparser LU factors

than S. chol (S(:, p)' * S(:, p)) also tends to be sparser than chol (S' * S). p

= ccolamd (S, 1) optimizes the ordering for lu (S(:, p)). The ordering is followed by a column elimination tree post-ordering. knobs is an optional 1-element to 5-element input vector, with a default value of [0 10 10 1 0] if not present or empty. Entries not present are set to their defaults.

knobs(1) if nonzero, the ordering is optimized for lu (S(:, p)). It will be a poor ordering for chol (S(:, p)' * S(:, p)). This is the most important knob for ccolamd.

knobs(2) if S is m-by-n, rows with more than max (16, knobs(2) * sqrt (n))

entries are ignored.

knobs(3) columns with more than max (16, knobs(3) * sqrt (min (m, n))) en-

tries are ignored and ordered last in the output permutation (subject to the cmember constraints).

knobs(4) if nonzero, aggressive absorption is performed.

knobs(5) if nonzero, statistics and knobs are printed.

cmember is an optional vector of length n. It defines the constraints on the column ordering. If cmember(j) = c, then column j is in constraint set c (c must be in the range 1 to n). In the output permutation p, all columns in set 1 appear first, followed by all columns in set 2, and so on. cmember = ones (1,n) if not present or empty.

ccolamd (S, [], 1 : n) returns 1 : n

p = ccolamd (S) is about the same as p = colamd (S). knobs and its default values differ. colamd always does aggressive absorption, and it finds an ordering suitable for both lu (S(:, p)) and chol (S(:, p)' * S(:, p)); it cannot optimize its ordering

for lu (S(:, p)) to the extent that ccolamd (S, 1) can.

stats is an optional 20-element output vector that provides data about the ordering and the validity of the input matrix S. Ordering statistics are in stats(1 : 3). stats(1) and stats(2) are the number of dense or empty rows and columns ignored by ccolamd and stats(3) is the number of garbage collections performed on the internal data structure used by ccolamd (roughly of size 2.2 * nnz (S) + 4 * m + 7 * n integers). stats(4 : 7) provide information if CCOLAMD was able to continue. The matrix is OK if stats(4) is zero, or 1 if invalid. stats(5) is the rightmost column index that is unsorted or contains duplicate entries, or zero if no such column exists. stats(6) is the last seen duplicate or out-of-order row index in the column index given by stats(5), or zero if no such row index exists. stats(7) is the number of duplicate or out-of-order row indices. stats(8 : 20) is always zero in the current version of ccolamd (reserved for future use). The authors of the code itself are S. Larimore, T. Davis (Univ. of Florida) and S. Rajamanickam in collaboration with J. Bilbert and E. Ng. Supported by the National Science Foundation (DMS-9504974, DMS-9803599, CCR-0203270), and a

grant from Sandia National Lab. See http://www.cise.ufl.edu/research/sparse

for ccolamd, csymamd, amd, colamd, symamd, and other related orderings.

See also: [colamd], page 504, [csymamd], page 505.

[Loadable Function] p = colamd (S) [Loadable Function] p = colamd (S, knobs) [Loadable Function] [p, stats] = colamd (S)

[Loadable Function] [p, stats] = colamd (S, knobs)

Column approximate minimum degree permutation. p = colamd (S) returns the column approximate minimum degree permutation vector for the sparse matrix S. For a non-symmetric matrix S, S(:,p) tends to have sparser LU factors than S. The Cholesky factorization of S(:,p)' * S(:,p) also tends to be sparser than that of S'

* S.

knobs is an optional one- to three-element input vector. If S is m-by-n, then rows with more than max(16,knobs(1)*sqrt(n)) entries are ignored. Columns with more than max (16,knobs(2)*sqrt(min(m,n))) entries are removed prior to ordering, and ordered last in the output permutation p. Only completely dense rows or columns are removed if knobs(1) and knobs(2) are < 0, respectively. If knobs(3) is nonzero, stats and knobs are printed. The default is knobs = [10 10 0]. Note that knobs differs from earlier versions of colamd. stats is an optional 20-element output vector that provides data about the ordering and the validity of the input matrix S. Ordering statistics are in stats(1:3). stats(1) and stats(2) are the number of dense or empty rows and columns ignored by colamd and stats(3) is the number of garbage collections performed on the internal data structure used by colamd (roughly of size 2.2 * nnz(S) + 4 * m + 7 * n integers).

Octave built-in functions are intended to generate valid sparse matrices, with no duplicate entries, with ascending row indices of the nonzeros in each column, with a non-negative number of entries in each column (!) and so on. If a matrix is invalid, then colamd may or may not be able to continue. If there are duplicate entries (a row index appears two or more times in the same column) or if the row indices in a column are out of order, then colamd can correct these errors by ignoring the duplicate entries and sorting each column of its internal copy of the matrix S (the input matrix S is not repaired, however). If a matrix is invalid in other ways then colamd cannot continue, an error message is printed, and no output arguments (p or stats) are returned. colamd is thus a simple way to check a sparse matrix to see if it’s valid. stats(4:7) provide information if COLAMD was able to continue. The matrix is OK if stats(4) is zero, or 1 if invalid. stats(5) is the rightmost column index that is unsorted or contains duplicate entries, or zero if no such column exists. stats(6) is the last seen duplicate or out-of-order row index in the column index given by stats(5), or zero if no such row index exists. stats(7) is the number of duplicate or out-of-order row indices. stats(8:20) is always zero in the current version of colamd (reserved for future use). The ordering is followed by a column elimination tree post-ordering. The authors of the code itself are Stefan I. Larimore and Timothy A. Davis davis@cise.ufl.edu, University of Florida. The algorithm was developed in collaboration with John Gilbert, Xerox PARC, and Esmond Ng, Oak Ridge National

Laboratory. (see http://www.cise.ufl.edu/research/sparse/colamd)

See also: [colperm], page 505, [symamd], page 506, [ccolamd], page 503.

[Function File] p = colperm (s)

Return the column permutations such that the columns of s (:, p) are ordered in terms of increase number of non-zero elements. If s is symmetric, then p is chosen such that s (p, p) orders the rows and columns with increasing number of non zeros elements.

[Loadable Function] p = csymamd (S) [Loadable Function] p = csymamd (S, knobs)

[Loadable Function] p = csymamd (S, knobs, cmember)

[Loadable Function] [p, stats] = csymamd (…)

For a symmetric positive definite matrix S, returns the permutation vector p such that S(p,p) tends to have a sparser Cholesky factor than S. Sometimes csymamd works well for symmetric indefinite matrices too. The matrix S is assumed to be symmetric; only the strictly lower triangular part is referenced. S must be square. The ordering is followed by an elimination tree post-ordering. knobs is an optional 1-element to 3-element input vector, with a default value of [10 1 0] if present or empty. Entries not present are set to their defaults.

knobs(1) If S is n-by-n, then rows and columns with more than max(16,knobs(1)*sqrt(n)) entries are ignored, and ordered last in the output permutation (subject to the cmember constraints).

knobs(2) If nonzero, aggressive absorption is performed. knobs(3) If nonzero, statistics and knobs are printed.

cmember is an optional vector of length n. It defines the constraints on the ordering. If cmember(j) = S, then row/column j is in constraint set c (c must be in the range 1 to n). In the output permutation p, rows/columns in set 1 appear first, followed by all rows/columns in set 2, and so on. cmember = ones (1,n) if not present or empty.

csymamd (S,[],1:n) returns 1:n.

p = csymamd (S) is about the same as p = symamd (S). knobs and its default values differ. stats(4:7) provide information if CCOLAMD was able to continue. The matrix is OK if stats(4) is zero, or 1 if invalid. stats(5) is the rightmost column index that is unsorted or contains duplicate entries, or zero if no such column exists. stats(6) is the last seen duplicate or out-of-order row index in the column index given by stats(5), or zero if no such row index exists. stats(7) is the number of duplicate or out-of-order row indices. stats(8:20) is always zero in the current version of ccolamd (reserved for future use). The authors of the code itself are S. Larimore, T. Davis (Uni of Florida) and S. Rajamanickam in collaboration with J. Bilbert and E. Ng. Supported by the National Science Foundation (DMS-9504974, DMS-9803599, CCR-0203270), and a grant

from Sandia National Lab. See http://www.cise.ufl.edu/research/sparse for

ccolamd, csymamd, amd, colamd, symamd, and other related orderings.

See also: [symamd], page 506, [ccolamd], page 503.

[Loadable Function] p = dmperm (S) [Loadable Function] [p, q, r, S] = dmperm (S)

Perform a Dulmage-Mendelsohn permutation of the sparse matrix S. With a single output argument dmperm performs the row permutations p such that S(p,:) has no zero elements on the diagonal. Called with two or more output arguments, returns the row and column permutations, such that S(p, q) is in block triangular form. The values of r and S define the boundaries of the blocks. If S is square then r == S. The method used is described in: A. Pothen & C.-J. Fan. Computing the Block Triangular Form of a Sparse Matrix. ACM Trans. Math. Software, 16(4):303-324, 1990.

See also: [colamd], page 504, [ccolamd], page 503.

[Loadable Function] p = symamd (S) [Loadable Function] p = symamd (S, knobs) [Loadable Function] [p, stats] = symamd (S)

[Loadable Function] [p, stats] = symamd (S, knobs)

For a symmetric positive definite matrix S, returns the permutation vector p such that S(p, p) tends to have a sparser Cholesky factor than S. Sometimes symamd works well for symmetric indefinite matrices too. The matrix S is assumed to be symmetric; only the strictly lower triangular part is referenced. S must be square. knobs is an optional one- to two-element input vector. If S is n-by-n, then rows and columns with more than max (16,knobs(1)*sqrt(n)) entries are removed prior to ordering, and ordered last in the output permutation p. No rows/columns are removed if knobs(1) < 0. If knobs (2) is nonzero, stats and knobs are printed. The default is knobs = [10 0]. Note that knobs differs from earlier versions of symamd.

stats is an optional 20-element output vector that provides data about the ordering and the validity of the input matrix S. Ordering statistics are in stats(1:3). stats(1) = stats(2) is the number of dense or empty rows and columns ignored by SYMAMD and stats(3) is the number of garbage collections performed on the internal data structure used by SYMAMD (roughly of size 8.4 * nnz (tril (S, -1)) + 9 * n integers).

Octave built-in functions are intended to generate valid sparse matrices, with no duplicate entries, with ascending row indices of the nonzeros in each column, with a non-negative number of entries in each column (!) and so on. If a matrix is invalid, then SYMAMD may or may not be able to continue. If there are duplicate entries (a row index appears two or more times in the same column) or if the row indices in a column are out of order, then SYMAMD can correct these errors by ignoring the duplicate entries and sorting each column of its internal copy of the matrix S (the input matrix S is not repaired, however). If a matrix is invalid in other ways then SYMAMD cannot continue, an error message is printed, and no output arguments (p or stats) are returned. SYMAMD is thus a simple way to check a sparse matrix to see if it’s valid.

stats(4:7) provide information if SYMAMD was able to continue. The matrix is OK if stats (4) is zero, or 1 if invalid. stats(5) is the rightmost column index that is unsorted or contains duplicate entries, or zero if no such column exists. stats(6) is the last seen duplicate or out-of-order row index in the column index given by stats(5), or zero if no such row index exists. stats(7) is the number of duplicate or out-of-order row indices. stats(8:20) is always zero in the current version of SYMAMD (reserved for future use).

The ordering is followed by a column elimination tree post-ordering.

The authors of the code itself are Stefan I. Larimore and Timothy A. Davis davis@cise.ufl.edu, University of Florida. The algorithm was developed in collaboration with John Gilbert, Xerox PARC, and Esmond Ng, Oak Ridge National

Laboratory. (see http://www.cise.ufl.edu/research/sparse/colamd)

See also: [colperm], page 505, [colamd], page 504.

[Loadable Function] p = symrcm (S)

Return the symmetric reverse Cuthill-McKee permutation of S. p is a permutation vector such that S(p, p) tends to have its diagonal elements closer to the diagonal than S. This is a good preordering for LU or Cholesky factorization of matrices that come from “long, skinny” problems. It works for both symmetric and asymmetric S.

The algorithm represents a heuristic approach to the NP-complete bandwidth minimization problem. The implementation is based in the descriptions found in

E. Cuthill, J. McKee. Reducing the Bandwidth of Sparse Symmetric Matrices. Proceedings of the 24th ACM National Conference, 157–172 1969, Brandon Press, New Jersey.

A. George, J.W.H. Liu. Computer Solution of Large Sparse Positive Definite Systems, Prentice Hall Series in Computational Mathematics, ISBN 0-13-165274-5, 1981.

See also: [colperm], page 505, [colamd], page 504, [symamd], page 506.

22.2 Linear Algebra on Sparse Matrices

Octave includes a polymorphic solver for sparse matrices, where the exact solver used to factorize the matrix, depends on the properties of the sparse matrix itself. Generally, the cost of determining the matrix type is small relative to the cost of factorizing the matrix itself, but in any case the matrix type is cached once it is calculated, so that it is not re-determined each time it is used in a linear equation.

The selection tree for how the linear equation is solve is

1. If the matrix is diagonal, solve directly and goto 8

2. If the matrix is a permuted diagonal, solve directly taking into account the permutations. Goto 8

3. If the matrix is square, banded and if the band density is less than that given by spparms ("bandden") continue, else goto 4. a. If the matrix is tridiagonal and the right-hand side is not sparse continue, else goto

3b.

1. If the matrix is Hermitian, with a positive real diagonal, attempt Cholesky factorization using lapack xPTSV.

2. If the above failed or the matrix is not Hermitian with a positive real diagonal use Gaussian elimination with pivoting using lapack xGTSV, and goto 8.

b. If the matrix is Hermitian with a positive real diagonal, attempt Cholesky factor-

ization using lapack xPBTRF. c. if the above failed or the matrix is not Hermitian with a positive real diagonal use

Gaussian elimination with pivoting using lapack xGBTRF, and goto 8.

4. If the matrix is upper or lower triangular perform a sparse forward or backward substitution, and goto 8

5. If the matrix is an upper triangular matrix with column permutations or lower triangular matrix with row permutations, perform a sparse forward or backward substitution, and goto 8

6. If the matrix is square, Hermitian with a real positive diagonal, attempt sparse Cholesky factorization using cholmod.

7. If the sparse Cholesky factorization failed or the matrix is not Hermitian with a real positive diagonal, and the matrix is square, factorize using umfpack.

8. If the matrix is not square, or any of the previous solvers flags a singular or near singular matrix, find a minimum norm solution using cxsparse^{3}.

The band density is defined as the number of non-zero values in the band divided by the total number of values in the full band. The banded matrix solvers can be entirely disabled by using spparms to set bandden to 1 (i.e., spparms ("bandden", 1)).

3 The cholmod, umfpack and cxsparse packages were written by Tim Davis and are available at

http://www.cise.ufl.edu/research/sparse/

The QR solver factorizes the problem with a Dulmage-Mendelsohn decomposition, to separate the problem into blocks that can be treated as over-determined, multiple well determined blocks, and a final over-determined block. For matrices with blocks of strongly connected nodes this is a big win as LU decomposition can be used for many blocks. It also significantly improves the chance of finding a solution to over-determined problems rather than just returning a vector of NaN’s.

All of the solvers above, can calculate an estimate of the condition number. This can be used to detect numerical stability problems in the solution and force a minimum norm solution to be used. However, for narrow banded, triangular or diagonal matrices, the cost of calculating the condition number is significant, and can in fact exceed the cost of factoring the matrix. Therefore the condition number is not calculated in these cases, and Octave relies on simpler techniques to detect singular matrices or the underlying lapack code in the case of banded matrices.

The user can force the type of the matrix with the matrix_type function. This overcomes the cost of discovering the type of the matrix. However, it should be noted that identifying the type of the matrix incorrectly will lead to unpredictable results, and so matrix_type should be used with care.

[Function File] n = normest (A) [Function File] n = normest (A, tol) [Function File] [n, c] = normest (…)

Estimate the 2-norm of the matrix A using a power series analysis. This is typically used for large matrices, where the cost of calculating norm (A) is prohibitive and an approximation to the 2-norm is acceptable. tol is the tolerance to which the 2-norm is calculated. By default tol is 1e-6. c returns the number of iterations needed for normest to converge.

[Function File] [est, v, w, iter] = onenormest (A, t)
[Function File] [est, v, w, iter] = onenormest (apply, apply_t, n, t)

Apply Higham and Tisseur’s randomized block 1-norm estimator to matrix A using t test vectors. If t exceeds 5, then only 5 test vectors are used. If the matrix is not explicit, e.g., when estimating the norm of inv (A) given an LU factorization, onenormest applies A and its conjugate transpose through a pair of functions apply and apply t, respectively, to a dense matrix of size n by t. The implicit version requires an explicit dimension n. Returns the norm estimate est, two vectors v and w related by norm (w, 1) = est * norm (v, 1), and the number of iterations iter. The number of iterations is limited to 10 and is at least 2. References:

• N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. SIMAX vol 21, no 4, pp 1185-

    1201. http://dx.doi.org/10.1137/S0895479899356080

• N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra.

    http://citeseer.ist.psu.edu/223007.html

See also: [condest], page 510, [norm], page 439, [cond], page 436.

[Function File] condest (A) [Function File] condest (A, t) [Function File] [est, v] = condest (…)

[Function File] [est, v] = condest (A, solve, solve_t, t)
[Function File] [est, v] = condest (apply, apply_t, solve, solve_t, n, t)

Estimate the 1-norm condition number of a matrix A using t test vectors using a randomized 1-norm estimator. If t exceeds 5, then only 5 test vectors are used. If the matrix is not explicit, e.g., when estimating the condition number of A given an LU factorization, condest uses the following functions:

apply A*x for a matrix x of size n by t.

apply t A'*x for a matrix x of size n by t.

solve A \ b for a matrix b of size n by t.

solve t A' \ b for a matrix b of size n by t.

The implicit version requires an explicit dimension n. condest uses a randomized algorithm to approximate the 1-norms. condest returns the 1-norm condition estimate est and a vector v satisfying norm

(A*v, 1) == norm (A, 1) * norm (v, 1) / est. When est is large, v is an approxi-

mate null vector. References:

• N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. SIMAX vol 21, no 4, pp 1185-

    1201. http://dx.doi.org/10.1137/S0895479899356080

• N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra.

    http://citeseer.ist.psu.edu/223007.html

See also: [cond], page 436, [norm], page 439, [onenormest], page 509.

[Built-in Function] spparms () [Built-in Function] vals = spparms () [Built-in Function] [keys, vals] = spparms () [Built-in Function] val = spparms (key)

[Built-in Function] spparms (vals)

[Built-in Function] spparms ("defaults")

[Built-in Function] spparms ("tight") [Built-in Function] spparms (key, val)

Query or set the parameters used by the sparse solvers and factorization functions. The first four calls above get information about the current settings, while the others change the current settings. The parameters are stored as pairs of keys and values, where the values are all floats and the keys are one of the following strings:

‘spumoni’ Printing level of debugging information of the solvers (default 0)

‘ths_rel’ Included for compatibility. Not used. (default 1)

‘ths_abs’ Included for compatibility. Not used. (default 1)

‘exact_d’ Included for compatibility. Not used. (default 0)

‘supernd’ Included for compatibility. Not used. (default 3)

‘rreduce’ Included for compatibility. Not used. (default 3)

‘wh_frac’ Included for compatibility. Not used. (default 0.5)

‘autommd’ Flag whether the LU/QR and the ’\’ and ’/’ operators will automatically use the sparsity preserving mmd functions (default 1)

‘autoamd’ Flag whether the LU and the ’\’ and ’/’ operators will automatically use the sparsity preserving amd functions (default 1)

‘piv_tol’ The pivot tolerance of the umfpack solvers (default 0.1)

‘sym_tol’ The pivot tolerance of the umfpack symmetric solvers (default 0.001)

‘bandden’ The density of non-zero elements in a banded matrix before it is treated by the lapack banded solvers (default 0.5)

‘umfpack’ Flag whether the umfpack or mmd solvers are used for the LU, ’\’ and ’/’ operations (default 1)

The value of individual keys can be set with spparms (key, val). The default values can be restored with the special keyword "defaults". The special keyword "tight" can be used to set the mmd solvers to attempt a sparser solution at the potential cost of longer running time.

See also: [chol], page 442, [colamd], page 504, [lu], page 444, [qr], page 446, [symamd], page 506.

[Loadable Function] p = sprank (S)

Calculate the structural rank of the sparse matrix S. Note that only the structure of the matrix is used in this calculation based on a Dulmage-Mendelsohn permutation to block triangular form. As such the numerical rank of the matrix S is bounded by

sprank (S) >= rank (S). Ignoring floating point errors sprank (S) == rank (S).

See also: [dmperm], page 506.

[Loadable Function] [count, h, parent, post, r] = symbfact (S)

[Loadable Function] [...] = symbfact (S, typ)

[Loadable Function] [...] = symbfact (S, typ, mode)

Perform a symbolic factorization analysis on the sparse matrix S. Where

S S is a complex or real sparse matrix.

typ Is the type of the factorization and can be one of

‘sym’ Factorize S. This is the default.

‘col’ Factorize S' * S.

‘row’ Factorize S * S'.

‘lo’ Factorize S' mode The default is to return the Cholesky factorization for r, and if mode is

'L', the conjugate transpose of the Cholesky factorization is returned. The conjugate transpose version is faster and uses less memory, but returns the same values for count, h, parent and post outputs.

The output variables are

count The row counts of the Cholesky factorization as determined by typ.

h The height of the elimination tree.

parent The elimination tree itself.

post A sparse boolean matrix whose structure is that of the Cholesky factorization as determined by typ.

For non square matrices, the user can also utilize the spaugment function to find a least squares solution to a linear equation.

[Function File] s = spaugment (A, c)

Create the augmented matrix of A.

This is given by

     [c * eye(m, m), A;
                 A', zeros(n, n)]

This is related to the least squares solution of A \ b, by

     s * [ r / c; x] = [ b, zeros(n, columns(b)) ]

where r is the residual error

     r = b - A * x

As the matrix s is symmetric indefinite it can be factorized with lu, and the minimum norm solution can therefore be found without the need for a qr factorization. As the residual error will be zeros (m, m) for underdetermined problems, and example can be

     m = 11; n = 10; mn = max (m, n);
     A = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],
                  [-1, 0, 1], m, n);
     x0 = A \ ones (m,1);
     s = spaugment (A);
     [L, U, P, Q] = lu (s);
     x1 = Q * (U \ (L \ (P  * [ones(m,1); zeros(n,1)])));
     x1 = x1(end - n + 1 : end);

To find the solution of an overdetermined problem needs an estimate of the residual error r and so it is more complex to formulate a minimum norm solution using the

spaugment function.

In general the left division operator is more stable and faster than using the spaugment function.

See also: [mldivide], page 139.

Finally, the function eigs can be used to calculate a limited number of eigenvalues and eigenvectors based on a selection criteria and likewise for svds which calculates a limited number of singular values and vectors.

[Function File] d = eigs (A) [Function File] d = eigs (A, k) [Function File] d = eigs (A, k, sigma) [Function File] d = eigs (A, k, sigma, opts) [Function File] d = eigs (A, B) [Function File] d = eigs (A, B, k) [Function File] d = eigs (A, B, k, sigma) [Function File] d = eigs (A, B, k, sigma, opts) [Function File] d = eigs (af, n) [Function File] d = eigs (af, n, B) [Function File] d = eigs (af, n, k) [Function File] d = eigs (af, n, B, k) [Function File] d = eigs (af, n, k, sigma) [Function File] d = eigs (af, n, B, k, sigma) [Function File] d = eigs (af, n, k, sigma, opts) [Function File] d = eigs (af, n, B, k, sigma, opts) [Function File] [V, d] = eigs (A, …) [Function File] [V, d] = eigs (af, n, …) [Function File] [V, d, flag] = eigs (A, …) [Function File] [V, d, flag] = eigs (af, n, …)

Calculate a limited number of eigenvalues and eigenvectors of A, based on a selection criteria. The number of eigenvalues and eigenvectors to calculate is given by k and defaults to 6. By default, eigs solve the equation Aν = λν, where λ is a scalar representing one of the eigenvalues, and ν is the corresponding eigenvector. If given the positive definite matrix B then eigs solves the general eigenvalue equation Aν = λBν. The argument sigma determines which eigenvalues are returned. sigma can be either a scalar or a string. When sigma is a scalar, the k eigenvalues closest to sigma are returned. If sigma is a string, it must have one of the following values.

"lm" Largest Magnitude (default).

"sm" Smallest Magnitude.

"la" Largest Algebraic (valid only for real symmetric problems).

"sa" Smallest Algebraic (valid only for real symmetric problems).

"be" Both Ends, with one more from the high-end if k is odd (valid only for real symmetric problems).

"lr" Largest Real part (valid only for complex or unsymmetric problems).

"sr" Smallest Real part (valid only for complex or unsymmetric problems).

"li" Largest Imaginary part (valid only for complex or unsymmetric problems).

"si" Smallest Imaginary part (valid only for complex or unsymmetric problems).

If opts is given, it is a structure defining possible options that eigs should use. The fields of the opts structure are:

issym If af is given, then flags whether the function af defines a symmetric problem. It is ignored if A is given. The default is false.

isreal If af is given, then flags whether the function af defines a real problem. It is ignored if A is given. The default is true.

tol Defines the required convergence tolerance, calculated as tol * norm (A). The default is eps.

maxit The maximum number of iterations. The default is 300.

p The number of Lanzcos basis vectors to use. More vectors will result in faster convergence, but a greater use of memory. The optimal value of p is problem dependent and should be in the range k to n. The default value is 2 * k.

v0 The starting vector for the algorithm. An initial vector close to the final vector will speed up convergence. The default is for arpack to randomly generate a starting vector. If specified, v0 must be an n-by-1 vector where

          n = rows (A)

disp The level of diagnostic printout (0|1|2). If disp is 0 then diagnostics are disabled. The default value is 0.

cholB Flag if chol (B) is passed rather than B. The default is false.

permB The permutation vector of the Cholesky factorization of B if cholB is true. That is chol (B(permB, permB)). The default is 1:n.

It is also possible to represent A by a function denoted af. af must be followed by a scalar argument n defining the length of the vector argument accepted by af. af can be a function handle, an inline function, or a string. When af is a string it holds the name of the function to use. af is a function of the form y = af (x) where the required return value of af is determined by the value of sigma. The four possible forms are

A * x if sigma is not given or is a string other than "sm".

A \ x if sigma is 0 or "sm".

(A - sigma * I) \ x

for the standard eigenvalue problem, where I is the identity matrix of the same size as A.

(A - sigma * B) \ x

for the general eigenvalue problem.

The return arguments of eigs depend on the number of return arguments requested. With a single return argument, a vector d of length k is returned containing the k eigenvalues that have been found. With two return arguments, V is a n-by-k matrix whose columns are the k eigenvectors corresponding to the returned eigenvalues. The eigenvalues themselves are returned in d in the form of a n-by-k matrix, where the elements on the diagonal are the eigenvalues. Given a third return argument flag, eigs returns the status of the convergence. If flag is 0 then all eigenvalues have converged. Any other value indicates a failure to converge. This function is based on the arpack package, written by R. Lehoucq, K. Maschhoff, D. Sorensen, and C. Yang. For more information see

http://www.caam.rice.edu/software/ARPACK/.

See also: [eig], page 437, [svds], page 515.

[Function File] s = svds (A) [Function File] s = svds (A, k) [Function File] s = svds (A, k, sigma) [Function File] s = svds (A, k, sigma, opts) [Function File] [u, s, v] = svds (…) [Function File] [u, s, v, flag] = svds (…)

Find a few singular values of the matrix A. The singular values are calculated using

     [m, n] = size (A);
     s = eigs ([sparse(m, m), A;
                          A', sparse(n, n)])

The eigenvalues returned by eigs correspond to the singular values of A. The number of singular values to calculate is given by k and defaults to 6. The argument sigma specifies which singular values to find. When sigma is the string 'L', the default, the largest singular values of A are found. Otherwise, sigma must be a real scalar and the singular values closest to sigma are found. As a corollary, sigma = 0 finds the smallest singular values. Note that for relatively small values of sigma, there is a chance that the requested number of singular values will not be found. In that case sigma should be increased. opts is a structure defining options that svds will pass to eigs. The possible fields of this structure are documented in eigs. By default, svds sets the following three fields:

tol The required convergence tolerance for the singular values. The default value is 1e-10. eigs is passed tol / sqrt(2).

maxit The maximum number of iterations. The default is 300.

disp The level of diagnostic printout (0|1|2). If disp is 0 then diagnostics are disabled. The default value is 0.

If more than one output is requested then svds will return an approximation of the singular value decomposition of A

     A_approx = u*s*v'

where A approx is a matrix of size A but only rank k. flag returns 0 if the algorithm has succesfully converged, and 1 otherwise. The test for convergence is

 norm (A*v - u*s, 1) <= tol * norm (A, 1)

svds is best for finding only a few singular values from a large sparse matrix. Otherwise, svd (full (A)) will likely be more efficient.

See also: [svd], page 450, [eigs], page 513.

22.3 Iterative Techniques Applied to Sparse Matrices

The left division \ and right division / operators, discussed in the previous section, use direct solvers to resolve a linear equation of the form x = A \ b or x = b / A. Octave equally includes a number of functions to solve sparse linear equations using iterative techniques.

[Function File] x = pcg (A, b, tol, maxit, m1, m2, x0, …)

[Function File] [x, flag, relres, iter, resvec, eigest] = pcg (. . .)

Solve the linear system of equations A * x = b by means of the Preconditioned Conjugate Gradient iterative method. The input arguments are

• A can be either a square (preferably sparse) matrix or a function handle, inline function or string containing the name of a function which computes A * x. In principle, A should be symmetric and positive definite; if pcg finds A not to be positive definite, a warning is printed and the flag output will be set.

• b is the right-hand side vector.

• tol is the required relative tolerance for the residual error, b - A * x. The iteration stops if norm (b - A * x) ≤ tol * norm (b). If tol is omitted or empty then a tolerance of 1e-6 is used.

• maxit is the maximum allowable number of iterations; if maxit is omitted or empty then a value of 20 is used.

• m = m1 * m2 is the (left) preconditioning matrix, so that the iteration is (theoretically) equivalent to solving by pcg P * x = m \ b, with P = m \ A. Note that a proper choice of the preconditioner may dramatically improve the overall performance of the method. Instead of matrices m1 and m2, the user may pass two functions which return the results of applying the inverse of m1 and m2 to a vector (usually this is the preferred way of using the preconditioner). If m1 is omitted or empty [] then no preconditioning is applied. If m2 is omitted, m = m1 will be used as a preconditioner.

• x0 is the initial guess. If x0 is omitted or empty then the function sets x0 to a zero vector by default.

The arguments which follow x0 are treated as parameters, and passed in a proper way to any of the functions (A or m) which are passed to pcg. See the examples below for further details. The output arguments are

• x is the computed approximation to the solution of A * x = b.

• flag reports on the convergence. A value of 0 means the solution converged and the tolerance criterion given by tol is satisfied. A value of 1 means that the maxit limit for the iteration count was reached. A value of 3 indicates that the (preconditioned) matrix was found not to be positive definite.

• relres is the ratio of the final residual to its initial value, measured in the Euclidean norm.

• iter is the actual number of iterations performed.

• resvec describes the convergence history of the method. resvec(i,1) is the Euclidean norm of the residual, and resvec(i,2) is the preconditioned residual norm, after the (i-1)-th iteration, i = 1, 2, ..., iter+1. The preconditioned residual norm is defined as norm (r) ^2 = r' * (m \ r) where r = b - A * x, see also the description of m. If eigest is not required, only resvec(:,1) is returned.

• eigest returns the estimate for the smallest eigest(1) and largest eigest(2) eigenvalues of the preconditioned matrix P = m \ A. In particular, if no preconditioning is used, the estimates for the extreme eigenvalues of A are returned. eigest(1) is an overestimate and eigest(2) is an underestimate, so that eigest(2) / eigest(1) is a lower bound for cond (P, 2), which nevertheless in the limit should theoretically be equal to the actual value of the condition number. The method which computes eigest works only for symmetric positive definite A and m, and the user is responsible for verifying this assumption.

Let us consider a trivial problem with a diagonal matrix (we exploit the sparsity of A)

 n = 10;
 A = diag (sparse (1:n));
 b = rand (n, 1);
 [l, u, p, q] = luinc (A, 1.e-3);

Example 1: Simplest use of pcg

 x = pcg (A, b)

Example 2: pcg with a function which computes A * x

 function y = apply_a (x)
   y = [1:N]' .* x;
 endfunction
 x = pcg ("apply_a", b)

Example 3: pcg with a preconditioner: l * u

 x = pcg (A, b, 1.e-6, 500, l*u)

Example 4: pcg with a preconditioner: l * u. Faster than Example 3 since lower and upper triangular matrices are easier to invert

 x = pcg (A, b, 1.e-6, 500, l, u)

Example 5: Preconditioned iteration, with full diagnostics. The preconditioner (quite strange, because even the original matrix A is trivial) is defined as a function

 function y = apply_m (x)
   k = floor (length (x) - 2);
   y = x;
   y(1:k) = x(1:k) ./ [1:k]';
 endfunction
 [x, flag, relres, iter, resvec, eigest] = ...
                    pcg (A, b, [], [], "apply_m");
 semilogy (1:iter+1, resvec);

Example 6: Finally, a preconditioner which depends on a parameter k.

 function y = apply_M (x, varargin)
   K = varargin{1};
   y = x;
   y(1:K) = x(1:K) ./ [1:K]';
 endfunction
 [x, flag, relres, iter, resvec, eigest] = ...
      pcg (A, b, [], [], "apply_m", [], [], 3)

References:

1. C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, 1995. (the base PCG algorithm)

2. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS 1996. (condition number estimate from PCG) Revised version of this book is available online at

http://www-users.cs.umn.edu/~saad/books.html

See also: [sparse], page 492, [pcr], page 518.

[Function File] x = pcr (A, b, tol, maxit, m, x0, …)

[Function File] [x, flag, relres, iter, resvec] = pcr (. . .)

Solve the linear system of equations A * x = b by means of the Preconditioned Conjugate Residuals iterative method. The input arguments are

• A can be either a square (preferably sparse) matrix or a function handle, inline function or string containing the name of a function which computes A * x. In principle A should be symmetric and non-singular; if pcr finds A to be numerically singular, you will get a warning message and the flag output parameter will be set.

• b is the right hand side vector.

• tol is the required relative tolerance for the residual error, b - A * x. The iter-

ation stops if norm (b - A * x) <= tol * norm (b - A * x0). If tol is empty or

is omitted, the function sets tol = 1e-6 by default.

• maxit is the maximum allowable number of iterations; if [] is supplied for maxit, or pcr has less arguments, a default value equal to 20 is used.

• m is the (left) preconditioning matrix, so that the iteration is (theoretically) equivalent to solving by pcr P * x = m \ b, with P = m \ A. Note that a proper choice of the preconditioner may dramatically improve the overall performance of the method. Instead of matrix m, the user may pass a function which returns the results of applying the inverse of m to a vector (usually this is the preferred way of using the preconditioner). If [] is supplied for m, or m is omitted, no preconditioning is applied.

• x0 is the initial guess. If x0 is empty or omitted, the function sets x0 to a zero vector by default.

The arguments which follow x0 are treated as parameters, and passed in a proper way to any of the functions (A or m) which are passed to pcr. See the examples below for further details. The output arguments are

• x is the computed approximation to the solution of A * x = b.

• flag reports on the convergence. flag = 0 means the solution converged and the tolerance criterion given by tol is satisfied. flag = 1 means that the maxit limit for the iteration count was reached. flag = 3 reports t pcr breakdown, see [1] for details.

• relres is the ratio of the final residual to its initial value, measured in the Euclidean norm.

• iter is the actual number of iterations performed.

• resvec describes the convergence history of the method, so that resvec (i) contains the Euclidean norms of the residual after the (i-1)-th iteration, i = 1,2,

..., iter+1.

Let us consider a trivial problem with a diagonal matrix (we exploit the sparsity of A)

 n = 10;
 A = sparse (diag (1:n));
 b = rand (N, 1);

Example 1: Simplest use of pcr

 x = pcr (A, b)

Example 2: pcr with a function which computes A * x.

 function y = apply_a (x)
   y = [1:10]' .* x;
 endfunction
 x = pcr ("apply_a", b)

Example 3: Preconditioned iteration, with full diagnostics. The preconditioner (quite strange, because even the original matrix A is trivial) is defined as a function

 function y = apply_m (x)
   k = floor (length (x) - 2);
   y = x;
   y(1:k) = x(1:k) ./ [1:k]';
 endfunction
 [x, flag, relres, iter, resvec] = ...
                    pcr (A, b, [], [], "apply_m")
 semilogy ([1:iter+1], resvec);

Example 4: Finally, a preconditioner which depends on a parameter k.

     function y = apply_m (x, varargin)
       k = varargin{1};
       y = x;
       y(1:k) = x(1:k) ./ [1:k]';
     endfunction
     [x, flag, relres, iter, resvec] = ...
                        pcr (A, b, [], [], "apply_m"', [], 3)

References:

[1] W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, section 9.5.4; Springer, 1994

See also: [sparse], page 492, [pcg], page 516.

The speed with which an iterative solver converges to a solution can be accelerated with the use of a pre-conditioning matrix M. In this case the linear equation M^-1 * x = M^-1 * A \ b is solved instead. Typical pre-conditioning matrices are partial factorizations of the original matrix.

[Built-in Function] [L, U, P, Q] = luinc (A, ’0’)

[Built-in Function] [L, U, P, Q] = luinc (A, droptol)

[Built-in Function] [L, U, P, Q] = luinc (A, opts)

Produce the incomplete LU factorization of the sparse matrix A. Two types of incomplete factorization are possible, and the type is determined by the second argument

to luinc.

Called with a second argument of '0', the zero-level incomplete LU factorization is produced. This creates a factorization of A where the position of the non-zero arguments correspond to the same positions as in the matrix A. Alternatively, the fill-in of the incomplete LU factorization can be controlled through the variable droptol or the structure opts. The umfpack multifrontal factorization code by Tim A. Davis is used for the incomplete LU factorization, (availability

http://www.cise.ufl.edu/research/sparse/umfpack/)

droptol determines the values below which the values in the LU factorization are dropped and replaced by zero. It must be a positive scalar, and any values in the factorization whose absolute value are less than this value are dropped, expect if leaving them increase the sparsity of the matrix. Setting droptol to zero results in a complete LU factorization which is the default. opts is a structure containing one or more of the fields

droptol The drop tolerance as above. If opts only contains droptol then this is equivalent to using the variable droptol.

milu A logical variable flagging whether to use the modified incomplete LU factorization. In the case that milu is true, the dropped values are subtracted from the diagonal of the matrix U of the factorization. The default is false. udiag A logical variable that flags whether zero elements on the diagonal of U

should be replaced with droptol to attempt to avoid singular factors. The default is false.

thresh Defines the pivot threshold in the interval [0,1]. Values outside that range are ignored.

All other fields in opts are ignored. The outputs from luinc are the same as for lu.

Given the string argument "vector", luinc returns the values of p q as vector values.

See also: [sparse], page 492, [lu], page 444.

22.4 Real Life Example using Sparse Matrices

A common application for sparse matrices is in the solution of Finite Element Models. Finite element models allow numerical solution of partial differential equations that do not have closed form solutions, typically because of the complex shape of the domain.

In order to motivate this application, we consider the boundary value Laplace equation. This system can model scalar potential fields, such as heat or electrical potential. Given a medium Ω with boundary ∂Ω. At all points on the ∂Ω the boundary conditions are known, and we wish to calculate the potential in Ω. Boundary conditions may specify the potential (Dirichlet boundary condition), its normal derivative across the boundary (Neumann boundary condition), or a weighted sum of the potential and its derivative (Cauchy boundary condition).

In a thermal model, we want to calculate the temperature in Ω and know the boundary temperature (Dirichlet condition) or heat flux (from which we can calculate the Neumann condition by dividing by the thermal conductivity at the boundary). Similarly, in an electrical model, we want to calculate the voltage in Ω and know the boundary voltage (Dirichlet) or current (Neumann condition after diving by the electrical conductivity). In an electrical model, it is common for much of the boundary to be electrically isolated; this is a Neumann boundary condition with the current equal to zero.

The simplest finite element models will divide Ω into simplexes (triangles in 2D, pyramids in 3D). We take as a 3-D example a cylindrical liquid filled tank with a small non-conductive ball from the EIDORS project^{4}. This is model is designed to reflect an application of electrical impedance tomography, where current patterns are applied to such a tank in order to image the internal conductivity distribution. In order to describe the FEM geometry, we have a matrix of vertices nodes and simplices elems.

The following example creates a simple rectangular 2-D electrically conductive medium with 10 V and 20 V imposed on opposite sides (Dirichlet boundary conditions). All other edges are electrically isolated.

4 EIDORS - Electrical Impedance Tomography and Diffuse optical Tomography Reconstruction Software

http://eidors3d.sourceforge.net

node_y = [1;1.2;1.5;1.8;2]*ones(1,11);
node_x = ones(5,1)*[1,1.05,1.1,1.2, ...
          1.3,1.5,1.7,1.8,1.9,1.95,2];
nodes = [node_x(:), node_y(:)];
[h,w] = size (node_x);
elems = [];
for idx = 1:w-1
  widx = (idx-1)*h;
  elems = [elems; ...
    widx+[(1:h-1);(2:h);h+(1:h-1)]'; ...
    widx+[(2:h);h+(2:h);h+(1:h-1)]' ];
endfor
E = size (elems,1); # No. of simplices
N = size (nodes,1); # No. of vertices
D = size (elems,2); # dimensions+1

This creates a N-by-2 matrix nodes and a E-by-3 matrix elems with values, which define finite element triangles:

nodes(1:7,:)'
 1.00 1.00 1.00 1.00 1.00 1.05 1.05 ...
 1.00 1.20 1.50 1.80 2.00 1.00 1.20 ...
elems(1:7,:)'
 1    2    3    4    2    3    4 ...
 2    3    4    5    7    8    9 ...
 6    7    8    9    6    7    8 ...

Using a first order FEM, we approximate the electrical conductivity distribution in Ω as constant on each simplex (represented by the vector conductivity). Based on the finite element geometry, we first calculate a system (or stiffness) matrix for each simplex (represented as 3-by-3 elements on the diagonal of the element-wise system matrix SE. Based on SE and a N-by-DE connectivity matrix C, representing the connections between simplices and vertices, the global connectivity matrix S is calculated.

## Element conductivity
conductivity = [1*ones(1,16), ...
      2*ones(1,48), 1*ones(1,16)];
## Connectivity matrix
C = sparse ((1:D*E), reshape (elems', ...
      D*E, 1), 1, D*E, N);
## Calculate system matrix
Siidx = floor ([0:D*E-1]'/D) * D * ...
      ones(1,D) + ones(D*E,1)*(1:D) ;
Sjidx = [1:D*E]'*ones (1,D);
Sdata = zeros (D*E,D);

dfact = factorial (D-1);
for j = 1:E
   a = inv ([ones(D,1), ...
       nodes(elems(j,:), :)]);
   const = conductivity(j) * 2 / ...
       dfact / abs (det (a));
   Sdata(D*(j-1)+(1:D),:) = const * ...
       a(2:D,:)' * a(2:D,:);
endfor
## Element-wise system matrix
SE = sparse(Siidx,Sjidx,Sdata);
## Global system matrix
S = C'* SE *C;

The system matrix acts like the conductivity S in Ohm’s law SV = I. Based on the Dirichlet and Neumann boundary conditions, we are able to solve for the voltages at each vertex V.

## Dirichlet boundary conditions
D_nodes = [1:5, 51:55];
D_value = [10*ones(1,5), 20*ones(1,5)];
V = zeros (N,1);
V(D_nodes) = D_value;
idx = 1:N; # vertices without Dirichlet
           # boundary condns
idx(D_nodes) = [];
## Neumann boundary conditions.  Note that
## N_value must be normalized by the
## boundary length and element conductivity
N_nodes = [];
N_value = [];
Q = zeros (N,1);
Q(N_nodes) = N_value;
V(idx) = S(idx,idx) \ ( Q(idx) - ...
          S(idx,D_nodes) * V(D_nodes));

Finally, in order to display the solution, we show each solved voltage value in the z-axis for each simplex vertex. See Figure 22.6.

elemx = elems(:,[1,2,3,1])';
xelems = reshape (nodes(elemx, 1), 4, E);
yelems = reshape (nodes(elemx, 2), 4, E);
velems = reshape (V(elemx), 4, E);
plot3 (xelems,yelems,velems,"k");
print "grid.eps";

524 GNU Octave

Figure 22.6: Example finite element model the showing triangular elements. The height of each vertex corresponds to the solution value.

23 Numerical Integration

Octave comes with several built-in functions for computing the integral of a function numerically (termed quadrature). These functions all solve 1-dimensional integration problems.

23.1 Functions of One Variable

Octave supports five different algorithms for computing the integral

∫_{a}^{b} f(x)dx

of a function f over the interval from a to b. These are

quad Numerical integration based on Gaussian quadrature.

quadv Numerical integration using an adaptive vectorized Simpson’s rule.

quadl Numerical integration using an adaptive Lobatto rule.

quadgk Numerical integration using an adaptive Gauss-Konrod rule.

quadcc Numerical integration using adaptive Clenshaw-Curtis rules.

trapz, cumtrapz

Numerical integration of data using the trapezoidal method.

The best quadrature algorithm to use depends on the integrand. If you have empirical data, rather than a function, the choice is trapz or cumtrapz. If you are uncertain about the characteristics of the integrand, quadcc will be the most robust as it can handle discontinuities, singularities, oscillatory functions, and infinite intervals. When the integrand is smooth quadgk may be the fastest of the algorithms.

Function  Characteristics
quad  Low accuracy with nonsmooth integrands
quadv  Medium accuracy with smooth integrands
quadl  Medium accuracy with smooth integrands. Slower than quadgk.
quadgk  Medium accuracy (1e^{−6}–1e^{−9}) with smooth integrands.

Handles oscillatory functions and infinite bounds

quadcc  Low to High accuracy with nonsmooth/smooth integrands

Handles oscillatory functions, singularities, and infinite bounds

Here is an example of using quad to integrate the function

f(x) = x sin(1/x)\sqrt{|1 − x|}

from x = 0 to x = 3.

This is a fairly difficult integration (plot the function over the range of integration to see why).

The first step is to define the function:

function y = f (x)
  y = x .* sin (1./x) .* sqrt (abs (1 - x));
endfunction

Note the use of the ‘dot’ forms of the operators. This is not necessary for the quad integrator, but is required by the other integrators. In any case, it makes it much easier to generate a set of points for plotting because it is possible to call the function with a vector argument to produce a vector result.

The second step is to call quad with the limits of integration:

[q, ier, nfun, err] = quad ("f", 0, 3)
     ⇒ 1.9819

     ⇒ 1
     ⇒ 5061
     ⇒ 1.1522e-07

Although quad returns a nonzero value for ier, the result is reasonably accurate (to see why, examine what happens to the result if you move the lower bound to 0.1, then 0.01, then 0.001, etc.).

The function "f" can be the string name of a function, a function handle, or an inline function. These options make it quite easy to do integration without having to fully define a function in an m-file. For example:

# Verify integral (x^3) = x^4/4
f = inline ("x.^3");
quadgk (f, 0, 1)
     ⇒ 0.25000
# Verify gamma function = (n-1)! for n = 4
f = @(x) x.^3 .* exp (-x);
quadcc (f, 0, Inf)
     ⇒ 6.0000

[Built-in Function] q = quad (f, a, b) [Built-in Function] q = quad (f, a, b, tol) [Built-in Function] q = quad (f, a, b, tol, sing) [Built-in Function] [q, ier, nfun, err] = quad (…)

Numerically evaluate the integral of f from a to b using Fortran routines from quadpack. f is a function handle, inline function, or a string containing the name of the function to evaluate. The function must have the form y = f (x) where y and x are scalars. a and b are the lower and upper limits of integration. Either or both may be infinite. The optional argument tol is a vector that specifies the desired accuracy of the result. The first element of the vector is the desired absolute tolerance, and the second element is the desired relative tolerance. To choose a relative test only, set the absolute tolerance to zero. To choose an absolute test only, set the relative tolerance to zero. Both tolerances default to sqrt (eps) or approximately 1.5e^{−8}. The optional argument sing is a vector of values at which the integrand is known to be singular.

The result of the integration is returned in q. ier contains an integer error code (0 indicates a successful integration). nfun indicates the number of function evaluations that were made, and err contains an estimate of the error in the solution. The function quad_options can set other optional parameters for quad. Note: because quad is written in Fortran it cannot be called recursively. This prevents its use in integrating over more than one variable by routines dblquad and

triplequad.

See also: [quad options], page 527, [quadv], page 527, [quadl], page 528, [quadgk], page 528, [quadcc], page 530, [trapz], page 531, [dblquad], page 533, [triplequad], page 534.

[Built-in Function] quad_options () [Built-in Function] val = quad_options (opt) [Built-in Function] quad_options (opt, val)

Query or set options for the function quad. When called with no arguments, the names of all available options and their current values are displayed. Given one argument, return the value of the corresponding option. When called with two arguments, quad_options set the option opt to value val. Options include

"absolute tolerance"

Absolute tolerance; may be zero for pure relative error test.

"relative tolerance"

Non-negative relative tolerance. If the absolute tolerance is zero, the relative tolerance must be greater than or equal to max (50*eps, 0.5e-28).

"single precision absolute tolerance"

Absolute tolerance for single precision; may be zero for pure relative error test.

"single precision relative tolerance"

Non-negative relative tolerance for single precision. If the absolute tolerance is zero, the relative tolerance must be greater than or equal to

          max (50*eps, 0.5e-28).

[Function File] q = quadv (f, a, b) [Function File] q = quadv (f, a, b, tol) [Function File] q = quadv (f, a, b, tol, trace) [Function File] q = quadv (f, a, b, tol, trace, p1, p2, …) [Function File] [q, nfun] = quadv (…)

Numerically evaluate the integral of f from a to b using an adaptive Simpson’s rule. f is a function handle, inline function, or string containing the name of the function to evaluate. quadv is a vectorized version of quad and the function defined by f must accept a scalar or vector as input and return a scalar, vector, or array as output. a and b are the lower and upper limits of integration. Both limits must be finite. The optional argument tol defines the tolerance used to stop the adaptation procedure. The default value is 1e^{−6}.

The algorithm used by quadv involves recursively subdividing the integration interval and applying Simpson’s rule on each subinterval. If trace is true then after computing each of these partial integrals display: (1) the total number of function evaluations, (2) the left end of the subinterval, (3) the length of the subinterval, (4) the approximation of the integral over the subinterval. Additional arguments p1, etc., are passed directly to the function f. To use default values for tol and trace, one may pass empty matrices ([]). The result of the integration is returned in q. nfun indicates the number of function evaluations that were made. Note: quadv is written in Octave’s scripting language and can be used recursively in dblquad and triplequad, unlike the similar quad function.

See also: [quad], page 526, [quadl], page 528, [quadgk], page 528, [quadcc], page 530, [trapz], page 531, [dblquad], page 533, [triplequad], page 534.

[Function File] q = quadl (f, a, b) [Function File] q = quadl (f, a, b, tol) [Function File] q = quadl (f, a, b, tol, trace) [Function File] q = quadl (f, a, b, tol, trace, p1, p2, …)

Numerically evaluate the integral of f from a to b using an adaptive Lobatto rule. f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be vectorized and return a vector of output values if given a vector of input values. a and b are the lower and upper limits of integration. Both limits must be finite. The optional argument tol defines the relative tolerance with which to perform the integration. The default value is eps. The algorithm used by quadl involves recursively subdividing the integration interval. If trace is defined then for each subinterval display: (1) the left end of the subinterval, (2) the length of the subinterval, (3) the approximation of the integral over the subinterval. Additional arguments p1, etc., are passed directly to the function f. To use default values for tol and trace, one may pass empty matrices ([]). Reference: W. Gander and W. Gautschi, Adaptive Quadrature - Revisited, BIT Vol.

40, No. 1, March 2000, pp. 84–101. http://www.inf.ethz.ch/personal/gander/

See also: [quad], page 526, [quadv], page 527, [quadgk], page 528, [quadcc], page 530, [trapz], page 531, [dblquad], page 533, [triplequad], page 534.

[Function File] q = quadgk (f, a, b) [Function File] q = quadgk (f, a, b, abstol)

[Function File] q = quadgk (f, a, b, abstol, trace)

[Function File] q = quadgk (f, a, b, prop, val, …) [Function File] [q, err] = quadgk (…)

Numerically evaluate the integral of f from a to b using adaptive Gauss-Konrod quadrature. f is a function handle, inline function, or string containing the name of the function to evaluate. The formulation is based on a proposal by L.F. Shampine, "Vectorized adaptive quadrature in matlab", Journal of Computational and Applied Mathematics, pp131-140, Vol 211, Issue 2, Feb 2008 where all function evaluations at an iteration are calculated with a single call to f. Therefore, the function f must be vectorized and must accept a vector of input values x and return an output vector representing the function evaluations at the given values of x. a and b are the lower and upper limits of integration. Either or both limits may be infinite or contain weak end singularities. Variable transformation will be used to treat any infinite intervals and weaken the singularities. For example:

     quadgk (@(x) 1 ./ (sqrt (x) .* (x + 1)), 0, Inf)

Note that the formulation of the integrand uses the element-by-element operator ./ and all user functions to quadgk should do the same. The optional argument tol defines the absolute tolerance used to stop the integration procedure. The default value is 1e^{−10}. The algorithm used by quadgk involves subdividing the integration interval and evaluating each subinterval. If trace is true then after computing each of these partial integrals display: (1) the number of subintervals at this step, (2) the current estimate of the error err, (3) the current estimate for the integral q. Alternatively, properties of quadgk can be passed to the function as pairs "prop", val. Valid properties are

AbsTol Define the absolute error tolerance for the quadrature. The default absolute tolerance is 1e-10.

RelTol Define the relative error tolerance for the quadrature. The default relative tolerance is 1e-5.

MaxIntervalCount

quadgk initially subdivides the interval on which to perform the quadrature into 10 intervals. Subintervals that have an unacceptable error are subdivided and re-evaluated. If the number of subintervals exceeds 650 subintervals at any point then a poor convergence is signaled and the current estimate of the integral is returned. The property "MaxIntervalCount" can be used to alter the number of subintervals that can exist before exiting.

WayPoints

Discontinuities in the first derivative of the function to integrate can be flagged with the "WayPoints" property. This forces the ends of a subinterval to fall on the breakpoints of the function and can result in significantly improved estimation of the error in the integral, faster computation, or both. For example,

               quadgk (@(x) abs (1 - x.^2), 0, 2, "Waypoints", 1)

signals the breakpoint in the integrand at x = 1.

Trace If logically true quadgk prints information on the convergence of the quadrature at each iteration.

If any of a, b, or waypoints is complex then the quadrature is treated as a contour integral along a piecewise continuous path defined by the above. In this case the integral is assumed to have no edge singularities. For example,

     quadgk (@(z) log (z), 1+1i, 1+1i, "WayPoints",
             [1-1i, -1,-1i, -1+1i])

integrates log (z) along the square defined by [1+1i, 1-1i, -1-1i, -1+1i] The result of the integration is returned in q. err is an approximate bound on the error in the integral abs (q - I), where I is the exact value of the integral.

See also: [quad], page 526, [quadv], page 527, [quadl], page 528, [quadcc], page 530, [trapz], page 531, [dblquad], page 533, [triplequad], page 534.

[Function File] q = quadcc (f, a, b) [Function File] q = quadcc (f, a, b, tol) [Function File] q = quadcc (f, a, b, tol, sing)

[Function File] [q, err, nr_points] = quadcc (. . .)

Numerically evaluate the integral of f from a to b using the doubly-adaptive Clenshaw-Curtis quadrature described by P. Gonnet in Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants. f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be vectorized and must return a vector of output values if given a vector of input values. For example,

     f = @(x) x .* sin (1./x) .* sqrt (abs (1 - x));

which uses the element-by-element “dot” form for all operators. a and b are the lower and upper limits of integration. Either or both limits may be infinite. quadcc handles an inifinite limit by substituting the variable of integration

with x = tan (pi/2*u).

The optional argument tol defines the relative tolerance used to stop the integration procedure. The default value is 1e^{−6}. The optional argument sing contains a list of points where the integrand has known singularities, or discontinuities in any of its derivatives, inside the integration interval. For the example above, which has a discontinuity at x=1, the call to quadcc would be as follows

     int = quadcc (f, a, b, 1.0e-6, [ 1 ]);

The result of the integration is returned in q. err is an estimate of the absolute integration error and nr points is the number of points at which the integrand was evaluated. If the adaptive integration did not converge, the value of err will be larger than the requested tolerance. Therefore, it is recommended to verify this value for difficult integrands. quadcc is capable of dealing with non-numeric values of the integrand such as NaN or Inf. If the integral diverges, and quadcc detects this, then a warning is issued and Inf or -Inf is returned. Note: quadcc is a general purpose quadrature algorithm and, as such, may be less efficient for a smooth or otherwise well-behaved integrand than other methods such

as quadgk.

The algorithm uses Clenshaw-Curtis quadrature rules of increasing degree in each interval and bisects the interval if either the function does not appear to be smooth or a rule of maximum degree has been reached. The error estimate is computed from the L2-norm of the difference between two successive interpolations of the integrand over the nodes of the respective quadrature rules. Reference: P. Gonnet, Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants, ACM Transactions on Mathematical Software, Vol. 37, Issue 3, Article No. 3, 2010.

See also: [quad], page 526, [quadv], page 527, [quadl], page 528, [quadgk], page 528, [trapz], page 531, [dblquad], page 533, [triplequad], page 534.

Sometimes one does not have the function, but only the raw (x, y) points from which to perform an integration. This can occur when collecting data in an experiment. The trapz function can integrate these values as shown in the following example where "data" has been collected on the cosine function over the range [0, pi/2).

x = 0:0.1:pi/2;  # Uniformly spaced points
y = cos (x);
trapz (x, y)
     ⇒ 0.99666

The answer is reasonably close to the exact value of 1. Ordinary quadrature is sensitive to the characteristics of the integrand. Empirical integration depends not just on the integrand, but also on the particular points chosen to represent the function. Repeating the example above with the sine function over the range [0, pi/2) produces far inferior results.

x = 0:0.1:pi/2;  # Uniformly spaced points
y = sin (x);
trapz (x, y)
     ⇒ 0.92849

However, a slightly different choice of data points can change the result significantly. The same integration, with the same number of points, but spaced differently produces a more accurate answer.

x = linspace (0, pi/2, 16);  # Uniformly spaced, but including endpoint
y = sin (x);
trapz (x, y)
     ⇒ 0.99909

In general there may be no way of knowing the best distribution of points ahead of time. Or the points may come from an experiment where there is no freedom to select the best distribution. In any case, one must remain aware of this issue when using trapz.

[Function File] q = trapz (y) [Function File] q = trapz (x, y) [Function File] q = trapz (…, dim)

Numerically evaluate the integral of points y using the trapezoidal method. trapz (y) computes the integral of y along the first non-singleton dimension. When the argument x is omitted an equally spaced x vector with unit spacing (1) is assumed. trapz (x, y) evaluates the integral with respect to the spacing in x and the values in y. This is useful if the points in y have been sampled unevenly. If the optional dim argument is given, operate along this dimension. If x is not specified then unit spacing will be used. To scale the integral to the correct value you must multiply by the actual spacing value (deltaX). As an example, the integral of x^{3} over the range [0, 1] is x^{4}/4 or 0.25. The following code uses trapz to calculate the integral in three different ways.

     x = 0:0.1:1;
     y = x.^3;
     q = trapz (y)
       ⇒ q = 2.525   # No scaling
     q * 0.1
       ⇒ q = 0.2525  # Approximation to integral by scaling
     trapz (x, y)
       ⇒ q = 0.2525  # Same result by specifying x

See also: [cumtrapz], page 532.

[Function File] q = cumtrapz (y) [Function File] q = cumtrapz (x, y) [Function File] q = cumtrapz (…, dim)

Cumulative numerical integration of points y using the trapezoidal method. cumtrapz (y) computes the cumulative integral of y along the first non-singleton dimension. Where trapz reports only the overall integral sum, cumtrapz reports the current partial sum value at each point of y. When the argument x is omitted an equally spaced x vector with unit spacing (1) is assumed. cumtrapz (x, y) evaluates the integral with respect to the spacing in x and the values in y. This is useful if the points in y have been sampled unevenly. If the optional dim argument is given, operate along this dimension. If x is not specified then unit spacing will be used. To scale the integral to the correct value you must multiply by the actual spacing value (deltaX).

See also: [trapz], page 531, [cumsum], page 414.

23.2 Orthogonal Collocation

[Built-in Function] [r, amat, bmat, q] = colloc (n, "left", "right")

Compute derivative and integral weight matrices for orthogonal collocation using the subroutines given in J. Villadsen and M. L. Michelsen, Solution of Differential Equation Models by Polynomial Approximation.

Here is an example of using colloc to generate weight matrices for solving the second order differential equation u′−αu′′ = 0 with the boundary conditions u(0) = 0 and u(1) = 1.

First, we can generate the weight matrices for n points (including the endpoints of the interval), and incorporate the boundary conditions in the right hand side (for a specific value of α).

n = 7;
alpha = 0.1;
[r, a, b] = colloc (n-2, "left", "right");
at = a(2:n-1,2:n-1);
bt = b(2:n-1,2:n-1);
rhs = alpha * b(2:n-1,n) - a(2:n-1,n);

Then the solution at the roots r is

u = [ 0; (at - alpha * bt) \ rhs; 1]
     ⇒ [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ]

23.3 Functions of Multiple Variables

Octave does not have built-in functions for computing the integral of functions of multiple variables directly. It is possible, however, to compute the integral of a function of multiple variables using the existing functions for one-dimensional integrals.

To illustrate how the integration can be performed, we will integrate the function

f(x, y) = sin(πxy)\sqrt{xy}

for x and y between 0 and 1.

The first approach creates a function that integrates f with respect to x, and then integrates that function with respect to y. Because quad is written in Fortran it cannot be called recursively. This means that quad cannot integrate a function that calls quad, and hence cannot be used to perform the double integration. Any of the other integrators, however, can be used which is what the following code demonstrates.

function q = g(y)
  q = ones (size (y));
  for i = 1:length (y)
    f = @(x) sin (pi*x.*y(i)) .* sqrt (x.*y(i));
    q(i) = quadgk (f, 0, 1);
  endfor
endfunction
I = quadgk ("g", 0, 1)
      ⇒ 0.30022

The above process can be simplified with the dblquad and triplequad functions for integrals over two and three variables. For example:

I = dblquad (@(x, y) sin (pi*x.*y) .* sqrt (x.*y), 0, 1, 0, 1)
      ⇒ 0.30022

[Function File] dblquad (f, xa, xb, ya, yb) [Function File] dblquad (f, xa, xb, ya, yb, tol)

[Function File] dblquad (f, xa, xb, ya, yb, tol, quadf)

[Function File] dblquad (f, xa, xb, ya, yb, tol, quadf, …)

Numerically evaluate the double integral of f. f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must have the form z = f(x, y) where x is a vector and y is a scalar. It should return a vector of the same length and orientation as x. xa, ya and xb, yb are the lower and upper limits of integration for x and y respectively. The underlying integrator determines whether infinite bounds are accepted. The optional argument tol defines the absolute tolerance used to integrate each subintegral. The default value is 1e^{−6}. The optional argument quadf specifies which underlying integrator function to use. Any choice but quad is available and the default is quadcc.

Additional arguments, are passed directly to f. To use the default value for tol or quadf one may pass ':' or an empty matrix ([]).

See also: [triplequad], page 534, [quad], page 526, [quadv], page 527, [quadl], page 528, [quadgk], page 528, [quadcc], page 530, [trapz], page 531.

[Function File] triplequad (f, xa, xb, ya, yb, za, zb)
[Function File] triplequad (f, xa, xb, ya, yb, za, zb, tol)
[Function File] triplequad (f, xa, xb, ya, yb, za, zb, tol, quadf)
[Function File] triplequad (f, xa, xb, ya, yb, za, zb, tol, quadf, . . .)

Numerically evaluate the triple integral of f. f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must have the form w = f(x, y, z) where either x or y is a vector and the remaining inputs are scalars. It should return a vector of the same length and orientation as x or y. xa, ya, za and xb, yb, zb are the lower and upper limits of integration for x, y, and z respectively. The underlying integrator determines whether infinite bounds are accepted. The optional argument tol defines the absolute tolerance used to integrate each subintegral. The default value is 1e^{−6}. The optional argument quadf specifies which underlying integrator function to use. Any choice but quad is available and the default is quadcc. Additional arguments, are passed directly to f. To use the default value for tol or quadf one may pass ':' or an empty matrix ([]).

See also: [dblquad], page 533, [quad], page 526, [quadv], page 527, [quadl], page 528, [quadgk], page 528, [quadcc], page 530, [trapz], page 531.

The above mentioned approach works, but is fairly slow, and that problem increases exponentially with the dimensionality of the integral. Another possible solution is to use Orthogonal Collocation as described in the previous section (see Section 23.2 [Orthogonal Collocation], page 532). The integral of a function f(x, y) for x and y between 0 and 1 can be approximated using n points by

∫_{0}^{1} ∫_{0}^{1} f(x, y)dxdy ≈ ∑_{i=1}^{n} ∑_{j=1}^{n} q_{i}q_{j}f(r_{i}, r_{j}),

where q and r is as returned by colloc (n). The generalization to more than two variables is straight forward. The following code computes the studied integral using n = 8 points.

f = @(x,y) sin (pi*x*y') .* sqrt (x*y');
n = 8;
[t, ~, ~, q] = colloc (n);
I = q'*f(t,t)*q;
      ⇒ 0.30022

It should be noted that the number of points determines the quality of the approximation. If the integration needs to be performed between a and b, instead of 0 and 1, then a change of variables is needed.

24 Differential Equations

Octave has built-in functions for solving ordinary differential equations, and differentialalgebraic equations. All solvers are based on reliable ODE routines written in Fortran.

24.1 Ordinary Differential Equations

The function lsode can be used to solve ODEs of the form

\frac{dx}{dt} = f(x, t)

using Hindmarsh’s ODE solver lsode.

[Built-in Function] [x, istate, msg] = lsode (fcn, x_0, t)
[Built-in Function] [x, istate, msg] = lsode (fcn, x_0, t, t_crit)

Solve the set of differential equations

\frac{dx}{dt} = f(x, t)

with

x(t_{0}) = x_{0}

The solution is returned in the matrix x, with each row corresponding to an element of the vector t. The first element of t should be t_{0} and should correspond to the initial state of the system x 0, so that the first row of the output is x 0. The first argument, fcn, is a string, inline, or function handle that names the function f to call to compute the vector of right hand sides for the set of equations. The function must have the form

     xdot = f (x, t)

in which xdot and x are vectors and t is a scalar. If fcn is a two-element string array or a two-element cell array of strings, inline functions, or function handles, the first element names the function f described above, and the second element names a function to compute the Jacobian of f. The Jacobian function must have the form

     jac = j (x, t)

in which jac is the matrix of partial derivatives

J = \frac{∂f_{i}}{∂x_{j}} = [ \begin{matrix}\frac{∂f_{1}}{∂x_{1}} & \frac{∂f_{1}}{∂x_{2}} & ⋯ & \frac{∂f_{1}}{∂x_{N}} \\ \frac{∂f_{2}}{∂x_{1}} & \frac{∂f_{2}}{∂x_{2}} & ⋯ & \frac{∂f_{2}}{∂x_{N}} \\ ⋮ & ⋮ & ... & ⋮ \\ \frac{∂f_{3}}{∂x_{1}} & \frac{∂f_{3}}{∂x_{2}} & ⋯ & \frac{∂f_{3}}{∂x_{N}} \end{matrix} ]

The second and third arguments specify the initial state of the system, x_{0}, and the initial value of the independent variable t_{0}.

The fourth argument is optional, and may be used to specify a set of times that the ODE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative.

After a successful computation, the value of istate will be 2 (consistent with the Fortran version of lsode).

If the computation is not successful, istate will be something other than 2 and msg will contain additional information.

You can use the function lsode_options to set optional parameters for lsode.

See also: [daspk], page 538, [dassl], page 541, [dasrt], page 543.

[Built-in Function] lsode_options ()

[Built-in Function] val = lsode_options (opt)

[Built-in Function] lsode_options (opt, val)

Query or set options for the function lsode. When called with no arguments, the names of all available options and their current values are displayed. Given one argument, return the value of the corresponding option. When called with two arguments, lsode_options set the option opt to value val.

Options include

"absolute tolerance"

Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector.

"relative tolerance"

Relative tolerance parameter. Unlike the absolute tolerance, this parameter may only be a scalar. The local error test applied at each integration step is

                 abs (local error in x(i)) <= ...
                     rtol * abs (y(i)) + atol(i)
"integration method"

A string specifying the method of integration to use to solve the ODE system. Valid values are

          "adams"
          "non-stiff"

No Jacobian used (even if it is available).

          "bdf"

"stiff" Use stiff backward differentiation formula (BDF) method. If a function to compute the Jacobian is not supplied, lsode will compute a finite difference approximation of the Jacobian matrix.

"initial step size"

The step size to be attempted on the first step (default is determined automatically).

"maximum order"

Restrict the maximum order of the solution method. If using the Adams method, this option must be between 1 and 12. Otherwise, it must be between 1 and 5, inclusive.

"maximum step size"

Setting the maximum stepsize will avoid passing over very large regions (default is not specified).

"minimum step size"

The minimum absolute step size allowed (default is 0).

"step limit"

Maximum number of steps allowed (default is 100000).

Here is an example of solving a set of three differential equations using lsode. Given the function

## oregonator differential equation
function xdot = f (x, t)
  xdot = zeros (3,1);
  xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \
            - 8.375e-06*x(1)^2);
  xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27;
  xdot(3) = 0.161*(x(1) - x(3));
endfunction

and the initial condition x0 = [ 4; 1.1; 4 ], the set of equations can be integrated using the command

t = linspace (0, 500, 1000);
y = lsode ("f", x0, t);

If you try this, you will see that the value of the result changes dramatically between t = 0 and 5, and again around t = 305. A more efficient set of output points might be

t = [0, logspace(-1, log10(303), 150), \
        logspace(log10(304), log10(500), 150)];

See Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman, editor, (1983) for more information about the inner workings of lsode.

An m-file for the differential equation used above is included with the Octave distribution in the examples directory under the name ‘oregonator.m’.

24.2 Differential-Algebraic Equations

The function daspk can be used to solve DAEs of the form

0 = f(ẋ, x, t), x(t = 0) = x_{0}, ẋ(t = 0) = ẋ_{0}

where ẋ = \frac{dx}{dt} is the derivative of x. The equation is solved using Petzold’s DAE solver daspk.

[Built-in Function] [x, xdot, istate, msg] = daspk (fcn, x_0, xdot_0, t,
   t_crit)

Solve the set of differential-algebraic equations

0 = f(x, ẋ, t)

with

x(t_{0}) = x_{0}, ẋ(t_{0}) = ẋ_{0}

The solution is returned in the matrices x and xdot, with each row in the result matrices corresponding to one of the elements in the vector t. The first element of t should be t_{0} and correspond to the initial state of the system x 0 and its derivative xdot 0, so that the first row of the output x is x 0 and the first row of the output xdot is xdot 0. The first argument, fcn, is a string, inline, or function handle that names the function f to call to compute the vector of residuals for the set of equations. It must have the form

     res = f (x, xdot, t)

in which x, xdot, and res are vectors, and t is a scalar. If fcn is a two-element string array or a two-element cell array of strings, inline functions, or function handles, the first element names the function f described above, and the second element names a function to compute the modified Jacobian

J = \frac{∂f}{∂x} + c\frac{∂f}{∂ẋ}

The modified Jacobian function must have the form

     jac = j (x, xdot, t, c)

The second and third arguments to daspk specify the initial condition of the states and their derivatives, and the fourth argument specifies a vector of output times at which the solution is desired, including the time corresponding to the initial condition. The set of initial states and derivatives are not strictly required to be consistent. If they are not consistent, you must use the daspk_options function to provide additional information so that daspk can compute a consistent starting point. The fifth argument is optional, and may be used to specify a set of times that the DAE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative. After a successful computation, the value of istate will be greater than zero (consistent with the Fortran version of daspk). If the computation is not successful, the value of istate will be less than zero and msg will contain additional information. You can use the function daspk_options to set optional parameters for daspk.

See also: [dassl], page 541.

[Built-in Function] daspk_options ()

[Built-in Function] val = daspk_options (opt)

[Built-in Function] daspk_options (opt, val)

Query or set options for the function daspk. When called with no arguments, the names of all available options and their current values are displayed. Given one argument, return the value of the corresponding option. When called with two arguments, daspk_options set the option opt to value val.

Options include

"absolute tolerance"

Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the relative tolerance must also be a vector of the same length.

"relative tolerance"

Relative tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the absolute tolerance must also be a vector of the same length. The local error test applied at each integration step is

       abs (local error in x(i))
            <= rtol(i) * abs (Y(i)) + atol(i)
"compute consistent initial condition"

Denoting the differential variables in the state vector by ‘Y_d’ and the algebraic variables by ‘Y_a’, ddaspk can solve one of two initialization problems:

1. Given Y d, calculate Y a and Y’ d

2. Given Y’, calculate Y.

In either case, initial values for the given components are input, and initial guesses for the unknown components must also be provided as input. Set this option to 1 to solve the first problem, or 2 to solve the second (the default is 0, so you must provide a set of initial conditions that are consistent). If this option is set to a nonzero value, you must also set the "algebraic variables" option to declare which variables in the problem are algebraic.

"use initial condition heuristics"

Set to a nonzero value to use the initial condition heuristics options described below.

"initial condition heuristics"

A vector of the following parameters that can be used to control the initial condition calculation.

MXNIT Maximum number of Newton iterations (default is 5).

MXNJ Maximum number of Jacobian evaluations (default is 6).

MXNH Maximum number of values of the artificial stepsize

          parameter to be tried if the "compute consistent initial

condition" option has been set to 1 (default is 5).

Note that the maximum total number of Newton iterations

          allowed is MXNIT*MXNJ*MXNH if the "compute consistent

initial condition" option has been set to 1 and MXNIT*MXNJ if it is set to 2.

LSOFF Set to a nonzero value to disable the linesearch algorithm (default is 0).

STPTOL Minimum scaled step in linesearch algorithm (default is eps^(2/3)).

EPINIT Swing factor in the Newton iteration convergence test. The test is applied to the residual vector, premultiplied by the approximate Jacobian. For convergence, the weighted RMS norm of this vector (scaled by the error weights) must be less than EPINIT*EPCON, where EPCON = 0.33 is the analogous test constant used in the time steps. The default is EPINIT = 0.01.

"print initial condition info"

Set this option to a nonzero value to display detailed information about the initial condition calculation (default is 0).

"exclude algebraic variables from error test"

Set to a nonzero value to exclude algebraic variables from the error test. You must also set the "algebraic variables" option to declare which variables in the problem are algebraic (default is 0).

"algebraic variables"

A vector of the same length as the state vector. A nonzero element indicates that the corresponding element of the state vector is an algebraic variable (i.e., its derivative does not appear explicitly in the equation set.

This option is required by the compute consistent initial condition"
and "exclude algebraic variables from error test" options.
"enforce inequality constraints"

Set to one of the following values to enforce the inequality constraints specified by the "inequality constraint types" option (default is 0).

1. To have constraint checking only in the initial condition calculation.

2. To enforce constraint checking during the integration.

3. To enforce both options 1 and 2.

"inequality constraint types"

A vector of the same length as the state specifying the type of inequality constraint. Each element of the vector corresponds to an element of the state and should be assigned one of the following codes

-2 Less than zero.

-1 Less than or equal to zero.

0 Not constrained.

1 Greater than or equal to zero.

2 Greater than zero.

          This option only has an effect if the "enforce inequality constraints"

option is nonzero.

"initial step size"

Differential-algebraic problems may occasionally suffer from severe scaling difficulties on the first step. If you know a great deal about the scaling of your problem, you can help to alleviate this problem by specifying an initial stepsize (default is computed automatically).

"maximum order"

Restrict the maximum order of the solution method. This option must be between 1 and 5, inclusive (default is 5).

"maximum step size"

Setting the maximum stepsize will avoid passing over very large regions (default is not specified).

Octave also includes dassl, an earlier version of daspk, and dasrt, which can be used to solve DAEs with constraints (stopping conditions).

[Built-in Function] [x, xdot, istate, msg] = dassl (fcn, x_0, xdot_0, t,
   t_crit)

Solve the set of differential-algebraic equations

0 = f(x, ẋ, t)

with

x(t_{0}) = x_{0}, ẋ(t_{0}) = ẋ_{0}

The solution is returned in the matrices x and xdot, with each row in the result matrices corresponding to one of the elements in the vector t. The first element of t should be t_{0} and correspond to the initial state of the system x 0 and its derivative xdot 0, so that the first row of the output x is x 0 and the first row of the output xdot is xdot 0. The first argument, fcn, is a string, inline, or function handle that names the function f to call to compute the vector of residuals for the set of equations. It must have the form

     res = f (x, xdot, t)

in which x, xdot, and res are vectors, and t is a scalar. If fcn is a two-element string array or a two-element cell array of strings, inline functions, or function handles, the first element names the function f described above, and the second element names a function to compute the modified Jacobian

J = \frac{∂f}{∂x} + c\frac{∂f}{∂ẋ}

The modified Jacobian function must have the form

     jac = j (x, xdot, t, c)

The second and third arguments to dassl specify the initial condition of the states and their derivatives, and the fourth argument specifies a vector of output times at which the solution is desired, including the time corresponding to the initial condition.

The set of initial states and derivatives are not strictly required to be consistent. In practice, however, dassl is not very good at determining a consistent set for you, so it is best if you ensure that the initial values result in the function evaluating to zero.

The fifth argument is optional, and may be used to specify a set of times that the DAE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative.

After a successful computation, the value of istate will be greater than zero (consistent with the Fortran version of dassl).

If the computation is not successful, the value of istate will be less than zero and msg will contain additional information.

You can use the function dassl_options to set optional parameters for dassl.

See also: [daspk], page 538, [dasrt], page 543, [lsode], page 535.

[Built-in Function] dassl_options ()

[Built-in Function] val = dassl_options (opt)

[Built-in Function] dassl_options (opt, val)

Query or set options for the function dassl. When called with no arguments, the names of all available options and their current values are displayed. Given one argument, return the value of the corresponding option. When called with two arguments, dassl_options set the option opt to value val.

Options include

"absolute tolerance"

Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the relative tolerance must also be a vector of the same length.

"relative tolerance"

Relative tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the absolute tolerance must also be a vector of the same length. The local error test applied at each integration step is

                 abs (local error in x(i))
                      <= rtol(i) * abs (Y(i)) + atol(i)
"compute consistent initial condition"

If nonzero, dassl will attempt to compute a consistent set of initial conditions. This is generally not reliable, so it is best to provide a consistent set and leave this option set to zero.

"enforce nonnegativity constraints"

If you know that the solutions to your equations will always be nonnegative, it may help to set this parameter to a nonzero value. However, it is probably best to try leaving this option set to zero first, and only setting it to a nonzero value if that doesn’t work very well.

"initial step size"

Differential-algebraic problems may occasionally suffer from severe scaling difficulties on the first step. If you know a great deal about the scaling of your problem, you can help to alleviate this problem by specifying an initial stepsize.

"maximum order"

Restrict the maximum order of the solution method. This option must be between 1 and 5, inclusive.

"maximum step size"

Setting the maximum stepsize will avoid passing over very large regions (default is not specified).

"step limit"

Maximum number of integration steps to attempt on a single call to the underlying Fortran code.

[Built-in Function] [x, xdot, t_out, istat, msg] = dasrt (fcn, [], x_0,
   xdot_0, t)

[Built-in Function] ... = dasrt (fcn, g, x_0, xdot_0, t)

[Built-in Function] ... = dasrt (fcn, [], x_0, xdot_0, t, t_crit)
[Built-in Function] ... = dasrt (fcn, g, x_0, xdot_0, t, t_crit)

Solve the set of differential-algebraic equations

0 = f(x, ẋ, t)

with

x(t_{0}) = x_{0}, ẋ(t_{0}) = ẋ_{0}

with functional stopping criteria (root solving). The solution is returned in the matrices x and xdot, with each row in the result matrices corresponding to one of the elements in the vector t out. The first element of t should be t_{0} and correspond to the initial state of the system x 0 and its derivative xdot 0, so that the first row of the output x is x 0 and the first row of the output xdot is xdot 0. The vector t provides an upper limit on the length of the integration. If the stopping condition is met, the vector t out will be shorter than t, and the final element of t out will be the point at which the stopping condition was met, and may not correspond to any element of the vector t. The first argument, fcn, is a string, inline, or function handle that names the function f to call to compute the vector of residuals for the set of equations. It must have the form

     res = f (x, xdot, t)

in which x, xdot, and res are vectors, and t is a scalar. If fcn is a two-element string array or a two-element cell array of strings, inline functions, or function handles, the first element names the function f described above, and the second element names a function to compute the modified Jacobian

J = \frac{∂f}{∂x} + c\frac{∂f}{∂ẋ}

The modified Jacobian function must have the form

     jac = j (x, xdot, t, c)

The optional second argument names a function that defines the constraint functions whose roots are desired during the integration. This function must have the form

     g_out = g (x, t)

and return a vector of the constraint function values. If the value of any of the constraint functions changes sign, dasrt will attempt to stop the integration at the point of the sign change. If the name of the constraint function is omitted, dasrt solves the same problem as

daspk or dassl.

Note that because of numerical errors in the constraint functions due to round-off and integration error, dasrt may return false roots, or return the same root at two or more nearly equal values of T. If such false roots are suspected, the user should consider smaller error tolerances or higher precision in the evaluation of the constraint functions. If a root of some constraint function defines the end of the problem, the input to dasrt should nevertheless allow integration to a point slightly past that root, so that dasrt can locate the root by interpolation. The third and fourth arguments to dasrt specify the initial condition of the states and their derivatives, and the fourth argument specifies a vector of output times at which the solution is desired, including the time corresponding to the initial condition. The set of initial states and derivatives are not strictly required to be consistent. In practice, however, dassl is not very good at determining a consistent set for you, so it is best if you ensure that the initial values result in the function evaluating to zero. The sixth argument is optional, and may be used to specify a set of times that the DAE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative. After a successful computation, the value of istate will be greater than zero (consistent with the Fortran version of dassl). If the computation is not successful, the value of istate will be less than zero and msg will contain additional information. You can use the function dasrt_options to set optional parameters for dasrt.

See also: [dasrt options], page 545, [daspk], page 538, [dasrt], page 543, [lsode], page 535.

[Built-in Function] dasrt_options ()

[Built-in Function] val = dasrt_options (opt)

[Built-in Function] dasrt_options (opt, val)

Query or set options for the function dasrt. When called with no arguments, the names of all available options and their current values are displayed. Given one argument, return the value of the corresponding option. When called with two arguments, dasrt_options set the option opt to value val. Options include

"absolute tolerance"

Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the relative tolerance must also be a vector of the same length.

"relative tolerance"

Relative tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the absolute tolerance must also be a vector of the same length. The local error test applied at each integration step is

       abs (local error in x(i)) <= ...
           rtol(i) * abs (Y(i)) + atol(i)
"initial step size"

Differential-algebraic problems may occasionally suffer from severe scaling difficulties on the first step. If you know a great deal about the scaling of your problem, you can help to alleviate this problem by specifying an initial stepsize.

"maximum order"

Restrict the maximum order of the solution method. This option must be between 1 and 5, inclusive.

"maximum step size"

Setting the maximum stepsize will avoid passing over very large regions.

"step limit"

Maximum number of integration steps to attempt on a single call to the underlying Fortran code.

See K. E. Brenan, et al., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland (1989) for more information about the implementation of dassl.

25 Optimization

Octave comes with support for solving various kinds of optimization problems. Specifically Octave can solve problems in Linear Programming, Quadratic Programming, Nonlinear Programming, and Linear Least Squares Minimization.

25.1 Linear Programming

Octave can solve Linear Programming problems using the glpk function. That is, Octave can solve

min_{x} c^{T} x

subject to the linear constraints Ax = b where x ≥ 0. The glpk function also supports variations of this problem.

[Function File] [xopt, fmin, errnum, extra] = glpk (c, A, b, lb, ub, ctype,
   vartype, sense, param)

Solve a linear program using the GNU glpk library. Given three arguments, glpk solves the following standard LP:

min_{x} C^{T} x

subject to

Ax = b x ≥ 0

but may also solve problems of the form

[min_{x} | max_{x}]C^{T} x

subject to

Ax[= | ≤ | ≥]b LB ≤ x ≤ UB

Input arguments:

c A column array containing the objective function coefficients.

A A matrix containing the constraints coefficients.

b A column array containing the right-hand side value for each constraint in the constraint matrix.

lb An array containing the lower bound on each of the variables. If lb is not supplied, the default lower bound for the variables is zero.

ub An array containing the upper bound on each of the variables. If ub is not supplied, the default upper bound is assumed to be infinite.

ctype An array of characters containing the sense of each constraint in the constraint matrix. Each element of the array may be one of the following values "F" A free (unbounded) constraint (the constraint is ignored). "U" An inequality constraint with an upper bound (A(i,:)*x <=

          b(i)).

"S" An equality constraint (A(i,:)*x = b(i)). "L" An inequality with a lower bound (A(i,:)*x >= b(i)). "D" An inequality constraint with both upper and lower bounds

          (A(i,:)*x >= -b(i) and (A(i,:)*x <= b(i)).

vartype A column array containing the types of the variables.

"C" A continuous variable. "I" An integer variable.

sense If sense is 1, the problem is a minimization. If sense is -1, the problem is

a maximization. The default value is 1.

param A structure containing the following parameters used to define the behavior of solver. Missing elements in the structure take on default values, so you only need to set the elements that you wish to change from the default. Integer parameters:

msglev (default: 1)

Level of messages output by solver routines:

          0 (GLP_MSG_OFF)

No output.

          1 (GLP_MSG_ERR)

Error and warning messages only.

          2 (GLP_MSG_ON)

Normal output.

          3 (GLP_MSG_ALL)

Full output (includes informational messages).

scale (default: 16)

Scaling option. The values can be combined with the bitwise OR operator and may be the following:

          1 (GLP_SF_GM)

Geometric mean scaling.

          16 (GLP_SF_EQ)

Equilibration scaling.

          32 (GLP_SF_2N)

Round scale factors to power of two.

          64 (GLP_SF_SKIP)

Skip if problem is well scaled.

Alternatively, a value of 128 (GLP_SF_AUTO) may be also specified, in which case the routine chooses the scaling options automatically.

dual (default: 1)

Simplex method option:

1 (GLP_PRIMAL)

Use two-phase primal simplex.

2 (GLP_DUALP)

Use two-phase dual simplex, and if it fails, switch to the primal simplex.

3 (GLP_DUAL)

Use two-phase dual simplex.

price (default: 34)

Pricing option (for both primal and dual simplex):

17 (GLP_PT_STD)

Textbook pricing.

34 (GLP_PT_PSE)

Steepest edge pricing.

itlim (default: intmax)

Simplex iterations limit. It is decreased by one each time when one simplex iteration has been performed, and reaching zero value signals the solver to stop the search.

outfrq (default: 200)

Output frequency, in iterations. This parameter specifies how frequently the solver sends information about the solution to the standard output.

branch (default: 4)

Branching technique option (for MIP only):

1 (GLP_BR_FFV)

First fractional variable.

2 (GLP_BR_LFV)

Last fractional variable.

3 (GLP_BR_MFV)

Most fractional variable.

4 (GLP_BR_DTH)

Heuristic by Driebeck and Tomlin.

5 (GLP_BR_PCH)

Hybrid pseudocost heuristic.

btrack (default: 4)

Backtracking technique option (for MIP only):

1 (GLP_BT_DFS)

Depth first search.

2 (GLP_BT_BFS)

Breadth first search.

3 (GLP_BT_BLB)

Best local bound.

4 (GLP_BT_BPH)

Best projection heuristic.

presol (default: 1)

If this flag is set, the simplex solver uses the built-in LP presolver. Otherwise the LP presolver is not used.

lpsolver (default: 1)

Select which solver to use. If the problem is a MIP problem this flag will be ignored.

1 Revised simplex method.

2 Interior point method.

rtest (default: 34)

Ratio test technique:

17 (GLP_RT_STD)

Standard ("textbook").

34 (GLP_RT_HAR)

Harris’ two-pass ratio test.

tmlim (default: intmax)

Searching time limit, in milliseconds.

outdly (default: 0)

Output delay, in seconds. This parameter specifies how long the solver should delay sending information about the solution to the standard output.

save (default: 0)

If this parameter is nonzero, save a copy of the problem in CPLEX LP format to the file ‘"outpb.lp"’. There is currently no way to change the name of the output file.

Real parameters:

tolbnd (default: 1e-7)

Relative tolerance used to check if the current basic solution is primal feasible. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

toldj (default: 1e-7)

Absolute tolerance used to check if the current basic solution is dual feasible. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

tolpiv (default: 1e-10)

Relative tolerance used to choose eligible pivotal elements of the simplex table. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

objll (default: -DBL_MAX)

Lower limit of the objective function. If the objective function reaches this limit and continues decreasing, the solver stops the search. This parameter is used in the dual simplex method only.

objul (default: +DBL_MAX)

Upper limit of the objective function. If the objective function reaches this limit and continues increasing, the solver stops the search. This parameter is used in the dual simplex only.

tolint (default: 1e-5)

Relative tolerance used to check if the current basic solution is integer feasible. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

tolobj (default: 1e-7)

Relative tolerance used to check if the value of the objective function is not better than in the best known integer feasible solution. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.

Output values:

xopt The optimizer (the value of the decision variables at the optimum).

fopt The optimum value of the objective function.

errnum Error code.

0 No error.

1 (GLP_EBADB)

Invalid basis.

2 (GLP_ESING)

Singular matrix.

3 (GLP_ECOND)

Ill-conditioned matrix.

4 (GLP_EBOUND)

Invalid bounds.

5 (GLP_EFAIL)

Solver failed.

6 (GLP_EOBJLL)

Objective function lower limit reached.

7 (GLP_EOBJUL)

Objective function upper limit reached.

8 (GLP_EITLIM)

Iterations limit exhausted.

9 (GLP_ETMLIM)

Time limit exhausted.

10 (GLP_ENOPFS)

No primal feasible solution.

11 (GLP_ENODFS)

No dual feasible solution.

12 (GLP_EROOT)

Root LP optimum not provided.

13 (GLP_ESTOP)

Search terminated by application.

14 (GLP_EMIPGAP)

Relative MIP gap tolerance reached.

15 (GLP_ENOFEAS)

No primal/dual feasible solution.

16 (GLP_ENOCVG)

No convergence.

17 (GLP_EINSTAB)

Numerical instability.

18 (GLP_EDATA)

Invalid data.

19 (GLP_ERANGE)

Result out of range.

extra A data structure containing the following fields: lambda Dual variables. redcosts Reduced Costs. time Time (in seconds) used for solving LP/MIP problem. status Status of the optimization.

          1 (GLP_UNDEF)

Solution status is undefined.

               2 (GLP_FEAS)

Solution is feasible.

               3 (GLP_INFEAS)

Solution is infeasible.

               4 (GLP_NOFEAS)

Problem has no feasible solution.

               5 (GLP_OPT)

Solution is optimal.

               6 (GLP_UNBND)

Problem has no unbounded solution.

Example:

c = [10, 6, 4]';
A = [ 1, 1, 1;
     10, 4, 5;
      2, 2, 6];
b = [100, 600, 300]';
lb = [0, 0, 0]';
ub = [];
ctype = "UUU";
vartype = "CCC";
s = -1;
param.msglev = 1;
param.itlim = 100;
[xmin, fmin, status, extra] = ...
   glpk (c, A, b, lb, ub, ctype, vartype, s, param);

25.2 Quadratic Programming

Octave can also solve Quadratic Programming problems, this is

min_{x} \frac{1}{2}x^{T} Hx + x^{T} q

subject to

Ax = b  lb ≤ x ≤ ub  A_{lb} ≤ A_{in} ≤ A_{ub}

[Function File] [x, obj, info, lambda] = qp (x0, H)
[Function File] [x, obj, info, lambda] = qp (x0, H, q)
[Function File] [x, obj, info, lambda] = qp (x0, H, q, A, b)
[Function File] [x, obj, info, lambda] = qp (x0, H, q, A, b, lb, ub)
[Function File] [x, obj, info, lambda] = qp (x0, H, q, A, b, lb, ub, A_lb,
A_in, A_ub)
[Function File] [x, obj, info, lambda] = qp (. . ., options)

Solve the quadratic program

min_{x} \frac{1}{2}x^{T} Hx + x^{T} q

subject to

Ax = b  lb ≤ x ≤ ub  A_{lb} ≤ A_{in} ≤ A_{ub}

using a null-space active-set method. Any bound (A, b, lb, ub, A lb, A ub) may be set to the empty matrix ([]) if not present. If the initial guess is feasible the algorithm is faster.

options An optional structure containing the following parameter(s) used to define the behavior of the solver. Missing elements in the structure take on default values, so you only need to set the elements that you wish to change from the default.

          MaxIter (default: 200)

Maximum number of iterations.

info Structure containing run-time information about the algorithm. The following fields are defined:

          solveiter

The number of iterations required to find the solution.

info An integer indicating the status of the solution.

0 The problem is feasible and convex. Global solution found.

1 The problem is not convex. Local solution found.

2 The problem is not convex and unbounded.

3 Maximum number of iterations reached.

6 The problem is infeasible.

[Function File] x = pqpnonneg (c, d) [Function File] x = pqpnonneg (c, d, x0)

[Function File] [x, minval] = pqpnonneg (. . .)
[Function File] [x, minval, exitflag] = pqpnonneg (. . .)
[Function File] [x, minval, exitflag, output] = pqpnonneg (. . .)
[Function File] [x, minval, exitflag, output, lambda] = pqpnonneg (. . .)

Minimize 1/2*x'*c*x + d'*x subject to x >= 0. c and d must be real, and c must be symmetric and positive definite. x0 is an optional initial guess for x. Outputs:

• minval The minimum attained model value, 1/2*xmin’*c*xmin + d’*xmin

• exitflag An indicator of convergence. 0 indicates that the iteration count was exceeded, and therefore convergence was not reached; >0 indicates that the algorithm converged. (The algorithm is stable and will converge given enough iterations.)

• output A structure with two fields:

• "algorithm": The algorithm used ("nnls")

• "iterations": The number of iterations taken.

• lambda Not implemented.

See also: [optimset], page 558, [lsqnonneg], page 558, [qp], page 553.

25.3 Nonlinear Programming

Octave can also perform general nonlinear minimization using a successive quadratic programming solver.

[Function File] [x, obj, info, iter, nf, lambda] = sqp (x0, phi)

[Function File] [...] = sqp (x0, phi, g) [Function File] [...] = sqp (x0, phi, g, h) [Function File] [...] = sqp (x0, phi, g, h, lb, ub)

[Function File] [...] = sqp (x0, phi, g, h, lb, ub, maxiter)
[Function File] [...] = sqp (x0, phi, g, h, lb, ub, maxiter, tol)

Solve the nonlinear program

min_{x} ϕ(x)

subject to

g(x) = 0  h(x) ≥ 0  lb ≤ x ≤ ub

using a sequential quadratic programming method.

The first argument is the initial guess for the vector x0.

The second argument is a function handle pointing to the objective function phi. The objective function must accept one vector argument and return a scalar.

The second argument may also be a 2- or 3-element cell array of function handles. The first element should point to the objective function, the second should point to a function that computes the gradient of the objective function, and the third should point to a function that computes the Hessian of the objective function. If the gradient function is not supplied, the gradient is computed by finite differences. If the Hessian function is not supplied, a BFGS update formula is used to approximate the Hessian.

When supplied, the gradient function phi{2} must accept one vector argument and return a vector. When supplied, the Hessian function phi{3} must accept one vector argument and return a matrix.

The third and fourth arguments g and h are function handles pointing to functions that compute the equality constraints and the inequality constraints, respectively. If the problem does not have equality (or inequality) constraints, then use an empty matrix ([]) for g (or h). When supplied, these equality and inequality constraint functions must accept one vector argument and return a vector.

The third and fourth arguments may also be 2-element cell arrays of function handles. The first element should point to the constraint function and the second should point to a function that computes the gradient of the constraint function:

(\frac{∂f(x)}{∂x_{1}}, \frac{∂f(x)}{∂x_{2}}, …, \frac{∂f(x)}{∂x_{N}})^{T}

The fifth and sixth arguments, lb and ub, contain lower and upper bounds on x. These must be consistent with the equality and inequality constraints g and h. If the arguments are vectors then x(i) is bound by lb(i) and ub(i). A bound can also be a scalar in which case all elements of x will share the same bound. If only one bound (lb, ub) is specified then the other will default to (-realmax, +realmax). The seventh argument maxiter specifies the maximum number of iterations. The default value is 100. The eighth argument tol specifies the tolerance for the stopping criteria. The default

value is sqrt (eps).

The value returned in info may be one of the following:

101 The algorithm terminated normally. Either all constraints meet the requested tolerance, or the stepsize, Δx, is less than tol * norm (x).

102 The BFGS update failed.

103 The maximum number of iterations was reached.

An example of calling sqp:

     function r = g (x)
       r = [ sumsq(x)-10;
             x(2)*x(3)-5*x(4)*x(5);
             x(1)^3+x(2)^3+1 ];
     endfunction
     function obj = phi (x)
       obj = exp (prod (x)) - 0.5*(x(1)^3+x(2)^3+1)^2;
     endfunction
     x0 = [-1.8; 1.7; 1.9; -0.8; -0.8];
     [x, obj, info, iter, nf, lambda] = sqp (x0, @phi, @g, [])
     x =
       -1.71714
        1.59571
        1.82725
       -0.76364
       -0.76364
     obj = 0.053950
     info = 101

     iter = 8
     nf = 10
     lambda =
       -0.0401627
        0.0379578
       -0.0052227

See also: [qp], page 553.

25.4 Linear Least Squares

Octave also supports linear least squares minimization. That is, Octave can find the parameter b such that the model y = xb fits data (x, y) as well as possible, assuming zero-mean Gaussian noise. If the noise is assumed to be isotropic the problem can be solved using the ‘\’ or ‘/’ operators, or the ols function. In the general case where the noise is assumed to be anisotropic the gls is needed.

[Function File] [beta, sigma, r] = ols (y, x)

Ordinary least squares estimation for the multivariate model y = xb + e with ē = 0, and cov(vec(e)) = kron (s, I) where y is a t × p matrix, x is a t × k matrix, b is a k × p matrix, and e is a t × p matrix. Each row of y and x is an observation and each column a variable. The return values beta, sigma, and r are defined as follows.

beta The OLS estimator for b. beta is calculated directly via (x^{T} x)^{−1}x^{T} y if the matrix x^{T} x is of full rank. Otherwise, beta = pinv (x) * y where pinv (x) denotes the pseudoinverse of x.

sigma The OLS estimator for the matrix s,

               sigma = (y-x*beta)'
                 * (y-x*beta)
                 / (t-rank(x))

r The matrix of OLS residuals, r = y - x*beta.

See also: [gls], page 557, [pinv], page 441.

[Function File] [beta, v, r] = gls (y, x, o)

Generalized least squares estimation for the multivariate model y = xb + e with ē = 0 and cov(vec(e)) = (s^{2})o, where y is a t × p matrix, x is a t × k matrix, b is a k × p matrix, e is a t × p matrix, and o is a tp × tp matrix. Each row of y and x is an observation and each column a variable. The return values beta, v, and r are defined as follows.

beta The GLS estimator for b.

v The GLS estimator for s^{2}.

r The matrix of GLS residuals, r = y − x ∗ beta.

See also: [ols], page 557.

[Function File] x = lsqnonneg (c, d) [Function File] x = lsqnonneg (c, d, x0)

[Function File] x = lsqnonneg (c, d, x0, options)
[Function File] [x, resnorm] = lsqnonneg (. . .)
[Function File] [x, resnorm, residual] = lsqnonneg (. . .)
[Function File] [x, resnorm, residual, exitflag] = lsqnonneg (. . .)
[Function File] [x, resnorm, residual, exitflag, output] = lsqnonneg

(…)

[Function File] [x, resnorm, residual, exitflag, output, lambda] =
   lsqnonneg (. . .)

Minimize norm (c*x - d) subject to x >= 0. c and d must be real. x0 is an optional initial guess for x. Currently, lsqnonneg recognizes these options: "MaxIter", "TolX". For a description of these options, see [optimset], page 558. Outputs:

• resnorm The squared 2-norm of the residual: norm (c*x-d)^2

• residual The residual: d-c*x

• exitflag An indicator of convergence. 0 indicates that the iteration count was exceeded, and therefore convergence was not reached; >0 indicates that the algorithm converged. (The algorithm is stable and will converge given enough iterations.)

• output

A structure with two fields:

• "algorithm": The algorithm used ("nnls")

• "iterations": The number of iterations taken.

• lambda Not implemented.

See also: [optimset], page 558, [pqpnonneg], page 554.

[Function File] optimset () [Function File] optimset (par, val, …) [Function File] optimset (old, par, val, …) [Function File] optimset (old, new)

Create options struct for optimization functions. Valid parameters are:

AutoScaling ComplexEqn Display Request verbose display of results from optimizations. Values are:

"off" [default] No display.

"iter" Display intermediate results for every loop iteration.

"final" Display the result of the final loop iteration.

"notify" Display the result of the final loop iteration if the function has failed to converge.

FinDiffType FunValCheck

When enabled, display an error if the objective function returns an invalid value (a complex number, NaN, or Inf). Must be set to "on" or "off" [default]. Note: the functions fzero and fminbnd correctly handle Inf values and only complex values or NaN will cause an error in this case.

GradObj When set to "on", the function to be minimized must return a second argument which is the gradient, or first derivative, of the function at the point x. If set to "off" [default], the gradient is computed via finite differences.

Jacobian When set to "on", the function to be minimized must return a second argument which is the Jacobian, or first derivative, of the function at the point x. If set to "off" [default], the Jacobian is computed via finite differences.

MaxFunEvals Maximum number of function evaluations before optimization stops. Must be a positive integer.

MaxIter Maximum number of algorithm iterations before optimization stops. Must be a positive integer.

OutputFcn A user-defined function executed once per algorithm iteration.

TolFun Termination criterion for the function output. If the difference in the calculated objective function between one algorithm iteration and the next is less than TolFun the optimization stops. Must be a positive scalar.

TolX Termination criterion for the function input. If the difference in x, the current search point, between one algorithm iteration and the next is less than TolX the optimization stops. Must be a positive scalar.

TypicalX

Updating

[Function File] optimget (options, parname)
[Function File] optimget (options, parname, default)

Return a specific option from a structure created by optimset. If parname is not a field of the options structure, return default if supplied, otherwise return an empty matrix.

26 Statistics

Octave has support for various statistical methods. This includes basic descriptive statistics, probability distributions, statistical tests, random number generation, and much more.

The functions that analyze data all assume that multi-dimensional data is arranged in a matrix where each row is an observation, and each column is a variable. Thus, the matrix defined by

a = [ 0.9, 0.7;
      0.1, 0.1;
      0.5, 0.4 ];

contains three observations from a two-dimensional distribution. While this is the default data arrangement, most functions support different arrangements.

It should be noted that the statistics functions don’t test for data containing NaN, NA, or Inf. These values need to be detected and dealt with explicitly. See [isnan], page 380, [isna], page 42, [isinf], page 380, [isfinite], page 380.

26.1 Descriptive Statistics

One principal goal of descriptive statistics is to represent the essence of a large data set concisely. Octave provides the mean, median, and mode functions which all summarize a data set with just a single number corresponding to the central tendency of the data.

[Function File] mean (x) [Function File] mean (x, dim) [Function File] mean (x, opt) [Function File] mean (x, dim, opt)

Compute the mean of the elements of the vector x.

mean(x) = x̄ = \frac{1}{N} ∑_{i=1}^{N} x_{i}

If x is a matrix, compute the mean for each column and return them in a row vector. The optional argument opt selects the type of mean to compute. The following options are recognized:

"a" Compute the (ordinary) arithmetic mean. [default]

"g" Compute the geometric mean.

"h" Compute the harmonic mean.

If the optional argument dim is given, operate along this dimension. Both dim and opt are optional. If both are supplied, either may appear first.

See also: [median], page 561, [mode], page 562.

[Function File] median (x) [Function File] median (x, dim)

Compute the median value of the elements of the vector x. If the elements of x are sorted, the median is defined as

median(x) = \{ \begin{matrix}x(⌈N/2⌉), & N odd; \\ (x(N/2) + x(N/2 + 1))/2, & N even. \end{matrix}

If x is a matrix, compute the median value for each column and return them in a row vector. If the optional dim argument is given, operate along this dimension.

See also: [mean], page 561, [mode], page 562.

[Function File] mode (x) [Function File] mode (x, dim) [Function File] [m, f, c] = mode (…)

Compute the most frequently occurring value in a dataset (mode). mode determines the frequency of values along the first non-singleton dimension and returns the value with the highest frequency. If two, or more, values have the same frequency mode returns the smallest. If the optional argument dim is given, operate along this dimension. The return variable f is the number of occurrences of the mode in in the dataset. The cell array c contains all of the elements with the maximum frequency.

See also: [mean], page 561, [median], page 561.

Using just one number, such as the mean, to represent an entire data set may not give an accurate picture of the data. One way to characterize the fit is to measure the dispersion of the data. Octave provides several functions for measuring dispersion.

[Function File] range (x) [Function File] range (x, dim)

Return the range, i.e., the difference between the maximum and the minimum of the input data. If x is a vector, the range is calculated over the elements of x. If x is a matrix, the range is calculated over each column of x. If the optional argument dim is given, operate along this dimension. The range is a quickly computed measure of the dispersion of a data set, but is less accurate than iqr if there are outlying data points.

See also: [iqr], page 562, [std], page 563.

[Function File] iqr (x) [Function File] iqr (x, dim)

Return the interquartile range, i.e., the difference between the upper and lower quartile of the input data. If x is a matrix, do the above for first non-singleton dimension of x. If the optional argument dim is given, operate along this dimension. As a measure of dispersion, the interquartile range is less affected by outliers than either range or std.

See also: [range], page 562, [std], page 563.

[Function File] meansq (x) [Function File] meansq (x, dim)

Compute the mean square of the elements of the vector x.

meansq(x) = \frac{∑_{i=1}^{N} x_{i}^{2}}{N}

where x̄ is the mean value of x. For matrix arguments, return a row vector containing the mean square of each column. If the optional argument dim is given, operate along this dimension.

See also: [var], page 563, [std], page 563, [moment], page 565.

[Function File] std (x) [Function File] std (x, opt) [Function File] std (x, opt, dim)

Compute the standard deviation of the elements of the vector x.

std(x) = σ = \sqrt{\frac{∑_{i=1}^{N}(x_{i} − x̄)^{2}}{N − 1}}

where x̄ is the mean value of x and N is the number of elements. If x is a matrix, compute the standard deviation for each column and return them in a row vector. The argument opt determines the type of normalization to use. Valid values are

0: normalize with N − 1, provides the square root of the best unbiased estimator of the variance [default]

1: normalize with N, this provides the square root of the second moment around the mean

If the optional argument dim is given, operate along this dimension.

See also: [var], page 563, [range], page 562, [iqr], page 562, [mean], page 561, [median], page 561.

In addition to knowing the size of a dispersion it is useful to know the shape of the data set. For example, are data points massed to the left or right of the mean? Octave provides several common measures to describe the shape of the data set. Octave can also calculate moments allowing arbitrary shape measures to be developed.

[Function File] var (x) [Function File] var (x, opt) [Function File] var (x, opt, dim)

Compute the variance of the elements of the vector x.

var(x) = σ^{2} = \frac{∑_{i=1}^{N}(x_{i} − x̄)^{2}}{N − 1}

where x̄ is the mean value of x. If x is a matrix, compute the variance for each column and return them in a row vector. The argument opt determines the type of normalization to use. Valid values are

0: normalize with N −1, provides the best unbiased estimator of the variance [default]

1: normalizes with N, this provides the second moment around the mean If N == 1 the value of opt is ignored and normalization by N is used. If the optional argument dim is given, operate along this dimension.

See also: [cov], page 571, [std], page 563, [skewness], page 564, [kurtosis], page 564, [moment], page 565.

[Function File] skewness (x) [Function File] skewness (x, flag) [Function File] skewness (x, flag, dim)

Compute the sample skewness of the elements of x:

skewness(x) = \frac{\frac{1}{N} ∑_{i=1}^{N}(x_{i} − x̄)^{3}}{σ^{3}} ,

where N is the length of x, x̄ its mean and σ its (uncorrected) standard deviation. The optional argument flag controls which normalization is used. If flag is equal to 1 (default value, used when flag is omitted or empty), return the sample skewness as defined above. If flag is equal to 0, return the adjusted skewness coefficient instead:

skewness(x) = \frac{\sqrt{N(N − 1)}}{N − 2} × \frac{\frac{1}{N} ∑_{i=1}^{N}(x_{i} − x̄)^{3}}{σ^{3}}

The adjusted skewness coefficient is obtained by replacing the sample second and third central moments by their bias-corrected versions. If x is a matrix, or more generally a multi-dimensional array, return the skewness along the first non-singleton dimension. If the optional dim argument is given, operate along this dimension.

See also: [var], page 563, [kurtosis], page 564, [moment], page 565.

[Function File] kurtosis (x) [Function File] kurtosis (x, flag) [Function File] kurtosis (x, flag, dim)

Compute the sample kurtosis of the elements of x:

κ_{1} = \frac{\frac{1}{N} ∑_{i=1}^{N}(x_{i} − x̄)^{4}}{σ^{4}} ,

where N is the length of x, x̄ its mean, and σ its (uncorrected) standard deviation. The optional argument flag controls which normalization is used. If flag is equal to 1 (default value, used when flag is omitted or empty), return the sample kurtosis as defined above. If flag is equal to 0, return the "bias-corrected" kurtosis coefficient instead:

κ_{0} = 3 + \frac{N−1}{(N−2)(N−3)} ((N + 1) κ_{1} − 3(N − 1))

The bias-corrected kurtosis coefficient is obtained by replacing the sample second and fourth central moments by their unbiased versions. It is an unbiased estimate of the population kurtosis for normal populations. If x is a matrix, or more generally a multi-dimensional array, return the kurtosis along the first non-singleton dimension. If the optional dim argument is given, operate along this dimension.

See also: [var], page 563, [skewness], page 564, [moment], page 565.

[Function File] moment (x, p) [Function File] moment (x, p, type) [Function File] moment (x, p, dim) [Function File] moment (x, p, type, dim) [Function File] moment (x, p, dim, type)

Compute the p-th central moment of the vector x.

\frac{∑_{i=1}^{N}(x_{i} − x̄)^{p}}{N}

If x is a matrix, return the row vector containing the p-th central moment of each column. The optional string type specifies the type of moment to be computed. Valid options are:

"c" Central Moment (default).

"a"

"ac" Absolute Central Moment. The moment about the mean ignoring sign defined as

\frac{∑_{i=1}^{N} |x_{i} − x̄|^{p}}{N}

"r" Raw Moment. The moment about zero defined as

moment(x) = \frac{∑_{i=1}^{N} x_{i}^{p}}{N}

"ar" Absolute Raw Moment. The moment about zero ignoring sign defined as

\frac{∑_{i=1}^{N} |x_{i}|^{p}}{N}

If the optional argument dim is given, operate along this dimension. If both type and dim are given they may appear in any order.

See also: [var], page 563, [skewness], page 564, [kurtosis], page 564.

[Function File] q = quantile (x) [Function File] q = quantile (x, p) [Function File] q = quantile (x, p, dim)

[Function File] q = quantile (x, p, dim, method)

For a sample, x, calculate the quantiles, q, corresponding to the cumulative probability values in p. All non-numeric values (NaNs) of x are ignored. If x is a matrix, compute the quantiles for each column and return them in a matrix, such that the i-th row of q contains the p(i)th quantiles of each column of x. If p is unspecified, return the quantiles for [0.00 0.25 0.50 0.75 1.00]. The optional argument dim determines the dimension along which the quantiles are calculated. If dim is omitted, and x is a vector or matrix, it defaults to 1 (column-wise quantiles). If x is an N-D array, dim defaults to the first non-singleton dimension.

The methods available to calculate sample quantiles are the nine methods used by R (http://www.r-project.org/). The default value is METHOD = 5. Discontinuous sample quantile methods 1, 2, and 3

1. Method 1: Inverse of empirical distribution function.

2. Method 2: Similar to method 1 but with averaging at discontinuities.

3. Method 3: SAS definition: nearest even order statistic.

Continuous sample quantile methods 4 through 9, where p(k) is the linear interpolation function respecting each methods’ representative cdf.

4. Method 4: p(k) = k / n. That is, linear interpolation of the empirical cdf.

5. Method 5: p(k) = (k - 0.5) / n. That is a piecewise linear function where the knots are the values midway through the steps of the empirical cdf.

6. Method 6: p(k) = k / (n + 1).

7. Method 7: p(k) = (k - 1) / (n - 1).

8. Method 8: p(k) = (k - 1/3) / (n + 1/3). The resulting quantile estimates are approximately median-unbiased regardless of the distribution of x.

9. Method 9: p(k) = (k - 3/8) / (n + 1/4). The resulting quantile estimates are approximately unbiased for the expected order statistics if x is normally distributed.

Hyndman and Fan (1996) recommend method 8. Maxima, S, and R (versions prior to 2.0.0) use 7 as their default. Minitab and SPSS use method 6. matlab uses method 5. References:

• Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

• Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages, American Statistician, 50, 361–365.

• R: A Language and Environment for Statistical Computing;

    http://cran.r-project.org/doc/manuals/fullrefman.pdf.

Examples:

      x = randi (1000, [10, 1]);  # Create empirical data in range 1-1000
      q = quantile (x, [0, 1]);   # Return minimum, maximum of distribution
      q = quantile (x, [0.25 0.5 0.75]); # Return quartiles of distribution

See also: [prctile], page 566.

[Function File] q = prctile (x) [Function File] q = prctile (x, p) [Function File] q = prctile (x, p, dim)

For a sample x, compute the quantiles, q, corresponding to the cumulative probability values, p, in percent. All non-numeric values (NaNs) of x are ignored. If x is a matrix, compute the percentiles for each column and return them in a matrix, such that the i-th row of y contains the p(i)th percentiles of each column of x. If p is unspecified, return the quantiles for [0 25 50 75 100]. The optional argument dim determines the dimension along which the percentiles are calculated. If dim is omitted, and x is a vector or matrix, it defaults to 1 (column-wise quantiles). When x is an N-D array, dim defaults to the first non-singleton dimension.

See also: [quantile], page 565.

A summary view of a data set can be generated quickly with the statistics function.

[Function File] statistics (x) [Function File] statistics (x, dim)

Return a vector with the minimum, first quartile, median, third quartile, maximum, mean, standard deviation, skewness, and kurtosis of the elements of the vector x.

If x is a matrix, calculate statistics over the first non-singleton dimension. If the optional argument dim is given, operate along this dimension.

See also: [min], page 416, [max], page 416, [median], page 561, [mean], page 561, [std], page 563, [skewness], page 564, [kurtosis], page 564.

26.2 Basic Statistical Functions

Octave supports various helpful statistical functions. Many are useful as initial steps to prepare a data set for further analysis. Others provide different measures from those of the basic descriptive statistics.

[Function File] center (x) [Function File] center (x, dim)

If x is a vector, subtract its mean. If x is a matrix, do the above for each column. If the optional argument dim is given, operate along this dimension.

See also: [zscore], page 567.

[Function File] [z, mu, sigma] = zscore (x)
[Function File] [z, mu, sigma] = zscore (x, opt)
[Function File] [z, mu, sigma] = zscore (x, opt, dim)

If x is a vector, subtract its mean and divide by its standard deviation. If the standard deviation is zero, divide by 1 instead. The optional parameter opt determines the normalization to use when computing the standard deviation and is the same as the corresponding parameter for std.

If x is a matrix, do the above along the first non-singleton dimension. If the third optional argument dim is given, operate along this dimension.

The mean and standard deviation along dim are given in mu and sigma respectively.

See also: [mean], page 561, [std], page 563, [center], page 567.

[Function File] n = histc (x, edges) [Function File] n = histc (x, edges, dim) [Function File] [n, idx] = histc (…)

Produce histogram counts.

When x is a vector, the function counts the number of elements of x that fall in the histogram bins defined by edges. This must be a vector of monotonically increasing values that define the edges of the histogram bins. n(k) contains the number of elements in x for which edges(k) <= x < edges(k+1). The final element of n contains the number of elements of x exactly equal to the last element of edges. When x is an N-dimensional array, the computation is carried out along dimension dim. If not specified dim defaults to the first non-singleton dimension. When a second output argument is requested an index matrix is also returned. The idx matrix has the same size as x. Each element of idx contains the index of the histogram bin in which the corresponding element of x was counted.

See also: [hist], page 266.

[Function File] c = nchoosek (n, k) [Function File] c = nchoosek (set, k)

Compute the binomial coefficient or all combinations of a set of items. If n is a scalar then calculate the binomial coefficient of n and k which is defined as

\binom{n}{k} = \frac{n(n − 1)(n − 2) ⋯ (n − k + 1)}{k!} = \frac{n!}{k!(n − k)!}

This is the number of combinations of n items taken in groups of size k. If the first argument is a vector, set, then generate all combinations of the elements of set, taken k at a time, with one row per combination. The result c has k columns

and nchoosek (length (set), k) rows.

For example: How many ways can three items be grouped into pairs?

     nchoosek (3, 2)
        ⇒ 3

What are the possible pairs?

     nchoosek (1:3, 2)
        ⇒ 1   2
            1   3
            2   3

nchoosek works only for non-negative, integer arguments. Use bincoeff for noninteger and negative scalar arguments, or for computing many binomial coefficients at once with vector inputs for n or k.

See also: [bincoeff], page 424, [perms], page 568.

[Function File] perms (v)

Generate all permutations of v, one row per permutation. The result has size factorial (n) * n, where n is the length of v. As an example, perms ([1, 2, 3]) returns the matrix

       1   2   3
       2   1   3
       1   3   2
       2   3   1
       3   1   2
       3   2   1

[Function File] ranks (x, dim)

Return the ranks of x along the first non-singleton dimension adjusted for ties. If the optional argument dim is given, operate along this dimension.

See also: [spearman], page 571, [kendall], page 572.

[Function File] run_count (x, n) [Function File] run_count (x, n, dim)

Count the upward runs along the first non-singleton dimension of x of length 1, 2, …, n-1 and greater than or equal to n.

If the optional argument dim is given then operate along this dimension.

[Function File] [count, value] = runlength (x)

Find the lengths of all sequences of common values. Return the vector of lengths and the value that was repeated.

     runlength ([2, 2, 0, 4, 4, 4, 0, 1, 1, 1, 1])
     ⇒ [2, 1, 3, 1, 4]

[Function File] probit (p)

For each component of p, return the probit (the quantile of the standard normal distribution) of p.

[Function File] logit (p)

For each component of p, return the logit of p defined as

logit(p) = log (\frac{p}{1 − p})

See also: [logistic cdf], page 577.

[Function File] cloglog (x)

Return the complementary log-log function of x, defined as

cloglog(x) = − log(− log(x))

[Function File] mahalanobis (x, y)

Return the Mahalanobis’ D-square distance between the multivariate samples x and y, which must have the same number of components (columns), but may have a different number of observations (rows).

[Function File] [t, l_x] = table (x)

[Function File] [t, l_x, l_y] = table (x, y)

Create a contingency table t from data vectors. The l x and l y vectors are the corresponding levels.

Currently, only 1- and 2-dimensional tables are supported.

26.3 Statistical Plots

Octave can create Quantile Plots (QQ-Plots), and Probability Plots (PP-Plots). These are simple graphical tests for determining if a data set comes from a certain distribution.

Note that Octave can also show histograms of data using the hist function as described in Section 15.2.1 [Two-Dimensional Plots], page 259.

[Function File] [q, s] = qqplot (x) [Function File] [q, s] = qqplot (x, y)

[Function File] [q, s] = qqplot (x, dist)
[Function File] [q, s] = qqplot (x, y, params)

[Function File] qqplot (…)

Perform a QQ-plot (quantile plot).

If F is the CDF of the distribution dist with parameters params and G its inverse, and x a sample vector of length n, the QQ-plot graphs ordinate s(i) = i-th largest element of x versus abscissa q(if) = G((i - 0.5)/n).

If the sample comes from F, except for a transformation of location and scale, the pairs will approximately follow a straight line.

If the second argument is a vector y the empirical CDF of y is used as dist.

The default for dist is the standard normal distribution. The optional argument params contains a list of parameters of dist. For example, for a quantile plot of the uniform distribution on [2,4] and x, use

     qqplot (x, "unif", 2, 4)

dist can be any string for which a function distinv or dist inv exists that calculates the inverse CDF of distribution dist.

If no output arguments are given, the data are plotted directly.

[Function File] [p, y] = ppplot (x, dist, params)

Perform a PP-plot (probability plot).

If F is the CDF of the distribution dist with parameters params and x a sample vector of length n, the PP-plot graphs ordinate y(i) = F (i-th largest element of x) versus abscissa p(i) = (i - 0.5)/n. If the sample comes from F, the pairs will approximately follow a straight line.

The default for dist is the standard normal distribution. The optional argument params contains a list of parameters of dist. For example, for a probability plot of the uniform distribution on [2,4] and x, use

     ppplot (x, "uniform", 2, 4)

dist can be any string for which a function dist cdf that calculates the CDF of distribution dist exists.

If no output arguments are given, the data are plotted directly.

26.4 Correlation and Regression Analysis

[Function File] cov (x) [Function File] cov (x, opt) [Function File] cov (x, y) [Function File] cov (x, y, opt)

Compute the covariance matrix. If each row of x and y is an observation, and each column is a variable, then the (i, j)-th entry of cov (x, y) is the covariance between the i-th variable in x and the j-th variable in y.

σ_{ij} = \frac{1}{N − 1} ∑_{i=1}^{N}(x_{i} − x̄)(y_{i} − ȳ)

where x̄ and ȳ are the mean values of x and y. If called with one argument, compute cov (x, x), the covariance between the columns of x. The argument opt determines the type of normalization to use. Valid values are

0: normalize with N − 1, provides the best unbiased estimator of the covariance [default]

1: normalize with N, this provides the second moment around the mean

matlab compatibility: Octave always computes the covariance matrix. For two inputs, however, matlab will calculate cov (x(:), y(:)) whenever the number of elements in x and y are equal. This will result in a scalar rather than a matrix output. Code relying on this odd definition will need to be changed when running in Octave.

See also: [corr], page 571.

[Function File] corr (x) [Function File] corr (x, y)

Compute matrix of correlation coefficients. If each row of x and y is an observation and each column is a variable, then the (i, j)-th entry of corr (x, y) is the correlation between the i-th variable in x and the j-th variable in y.

corr(x, y) = \frac{cov(x, y)}{std(x)std(y)}

If called with one argument, compute corr (x, x), the correlation between the columns of x.

See also: [cov], page 571.

[Function File] spearman (x) [Function File] spearman (x, y)

Compute Spearman’s rank correlation coefficient rho. For two data vectors x and y, Spearman’s rho is the correlation coefficient of the ranks of x and y.

If x and y are drawn from independent distributions, rho has zero mean and variance 1 / (n - 1), and is asymptotically normally distributed.

spearman (x) is equivalent to spearman (x, x).

See also: [ranks], page 569, [kendall], page 572.

[Function File] kendall (x) [Function File] kendall (x, y)

Compute Kendall’s tau. For two data vectors x, y of common length n, Kendall’s tau is the correlation of the signs of all rank differences of x and y; i.e., if both x and y have distinct entries, then

τ = \frac{1}{n(n − 1)} ∑_{i,j} sign(q_{i} − q_{j})sign(r_{i} − r_{j})

in which the q_{i} and r_{i} are the ranks of x and y, respectively. If x and y are drawn from independent distributions, Kendall’s tau is asymptotically normal with mean 0 and variance \frac{2(2n+5)}{9n(n−1)}.

kendall (x) is equivalent to kendall (x, x).

See also: [ranks], page 569, [spearman], page 571.

[Function File] [theta, beta, dev, dl, d2l, p] = logistic_regression
   (y, x, print, theta, beta)

Perform ordinal logistic regression. Suppose y takes values in k ordered categories, and let gamma_i (x) be the cumulative probability that y falls in one of the first i categories given the covariate x. Then

     [theta, beta] = logistic_regression (y, x)

fits the model

     logit (gamma_i (x)) = theta_i - beta' * x,   i = 1 ... k-1

The number of ordinal categories, k, is taken to be the number of distinct values of round (y). If k equals 2, y is binary and the model is ordinary logistic regression. The matrix x is assumed to have full column rank. Given y only, theta = logistic_regression (y) fits the model with baseline logit odds only. The full form is

     [theta, beta, dev, dl, d2l, gamma]
        = logistic_regression (y, x, print, theta, beta)

in which all output arguments and all input arguments except y are optional. Setting print to 1 requests summary information about the fitted model to be displayed. Setting print to 2 requests information about convergence at each iteration. Other values request no information to be displayed. The input arguments theta and beta give initial estimates for theta and beta. The returned value dev holds minus twice the log-likelihood. The returned values dl and d2l are the vector of first and the matrix of second derivatives of the log-likelihood with respect to theta and beta. p holds estimates for the conditional distribution of y given x.

26.5 Distributions

Octave has functions for computing the Probability Density Function (PDF), the Cumulative Distribution function (CDF), and the quantile (the inverse of the CDF) for a large number of distributions.

The following table summarizes the supported distributions (in alphabetical order).

Distribution  PDF  CDF  Quantile
Beta  betapdf  betacdf  betainv
Binomial  binopdf  binocdf  binoinv
Cauchy  cauchy pdf  cauchy cdf  cauchy inv
Chi-Square  chi2pdf  chi2cdf  chi2inv
Univariate Discrete  discrete pdf  discrete cdf  discrete inv
Empirical  empirical pdf empirical cdf empirical inv
Exponential  exppdf  expcdf  expinv
F  fpdf  fcdf  finv
Gamma  gampdf  gamcdf  gaminv
Geometric  geopdf  geocdf  geoinv
Hypergeometric  hygepdf  hygecdf  hygeinv
Kolmogorov Smirnov Not Available kolmogorov  Not Available

smirnov cdf

Laplace  laplace pdf  laplace cdf  laplace inv
Logistic  logistic pdf  logistic cdf  logistic inv
Log-Normal  lognpdf  logncdf  logninv
Univariate Normal  normpdf  normcdf  norminv
Pascal  nbinpdf  nbincdf  nbininv
Poisson  poisspdf  poisscdf  poissinv
Standard Normal  stdnormal pdf stdnormal cdf stdnormal inv
t (Student)  tpdf  tcdf  tinv
Uniform Discrete  unidpdf  unidcdf  unidinv
Uniform  unifpdf  unifcdf  unifinv
Weibull  wblpdf  wblcdf  wblinv

[Function File] betapdf (x, a, b)

For each element of x, compute the probability density function (PDF) at x of the Beta distribution with parameters a and b.

[Function File] betacdf (x, a, b)

For each element of x, compute the cumulative distribution function (CDF) at x of the Beta distribution with parameters a and b.

[Function File] betainv (x, a, b)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Beta distribution with parameters a and b.

[Function File] binopdf (x, n, p)

For each element of x, compute the probability density function (PDF) at x of the binomial distribution with parameters n and p, where n is the number of trials and p is the probability of success.

[Function File] binocdf (x, n, p)

For each element of x, compute the cumulative distribution function (CDF) at x of the binomial distribution with parameters n and p, where n is the number of trials and p is the probability of success.

[Function File] binoinv (x, n, p)

For each element of x, compute the quantile (the inverse of the CDF) at x of the binomial distribution with parameters n and p, where n is the number of trials and p is the probability of success.

[Function File] cauchy_pdf (x)

[Function File] cauchy_pdf (x, location, scale)

For each element of x, compute the probability density function (PDF) at x of the Cauchy distribution with location parameter location and scale parameter scale > 0. Default values are location = 0, scale = 1.

[Function File] cauchy_cdf (x)

[Function File] cauchy_cdf (x, location, scale)

For each element of x, compute the cumulative distribution function (CDF) at x of the Cauchy distribution with location parameter location and scale parameter scale. Default values are location = 0, scale = 1.

[Function File] cauchy_inv (x)

[Function File] cauchy_inv (x, location, scale)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Cauchy distribution with location parameter location and scale parameter scale. Default values are location = 0, scale = 1.

[Function File] chi2pdf (x, n)

For each element of x, compute the probability density function (PDF) at x of the chi-square distribution with n degrees of freedom.

[Function File] chi2cdf (x, n)

For each element of x, compute the cumulative distribution function (CDF) at x of the chi-square distribution with n degrees of freedom.

[Function File] chi2inv (x, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the chi-square distribution with n degrees of freedom.

[Function File] discrete_pdf (x, v, p)

For each element of x, compute the probability density function (PDF) at x of a univariate discrete distribution which assumes the values in v with probabilities p.

[Function File] discrete_cdf (x, v, p)

For each element of x, compute the cumulative distribution function (CDF) at x of a univariate discrete distribution which assumes the values in v with probabilities p.

[Function File] discrete_inv (x, v, p)

For each element of x, compute the quantile (the inverse of the CDF) at x of the univariate distribution which assumes the values in v with probabilities p.

[Function File] empirical_pdf (x, data)

For each element of x, compute the probability density function (PDF) at x of the empirical distribution obtained from the univariate sample data.

[Function File] empirical_cdf (x, data)

For each element of x, compute the cumulative distribution function (CDF) at x of the empirical distribution obtained from the univariate sample data.

[Function File] empirical_inv (x, data)

For each element of x, compute the quantile (the inverse of the CDF) at x of the empirical distribution obtained from the univariate sample data.

[Function File] exppdf (x, lambda)

For each element of x, compute the probability density function (PDF) at x of the exponential distribution with mean lambda.

[Function File] expcdf (x, lambda)

For each element of x, compute the cumulative distribution function (CDF) at x of the exponential distribution with mean lambda. The arguments can be of common size or scalars.

[Function File] expinv (x, lambda)

For each element of x, compute the quantile (the inverse of the CDF) at x of the exponential distribution with mean lambda.

[Function File] fpdf (x, m, n)

For each element of x, compute the probability density function (PDF) at x of the F distribution with m and n degrees of freedom.

[Function File] fcdf (x, m, n)

For each element of x, compute the cumulative distribution function (CDF) at x of the F distribution with m and n degrees of freedom.

[Function File] finv (x, m, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the F distribution with m and n degrees of freedom.

[Function File] gampdf (x, a, b)

For each element of x, return the probability density function (PDF) at x of the Gamma distribution with shape parameter a and scale b.

[Function File] gamcdf (x, a, b)

For each element of x, compute the cumulative distribution function (CDF) at x of the Gamma distribution with shape parameter a and scale b.

[Function File] gaminv (x, a, b)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Gamma distribution with shape parameter a and scale b.

[Function File] geopdf (x, p)

For each element of x, compute the probability density function (PDF) at x of the geometric distribution with parameter p.

The geometric distribution models the number of failures (x-1) of a Bernoulli trial with probability p before the first success (x).

[Function File] geocdf (x, p)

For each element of x, compute the cumulative distribution function (CDF) at x of the geometric distribution with parameter p.

The geometric distribution models the number of failures (x-1) of a Bernoulli trial with probability p before the first success (x).

[Function File] geoinv (x, p)

For each element of x, compute the quantile (the inverse of the CDF) at x of the geometric distribution with parameter p.

The geometric distribution models the number of failures (x-1) of a Bernoulli trial with probability p before the first success (x).

[Function File] hygepdf (x, t, m, n)

Compute the probability density function (PDF) at x of the hypergeometric distribution with parameters t, m, and n. This is the probability of obtaining x marked items when randomly drawing a sample of size n without replacement from a population of total size t containing m marked items.

The parameters t, m, and n must be positive integers with m and n not greater than t.

[Function File] hygecdf (x, t, m, n)

Compute the cumulative distribution function (CDF) at x of the hypergeometric distribution with parameters t, m, and n. This is the probability of obtaining not more than x marked items when randomly drawing a sample of size n without replacement from a population of total size t containing m marked items.

The parameters t, m, and n must be positive integers with m and n not greater than t.

[Function File] hygeinv (x, t, m, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the hypergeometric distribution with parameters t, m, and n. This is the probability of obtaining x marked items when randomly drawing a sample of size n without replacement from a population of total size t containing m marked items.

The parameters t, m, and n must be positive integers with m and n not greater than t.

[Function File] kolmogorov_smirnov_cdf (x, tol)

Return the cumulative distribution function (CDF) at x of the Kolmogorov-Smirnov distribution,

Q(x) = ∑_{k=−∞}^{∞}(−1)^{k} exp(−2k^{2}x^{2})

for x > 0. The optional parameter tol specifies the precision up to which the series should be evaluated; the default is tol = eps.

[Function File] laplace_pdf (x)

For each element of x, compute the probability density function (PDF) at x of the Laplace distribution.

[Function File] laplace_cdf (x)

For each element of x, compute the cumulative distribution function (CDF) at x of the Laplace distribution.

[Function File] laplace_inv (x)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Laplace distribution.

[Function File] logistic_pdf (x)

For each element of x, compute the PDF at x of the logistic distribution.

[Function File] logistic_cdf (x)

For each element of x, compute the cumulative distribution function (CDF) at x of the logistic distribution.

[Function File] logistic_inv (x)

For each element of x, compute the quantile (the inverse of the CDF) at x of the logistic distribution.

[Function File] lognpdf (x) [Function File] lognpdf (x, mu, sigma)

For each element of x, compute the probability density function (PDF) at x of the lognormal distribution with parameters mu and sigma. If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma. Default values are mu = 0, sigma = 1.

[Function File] logncdf (x) [Function File] logncdf (x, mu, sigma)

For each element of x, compute the cumulative distribution function (CDF) at x of the lognormal distribution with parameters mu and sigma. If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma. Default values are mu = 0, sigma = 1.

[Function File] logninv (x) [Function File] logninv (x, mu, sigma)

For each element of x, compute the quantile (the inverse of the CDF) at x of the lognormal distribution with parameters mu and sigma. If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma. Default values are mu = 0, sigma = 1.

[Function File] nbinpdf (x, n, p)

For each element of x, compute the probability density function (PDF) at x of the negative binomial distribution with parameters n and p. When n is integer this is the Pascal distribution. When n is extended to real numbers this is the Polya distribution. The number of failures in a Bernoulli experiment with success probability p before the n-th success follows this distribution.

[Function File] nbincdf (x, n, p)

For each element of x, compute the cumulative distribution function (CDF) at x of the negative binomial distribution with parameters n and p. When n is integer this is the Pascal distribution. When n is extended to real numbers this is the Polya distribution. The number of failures in a Bernoulli experiment with success probability p before the n-th success follows this distribution.

[Function File] nbininv (x, n, p)

For each element of x, compute the quantile (the inverse of the CDF) at x of the negative binomial distribution with parameters n and p. When n is integer this is the Pascal distribution. When n is extended to real numbers this is the Polya distribution. The number of failures in a Bernoulli experiment with success probability p before the n-th success follows this distribution.

[Function File] normpdf (x) [Function File] normpdf (x, mu, sigma)

For each element of x, compute the probability density function (PDF) at x of the normal distribution with mean mu and standard deviation sigma. Default values are mu = 0, sigma = 1.

[Function File] normcdf (x) [Function File] normcdf (x, mu, sigma)

For each element of x, compute the cumulative distribution function (CDF) at x of the normal distribution with mean mu and standard deviation sigma. Default values are mu = 0, sigma = 1.

[Function File] norminv (x) [Function File] norminv (x, mu, sigma)

For each element of x, compute the quantile (the inverse of the CDF) at x of the normal distribution with mean mu and standard deviation sigma.

Default values are mu = 0, sigma = 1.

[Function File] poisspdf (x, lambda)

For each element of x, compute the probability density function (PDF) at x of the Poisson distribution with parameter lambda.

[Function File] poisscdf (x, lambda)

For each element of x, compute the cumulative distribution function (CDF) at x of the Poisson distribution with parameter lambda.

[Function File] poissinv (x, lambda)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Poisson distribution with parameter lambda.

[Function File] stdnormal_pdf (x)

For each element of x, compute the probability density function (PDF) at x of the standard normal distribution (mean = 0, standard deviation = 1).

[Function File] stdnormal_cdf (x)

For each element of x, compute the cumulative distribution function (CDF) at x of the standard normal distribution (mean = 0, standard deviation = 1).

[Function File] stdnormal_inv (x)

For each element of x, compute the quantile (the inverse of the CDF) at x of the standard normal distribution (mean = 0, standard deviation = 1).

[Function File] tpdf (x, n)

For each element of x, compute the probability density function (PDF) at x of the t (Student) distribution with n degrees of freedom.

[Function File] tcdf (x, n)

For each element of x, compute the cumulative distribution function (CDF) at x of the t (Student) distribution with n degrees of freedom, i.e., PROB (t(n) ≤ x).

[Function File] tinv (x, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the t (Student) distribution with n degrees of freedom. This function is analogous to looking in a table for the t-value of a single-tailed distribution.

[Function File] unidpdf (x, n)

For each element of x, compute the probability density function (PDF) at x of a discrete uniform distribution which assumes the integer values 1–n with equal probability. Warning: The underlying implementation uses the double class and will only be accurate for n ≤ bitmax (2^{53} − 1 on IEEE-754 compatible systems).

[Function File] unidcdf (x, n)

For each element of x, compute the cumulative distribution function (CDF) at x of a discrete uniform distribution which assumes the integer values 1–n with equal probability.

[Function File] unidinv (x, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the discrete uniform distribution which assumes the integer values 1–n with equal probability.

[Function File] unifpdf (x) [Function File] unifpdf (x, a, b)

For each element of x, compute the probability density function (PDF) at x of the uniform distribution on the interval [a, b]. Default values are a = 0, b = 1.

[Function File] unifcdf (x) [Function File] unifcdf (x, a, b)

For each element of x, compute the cumulative distribution function (CDF) at x of the uniform distribution on the interval [a, b]. Default values are a = 0, b = 1.

[Function File] unifinv (x) [Function File] unifinv (x, a, b)

For each element of x, compute the quantile (the inverse of the CDF) at x of the uniform distribution on the interval [a, b]. Default values are a = 0, b = 1.

[Function File] wblpdf (x) [Function File] wblpdf (x, scale) [Function File] wblpdf (x, scale, shape)

Compute the probability density function (PDF) at x of the Weibull distribution with scale parameter scale and shape parameter shape which is given by

\frac{shape}{scale^{shape}} · x^{shape−1} · e^{−( \frac{x}{scale})^{shape}}

for x ≥ 0. Default values are scale = 1, shape = 1.

[Function File] wblcdf (x) [Function File] wblcdf (x, scale) [Function File] wblcdf (x, scale, shape)

Compute the cumulative distribution function (CDF) at x of the Weibull distribution with scale parameter scale and shape parameter shape, which is

1 − e^{−( \frac{x}{scale})^{shape}}

for x ≥ 0.

[Function File] wblinv (x) [Function File] wblinv (x, scale) [Function File] wblinv (x, scale, shape)

Compute the quantile (the inverse of the CDF) at x of the Weibull distribution with scale parameter scale and shape parameter shape. Default values are scale = 1, shape = 1.

26.6 Tests

Octave can perform many different statistical tests. The following table summarizes the available tests.

Hypothesis  Test Functions
Equal mean values  anova, hotelling test2, t test 2,

welch test, wilcoxon test, z test 2

Equal medians  kruskal wallis test, sign test
Equal variances  bartlett test, manova, var test
Equal distributions  chisquare test homogeneity,

kolmogorov smirnov test 2, u test

Equal marginal frequencies  mcnemar test
Equal success probabilities  prop test 2
Independent observations  chisquare test independence,

run test

Uncorrelated observations  cor test
Given mean value  hotelling test, t test, z test
Observations from distribution kolmogorov smirnov test
Regression  f test regression, t test regression

The tests return a p-value that describes the outcome of the test. Assuming that the test hypothesis is true, the p-value is the probability of obtaining a worse result than the observed one. So large p-values corresponds to a successful test. Usually a test hypothesis is accepted if the p-value exceeds 0.05.

[Function File] [pval, f, df_b, df_w] = anova (y, g)

Perform a one-way analysis of variance (ANOVA). The goal is to test whether the population means of data taken from k different groups are all equal. Data may be given in a single vector y with groups specified by a corresponding vector of group labels g (e.g., numbers from 1 to k). This is the general form which does not impose any restriction on the number of data in each group or the group labels. If y is a matrix and g is omitted, each column of y is treated as a group. This form is only appropriate for balanced ANOVA in which the numbers of samples from each group are all equal. Under the null of constant means, the statistic f follows an F distribution with df b and df w degrees of freedom. The p-value (1 minus the CDF of this distribution at f ) is returned in pval. If no output argument is given, the standard one-way ANOVA table is printed.

[Function File] [pval, chisq, df] = bartlett_test (x1, . . .)

Perform a Bartlett test for the homogeneity of variances in the data vectors x1, x2, …, xk, where k > 1. Under the null of equal variances, the test statistic chisq approximately follows a chi-square distribution with df degrees of freedom. The p-value (1 minus the CDF of this distribution at chisq) is returned in pval.

If no output argument is given, the p-value is displayed.

[Function File] [pval, chisq, df] = chisquare_test_homogeneity (x, y, c)

Given two samples x and y, perform a chisquare test for homogeneity of the null hypothesis that x and y come from the same distribution, based on the partition induced by the (strictly increasing) entries of c.

For large samples, the test statistic chisq approximately follows a chisquare distribution with df = length (c) degrees of freedom.

The p-value (1 minus the CDF of this distribution at chisq) is returned in pval.

If no output argument is given, the p-value is displayed.

[Function File] [pval, chisq, df] = chisquare_test_independence (x)

Perform a chi-square test for independence based on the contingency table x. Under the null hypothesis of independence, chisq approximately has a chi-square distribution with df degrees of freedom.

The p-value (1 minus the CDF of this distribution at chisq) of the test is returned in pval.

If no output argument is given, the p-value is displayed.

[Function File] cor_test (x, y, alt, method)

Test whether two samples x and y come from uncorrelated populations.

The optional argument string alt describes the alternative hypothesis, and can be "!=" or "<>" (non-zero), ">" (greater than 0), or "<" (less than 0). The default is the two-sided case.

The optional argument string method specifies which correlation coefficient to use for testing. If method is "pearson" (default), the (usual) Pearson’s product moment correlation coefficient is used. In this case, the data should come from a bivariate normal distribution. Otherwise, the other two methods offer nonparametric alternatives. If method is "kendall", then Kendall’s rank correlation tau is used. If method is "spearman", then Spearman’s rank correlation rho is used. Only the first character is necessary.

The output is a structure with the following elements:

pval The p-value of the test.

stat The value of the test statistic.

dist The distribution of the test statistic.

params The parameters of the null distribution of the test statistic.

alternative The alternative hypothesis.

method The method used for testing.

If no output argument is given, the p-value is displayed.

[Function File] [pval, f, df_num, df_den] = f_test_regression (y, x, rr,
   r)

Perform an F test for the null hypothesis rr * b = r in a classical normal regression model y = X * b + e. Under the null, the test statistic f follows an F distribution with df num and df den degrees of freedom. The p-value (1 minus the CDF of this distribution at f ) is returned in pval. If not given explicitly, r = 0. If no output argument is given, the p-value is displayed.

[Function File] [pval, tsq] = hotelling_test (x, m)

For a sample x from a multivariate normal distribution with unknown mean and covariance matrix, test the null hypothesis that mean (x) == m. Hotelling’s T^{2} is returned in tsq. Under the null, (n − p)T^{2}/(p(n − 1)) has an F distribution with p and n − p degrees of freedom, where n and p are the numbers of samples and variables, respectively. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, tsq] = hotelling_test_2 (x, y)

For two samples x from multivariate normal distributions with the same number of variables (columns), unknown means and unknown equal covariance matrices, test the null hypothesis mean (x) == mean (y). Hotelling’s two-sample T^{2} is returned in tsq. Under the null,

\frac{n_{x} + n_{y} − p − 1)T^{2}}{p(n_{x} + n_{y} − 2)}

has an F distribution with p and n_{x} + n_{y} − p − 1 degrees of freedom, where n_{x} and n_{y} are the sample sizes and p is the number of variables. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, ks] = kolmogorov_smirnov_test (x, dist, params,
   alt)

Perform a Kolmogorov-Smirnov test of the null hypothesis that the sample x comes from the (continuous) distribution dist. I.e., if F and G are the CDFs corresponding to the sample and dist, respectively, then the null is that F == G. The optional argument params contains a list of parameters of dist. For example, to test whether a sample x comes from a uniform distribution on [2,4], use

     kolmogorov_smirnov_test (x, "unif", 2, 4)

dist can be any string for which a function dist cdf that calculates the CDF of distribution dist exists. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative F != G. In this case, the test statistic ks follows a two-sided Kolmogorov-Smirnov distribution. If alt is ">", the one-sided alternative F > G is considered. Similarly for "<", the one-sided alternative F > G is considered. In this case, the test statistic ks has a one-sided Kolmogorov-Smirnov distribution. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value is displayed.

[Function File] [pval, ks, d] = kolmogorov_smirnov_test_2 (x, y, alt)

Perform a 2-sample Kolmogorov-Smirnov test of the null hypothesis that the samples x and y come from the same (continuous) distribution. I.e., if F and G are the CDFs corresponding to the x and y samples, respectively, then the null is that F == G. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative F != G. In this case, the test statistic ks follows a two-sided Kolmogorov-Smirnov distribution. If alt is ">", the one-sided alternative F > G is considered. Similarly for "<", the one-sided alternative F < G is considered. In this case, the test statistic ks has a one-sided Kolmogorov-Smirnov distribution. The default is the two-sided case. The p-value of the test is returned in pval. The third returned value, d, is the test statistic, the maximum vertical distance between the two cumulative distribution functions. If no output argument is given, the p-value is displayed.

[Function File] [pval, k, df] = kruskal_wallis_test (x1, . . .)

Perform a Kruskal-Wallis one-factor analysis of variance. Suppose a variable is observed for k > 1 different groups, and let x1, …, xk be the corresponding data vectors. Under the null hypothesis that the ranks in the pooled sample are not affected by the group memberships, the test statistic k is approximately chi-square with df = k - 1 degrees of freedom. If the data contains ties (some value appears more than once) k is divided by 1 - sum ties / (n^3 - n) where sum ties is the sum of t^2 - t over each group of ties where t is the number of ties in the group and n is the total number of values in the input data. For more info on this adjustment see William H. Kruskal and W. Allen Wallis, Use of Ranks in One-Criterion Variance Analysis, Journal of the American Statistical Association, Vol. 47, No. 260 (Dec 1952). The p-value (1 minus the CDF of this distribution at k) is returned in pval. If no output argument is given, the p-value is displayed.

[Function File] manova (x, g)

Perform a one-way multivariate analysis of variance (MANOVA). The goal is to test whether the p-dimensional population means of data taken from k different groups are all equal. All data are assumed drawn independently from p-dimensional normal distributions with the same covariance matrix.

The data matrix is given by x. As usual, rows are observations and columns are variables. The vector g specifies the corresponding group labels (e.g., numbers from 1 to k). The LR test statistic (Wilks’ Lambda) and approximate p-values are computed and displayed.

[Function File] [pval, chisq, df] = mcnemar_test (x)

For a square contingency table x of data cross-classified on the row and column variables, McNemar’s test can be used for testing the null hypothesis of symmetry of the classification probabilities. Under the null, chisq is approximately distributed as chisquare with df degrees of freedom. The p-value (1 minus the CDF of this distribution at chisq) is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, z] = prop_test_2 (x1, n1, x2, n2, alt)

If x1 and n1 are the counts of successes and trials in one sample, and x2 and n2 those in a second one, test the null hypothesis that the success probabilities p1 and p2 are the same. Under the null, the test statistic z approximately follows a standard normal distribution. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative p1 != p2. If alt is ">", the one-sided alternative p1 > p2 is used. Similarly for "<", the one-sided alternative p1 < p2 is used. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, chisq] = run_test (x)

Perform a chi-square test with 6 degrees of freedom based on the upward runs in the columns of x. Can be used to test whether x contains independent data. The p-value of the test is returned in pval. If no output argument is given, the p-value is displayed.

[Function File] [pval, b, n] = sign_test (x, y, alt)

For two matched-pair samples x and y, perform a sign test of the null hypothesis PROB (x > y) == PROB (x < y) == 1/2. Under the null, the test statistic b roughly follows a binomial distribution with parameters n = sum (x != y) and p = 1/2. With the optional argument alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null hypothesis is tested against the two-sided alternative PROB (x < y) != 1/2. If alt is ">", the one-sided alternative PROB (x > y) > 1/2 ("x is stochastically greater than y") is considered. Similarly for "<", the one-sided alternative PROB (x > y) < 1/2 ("x is stochastically less than y") is considered. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, t, df] = t_test (x, m, alt)

For a sample x from a normal distribution with unknown mean and variance, perform a t-test of the null hypothesis mean (x) == m. Under the null, the test statistic t follows a Student distribution with df = length (x) - 1 degrees of freedom. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x) != m. If alt is ">", the one-sided alternative mean (x) > m is considered. Similarly for "<", the one-sided alternative mean (x) < m is considered. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, t, df] = t_test_2 (x, y, alt)

For two samples x and y from normal distributions with unknown means and unknown equal variances, perform a two-sample t-test of the null hypothesis of equal means. Under the null, the test statistic t follows a Student distribution with df degrees of freedom. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x) != mean (y). If alt is ">", the one-sided alternative mean (x) > mean (y) is used. Similarly for "<", the one-sided alternative mean (x) < mean (y) is used. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, t, df] = t_test_regression (y, x, rr, r, alt)

Perform a t test for the null hypothesis rr * b = r in a classical normal regression model y = x * b + e. Under the null, the test statistic t follows a t distribution with df degrees of freedom. If r is omitted, a value of 0 is assumed. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative rr * b != r. If alt is ">", the one-sided alternative rr * b > r is used. Similarly for "<", the one-sided alternative rr * b < r is used. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, z] = u_test (x, y, alt)

For two samples x and y, perform a Mann-Whitney U-test of the null hypothesis PROB (x > y) == 1/2 == PROB (x < y). Under the null, the test statistic z approximately follows a standard normal distribution. Note that this test is equivalent to the Wilcoxon rank-sum test. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative PROB (x > y) != 1/2. If alt is ">", the one-sided alternative PROB (x > y) > 1/2 is considered.

Similarly for "<", the one-sided alternative PROB (x > y) < 1/2 is considered. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, f, df_num, df_den] = var_test (x, y, alt)

For two samples x and y from normal distributions with unknown means and unknown variances, perform an F-test of the null hypothesis of equal variances. Under the null, the test statistic f follows an F-distribution with df num and df den degrees of freedom. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative var (x) != var (y). If alt is ">", the one-sided alternative var (x) > var (y) is used. Similarly for "<", the one-sided alternative var (x) > var (y) is used. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, t, df] = welch_test (x, y, alt)

For two samples x and y from normal distributions with unknown means and unknown and not necessarily equal variances, perform a Welch test of the null hypothesis of equal means. Under the null, the test statistic t approximately follows a Student distribution with df degrees of freedom. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x) != m. If alt is ">", the one-sided alternative mean(x) > m is considered. Similarly for "<", the one-sided alternative mean(x) < m is considered. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, z] = wilcoxon_test (x, y, alt)

For two matched-pair sample vectors x and y, perform a Wilcoxon signed-rank test of the null hypothesis PROB (x > y) == 1/2. Under the null, the test statistic z approximately follows a standard normal distribution when n > 25. Caution: This function assumes a normal distribution for z and thus is invalid for n ≤ 25. With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative PROB (x > y) != 1/2. If alt is ">", the one-sided alternative PROB (x > y) > 1/2 is considered. Similarly for "<", the one-sided alternative PROB (x > y) < 1/2 is considered. The default is the two-sided case. The p-value of the test is returned in pval. If no output argument is given, the p-value of the test is displayed.

[Function File] [pval, z] = z_test (x, m, v, alt)

Perform a Z-test of the null hypothesis mean (x) == m for a sample x from a normal distribution with unknown mean and known variance v. Under the null, the test statistic z follows a standard normal distribution.

With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x) != m. If alt is ">", the one-sided alternative mean (x) > m is considered. Similarly for "<", the one-sided alternative mean (x) < m is considered. The default is the two-sided case.

The p-value of the test is returned in pval.

If no output argument is given, the p-value of the test is displayed along with some information.

[Function File] [pval, z] = z_test_2 (x, y, v_x, v_y, alt)

For two samples x and y from normal distributions with unknown means and known variances v x and v y, perform a Z-test of the hypothesis of equal means. Under the null, the test statistic z follows a standard normal distribution.

With the optional argument string alt, the alternative of interest can be selected. If alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x) != mean (y). If alt is ">", the one-sided alternative mean (x) > mean (y) is used. Similarly for "<", the one-sided alternative mean (x) < mean (y) is used. The default is the two-sided case.

The p-value of the test is returned in pval.

If no output argument is given, the p-value of the test is displayed along with some information.

26.7 Random Number Generation

Octave can generate random numbers from a large number of distributions. The random number generators are based on the random number generators described in Section 16.3 [Special Utility Matrices], page 391.

The following table summarizes the available random number generators (in alphabetical order).

Distribution  Function
Beta Distribution  betarnd
Binomial Distribution  binornd
Cauchy Distribution  cauchy rnd
Chi-Square Distribution  chi2rnd
Univariate Discrete Distribution discrete rnd
Empirical Distribution  empirical rnd
Exponential Distribution  exprnd
F Distribution  frnd
Gamma Distribution  gamrnd
Geometric Distribution  geornd
Hypergeometric Distribution  hygernd
Laplace Distribution  laplace rnd
Logistic Distribution  logistic rnd
Log-Normal Distribution  lognrnd
Pascal Distribution  nbinrnd
Univariate Normal Distribution normrnd
Poisson Distribution  poissrnd
Standard Normal Distribution  stdnormal rnd
t (Student) Distribution  trnd
Univariate Discrete Distribution unidrnd
Uniform Distribution  unifrnd
Weibull Distribution  wblrnd
Wiener Process  wienrnd

[Function File] betarnd (a, b) [Function File] betarnd (a, b, r) [Function File] betarnd (a, b, r, c, …) [Function File] betarnd (a, b, [sz])

Return a matrix of random samples from the Beta distribution with parameters a and b. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of a and b.

[Function File] binornd (n, p) [Function File] binornd (n, p, r) [Function File] binornd (n, p, r, c, …) [Function File] binornd (n, p, [sz])

Return a matrix of random samples from the binomial distribution with parameters n and p, where n is the number of trials and p is the probability of success. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz.

If no size arguments are given then the result matrix is the common size of n and p.

[Function File] cauchy_rnd (location, scale)
[Function File] cauchy_rnd (location, scale, r)
[Function File] cauchy_rnd (location, scale, r, c, . . .)
[Function File] cauchy_rnd (location, scale, [sz])

Return a matrix of random samples from the Cauchy distribution with parameters location and scale. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of location and scale.

[Function File] chi2rnd (n) [Function File] chi2rnd (n, r) [Function File] chi2rnd (n, r, c, …) [Function File] chi2rnd (n, [sz])

Return a matrix of random samples from the chi-square distribution with n degrees of freedom. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the size of n.

[Function File] discrete_rnd (v, p) [Function File] discrete_rnd (v, p, r) [Function File] discrete_rnd (v, p, r, c, …) [Function File] discrete_rnd (v, p, [sz])

Return a matrix of random samples from the univariate distribution which assumes the values in v with probabilities p. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of v and p.

[Function File] empirical_rnd (data)
[Function File] empirical_rnd (data, r)

[Function File] empirical_rnd (data, r, c, …)

[Function File] empirical_rnd (data, [sz])

Return a matrix of random samples from the empirical distribution obtained from the univariate sample data. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is a random ordering of the sample data.

[Function File] exprnd (lambda) [Function File] exprnd (lambda, r) [Function File] exprnd (lambda, r, c, …) [Function File] exprnd (lambda, [sz])

Return a matrix of random samples from the exponential distribution with mean lambda. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the size of lambda.

[Function File] frnd (m, n) [Function File] frnd (m, n, r) [Function File] frnd (m, n, r, c, …) [Function File] frnd (m, n, [sz])

Return a matrix of random samples from the F distribution with m and n degrees of freedom. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of m and n.

[Function File] gamrnd (a, b) [Function File] gamrnd (a, b, r) [Function File] gamrnd (a, b, r, c, …) [Function File] gamrnd (a, b, [sz])

Return a matrix of random samples from the Gamma distribution with shape parameter a and scale b. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of a and b.

[Function File] geornd (p) [Function File] geornd (p, r) [Function File] geornd (p, r, c, …) [Function File] geornd (p, [sz])

Return a matrix of random samples from the geometric distribution with parameter p.

When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the size of p. The geometric distribution models the number of failures (x-1) of a Bernoulli trial with probability p before the first success (x).

[Function File] hygernd (t, m, n) [Function File] hygernd (t, m, n, r) [Function File] hygernd (t, m, n, r, c, …) [Function File] hygernd (t, m, n, [sz])

Return a matrix of random samples from the hypergeometric distribution with parameters t, m, and n. The parameters t, m, and n must be positive integers with m and n not greater than t. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of t, m, and n.

[Function File] laplace_rnd (r) [Function File] laplace_rnd (r, c, …) [Function File] laplace_rnd ([sz])

Return a matrix of random samples from the Laplace distribution. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz.

[Function File] logistic_rnd (r) [Function File] logistic_rnd (r, c, …) [Function File] logistic_rnd ([sz])

Return a matrix of random samples from the logistic distribution. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz.

[Function File] lognrnd (mu, sigma) [Function File] lognrnd (mu, sigma, r) [Function File] lognrnd (mu, sigma, r, c, …) [Function File] lognrnd (mu, sigma, [sz])

Return a matrix of random samples from the lognormal distribution with parameters mu and sigma.

When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of mu and sigma.

[Function File] nbinrnd (n, p) [Function File] nbinrnd (n, p, r) [Function File] nbinrnd (n, p, r, c, …) [Function File] nbinrnd (n, p, [sz])

Return a matrix of random samples from the negative binomial distribution with parameters n and p. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of n and p.

[Function File] normrnd (mu, sigma) [Function File] normrnd (mu, sigma, r) [Function File] normrnd (mu, sigma, r, c, …) [Function File] normrnd (mu, sigma, [sz])

Return a matrix of random samples from the normal distribution with parameters mean mu and standard deviation sigma. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of mu and sigma.

[Function File] poissrnd (lambda) [Function File] poissrnd (lambda, r) [Function File] poissrnd (lambda, r, c, …) [Function File] poissrnd (lambda, [sz])

Return a matrix of random samples from the Poisson distribution with parameter lambda. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the size of lambda.

[Function File] stdnormal_rnd (r) [Function File] stdnormal_rnd (r, c, …)

[Function File] stdnormal_rnd ([sz])

Return a matrix of random samples from the standard normal distribution (mean = 0, standard deviation = 1). When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz.

[Function File] trnd (n) [Function File] trnd (n, r) [Function File] trnd (n, r, c, …) [Function File] trnd (n, [sz])

Return a matrix of random samples from the t (Student) distribution with n degrees of freedom. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the size of n.

[Function File] unidrnd (n) [Function File] unidrnd (n, r) [Function File] unidrnd (n, r, c, …) [Function File] unidrnd (n, [sz])

Return a matrix of random samples from the discrete uniform distribution which assumes the integer values 1–n with equal probability. n may be a scalar or a multidimensional array. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the size of n.

[Function File] unifrnd (a, b) [Function File] unifrnd (a, b, r) [Function File] unifrnd (a, b, r, c, …) [Function File] unifrnd (a, b, [sz])

Return a matrix of random samples from the uniform distribution on [a, b]. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of a and b.

[Function File] wblrnd (scale, shape) [Function File] wblrnd (scale, shape, r) [Function File] wblrnd (scale, shape, r, c, …)

[Function File] wblrnd (scale, shape, [sz])

Return a matrix of random samples from the Weibull distribution with parameters scale and shape. When called with a single size argument, return a square matrix with the dimension specified. When called with more than one scalar argument the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a vector of dimensions sz. If no size arguments are given then the result matrix is the common size of scale and shape.

[Function File] wienrnd (t, d, n)

Return a simulated realization of the d-dimensional Wiener Process on the interval [0, t]. If d is omitted, d = 1 is used. The first column of the return matrix contains time, the remaining columns contain the Wiener process. The optional parameter n gives the number of summands used for simulating the process over an interval of length 1. If n is omitted, n = 1000 is used.

27 Sets

Octave has a limited number of functions for managing sets of data, where a set is defined as a collection of unique elements. In Octave a set is represented as a vector of numbers.

[Function File] unique (x) [Function File] unique (x, "rows") [Function File] unique (…, "first") [Function File] unique (…, "last") [Function File] [y, i, j] = unique (…)

Return the unique elements of x, sorted in ascending order. If the input x is a vector then the output is also a vector with the same orientation (row or column) as the input. For a matrix input the output is always a column vector. x may also be a cell array of strings.

If the optional argument "rows" is supplied, return the unique rows of x, sorted in ascending order.

If requested, return index vectors i and j such that x(i)==y and y(j)==x.

Additionally, if i is a requested output then one of "first" or "last" may be given as an input. If "last" is specified, return the highest possible indices in i, otherwise, if "first" is specified, return the lowest. The default is "last".

See also: [union], page 598, [intersect], page 598, [setdiff], page 598, [setxor], page 599, [ismember], page 597.

27.1 Set Operations

Octave supports the basic set operations. That is, Octave can compute the union, intersection, and difference of two sets. Octave also supports the Exclusive Or set operation, and membership determination. The functions for set operations all work in pretty much the same way. As an example, assume that x and y contains two sets, then

union (x, y)

computes the union of the two sets.

[Function File] tf = ismember (A, s)

[Function File] [tf, S_idx] = ismember (A, s)
[Function File] [tf, S_idx] = ismember (A, s, "rows")

Return a logical matrix tf with the same shape as A which is true (1) if A(i,j) is in s and false (0) if it is not. If a second output argument is requested, the index into s of each of the matching elements is also returned.

     a = [3, 10, 1];
     s = [0:9];
     [tf, s_idx] = ismember (a, s)
          ⇒ tf = [1, 0, 1]

          ⇒ s_idx = [4, 0, 2]

The inputs, A and s, may also be cell arrays.

     a = {"abc"};
     s = {"abc", "def"};
     [tf, s_idx] = ismember (a, s)
          ⇒ tf = [1, 0]

          ⇒ s_idx = [1, 0]

With the optional third argument "rows", and matrices A and s with the same number of columns, compare rows in A with the rows in s.

     a = [1:3; 5:7; 4:6];
     s = [0:2; 1:3; 2:4; 3:5; 4:6];
     [tf, s_idx] = ismember (a, s, "rows")
          ⇒ tf = logical ([1; 0; 1])
          ⇒ s_idx = [2; 0; 5];

See also: [unique], page 597, [union], page 598, [intersect], page 598, [setxor], page 599, [setdiff], page 598.

[Function File] union (a, b) [Function File] union (a, b, "rows")

[Function File] [c, ia, ib] = union (a, b)

Return the set of elements that are in either of the sets a and b. a, b may be cell arrays of strings. For example:

     union ([1, 2, 4], [2, 3, 5]) ⇒ [1, 2, 3, 4, 5]

If the optional third input argument is the string "rows" then each row of the matrices a and b will be considered as a single set element. For example:

     union ([1, 2; 2, 3], [1, 2; 3, 4], "rows")
        ⇒ 1   2
            2   3
            3   4

The optional outputs ia and ib are index vectors such that a(ia) and b(ib) are disjoint sets whose union is c.

See also: [intersect], page 598, [setdiff], page 598, [unique], page 597.

[Function File] intersect (a, b)

[Function File] [c, ia, ib] = intersect (a, b)

Return the elements in both a and b, sorted in ascending order. If a and b are both column vectors return a column vector, otherwise return a row vector. a, b may be cell arrays of string(s).

Return index vectors ia and ib such that a(ia)==c and b(ib)==c.

See also: [unique], page 597, [union], page 598, [setxor], page 599, [setdiff], page 598, [ismember], page 597.

[Function File] setdiff (a, b) [Function File] setdiff (a, b, "rows")

[Function File] [c, i] = setdiff (a, b)

Return the elements in a that are not in b, sorted in ascending order. If a and b are both column vectors return a column vector, otherwise return a row vector. a, b may be cell arrays of string(s). Given the optional third argument "rows", return the rows in a that are not in b, sorted in ascending order by rows. If requested, return i such that c = a(i).

See also: [unique], page 597, [union], page 598, [intersect], page 598, [setxor], page 599, [ismember], page 597.

[Function File] setxor (a, b) [Function File] setxor (a, b, "rows")

[Function File] [c, ia, ib] = setxor (a, b)

Return the elements exclusive to a or b, sorted in ascending order. If a and b are both column vectors return a column vector, otherwise return a row vector. a, b may be cell arrays of string(s). With three output arguments, return index vectors ia and ib such that a(ia) and b(ib) are disjoint sets whose union is c.

See also: [unique], page 597, [union], page 598, [intersect], page 598, [setdiff], page 598, [ismember], page 597.

[Function File] powerset (a) [Function File] powerset (a, "rows")

Compute the powerset (all subsets) of the set a. The set a must be a numerical matrix or a cell array of strings. The output will always be a cell array of either vectors or strings. With the optional second argument "rows", each row of the set a is considered one element of the set. As a result, a must then be a numerical 2-D matrix.

See also: [unique], page 597, [union], page 598, [setxor], page 599, [setdiff], page 598, [ismember], page 597.

28 Polynomial Manipulations

In Octave, a polynomial is represented by its coefficients (arranged in descending order). For example, a vector c of length N + 1 corresponds to the following polynomial of order N

p(x) = c_{1}x^{N} + … + c_{N}x + c_{N+1}.

28.1 Evaluating Polynomials

The value of a polynomial represented by the vector c can be evaluated at the point x very easily, as the following example shows:

N = length (c) - 1;
val = dot (x.^(N:-1:0), c);

While the above example shows how easy it is to compute the value of a polynomial, it isn’t the most stable algorithm. With larger polynomials you should use more elegant algorithms, such as Horner’s Method, which is exactly what the Octave function polyval does.

In the case where x is a square matrix, the polynomial given by c is still well-defined. As when x is a scalar the obvious implementation is easily expressed in Octave, but also in this case more elegant algorithms perform better. The polyvalm function provides such an algorithm.

[Function File] y = polyval (p, x) [Function File] y = polyval (p, x, [], mu) [Function File] [y, dy] = polyval (p, x, s)

[Function File] [y, dy] = polyval (p, x, s, mu)

Evaluate the polynomial p at the specified values of x. When mu is present, evaluate the polynomial for (x-mu(1))/mu(2). If x is a vector or matrix, the polynomial is evaluated for each of the elements of x.

In addition to evaluating the polynomial, the second output represents the prediction interval, y +/- dy, which contains at least 50% of the future predictions. To calculate the prediction interval, the structured variable s, originating from polyfit, must be supplied.

See also: [polyvalm], page 601, [polyaffine], page 606, [polyfit], page 607, [roots], page 602, [poly], page 615.

[Function File] polyvalm (c, x)

Evaluate a polynomial in the matrix sense.

polyvalm (c, x) will evaluate the polynomial in the matrix sense, i.e., matrix multiplication is used instead of element by element multiplication as used in polyval.

The argument x must be a square matrix.

See also: [polyval], page 601, [roots], page 602, [poly], page 615.

28.2 Finding Roots

Octave can find the roots of a given polynomial. This is done by computing the companion matrix of the polynomial (see the compan function for a definition), and then finding its eigenvalues.

[Function File] roots (v)

For a vector v with N components, return the roots of the polynomial

v_{1}z^{N−1} + ⋯ + v_{N−1}z + v_{N}.

As an example, the following code finds the roots of the quadratic polynomial

p(x) = x^{2} − 5.

     c = [1, 0, -5];
     roots (c)
     ⇒ 2.2361
     ⇒ -2.2361

Note that the true result is ±\sqrt{5} which is roughly ±2.2361.

See also: [poly], page 615, [compan], page 602, [fzero], page 475.

[Function File] z = polyeig (C0, C1, …, Cl) [Function File] [v, z] = polyeig (C0, C1, …, Cl)

Solve the polynomial eigenvalue problem of degree l. Given an n*n matrix polynomial C(s) = C0 + C1 s + ... + Cl s^l polyeig solves the eigenvalue problem (C0 + C1 + ... + Cl)v = 0. Note that the eigenvalues z are the zeros of the matrix polynomial. z is an lxn vector and v is an (n x n)l matrix with columns that correspond to the eigenvectors.

See also: [eig], page 437, [eigs], page 513, [compan], page 602.

[Function File] compan (c)

Compute the companion matrix corresponding to polynomial coefficient vector c. The companion matrix is

A = [ \begin{matrix}−c_{2}/c_{1} & −c_{3}/c_{1} & ⋯ & −c_{N}/c_{1} & −c_{N+1}/c_{1} \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix} ] .

The eigenvalues of the companion matrix are equal to the roots of the polynomial.

See also: [roots], page 602, [poly], page 615, [eig], page 437.

[Function File] [multp, idxp] = mpoles (p)
[Function File] [multp, idxp] = mpoles (p, tol)
[Function File] [multp, idxp] = mpoles (p, tol, reorder)

Identify unique poles in p and their associated multiplicity. The output is ordered from largest pole to smallest pole.

If the relative difference of two poles is less than tol then they are considered to be multiples. The default value for tol is 0.001. If the optional parameter reorder is zero, poles are not sorted. The output multp is a vector specifying the multiplicity of the poles. multp(n) refers to the multiplicity of the Nth pole p(idxp(n)). For example:

     p = [2 3 1 1 2];
     [m, n] = mpoles (p)
        ⇒ m = [1; 1; 2; 1; 2]
        ⇒ n = [2; 5; 1; 4; 3]
        ⇒ p(n) = [3, 2, 2, 1, 1]

See also: [residue], page 604, [poly], page 615, [roots], page 602, [conv], page 603, [deconv], page 604.

28.3 Products of Polynomials

[Function File] conv (a, b) [Function File] conv (a, b, shape)

Convolve two vectors a and b. The output convolution is a vector with length equal to length (a) + length (b)

- 1. When a and b are the coefficient vectors of two polynomials, the convolution represents the coefficient vector of the product polynomial. The optional shape argument may be

shape = "full"

Return the full convolution. (default)

shape = "same"

Return the central part of the convolution with the same size as a.

See also: [deconv], page 604, [conv2], page 604, [convn], page 603, [fftconv], page 642.

[Built-in Function] C = convn (A, B) [Built-in Function] C = convn (A, B, shape)

Return the n-D convolution of A and B. The size of the result is determined by the optional shape argument which takes the following values

shape = "full"

Return the full convolution. (default)

shape = "same"

Return central part of the convolution with the same size as A. The central part of the convolution begins at the indices floor ([size(B)/2]

          + 1).
shape = "valid"

Return only the parts which do not include zero-padded edges. The size

          of the result is max (size (A) - size (B) + 1, 0).

See also: [conv2], page 604, [conv], page 603.

[Function File] deconv (y, a)

Deconvolve two vectors.

[b, r] = deconv (y, a) solves for b and r such that y = conv (a, b) + r.

If y and a are polynomial coefficient vectors, b will contain the coefficients of the polynomial quotient and r will be a remainder polynomial of lowest order.

See also: [conv], page 603, [residue], page 604.

[Built-in Function] conv2 (A, B) [Built-in Function] conv2 (v1, v2, m) [Built-in Function] conv2 (…, shape)

Return the 2-D convolution of A and B. The size of the result is determined by the optional shape argument which takes the following values

shape = "full"

Return the full convolution. (default)

shape = "same"

Return the central part of the convolution with the same size as A. The central part of the convolution begins at the indices floor ([size(B)/2]

          + 1).
shape = "valid"

Return only the parts which do not include zero-padded edges. The size

          of the result is max (size (A) - size (B) + 1, 0).

When the third argument is a matrix, return the convolution of the matrix m by the vector v1 in the column direction and by the vector v2 in the row direction.

See also: [conv], page 603, [convn], page 603.

[Function File] q = polygcd (b, a) [Function File] q = polygcd (b, a, tol)

Find the greatest common divisor of two polynomials. This is equivalent to the polynomial found by multiplying together all the common roots. Together with deconv, you can reduce a ratio of two polynomials. The tolerance tol defaults to sqrt (eps).

Caution: This is a numerically unstable algorithm and should not be used on large polynomials.

Example code:

     polygcd (poly (1:8), poly (3:12)) - poly (3:8)
     ⇒ [ 0, 0, 0, 0, 0, 0, 0 ]
     deconv (poly (1:8), polygcd (poly (1:8), poly (3:12))) - poly (1:2)
     ⇒ [ 0, 0, 0 ]

See also: [poly], page 615, [roots], page 602, [conv], page 603, [deconv], page 604, [residue], page 604.

[Function File] [r, p, k, e] = residue (b, a)

[Function File] [b, a] = residue (r, p, k)

[Function File] [b, a] = residue (r, p, k, e)

The first calling form computes the partial fraction expansion for the quotient of the polynomials, b and a.

\frac{B(s)}{A(s)} = ∑_{m=1}^{M} \frac{r_{m}}{(s − p_{m})_{m}^{e}} + ∑_{i=1}^{N} k_{i}s^{N−i}.

where M is the number of poles (the length of the r, p, and e), the k vector is a polynomial of order N − 1 representing the direct contribution, and the e vector specifies the multiplicity of the m-th residue’s pole. For example,

     b = [1, 1, 1];
     a = [1, -5, 8, -4];
     [r, p, k, e] = residue (b, a)
        ⇒ r = [-2; 7; 3]
        ⇒ p = [2; 2; 1]
        ⇒ k = [](0x0)
        ⇒ e = [1; 2; 1]

which represents the following partial fraction expansion

\frac{s^{2} + s + 1}{s^{3} − 5s^{2} + 8s − 4} = \frac{−2}{s − 2} + \frac{7}{(s − 2)^{2}} + \frac{3}{s − 1}

The second calling form performs the inverse operation and computes the reconstituted quotient of polynomials, b(s)/a(s), from the partial fraction expansion; represented by the residues, poles, and a direct polynomial specified by r, p and k, and the pole multiplicity e. If the multiplicity, e, is not explicitly specified the multiplicity is determined by the function mpoles. For example:

     r = [-2; 7; 3];
     p = [2; 2; 1];
     k = [1, 0];
     [b, a] = residue (r, p, k)
        ⇒ b = [1, -5, 9, -3, 1]
        ⇒ a = [1, -5, 8, -4]
     where mpoles is used to determine e = [1; 2; 1]

Alternatively the multiplicity may be defined explicitly, for example,

     r = [7; 3; -2];
     p = [2; 1; 2];
     k = [1, 0];
     e = [2; 1; 1];
     [b, a] = residue (r, p, k, e)
        ⇒ b = [1, -5, 9, -3, 1]
        ⇒ a = [1, -5, 8, -4]

which represents the following partial fraction expansion

\frac{−2}{s − 2} + \frac{7}{(s − 2)^{2}} + \frac{3}{s − 1} + s = \frac{s^{4} − 5s^{3} + 9s^{2} − 3s + 1}{s^{3} − 5s^{2} + 8s − 4}

See also: [mpoles], page 602, [poly], page 615, [roots], page 602, [conv], page 603, [deconv], page 604.

28.4 Derivatives / Integrals / Transforms

Octave comes with functions for computing the derivative and the integral of a polynomial. The functions polyder and polyint both return new polynomials describing the result. As an example we’ll compute the definite integral of p(x) = x^{2} + 1 from 0 to 3.

c = [1, 0, 1];
integral = polyint (c);
area = polyval (integral, 3) - polyval (integral, 0)
⇒ 12

[Function File] polyder (p) [Function File] [k] = polyder (a, b) [Function File] [q, d] = polyder (b, a)

Return the coefficients of the derivative of the polynomial whose coefficients are given by the vector p. If a pair of polynomials is given, return the derivative of the product a∗b. If two inputs and two outputs are given, return the derivative of the polynomial quotient b/a. The quotient numerator is in q and the denominator in d.

See also: [polyint], page 606, [polyval], page 601, [polyreduce], page 616.

[Function File] polyint (p) [Function File] polyint (p, k)

Return the coefficients of the integral of the polynomial whose coefficients are represented by the vector p. The variable k is the constant of integration, which by default is set to zero.

See also: [polyder], page 606, [polyval], page 601.

[Function File] polyaffine (f, mu)

Return the coefficients of the polynomial vector f after an affine transformation. If f is the vector representing the polynomial f(x), then g = polyaffine (f, mu) is the vector representing:

     g(x) = f( (x - mu(1)) / mu(2) )

See also: [polyval], page 601, [polyfit], page 607.

28.5 Polynomial Interpolation

Octave comes with good support for various kinds of interpolation, most of which are described in Chapter 29 [Interpolation], page 617. One simple alternative to the functions described in the aforementioned chapter, is to fit a single polynomial, or a piecewise polynomial (spline) to some given data points. To avoid a highly fluctuating polynomial, one most often wants to fit a low-order polynomial to data. This usually means that it is necessary to fit the polynomial in a least-squares sense, which just is what the polyfit function does.

[Function File] p = polyfit (x, y, n) [Function File] [p, s] = polyfit (x, y, n)

[Function File] [p, s, mu] = polyfit (x, y, n)

Return the coefficients of a polynomial p(x) of degree n that minimizes the leastsquares-error of the fit to the points [x, y]. If n is a logical vector, it is used as a mask to selectively force the corresponding polynomial coefficients to be used or ignored. The polynomial coefficients are returned in a row vector. The optional output s is a structure containing the following fields:

‘R’ Triangular factor R from the QR decomposition.

‘X’ The Vandermonde matrix used to compute the polynomial coefficients.

‘C’ The unscaled covariance matrix, formally equal to the inverse of x’*x, but computed in a way minimizing roundoff error propagation.

‘df’ The degrees of freedom.

‘normr’ The norm of the residuals.

‘yf’ The values of the polynomial for each value of x.

The second output may be used by polyval to calculate the statistical error limits of the predicted values. In particular, the standard deviation of p coefficients is given by

sqrt (diag (s.C)/s.df)*s.normr.

When the third output, mu, is present the coefficients, p, are associated with a polynomial in xhat = (x-mu(1))/mu(2). Where mu(1) = mean (x), and mu(2) = std (x). This linear transformation of x improves the numerical stability of the fit.

See also: [polyval], page 601, [polyaffine], page 606, [roots], page 602, [vander], page 406, [zscore], page 567.

In situations where a single polynomial isn’t good enough, a solution is to use several polynomials pieced together. The function splinefit fits a peicewise polynomial (spline) to a set of data.

[Function File] pp = splinefit (x, y, breaks)

[Function File] pp = splinefit (x, y, p) [Function File] pp = splinefit (…, "periodic", periodic) [Function File] pp = splinefit (…, "robust", robust) [Function File] pp = splinefit (…, "beta", beta) [Function File] pp = splinefit (…, "order", order) [Function File] pp = splinefit (…, "constraints", constraints)

Fit a piecewise cubic spline with breaks (knots) breaks to the noisy data, x and y. x is a vector, and y is a vector or N-D array. If y is an N-D array, then x(j) is matched to y(:,…,:,j).

The fitted spline is returned as a piecewise polynomial, pp, and may be evaluated using ppval.

p is a positive integer defining the number of intervals along x, and p+1 is the number of breaks. The number of points in each interval differ by no more than 1.

The optional property periodic is a logical value which specifies whether a periodic boundary condition is applied to the spline. The length of the period is max (breaks)

- min (breaks). The default value is false.

The optional property robust is a logical value which specifies if robust fitting is to be applied to reduce the influence of outlying data points. Three iterations of weighted least squares are performed. Weights are computed from previous residuals. The sensitivity of outlier identification is controlled by the property beta. The value of beta is stricted to the range, 0 < beta < 1. The default value is beta = 1/2. Values close to 0 give all data equal weighting. Increasing values of beta reduce the influence of outlying data. Values close to unity may cause instability or rank deficiency.

The splines are constructed of polynomials with degree order. The default is a cubic, order=3. A spline with P pieces has P+order degrees of freedom. With periodic boundary conditions the degrees of freedom are reduced to P.

The optional property, constaints, is a structure specifying linear constraints on the fit. The structure has three fields, "xc", "yc", and "cc".

"xc" Vector of the x-locations of the constraints.

"yc" Constraining values at the locations xc. The default is an array of zeros.

"cc" Coefficients (matrix). The default is an array of ones. The number of rows is limited to the order of the piecewise polynomials, order.

Constraints are linear combinations of derivatives of order 0 to order-1 according to

cc(1, j) · y(xc(j)) + cc(2, j) · y′(xc(j)) + ... = yc(:, …,:, j).

See also: [interp1], page 617, [unmkpp], page 614, [ppval], page 614, [spline], page 620, [pchip], page 647, [ppder], page 615, [ppint], page 615, [ppjumps], page 615.

The number of breaks (or knots) used to construct the piecewise polynomial is a significant factor in suppressing the noise present in the input data, x and y. This is demonstrated by the example below.

x = 2 * pi * rand (1, 200);
y = sin (x) + sin (2 * x) + 0.2 * randn (size (x));
## Uniform breaks
breaks = linspace (0, 2 * pi, 41); % 41 breaks, 40 pieces
pp1 = splinefit (x, y, breaks);
## Breaks interpolated from data
pp2 = splinefit (x, y, 10);  % 11 breaks, 10 pieces
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
plot (x, y, ".", xx, [y1; y2])
axis tight
ylim auto
legend ({"data", "41 breaks, 40 pieces", "11 breaks, 10 pieces"})

The result of which can be seen in Figure 28.1.

Figure 28.1: Comparison of a fitting a piecewise polynomial with 41 breaks to one with 11 breaks. The fit with the large number of breaks exhibits a fast ripple that is not present in the underlying function.

The piecewise polynomial fit, provided by splinefit, has continuous derivatives up to the order-1. For example, a cubic fit has continuous first and second derivatives. This is demonstrated by the code

## Data (200 points)
x = 2 * pi * rand (1, 200);
y = sin (x) + sin (2 * x) + 0.1 * randn (size (x));
## Piecewise constant
pp1 = splinefit (x, y, 8, "order", 0);
## Piecewise linear
pp2 = splinefit (x, y, 8, "order", 1);

## Piecewise quadratic
pp3 = splinefit (x, y, 8, "order", 2);
## Piecewise cubic
pp4 = splinefit (x, y, 8, "order", 3);
## Piecewise quartic
pp5 = splinefit (x, y, 8, "order", 4);
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
y3 = ppval (pp3, xx);
y4 = ppval (pp4, xx);
y5 = ppval (pp5, xx);
plot (x, y, ".", xx, [y1; y2; y3; y4; y5])
axis tight
ylim auto
legend ({"data", "order 0", "order 1", "order 2", "order 3", "order 4"})

The result of which can be seen in Figure 28.2.

Figure 28.2: Comparison of a piecewise constant, linear, quadratic, cubic, and quartic polynomials with 8 breaks to noisy data. The higher order solutions more accurately represent the underlying function, but come with the expense of computational complexity.

When the underlying function to provide a fit to is periodic, splinefit is able to apply the boundary conditions needed to manifest a periodic fit. This is demonstrated by the code below.

## Data (100 points)
x = 2 * pi * [0, (rand (1, 98)), 1];
y = sin (x) - cos (2 * x) + 0.2 * randn (size (x));
## No constraints
pp1 = splinefit (x, y, 10, "order", 5);
## Periodic boundaries
pp2 = splinefit (x, y, 10, "order", 5, "periodic", true);
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
plot (x, y, ".", xx, [y1; y2])
axis tight
ylim auto
legend ({"data", "no constraints", "periodic"})

The result of which can be seen in Figure 28.3.

Figure 28.3: Comparison of piecewise polynomial fits to a noisy periodic function with, and without, periodic boundary conditions.

More complex constraints may be added as well. For example, the code below illustrates a periodic fit with values that have been clamped at the endpoints, and a second periodic fit which is hinged at the endpoints.

## Data (200 points)
x = 2 * pi * rand (1, 200);
y = sin (2 * x) + 0.1 * randn (size (x));
## Breaks
breaks = linspace (0, 2 * pi, 10);
## Clamped endpoints, y = y' = 0
xc = [0, 0, 2*pi, 2*pi];
cc = [(eye (2)), (eye (2))];

con = struct ("xc", xc, "cc", cc);
pp1 = splinefit (x, y, breaks, "constraints", con);
## Hinged periodic endpoints, y = 0
con = struct ("xc", 0);
pp2 = splinefit (x, y, breaks, "constraints", con, "periodic", true);
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
plot (x, y, ".", xx, [y1; y2])
axis tight
ylim auto
legend ({"data", "clamped", "hinged periodic"})

The result of which can be seen in Figure 28.4.

Figure 28.4: Comparison of two periodic piecewise cubic fits to a noisy periodic signal. One fit has its endpoints clamped and the second has its endpoints hinged.

The splinefit function also provides the convenience of a robust fitting, where the effect of outlying data is reduced. In the example below, three different fits are provided. Two with differing levels of outlier suppression and a third illustrating the non-robust solution.

## Data
x = linspace (0, 2*pi, 200);
y = sin (x) + sin (2 * x) + 0.05 * randn (size (x));
## Add outliers
x = [x, linspace(0,2*pi,60)];
y = [y, -ones(1,60)];
## Fit splines with hinged conditions
con = struct ("xc", [0, 2*pi]);
## Robust fitting, beta = 0.25
pp1 = splinefit (x, y, 8, "constraints", con, "beta", 0.25);

## Robust fitting, beta = 0.75
pp2 = splinefit (x, y, 8, "constraints", con, "beta", 0.75);
## No robust fitting
pp3 = splinefit (x, y, 8, "constraints", con);
## Plot
xx = linspace (0, 2*pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
y3 = ppval (pp3, xx);
plot (x, y, ".", xx, [y1; y2; y3])
legend ({"data with outliers","robust, beta = 0.25", ...
         "robust, beta = 0.75", "no robust fitting"})
axis tight
ylim auto

The result of which can be seen in Figure 28.5.

Figure 28.5: Comparison of two different levels of robust fitting (beta = 0.25 and 0.75) to noisy data combined with outlying data. A conventional fit, without robust fitting (beta = 0) is also included.

The function, ppval, evaluates the piecewise polynomials, created by mkpp or other means, and unmkpp returns detailed information about the piecewise polynomial.

The following example shows how to combine two linear functions and a quadratic into one function. Each of these functions is expressed on adjoined intervals.

x = [-2, -1, 1, 2];
p = [ 0,  1, 0;
      1, -2, 1;
      0, -1, 1 ];
pp = mkpp (x, p);
xi = linspace (-2, 2, 50);
yi = ppval (pp, xi);
plot (xi, yi);
[Function File] pp = mkpp (breaks, coefs)
[Function File] pp = mkpp (breaks, coefs, d)

Construct a piecewise polynomial (pp) structure from sample points breaks and coefficients coefs. breaks must be a vector of strictly increasing values. The number of intervals is given by ni = length (breaks) - 1. When m is the polynomial order coefs must be of size: ni x m + 1. The i-th row of coefs, coefs (i,:), contains the coefficients for the polynomial over the i-th interval, ordered from highest (m) to lowest (0). coefs may also be a multi-dimensional array, specifying a vector-valued or array-valued polynomial. In that case the polynomial order is defined by the length of the last dimension of coefs. The size of first dimension(s) are given by the scalar or vector d. If d is not given it is set to 1. In any case coefs is reshaped to a 2-D matrix of size

[ni*prod(d m)]

See also: [unmkpp], page 614, [ppval], page 614, [spline], page 620, [pchip], page 647, [ppder], page 615, [ppint], page 615, [ppjumps], page 615.

[Function File] [x, p, n, k, d] = unmkpp (pp)

Extract the components of a piecewise polynomial structure pp. The components are:

x Sample points.

p Polynomial coefficients for points in sample interval. p (i, :) contains the coefficients for the polynomial over interval i ordered from highest to lowest. If d > 1, p (r, i, :) contains the coefficients for the r-th polynomial defined on interval i.

n Number of polynomial pieces.

k Order of the polynomial plus 1.

d Number of polynomials defined for each interval.

See also: [mkpp], page 614, [ppval], page 614, [spline], page 620, [pchip], page 647.

[Function File] yi = ppval (pp, xi)

Evaluate the piecewise polynomial structure pp at the points xi. If pp describes a scalar polynomial function, the result is an array of the same shape as xi. Otherwise, the size of the result is [pp.dim, length(xi)] if xi is a vector, or [pp.dim, size(xi)] if it is a multi-dimensional array.

See also: [mkpp], page 614, [unmkpp], page 614, [spline], page 620, [pchip], page 647.

[Function File] ppd = ppder (pp) [Function File] ppd = ppder (pp, m)

Compute the piecewise m-th derivative of a piecewise polynomial struct pp. If m is omitted the first derivative is calculated.

See also: [mkpp], page 614, [ppval], page 614, [ppint], page 615.

[Function File] ppi = ppint (pp) [Function File] ppi = ppint (pp, c)

Compute the integral of the piecewise polynomial struct pp. c, if given, is the constant of integration.

See also: [mkpp], page 614, [ppval], page 614, [ppder], page 615.

[Function File] jumps = ppjumps (pp)

Evaluate the boundary jumps of a piecewise polynomial. If there are n intervals, and the dimensionality of pp is d, the resulting array has dimensions [d, n-1].

See also: [mkpp], page 614.

28.6 Miscellaneous Functions

[Function File] poly (A) [Function File] poly (x)

If A is a square N-by-N matrix, poly (A) is the row vector of the coefficients of det (z * eye (N) - A), the characteristic polynomial of A. For example, the following code finds the eigenvalues of A which are the roots of poly (A).

     roots (poly (eye (3)))
         ⇒ 1.00001 + 0.00001i
            1.00001 - 0.00001i
            0.99999 + 0.00000i

In fact, all three eigenvalues are exactly 1 which emphasizes that for numerical performance the eig function should be used to compute eigenvalues. If x is a vector, poly (x) is a vector of the coefficients of the polynomial whose roots are the elements of x. That is, if c is a polynomial, then the elements of d = roots (poly (c)) are contained in c. The vectors c and d are not identical, however, due to sorting and numerical errors.

See also: [roots], page 602, [eig], page 437.

[Function File] polyout (c) [Function File] polyout (c, x) [Function File] str = polyout (…)

Write formatted polynomial

c(x) = c_{1}x^{n} + … + c_{n}x + c_{n+1}

and return it as a string or write it to the screen (if nargout is zero). x defaults to the string "s".

See also: [polyreduce], page 616.

[Function File] polyreduce (c)

Reduce a polynomial coefficient vector to a minimum number of terms by stripping off any leading zeros.

See also: [polyout], page 615.

29 Interpolation

29.1 One-dimensional Interpolation

Octave supports several methods for one-dimensional interpolation, most of which are described in this section. Section 28.5 [Polynomial Interpolation], page 606 and Section 30.4 [Interpolation on Scattered Data], page 637 describe additional methods.

[Function File] yi = interp1 (x, y, xi) [Function File] yi = interp1 (y, xi) [Function File] yi = interp1 (…, method) [Function File] yi = interp1 (…, extrap) [Function File] yi = interp1 (…, "left") [Function File] yi = interp1 (…, "right") [Function File] pp = interp1 (…, "pp")

One-dimensional interpolation. Interpolate input data to determine the value of yi at the points xi. If not specified, x is taken to be the indices of y. If y is a matrix or an N-dimensional array, the interpolation is performed on each column of y. Method is one of:

"nearest"

Return the nearest neighbor

"linear" Linear interpolation from nearest neighbors

"pchip" Piecewise cubic Hermite interpolating polynomial

"cubic" Cubic interpolation (same as pchip)

"spline" Cubic spline interpolation—smooth first and second derivatives throughout the curve

Adding ’*’ to the start of any method above forces interp1 to assume that x is uniformly spaced, and only x(1) and x(2) are referenced. This is usually faster, and is never slower. The default method is "linear". If extrap is the string "extrap", then extrapolate values beyond the endpoints using the current method. If extrap is a number, then replace values beyond the endpoints with that number. When unspecified, extrap defaults to NA. If the string argument "pp" is specified, then xi should not be supplied and interp1 returns a piecewise polynomial object. This object can later be used with ppval to evaluate the interpolation. There is an equivalence, such that ppval (interp1 (x,

y, method, "pp"), xi) == interp1 (x, y, xi, method, "extrap").

Duplicate points in x specify a discontinuous interpolant. There may be at most 2 consecutive points with the same value. If x is increasing, the default discontinuous interpolant is right-continuous. If x is decreasing, the default discontinuous interpolant is left-continuous. The continuity condition of the interpolant may be specified by using the options, "left" or "right", to select a left-continuous or right-continuous interpolant, respectively. Discontinuous interpolation is only allowed for "nearest" and "linear" methods; in all other cases, the x-values must be unique.

An example of the use of interp1 is

     xf = [0:0.05:10];
     yf = sin (2*pi*xf/5);
     xp = [0:10];
     yp = sin (2*pi*xp/5);
     lin = interp1 (xp, yp, xf);
     spl = interp1 (xp, yp, xf, "spline");
     cub = interp1 (xp, yp, xf, "cubic");
     near = interp1 (xp, yp, xf, "nearest");
     plot (xf, yf, "r", xf, lin, "g", xf, spl, "b",
           xf, cub, "c", xf, near, "m", xp, yp, "r*");
     legend ("original", "linear", "spline", "cubic", "nearest");

See also: [interpft], page 619, [interp2], page 621, [interp3], page 622, [interpn], page 623.

There are some important differences between the various interpolation methods. The "spline" method enforces that both the first and second derivatives of the interpolated values have a continuous derivative, whereas the other methods do not. This means that the results of the "spline" method are generally smoother. If the function to be interpolated is in fact smooth, then "spline" will give excellent results. However, if the function to be evaluated is in some manner discontinuous, then "pchip" interpolation might give better results.

This can be demonstrated by the code

t = -2:2;
dt = 1;
ti =-2:0.025:2;
dti = 0.025;
y = sign (t);
ys = interp1 (t,y,ti,"spline");
yp = interp1 (t,y,ti,"pchip");
ddys = diff (diff (ys)./dti) ./ dti;
ddyp = diff (diff (yp)./dti) ./ dti;
figure (1);
plot (ti,ys,"r-", ti,yp,"g-");
legend ("spline", "pchip", 4);
figure (2);
plot (ti,ddys,"r+", ti,ddyp,"g*");
legend ("spline", "pchip");

The result of which can be seen in Figure 29.1 and Figure 29.2.

Figure 29.1: Comparison of "pchip" and "spline" interpolation methods for a step function

Figure 29.2: Comparison of the second derivative of the "pchip" and "spline" interpolation methods for a step function

Fourier interpolation, is a resampling technique where a signal is converted to the frequency domain, padded with zeros and then reconverted to the time domain.

[Function File] interpft (x, n) [Function File] interpft (x, n, dim)

Fourier interpolation. If x is a vector, then x is resampled with n points. The data in x is assumed to be equispaced. If x is a matrix or an N-dimensional array, the interpolation is performed on each column of x. If dim is specified, then interpolate along the dimension dim. interpft assumes that the interpolated function is periodic, and so assumptions are made about the endpoints of the interpolation.

See also: [interp1], page 617.

There are two significant limitations on Fourier interpolation. First, the function signal is assumed to be periodic, and so non-periodic signals will be poorly represented at the edges. Second, both the signal and its interpolation are required to be sampled at equispaced points. An example of the use of interpft is

t = 0 : 0.3 : pi; dt = t(2)-t(1);
n = length (t); k = 100;
ti = t(1) + [0 : k-1]*dt*n/k;
y = sin (4*t + 0.3) .* cos (3*t - 0.1);
yp = sin (4*ti + 0.3) .* cos (3*ti - 0.1);
plot (ti, yp, "g", ti, interp1 (t, y, ti, "spline"), "b", ...
      ti, interpft (y, k), "c", t, y, "r+");
legend ("sin(4t+0.3)cos(3t-0.1)", "spline", "interpft", "data");

which demonstrates the poor behavior of Fourier interpolation for non-periodic functions, as can be seen in Figure 29.3.

Figure 29.3: Comparison of interp1 and interpft for non-periodic data

In addition, the support functions spline and lookup that underlie the interp1 function can be called directly.

[Function File] pp = spline (x, y) [Function File] yi = spline (x, y, xi)

Return the cubic spline interpolant of points x and y. When called with two arguments, return the piecewise polynomial pp that may be used with ppval to evaluate the polynomial at specific points. When called with a third input argument, spline evaluates the spline at the points xi. The third calling

form spline (x, y, xi) is equivalent to ppval (spline (x, y), xi).

The variable x must be a vector of length n. y can be either a vector or array. If y is a vector it must have a length of either n or n + 2. If the length of y is n, then the "not-a-knot" end condition is used. If the length of y is n + 2, then the first and last values of the vector y are the values of the first derivative of the cubic spline at the endpoints. If y is an array, then the size of y must have the form

[s_{1}, s_{2}, ⋯, s_{k}, n]

or

[s_{1}, s_{2}, ⋯, s_{k}, n + 2].

The array is reshaped internally to a matrix where the leading dimension is given by

s_{1}s_{2} ⋯ s_{k}

and each row of this matrix is then treated separately. Note that this is exactly opposite to interp1 but is done for matlab compatibility.

See also: [pchip], page 647, [ppval], page 614, [mkpp], page 614, [unmkpp], page 614.

29.2 Multi-dimensional Interpolation

There are three multi-dimensional interpolation functions in Octave, with similar capabilities. Methods using Delaunay tessellation are described in Section 30.4 [Interpolation on Scattered Data], page 637.

[Function File] zi = interp2 (x, y, z, xi, yi) [Function File] zi = interp2 (Z, xi, yi) [Function File] zi = interp2 (Z, n) [Function File] zi = interp2 (…, method)

[Function File] zi = interp2 (. . ., method, extrapval)

Two-dimensional interpolation. x, y and z describe a surface function. If x and y are vectors their length must correspondent to the size of z. x and y must be monotonic. If they are matrices they must have the meshgrid format.

interp2 (x, y, Z, xi, yi, ...)

Returns a matrix corresponding to the points described by the matrices xi, yi. If the last argument is a string, the interpolation method can be specified. The method can be "linear", "nearest" or "cubic". If it is omitted "linear" interpolation is assumed.

interp2 (z, xi, yi)
          Assumes x = 1:rows (z) and y = 1:columns (z)
interp2 (z, n)

Interleaves the matrix z n-times. If n is omitted a value of n = 1 is assumed.

The variable method defines the method to use for the interpolation. It can take one of the following values

"nearest"

Return the nearest neighbor.

"linear" Linear interpolation from nearest neighbors.

"pchip" Piecewise cubic Hermite interpolating polynomial.

"cubic" Cubic interpolation from four nearest neighbors.

"spline" Cubic spline interpolation—smooth first and second derivatives throughout the curve.

If a scalar value extrapval is defined as the final value, then values outside the mesh as set to this value. Note that in this case method must be defined as well. If extrapval is not defined then NA is assumed.

See also: [interp1], page 617.

[Function File] vi = interp3 (x, y, z, v, xi, yi, zi) [Function File] vi = interp3 (v, xi, yi, zi) [Function File] vi = interp3 (v, m) [Function File] vi = interp3 (v) [Function File] vi = interp3 (…, method)

[Function File] vi = interp3 (. . ., method, extrapval)

Perform 3-dimensional interpolation. Each element of the 3-dimensional array v represents a value at a location given by the parameters x, y, and z. The parameters x, x, and z are either 3-dimensional arrays of the same size as the array v in the "meshgrid" format or vectors. The parameters xi, etc. respect a similar format to x, etc., and they represent the points at which the array vi is interpolated. If x, y, z are omitted, they are assumed to be x = 1 : size (v, 2), y = 1 : size (v,

1) and z = 1 : size (v, 3). If m is specified, then the interpolation adds a point half way between each of the interpolation points. This process is performed m times. If only v is specified, then m is assumed to be 1. Method is one of:

"nearest"

Return the nearest neighbor.

"linear" Linear interpolation from nearest neighbors.

"cubic" Cubic interpolation from four nearest neighbors (not implemented yet).

"spline" Cubic spline interpolation—smooth first and second derivatives throughout the curve.

The default method is "linear". If extrap is the string "extrap", then extrapolate values beyond the endpoints. If extrap is a number, replace values beyond the endpoints with that number. If extrap is missing, assume NA.

See also: [interp1], page 617, [interp2], page 621, [spline], page 620, [meshgrid], page 302.

[Function File] vi = interpn (x1, x2, …, v, y1, y2, …) [Function File] vi = interpn (v, y1, y2, …) [Function File] vi = interpn (v, m) [Function File] vi = interpn (v) [Function File] vi = interpn (…, method)

[Function File] vi = interpn (. . ., method, extrapval)

Perform n-dimensional interpolation, where n is at least two. Each element of the n-dimensional array v represents a value at a location given by the parameters x1, x2, …, xn. The parameters x1, x2, …, xn are either n-dimensional arrays of the same size as the array v in the "ndgrid" format or vectors. The parameters y1, etc. respect a similar format to x1, etc., and they represent the points at which the array vi is interpolated. If x1, …, xn are omitted, they are assumed to be x1 = 1 : size (v, 1), etc. If m is specified, then the interpolation adds a point half way between each of the interpolation points. This process is performed m times. If only v is specified, then m is assumed to be 1. Method is one of:

"nearest"

Return the nearest neighbor.

"linear" Linear interpolation from nearest neighbors.

"cubic" Cubic interpolation from four nearest neighbors (not implemented yet).

"spline" Cubic spline interpolation—smooth first and second derivatives throughout the curve.

The default method is "linear". If extrapval is the scalar value, use it to replace the values beyond the endpoints with that number. If extrapval is missing, assume NA.

See also: [interp1], page 617, [interp2], page 621, [spline], page 620, [ndgrid], page 303.

A significant difference between interpn and the other two multi-dimensional interpolation functions is the fashion in which the dimensions are treated. For interp2 and interp3, the y-axis is considered to be the columns of the matrix, whereas the x-axis corresponds to the rows of the array. As Octave indexes arrays in column major order, the first dimension of any array is the columns, and so interpn effectively reverses the ’x’ and ’y’ dimensions. Consider the example,

x = y = z = -1:1;
f = @(x,y,z) x.^2 - y - z.^2;
[xx, yy, zz] = meshgrid (x, y, z);
v = f (xx,yy,zz);
xi = yi = zi = -1:0.1:1;
[xxi, yyi, zzi] = meshgrid (xi, yi, zi);
vi = interp3 (x, y, z, v, xxi, yyi, zzi, "spline");
[xxi, yyi, zzi] = ndgrid (xi, yi, zi);
vi2 = interpn (x, y, z, v, xxi, yyi, zzi, "spline");
mesh (zi, yi, squeeze (vi2(1,:,:)));

where vi and vi2 are identical. The reversal of the dimensions is treated in the meshgrid and ndgrid functions respectively. The result of this code can be seen in Figure 29.4.

Figure 29.4: Demonstration of the use of interpn In additional the support function bicubic that underlies the cubic interpolation of

interp2 function can be called directly.

[Function File] zi = bicubic (x, y, z, xi, yi, extrapval)

Return a matrix zi corresponding to the bicubic interpolations at xi and yi of the data supplied as x, y and z. Points outside the grid are set to extrapval.

See http://wiki.woodpecker.org.cn/moin/Octave/Bicubic for further informa-

tion.

See also: [interp2], page 621.

30 Geometry

Much of the geometry code in Octave is based on the Qhull library^{1}. Some of the documentation for Qhull, particularly for the options that can be passed to delaunay, voronoi and convhull, etc., is relevant to Octave users.

30.1 Delaunay Triangulation

The Delaunay triangulation is constructed from a set of circum-circles. These circum-circles are chosen so that there are at least three of the points in the set to triangulation on the circumference of the circum-circle. None of the points in the set of points falls within any of the circum-circles.

In general there are only three points on the circumference of any circum-circle. However, in some cases, and in particular for the case of a regular grid, 4 or more points can be on a single circum-circle. In this case the Delaunay triangulation is not unique.

[Function File] delaunay (x, y) [Function File] delaunay (x) [Function File] delaunay (…, options) [Function File] tri = delaunay (…)

Compute the Delaunay triangulation for a 2-D set of points. The return value tri is a set of triangles which satisfies the Delaunay circum-circle criterion, i.e., only a single data point from [x, y] is within the circum-circle of the defining triangle. The input x may also be a matrix with two columns where the first column contains x-data and the second y-data. The set of triangles tri is a matrix of size [n, 3]. Each row defines a triangle and the three columns are the three vertices of the triangle. The value of tri(i,j) is an index into x and y for the location of the j-th vertex of the i-th triangle. The optional last argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the

Qhull library for details http://www.qhull.org/html/qh-quick.htm#options. The
default options are {"Qt", "Qbb", "Qc", "Qz"}.

If options is not present or [] then the default arguments are used. Otherwise, options replaces the default argument list. To append user options to the defaults it is necessary to repeat the default arguments in options. Use a null string to pass no arguments. If no output argument is specified the resulting Delaunay triangulation is plotted along with the original set of points.

     x = rand (1, 10);
     y = rand (1, 10);
     T = delaunay (x, y);
     VX = [ x(T(:,1)); x(T(:,2)); x(T(:,3)); x(T(:,1)) ];
     VY = [ y(T(:,1)); y(T(:,2)); y(T(:,3)); y(T(:,1)) ];
     axis ([0,1,0,1]);
     plot (VX, VY, "b", x, y, "r*");

1 Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., The Quickhull Algorithm for Convex Hulls, ACM

Trans. on Mathematical Software, 22(4):469–483, Dec 1996, http://www.qhull.org

See also: [delaunay3], page 626, [delaunayn], page 626, [convhull], page 635, [voronoi], page 632, [triplot], page 627, [trimesh], page 628, [trisurf], page 628.

The 3- and N-dimensional extension of the Delaunay triangulation are given by delaunay3 and delaunayn respectively. delaunay3 returns a set of tetrahedra that satisfy the Delaunay circum-circle criteria. Similarly, delaunayn returns the N-dimensional simplex satisfying the Delaunay circum-circle criteria. The N-dimensional extension of a triangulation is called a tessellation.

[Function File] tetr = delaunay3 (x, y, z)

[Function File] tetr = delaunay3 (x, y, z, options)

Compute the Delaunay triangulation for a 3-D set of points. The return value tetr is a set of tetrahedrons which satisfies the Delaunay circum-circle criterion, i.e., only a single data point from [x, y, z] is within the circum-circle of the defining tetrahedron. The set of tetrahedrons tetr is a matrix of size [n, 4]. Each row defines a tetrahedron and the four columns are the four vertices of the tetrahedron. The value of tetr(i,j) is an index into x, y, z for the location of the j-th vertex of the i-th tetrahedron. An optional fourth argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the

Qhull library for details http://www.qhull.org/html/qh-quick.htm#options. The
default options are {"Qt", "Qbb", "Qc", "Qz"}.

If options is not present or [] then the default arguments are used. Otherwise, options replaces the default argument list. To append user options to the defaults it is necessary to repeat the default arguments in options. Use a null string to pass no arguments.

See also: [delaunay], page 625, [delaunayn], page 626, [convhull], page 635, [voronoi], page 632, [tetramesh], page 629.

[Function File] T = delaunayn (pts)

[Function File] T = delaunayn (pts, options)

Compute the Delaunay triangulation for an N-dimensional set of points. The Delaunay triangulation is a tessellation of the convex hull of a set of points such that no N-sphere defined by the N-triangles contains any other points from the set. The input matrix pts of size [n, dim] contains n points in a space of dimension dim. The return matrix T has size [m, dim+1]. Each row of T contains a set of indices back into the original set of points pts which describes a simplex of dimension dim. For example, a 2-D simplex is a triangle and 3-D simplex is a tetrahedron. An optional second argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the

Qhull library for details http://www.qhull.org/html/qh-quick.htm#options. The

default options depend on the dimension of the input:

  • 2-D and 3-D: options = {"Qt", "Qbb", "Qc", "Qz"}
  • 4-D and higher: options = {"Qt", "Qbb", "Qc", "Qx"}

If options is not present or [] then the default arguments are used. Otherwise, options replaces the default argument list. To append user options to the defaults it is necessary to repeat the default arguments in options. Use a null string to pass no arguments.

See also: [delaunay], page 625, [delaunay3], page 626, [convhulln], page 636, [voronoin], page 633, [trimesh], page 628, [tetramesh], page 629.

An example of a Delaunay triangulation of a set of points is

rand ("state", 2);
x = rand (10, 1);
y = rand (10, 1);
T = delaunay (x, y);
X = [ x(T(:,1)); x(T(:,2)); x(T(:,3)); x(T(:,1)) ];
Y = [ y(T(:,1)); y(T(:,2)); y(T(:,3)); y(T(:,1)) ];
axis ([0, 1, 0, 1]);
plot (X, Y, "b", x, y, "r*");

The result of which can be seen in Figure 30.1.

Figure 30.1: Delaunay triangulation of a random set of points

30.1.1 Plotting the Triangulation

Octave has the functions triplot, trimesh, and trisurf to plot the Delaunay triangulation of a 2-dimensional set of points. tetramesh will plot the triangulation of a 3-dimensional set of points.

[Function File] triplot (tri, x, y)

[Function File] triplot (tri, x, y, linespec)

[Function File] h = triplot (…)

Plot a 2-D triangular mesh. tri is typically the output of a Delaunay triangulation over the grid of x, y. Every row of tri represents one triangle and contains three indices into [x, y] which are the vertices of the triangles in the x-y plane.

The linestyle to use for the plot can be defined with the argument linespec of the same format as the plot command.

The optional return value h is a graphics handle to the created patch object.

See also: [plot], page 260, [trimesh], page 628, [trisurf], page 628, [delaunay], page 625.

[Function File] trimesh (tri, x, y, z, c) [Function File] trimesh (tri, x, y, z) [Function File] trimesh (tri, x, y) [Function File] trimesh (…, prop, val, …) [Function File] h = trimesh (…)

Plot a 3-D triangular wireframe mesh.

In contrast to mesh, which plots a mesh using rectangles, trimesh plots the mesh using triangles.

tri is typically the output of a Delaunay triangulation over the grid of x, y. Every row of tri represents one triangle and contains three indices into [x, y] which are the vertices of the triangles in the x-y plane. z determines the height above the plane of each vertex. If no z input is given then the triangles are plotted as a 2-D figure.

The color of the trimesh is computed by linearly scaling the z values to fit the range of the current colormap. Use caxis and/or change the colormap to control the appearance.

Optionally, the color of the mesh can be specified independently of z by supplying a color matrix, c. If z has N elements, then c should be an Nx1 vector for colormap data or an Nx3 matrix for RGB data.

Any property/value pairs are passed directly to the underlying patch object.

The optional return value h is a graphics handle to the created patch object.

See also: [mesh], page 292, [tetramesh], page 629, [triplot], page 627, [trisurf], page 628, [delaunay], page 625, [patch], page 337, [hidden], page 294.

[Function File] trisurf (tri, x, y, z, c) [Function File] trisurf (tri, x, y, z) [Function File] trisurf (…, prop, val, …) [Function File] h = trisurf (…)

Plot a 3-D triangular surface.

In contrast to surf, which plots a surface mesh using rectangles, trisurf plots the mesh using triangles.

tri is typically the output of a Delaunay triangulation over the grid of x, y. Every row of tri represents one triangle and contains three indices into [x, y] which are the vertices of the triangles in the x-y plane. z determines the height above the plane of each vertex.

The color of the trimesh is computed by linearly scaling the z values to fit the range of the current colormap. Use caxis and/or change the colormap to control the appearance.

Optionally, the color of the mesh can be specified independently of z by supplying a color matrix, c. If z has N elements, then c should be an Nx1 vector for colormap data or an Nx3 matrix for RGB data.

Any property/value pairs are passed directly to the underlying patch object.

The optional return value h is a graphics handle to the created patch object.

See also: [surf], page 295, [triplot], page 627, [trimesh], page 628, [delaunay], page 625, [patch], page 337, [shading], page 306.

[Function File] tetramesh (T, X) [Function File] tetramesh (T, X, C) [Function File] tetramesh (…, property, val, …) [Function File] h = tetramesh (…)

Display the tetrahedrons defined in the m-by-4 matrix T as 3-D patches.

T is typically the output of a Delaunay triangulation of a 3-D set of points. Every row of T contains four indices into the n-by-3 matrix X of the vertices of a tetrahedron. Every row in X represents one point in 3-D space.

The vector C specifies the color of each tetrahedron as an index into the current colormap. The default value is 1:m where m is the number of tetrahedrons; the indices are scaled to map to the full range of the colormap. If there are more tetrahedrons than colors in the colormap then the values in C are cyclically repeated.

Calling tetramesh (..., "property", "value", ...) passes all property/value

pairs directly to the patch function as additional arguments.

The optional return value h is a vector of patch handles where each handle represents one tetrahedron in the order given by T. A typical use case for h is to turn the respective patch "visible" property "on" or "off".

Type demo tetramesh to see examples on using tetramesh.

See also: [trimesh], page 628, [delaunay3], page 626, [delaunayn], page 626, [patch], page 337.

The difference between triplot, and trimesh or triplot, is that the former only plots the 2-dimensional triangulation itself, whereas the second two plot the value of a function f (x, y). An example of the use of the triplot function is

rand ("state", 2)
x = rand (20, 1);
y = rand (20, 1);
tri = delaunay (x, y);
triplot (tri, x, y);

which plots the Delaunay triangulation of a set of random points in 2-dimensions. The output of the above can be seen in Figure 30.2.

Figure 30.2: Delaunay triangulation of a random set of points

30.1.2 Identifying Points in Triangulation

It is often necessary to identify whether a particular point in the N-dimensional space is within the Delaunay tessellation of a set of points in this N-dimensional space, and if so which N-simplex contains the point and which point in the tessellation is closest to the desired point. The functions tsearch and dsearch perform this function in a triangulation, and tsearchn and dsearchn in an N-dimensional tessellation.

To identify whether a particular point represented by a vector p falls within one of the simplices of an N-simplex, we can write the Cartesian coordinates of the point in a parametric form with respect to the N-simplex. This parametric form is called the Barycentric Coordinates of the point. If the points defining the N-simplex are given by N + 1 vectors t(i,:), then the Barycentric coordinates defining the point p are given by

     p = sum (beta(1:N+1) * t(1:N+1),:)

where there are N + 1 values beta(i) that together as a vector represent the Barycentric coordinates of the point p. To ensure a unique solution for the values of beta(i) an additional criteria of

     sum (beta(1:N+1)) == 1

is imposed, and we can therefore write the above as

     p - t(end, :) = beta(1:end-1) * (t(1:end-1, :)
           - ones (N, 1) * t(end, :)

Solving for beta we can then write

     beta(1:end-1) = (p - t(end, :)) / (t(1:end-1, :)
           - ones (N, 1) * t(end, :))
     beta(end) = sum (beta(1:end-1))

which gives the formula for the conversion of the Cartesian coordinates of the point p to the Barycentric coordinates beta. An important property of the Barycentric coordinates is that for all points in the N-simplex

0 <= beta(i) <= 1

Therefore, the test in tsearch and tsearchn essentially only needs to express each point in terms of the Barycentric coordinates of each of the simplices of the N-simplex and test the values of beta. This is exactly the implementation used in tsearchn. tsearch is optimized for 2-dimensions and the Barycentric coordinates are not explicitly formed.

[Loadable Function] idx = tsearch (x, y, t, xi, yi)

Search for the enclosing Delaunay convex hull. For t = delaunay (x, y), finds the index in t containing the points (xi, yi). For points outside the convex hull, idx is NaN.

See also: [delaunay], page 625, [delaunayn], page 626.

[Function File] [idx, p] = tsearchn (x, t, xi)

Search for the enclosing Delaunay convex hull. For t = delaunayn (x), finds the index in t containing the points xi. For points outside the convex hull, idx is NaN. If requested tsearchn also returns the Barycentric coordinates p of the enclosing triangles.

See also: [delaunay], page 625, [delaunayn], page 626.

An example of the use of tsearch can be seen with the simple triangulation

x = [-1; -1; 1; 1];
y = [-1; 1; -1; 1];
tri = [1, 2, 3; 2, 3, 1];

consisting of two triangles defined by tri. We can then identify which triangle a point falls in like

tsearch (x, y, tri, -0.5, -0.5)
⇒ 1
tsearch (x, y, tri, 0.5, 0.5)
⇒ 2

and we can confirm that a point doesn’t lie within one of the triangles like

tsearch (x, y, tri, 2, 2)
⇒ NaN

The dsearch and dsearchn find the closest point in a tessellation to the desired point. The desired point does not necessarily have to be in the tessellation, and even if it the returned point of the tessellation does not have to be one of the vertexes of the N-simplex within which the desired point is found.

[Function File] idx = dsearch (x, y, tri, xi, yi)
[Function File] idx = dsearch (x, y, tri, xi, yi, s)

Return the index idx or the closest point in x, y to the elements [xi(:), yi(:)]. The variable s is accepted for compatibility but is ignored.

See also: [dsearchn], page 631, [tsearch], page 631.

[Function File] idx = dsearchn (x, tri, xi)
[Function File] idx = dsearchn (x, tri, xi, outval)

[Function File] idx = dsearchn (x, xi)

[Function File] [idx, d] = dsearchn (…)

Return the index idx or the closest point in x to the elements xi. If outval is supplied, then the values of xi that are not contained within one of the simplices tri are set to outval. Generally, tri is returned from delaunayn (x).

See also: [dsearch], page 631, [tsearch], page 631.

An example of the use of dsearch, using the above values of x, y and tri is

dsearch (x, y, tri, -2, -2)
⇒ 1

If you wish the points that are outside the tessellation to be flagged, then dsearchn can be used as

dsearchn ([x, y], tri, [-2, -2], NaN)
⇒ NaN
dsearchn ([x, y], tri, [-0.5, -0.5], NaN)
⇒ 1

where the point outside the tessellation are then flagged with NaN.

30.2 Voronoi Diagrams

A Voronoi diagram or Voronoi tessellation of a set of points s in an N-dimensional space, is the tessellation of the N-dimensional space such that all points in v(p), a partitions of the tessellation where p is a member of s, are closer to p than any other point in s. The Voronoi diagram is related to the Delaunay triangulation of a set of points, in that the vertexes of the Voronoi tessellation are the centers of the circum-circles of the simplices of the Delaunay tessellation.

[Function File] voronoi (x, y) [Function File] voronoi (x, y, options)

[Function File] voronoi (…, "linespec")

[Function File] voronoi (hax, …) [Function File] h = voronoi (…) [Function File] [vx, vy] = voronoi (…)

Plot the Voronoi diagram of points (x, y). The Voronoi facets with points at infinity are not drawn.

If "linespec" is given it is used to set the color and line style of the plot. If an axis graphics handle hax is supplied then the Voronoi diagram is drawn on the specified axis rather than in a new figure.

The options argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the Qhull library

for details http://www.qhull.org/html/qh-quick.htm#options.

If a single output argument is requested then the Voronoi diagram will be plotted and a graphics handle h to the plot is returned. [vx, vy] = voronoi (…) returns the Voronoi vertices instead of plotting the diagram.

     x = rand (10, 1);
     y = rand (size (x));
     h = convhull (x, y);
     [vx, vy] = voronoi (x, y);
     plot (vx, vy, "-b", x, y, "o", x(h), y(h), "-g");
     legend ("", "points", "hull");

See also: [voronoin], page 633, [delaunay], page 625, [convhull], page 635.

[Function File] [C, F] = voronoin (pts)
[Function File] [C, F] = voronoin (pts, options)

Compute N-dimensional Voronoi facets. The input matrix pts of size [n, dim] contains n points in a space of dimension dim. C contains the points of the Voronoi facets. The list F contains, for each facet, the indices of the Voronoi points.

An optional second argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the Qhull

library for details http://www.qhull.org/html/qh-quick.htm#options.

The default options depend on the dimension of the input:

• 2-D and 3-D: options = {"Qbb"}

• 4-D and higher: options = {"Qbb", "Qx"}

If options is not present or [] then the default arguments are used. Otherwise, options replaces the default argument list. To append user options to the defaults it is necessary to repeat the default arguments in options. Use a null string to pass no arguments.

See also: [voronoi], page 632, [convhulln], page 636, [delaunayn], page 626.

An example of the use of voronoi is

rand ("state",9);
x = rand (10,1);
y = rand (10,1);
tri = delaunay (x, y);
[vx, vy] = voronoi (x, y, tri);
triplot (tri, x, y, "b");
hold on;
plot (vx, vy, "r");

The result of which can be seen in Figure 30.3. Note that the circum-circle of one of the triangles has been added to this figure, to make the relationship between the Delaunay tessellation and the Voronoi diagram clearer.

Figure 30.3: Delaunay triangulation and Voronoi diagram of a random set of points Additional information about the size of the facets of a Voronoi diagram, and which

points of a set of points is in a polygon can be had with the polyarea and inpolygon functions respectively.

[Function File] polyarea (x, y) [Function File] polyarea (x, y, dim)

Determine area of a polygon by triangle method. The variables x and y define the vertex pairs, and must therefore have the same shape. They can be either vectors or arrays. If they are arrays then the columns of x and y are treated separately and an area returned for each. If the optional dim argument is given, then polyarea works along this dimension of the arrays x and y.

An example of the use of polyarea might be

rand ("state", 2);
x = rand (10, 1);
y = rand (10, 1);
[c, f] = voronoin ([x, y]);
af = zeros (size (f));
for i = 1 : length (f)
  af(i) = polyarea (c (f {i, :}, 1), c (f {i, :}, 2));
endfor

Facets of the Voronoi diagram with a vertex at infinity have infinity area. A simplified version of polyarea for rectangles is available with rectint

[Function File] area = rectint (a, b)

Compute the area of intersection of rectangles in a and rectangles in b. Rectangles are defined as [x y width height] where x and y are the minimum values of the two orthogonal dimensions.

If a or b are matrices, then the output, area, is a matrix where the i-th row corresponds to the i-th row of a and the j-th column corresponds to the j-th row of b.

See also: [polyarea], page 634.

[Function File] [in, on] = inpolygon (x, y, xv, yv)

For a polygon defined by vertex points (xv, yv), determine if the points (x, y) are inside or outside the polygon. The variables x, y, must have the same dimension. The optional output on gives the points that are on the polygon.

An example of the use of inpolygon might be

randn ("state", 2);
x = randn (100, 1);
y = randn (100, 1);
vx = cos (pi * [-1 : 0.1: 1]);
vy = sin (pi * [-1 : 0.1 : 1]);
in = inpolygon (x, y, vx, vy);
plot (vx, vy, x(in), y(in), "r+", x(!in), y(!in), "bo");
axis ([-2, 2, -2, 2]);

The result of which can be seen in Figure 30.4.

Figure 30.4: Demonstration of the inpolygon function to determine the points inside a polygon

30.3 Convex Hull

The convex hull of a set of points is the minimum convex envelope containing all of the points. Octave has the functions convhull and convhulln to calculate the convex hull of 2-dimensional and N-dimensional sets of points.

[Function File] H = convhull (x, y)

[Function File] H = convhull (x, y, options)

Compute the convex hull of the set of points defined by the arrays x and y. The hull H is an index vector into the set of points and specifies which points form the enclosing hull.

An optional third argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the

Qhull library for details http://www.qhull.org/html/qh-quick.htm#options. The

default option is {"Qt"}.

If options is not present or [] then the default arguments are used. Otherwise, options replaces the default argument list. To append user options to the defaults it is necessary to repeat the default arguments in options. Use a null string to pass no arguments.

See also: [convhulln], page 636, [delaunay], page 625, [voronoi], page 632.

[Loadable Function] h = convhulln (pts)

[Loadable Function] h = convhulln (pts, options)

[Loadable Function] [h, v] = convhulln (…)

Compute the convex hull of the set of points pts which is a matrix of size [n, dim] containing n points in a space of dimension dim. The hull h is an index vector into the set of points and specifies which points form the enclosing hull.

An optional second argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the

Qhull library for details http://www.qhull.org/html/qh-quick.htm#options. The

default options depend on the dimension of the input:

• 2D, 3D, 4D: options = {"Qt"}

• 5D and higher: options = {"Qt", "Qx"}

If options is not present or [] then the default arguments are used. Otherwise, options replaces the default argument list. To append user options to the defaults it is necessary to repeat the default arguments in options. Use a null string to pass no arguments.

If the second output v is requested the volume of the enclosing convex hull is calculated.

See also: [convhull], page 635, [delaunayn], page 626, [voronoin], page 633.

An example of the use of convhull is

x = -3:0.05:3;
y = abs (sin (x));
k = convhull (x, y);
plot (x(k), y(k), "r-", x, y, "b+");
axis ([-3.05, 3.05, -0.05, 1.05]);

The output of the above can be seen in Figure 30.5.

Figure 30.5: The convex hull of a simple set of points

30.4 Interpolation on Scattered Data

An important use of the Delaunay tessellation is that it can be used to interpolate from scattered data to an arbitrary set of points. To do this the N-simplex of the known set of points is calculated with delaunay, delaunay3 or delaunayn. Then the simplices in to which the desired points are found are identified. Finally the vertices of the simplices are used to interpolate to the desired points. The functions that perform this interpolation are

griddata, griddata3 and griddatan.

[Function File] zi = griddata (x, y, z, xi, yi)

[Function File] zi = griddata (x, y, z, xi, yi, method)
[Function File] [xi, yi, zi] = griddata (. . .)

Generate a regular mesh from irregular data using interpolation. The function is defined by z = f (x, y). Inputs x, y, z are vectors of the same length or x, y are vectors and z is matrix. The interpolation points are all (xi, yi). If xi, yi are vectors then they are made into a 2-D mesh. The interpolation method can be "nearest", "cubic" or "linear". If method is omitted it defaults to "linear".

See also: [griddata3], page 637, [griddatan], page 638, [delaunay], page 625.

[Function File] vi = griddata3 (x, y, z, v, xi, yi, zi)
[Function File] vi = griddata3 (x, y, z, v, xi, yi, zi, method)
[Function File] vi = griddata3 (x, y, z, v, xi, yi, zi, method, options)

Generate a regular mesh from irregular data using interpolation. The function is defined by v = f (x, y, z). The interpolation points are specified by xi, yi, zi. The interpolation method can be "nearest" or "linear". If method is omitted it defaults to "linear".

The optional argument options is passed directly to Qhull when computing the Delaunay triangulation used for interpolation. See delaunayn for information on the defaults and how to pass different values.

See also: [griddata], page 637, [griddatan], page 638, [delaunayn], page 626.

[Function File] yi = griddatan (x, y, xi)

[Function File] yi = griddatan (x, y, xi, method)
[Function File] yi = griddatan (x, y, xi, method, options)

Generate a regular mesh from irregular data using interpolation. The function is defined by y = f (x). The interpolation points are all xi. The interpolation method can be "nearest" or "linear". If method is omitted it defaults to "linear". The optional argument options is passed directly to Qhull when computing the Delaunay triangulation used for interpolation. See delaunayn for information on the defaults and how to pass different values.

See also: [griddata], page 637, [griddata3], page 637, [delaunayn], page 626.

An example of the use of the griddata function is

rand ("state", 1);
x = 2*rand (1000,1) - 1;
y = 2*rand (size (x)) - 1;
z = sin (2*(x.^2+y.^2));
[xx,yy] = meshgrid (linspace (-1,1,32));
griddata (x,y,z,xx,yy);

that interpolates from a random scattering of points, to a uniform grid. The output of the above can be seen in Figure 30.6.

Figure 30.6: Interpolation from a scattered data to a regular grid

31 Signal Processing

This chapter describes the signal processing and fast Fourier transform functions available in Octave. Fast Fourier transforms are computed with the fftw or fftpack libraries depending on how Octave is built.

[Built-in Function] fft (x) [Built-in Function] fft (x, n) [Built-in Function] fft (x, n, dim)

Compute the discrete Fourier transform of A using a Fast Fourier Transform (FFT) algorithm. The FFT is calculated along the first non-singleton dimension of the array. Thus if x is a matrix, fft (x) computes the FFT for each column of x. If called with two arguments, n is expected to be an integer specifying the number of elements of x to use, or an empty matrix to specify that its value should be ignored. If n is larger than the dimension along which the FFT is calculated, then x is resized and padded with zeros. Otherwise, if n is smaller than the dimension along which the FFT is calculated, then x is truncated. If called with three arguments, dim is an integer specifying the dimension of the matrix along which the FFT is performed

See also: [ifft], page 639, [fft2], page 639, [fftn], page 640, [fftw], page 640.

[Built-in Function] ifft (x) [Built-in Function] ifft (x, n) [Built-in Function] ifft (x, n, dim)

Compute the inverse discrete Fourier transform of A using a Fast Fourier Transform (FFT) algorithm. The inverse FFT is calculated along the first non-singleton dimension of the array. Thus if x is a matrix, fft (x) computes the inverse FFT for each column of x. If called with two arguments, n is expected to be an integer specifying the number of elements of x to use, or an empty matrix to specify that its value should be ignored. If n is larger than the dimension along which the inverse FFT is calculated, then x is resized and padded with zeros. Otherwise, if n is smaller than the dimension along which the inverse FFT is calculated, then x is truncated. If called with three arguments, dim is an integer specifying the dimension of the matrix along which the inverse FFT is performed

See also: [fft], page 639, [ifft2], page 640, [ifftn], page 640, [fftw], page 640.

[Built-in Function] fft2 (A) [Built-in Function] fft2 (A, m, n)

Compute the two-dimensional discrete Fourier transform of A using a Fast Fourier Transform (FFT) algorithm. The optional arguments m and n may be used specify the number of rows and columns of A to use. If either of these is larger than the size of A, A is resized and padded with zeros.

If A is a multi-dimensional matrix, each two-dimensional sub-matrix of A is treated separately.

See also: ifft2, fft, fftn, fftw.

[Built-in Function] ifft2 (A) [Built-in Function] ifft2 (A, m, n)

Compute the inverse two-dimensional discrete Fourier transform of A using a Fast Fourier Transform (FFT) algorithm. The optional arguments m and n may be used specify the number of rows and columns of A to use. If either of these is larger than the size of A, A is resized and padded with zeros. If A is a multi-dimensional matrix, each two-dimensional sub-matrix of A is treated separately

See also: fft2, ifft, ifftn, fftw.

[Built-in Function] fftn (A) [Built-in Function] fftn (A, size)

Compute the N-dimensional discrete Fourier transform of A using a Fast Fourier Transform (FFT) algorithm. The optional vector argument size may be used specify the dimensions of the array to be used. If an element of size is smaller than the corresponding dimension of A, then the dimension of A is truncated prior to performing the FFT. Otherwise, if an element of size is larger than the corresponding dimension then A is resized and padded with zeros.

See also: [ifftn], page 640, [fft], page 639, [fft2], page 639, [fftw], page 640.

[Built-in Function] ifftn (A) [Built-in Function] ifftn (A, size)

Compute the inverse N-dimensional discrete Fourier transform of A using a Fast Fourier Transform (FFT) algorithm. The optional vector argument size may be used specify the dimensions of the array to be used. If an element of size is smaller than the corresponding dimension of A, then the dimension of A is truncated prior to performing the inverse FFT. Otherwise, if an element of size is larger than the corresponding dimension then A is resized and padded with zeros.

See also: [fftn], page 640, [ifft], page 639, [ifft2], page 640, [fftw], page 640.

Octave uses the fftw libraries to perform FFT computations. When Octave starts up and initializes the fftw libraries, they read a system wide file (on a Unix system, it is typically ‘/etc/fftw/wisdom’) that contains information useful to speed up FFT computations. This information is called the wisdom. The system-wide file allows wisdom to be shared between all applications using the fftw libraries.

Use the fftw function to generate and save wisdom. Using the utilities provided together with the fftw libraries (fftw-wisdom on Unix systems), you can even add wisdom generated by Octave to the system-wide wisdom file.

[Loadable Function] method = fftw ("planner") [Loadable Function] fftw ("planner", method) [Loadable Function] wisdom = fftw ("dwisdom") [Loadable Function] fftw ("dwisdom", wisdom) [Loadable Function] fftw ("threads", nthreads) [Loadable Function] nthreads = fftw ("threads")

Manage fftw wisdom data. Wisdom data can be used to significantly accelerate the calculation of the FFTs, but implies an initial cost in its calculation. When the fftw libraries are initialized, they read a system wide wisdom file (typically in ‘/etc/fftw/wisdom’), allowing wisdom to be shared between applications other than Octave. Alternatively, the fftw function can be used to import wisdom. For example,

     wisdom = fftw ("dwisdom")

will save the existing wisdom used by Octave to the string wisdom. This string can then be saved to a file and restored using the save and load commands respectively. This existing wisdom can be re-imported as follows

     fftw ("dwisdom", wisdom)

If wisdom is an empty string, then the wisdom used is cleared. During the calculation of Fourier transforms further wisdom is generated. The fashion in which this wisdom is generated is also controlled by the fftw function. There are five different manners in which the wisdom can be treated:

"estimate"

Specifies that no run-time measurement of the optimal means of calculating a particular is performed, and a simple heuristic is used to pick a (probably sub-optimal) plan. The advantage of this method is that there is little or no overhead in the generation of the plan, which is appropriate for a Fourier transform that will be calculated once.

"measure"

In this case a range of algorithms to perform the transform is considered and the best is selected based on their execution time.

"patient"

Similar to "measure", but a wider range of algorithms is considered.

"exhaustive"

Like "measure", but all possible algorithms that may be used to treat the transform are considered.

"hybrid" As run-time measurement of the algorithm can be expensive, this is a compromise where "measure" is used for transforms up to the size of 8192 and beyond that the "estimate" method is used.

The default method is "estimate". The current method can be queried with

     method = fftw ("planner")

or set by using

     fftw ("planner", method)

Note that calculated wisdom will be lost when restarting Octave. However, the wisdom data can be reloaded if it is saved to a file as described above. Saved wisdom files should not be used on different platforms since they will not be efficient and the point of calculating the wisdom is lost. The number of threads used for computing the plans and executing the transforms can be set with

     fftw ("threads", NTHREADS)

Note that octave must be compiled with multi-threaded fftw support for this feature. The number of processors available to the current process is used per default.

See also: [fft], page 639, [ifft], page 639, [fft2], page 639, [ifft2], page 640, [fftn], page 640, [ifftn], page 640.

[Function File] fftconv (x, y) [Function File] fftconv (x, y, n)

Convolve two vectors using the FFT for computation.

c = fftconv (x, y) returns a vector of length equal to length (x) + length (y) -

1. If x and y are the coefficient vectors of two polynomials, the returned value is the coefficient vector of the product polynomial. The computation uses the FFT by calling the function fftfilt. If the optional argument n is specified, an N-point FFT is used.

See also: [deconv], page 604, [conv], page 603, [conv2], page 604.

[Function File] fftfilt (b, x) [Function File] fftfilt (b, x, n)

With two arguments, fftfilt filters x with the FIR filter b using the FFT. Given the optional third argument, n, fftfilt uses the overlap-add method to filter x with b using an N-point FFT. The FFT size must be an even power of 2 and must be greater than or equal to the length of b. If the specified n does not meet these criteria, it is automatically adjusted to the nearest value that does. If x is a matrix, filter each column of the matrix.

See also: [filter], page 642, [filter2], page 643.

[Built-in Function] y = filter (b, a, x) [Built-in Function] [y, sf] = filter (b, a, x, si) [Built-in Function] [y, sf] = filter (b, a, x, [], dim) [Built-in Function] [y, sf] = filter (b, a, x, si, dim)

Return the solution to the following linear, time-invariant difference equation:

∑_{k=0}^{N} a_{k+1}y_{n−k} = ∑_{k=0}^{M} b_{k+1}x_{n−k}, 1 ≤ n ≤ P

where a ∈ ℜ^{N−1}, b ∈ ℜ^{M−1}, and x ∈ ℜ^{P}. The result is calculated over the first non-singleton dimension of x or over dim if supplied. An equivalent form of the equation is:

y_{n} = − ∑_{k=1}^{N} c_{k+1}y_{n−k} + ∑_{k=0}^{M} d_{k+1}x_{n−k}, 1 ≤ n ≤ P

where c = a/a_{1} and d = b/a_{1}. If the fourth argument si is provided, it is taken as the initial state of the system and the final state is returned as sf. The state vector is a column vector whose length is equal to the length of the longest coefficient vector minus one. If si is not supplied, the initial state vector is set to all zeros. In terms of the Z Transform, y is the result of passing the discrete- time signal x through a system characterized by the following rational system function:

H(z) = \frac{∑_{k=0}^{M} d_{k+1}z^{−k}}{1 + ∑_{k+1}^{N} c_{k+1}z^{−k}}

See also: [filter2], page 643, [fftfilt], page 642, [freqz], page 643.

[Function File] y = filter2 (b, x) [Function File] y = filter2 (b, x, shape)

Apply the 2-D FIR filter b to x. If the argument shape is specified, return an array of the desired shape. Possible values are:

"full" pad x with zeros on all sides before filtering.

"same" unpadded x (default)

"valid" trim x after filtering so edge effects are no included.

Note this is just a variation on convolution, with the parameters reversed and b rotated 180 degrees.

See also: [conv2], page 604.

[Function File] [h, w] = freqz (b, a, n, "whole") [Function File] [h, w] = freqz (b) [Function File] [h, w] = freqz (b, a) [Function File] [h, w] = freqz (b, a, n) [Function File] h = freqz (b, a, w) [Function File] [...] = freqz (…, Fs) [Function File] freqz (…)

Return the complex frequency response h of the rational IIR filter whose numerator and denominator coefficients are b and a, respectively. The response is evaluated at n angular frequencies between 0 and 2π. The output value w is a vector of the frequencies. If a is omitted, the denominator is assumed to be 1 (this corresponds to a simple FIR filter). If n is omitted, a value of 512 is assumed. For fastest computation, n should factor into a small number of small primes. If the fourth argument, "whole", is omitted the response is evaluated at frequencies between 0 and π.

freqz (b, a, w)

Evaluate the response at the specific frequencies in the vector w. The values for w are measured in radians.

[...] = freqz (..., Fs)

Return frequencies in Hz instead of radians assuming a sampling rate Fs. If you are evaluating the response at specific frequencies w, those frequencies should be requested in Hz rather than radians.

freqz (...)

Plot the magnitude and phase response of h rather than returning them.

See also: [freqz plot], page 644.

[Function File] freqz_plot (w, h)

[Function File] freqz_plot (w, h, freq_norm)

Plot the magnitude and phase response of h. If the optional freq norm argument is true, the frequency vector w is in units of normalized radians. If freq norm is false, or not given, then w is measured in Hertz.

See also: [freqz], page 643.

[Function File] sinc (x)

Return sin(πx)/(πx).

[Function File] b = unwrap (x) [Function File] b = unwrap (x, tol) [Function File] b = unwrap (x, tol, dim)

Unwrap radian phases by adding multiples of 2*pi as appropriate to remove jumps greater than tol. tol defaults to pi. Unwrap will work along the dimension dim. If dim is unspecified it defaults to the first non-singleton dimension.

[Function File] [a, b] = arch_fit (y, x, p, iter, gamma, a0, b0)

Fit an ARCH regression model to the time series y using the scoring algorithm in Engle’s original ARCH paper. The model is

     y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t),
     h(t) = a(1) + a(2) * e(t-1)^2 + ... + a(p+1) * e(t-p)^2

in which e(t) is N(0, h(t)), given a time-series vector y up to time t − 1 and a matrix of (ordinary) regressors x up to t. The order of the regression of the residual variance is specified by p. If invoked as arch_fit (y, k, p) with a positive integer k, fit an ARCH(k, p) process, i.e., do the above with the t-th row of x given by

     [1, y(t-1), ..., y(t-k)]

Optionally, one can specify the number of iterations iter, the updating factor gamma, and initial values a0 and b0 for the scoring algorithm.

[Function File] arch_rnd (a, b, t)

Simulate an ARCH sequence of length t with AR coefficients b and CH coefficients a. I.e., the result y(t) follows the model

      y(t) = b(1) + b(2) * y(t-1) + ... + b(lb) * y(t-lb+1) + e(t),

where e(t), given y up to time t − 1, is N(0, h(t)), with

      h(t) = a(1) + a(2) * e(t-1)^2 + ... + a(la) * e(t-la+1)^2
[Function File] [pval, lm] = arch_test (y, x, p)

For a linear regression model

     y = x * b + e

perform a Lagrange Multiplier (LM) test of the null hypothesis of no conditional heteroscedascity against the alternative of CH(p). I.e., the model is

     y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t),

given y up to t − 1 and x up to t, e(t) is N(0, h(t)) with

     h(t) = v + a(1) * e(t-1)^2 + ... + a(p) * e(t-p)^2,

and the null is a(1) == … == a(p) == 0. If the second argument is a scalar integer, k, perform the same test in a linear autoregression model of order k, i.e., with

     [1, y(t-1), ..., y(t-k)]

as the t-th row of x. Under the null, LM approximately has a chisquare distribution with p degrees of freedom and pval is the p-value (1 minus the CDF of this distribution at LM) of the test. If no output argument is given, the p-value is displayed.

[Function File] arma_rnd (a, b, v, t, n)

Return a simulation of the ARMA model

     x(n) = a(1) * x(n-1) + ... + a(k) * x(n-k)
          + e(n) + b(1) * e(n-1) + ... + b(l) * e(n-l)

in which k is the length of vector a, l is the length of vector b and e is Gaussian white noise with variance v. The function returns a vector of length t. The optional parameter n gives the number of dummy x(i) used for initialization, i.e., a sequence of length t+n is generated and x(n+1:t+n) is returned. If n is omitted, n = 100 is used.

[Function File] autoreg_matrix (y, k)

Given a time series (vector) y, return a matrix with ones in the first column and the first k lagged values of y in the other columns. I.e., for t > k, [1, y(t-1), ..., y(t-k)] is the t-th row of the result. The resulting matrix may be used as a regressor matrix in autoregressions.

[Function File] bartlett (m)

Return the filter coefficients of a Bartlett (triangular) window of length m. For a definition of the Bartlett window, see e.g., A. V. Oppenheim & R. W. Schafer, Discrete-Time Signal Processing.

[Function File] blackman (m)

Return the filter coefficients of a Blackman window of length m. For a definition of the Blackman window, see e.g., A. V. Oppenheim & R. W. Schafer, Discrete-Time Signal Processing.

[Function File] detrend (x, p)

If x is a vector, detrend (x, p) removes the best fit of a polynomial of order p from the data x. If x is a matrix, detrend (x, p) does the same for each column in x. The second argument is optional. If it is not specified, a value of 1 is assumed. This corresponds to removing a linear trend. The order of the polynomial can also be given as a string, in which case p must be either "constant" (corresponds to p=0) or "linear" (corresponds to p=1).

See also: [polyfit], page 607.

[Function File] [d, dd] = diffpara (x, a, b)

Return the estimator d for the differencing parameter of an integrated time series. The frequencies from [2 ∗ pi ∗ a/t, 2 ∗ pi ∗ b/T] are used for the estimation. If b is omitted, the interval [2 ∗ pi/T, 2 ∗ pi ∗ a/T] is used. If both b and a are omitted then a = 0.5 ∗ sqrt(T) and b = 1.5 ∗ sqrt(T) is used, where T is the sample size. If x is a matrix, the differencing parameter of each column is estimated. The estimators for all frequencies in the intervals described above is returned in dd. The value of d is simply the mean of dd. Reference: P.J. Brockwell & R.A. Davis. Time Series: Theory and Methods. Springer 1987.

[Function File] durbinlevinson (c, oldphi, oldv)

Perform one step of the Durbin-Levinson algorithm. The vector c specifies the autocovariances [gamma_0, ..., gamma_t] from lag 0 to t, oldphi specifies the coefficients based on c(t-1) and oldv specifies the corresponding error. If oldphi and oldv are omitted, all steps from 1 to t of the algorithm are performed.

[Function File] fftshift (x) [Function File] fftshift (x, dim)

Perform a shift of the vector x, for use with the fft and ifft functions, in order the move the frequency 0 to the center of the vector or matrix. If x is a vector of N elements corresponding to N time samples spaced by dt, then fftshift (fft (x)) corresponds to frequencies

     f = [ -(ceil((N-1)/2):-1:1)*df 0 (1:floor((N-1)/2))*df ]

where df = 1 / dt. If x is a matrix, the same holds for rows and columns. If x is an array, then the same holds along each dimension. The optional dim argument can be used to limit the dimension along which the permutation occurs.

[Function File] ifftshift (x) [Function File] ifftshift (x, dim)

Undo the action of the fftshift function. For even length x, fftshift is its own inverse, but odd lengths differ slightly.

[Function File] fractdiff (x, d)

Compute the fractional differences (1 − L)^{d}x where L denotes the lag-operator and d is greater than -1.

[Function File] hamming (m)

Return the filter coefficients of a Hamming window of length m. For a definition of the Hamming window, see e.g., A. V. Oppenheim & R. W. Schafer, Discrete-Time Signal Processing.

[Function File] hanning (m)

Return the filter coefficients of a Hanning window of length m. For a definition of this window type, see e.g., A. V. Oppenheim & R. W. Schafer, Discrete-Time Signal Processing.

[Function File] hurst (x)

Estimate the Hurst parameter of sample x via the rescaled range statistic. If x is a matrix, the parameter is estimated for every single column.

[Function File] pp = pchip (x, y) [Function File] yi = pchip (x, y, xi)

Return the Piecewise Cubic Hermite Interpolating Polynomial (pchip) of points x and y. If called with two arguments, return the piecewise polynomial pp that may be used with ppval to evaluate the polynomial at specific points. When called with a third input argument, pchip evaluates the pchip polynomial at the points xi. The third calling form is equivalent to ppval (pchip (x, y), xi). The variable x must be a strictly monotonic vector (either increasing or decreasing) of length n. y can be either a vector or array. If y is a vector then it must be the same length n as x. If y is an array then the size of y must have the form

[s_{1}, s_{2}, ⋯, s_{k}, n]

The array is reshaped internally to a matrix where the leading dimension is given by

s_{1}s_{2} ⋯ s_{k}

and each row of this matrix is then treated separately. Note that this is exactly opposite to interp1 but is done for matlab compatibility.

See also: [spline], page 620, [ppval], page 614, [mkpp], page 614, [unmkpp], page 614.

[Function File] [Pxx, w] = periodogram (x)

For a data matrix x from a sample of size n, return the periodogram. The angular frequency is returned in w.

[Pxx,w] = periodogram (x). [Pxx,w] = periodogram (x,win). [Pxx,w] = periodogram (x,win,nfft). [Pxx,f] = periodogram (x,win,nfft,Fs). [Pxx,f] = periodogram (x,win,nfft,Fs,"range").

• x: data; if real-valued a one-sided spectrum is estimated, if complex-valued or range indicates "twosided", the full spectrum is estimated.

• win: weight data with window, x.*win is used for further computation, if window is empty, a rectangular window is used.

• nfft: number of frequency bins, default max (256, 2.^ceil (log2 (length (x)))).

• Fs: sampling rate, default 1.

• range: "onesided" computes spectrum from [0..nfft/2+1]. "twosided" computes spectrum from [0..nfft-1]. These strings can appear at any position in the list input arguments after window.

• Pxx: one-, or two-sided power spectrum.

• w: angular frequency [0..2*pi) (two-sided) or [0..pi] one-sided.

• f: frequency [0..Fs) (two-sided) or [0..Fs/2] one-sided.

[Function File] sinetone (freq, rate, sec, ampl)

Return a sinetone of frequency freq with length of sec seconds at sampling rate rate and with amplitude ampl. The arguments freq and ampl may be vectors of common size. Defaults are rate = 8000, sec = 1 and ampl = 64.

[Function File] sinewave (m, n, d)

Return an m-element vector with i-th element given by sin (2 * pi * (i+d-1) / n). The default value for d is 0 and the default value for n is m.

[Function File] spectral_adf (c) [Function File] spectral_adf (c, win) [Function File] spectral_adf (c, win, b)

Return the spectral density estimator given a vector of autocovariances c, window name win, and bandwidth, b. The window name, e.g., "triangle" or "rectangle" is used to search for a function called win_lw. If win is omitted, the triangle window is used. If b is omitted, 1 / sqrt (length (x)) is used.

See also: [spectral xdf], page 648.

[Function File] spectral_xdf (x) [Function File] spectral_xdf (x, win) [Function File] spectral_xdf (x, win, b)

Return the spectral density estimator given a data vector x, window name win, and bandwidth, b.

The window name, e.g., "triangle" or "rectangle" is used to search for a function called win_sw. If win is omitted, the triangle window is used. If b is omitted, 1 / sqrt (length (x)) is used.

See also: [spectral adf], page 648.

[Function File] spencer (x)

Return Spencer’s 15 point moving average of each column of x.

[Function File] y = stft (x) [Function File] y = stft (x, win_size)

[Function File] y = stft (x, win_size, inc)
[Function File] y = stft (x, win_size, inc, num_coef)
[Function File] y = stft (x, win_size, inc, num_coef, win_type)

[Function File] [y, c] = stft (…)

Compute the short-time Fourier transform of the vector x with num coef coefficients by applying a window of win size data points and an increment of inc points. Before computing the Fourier transform, one of the following windows is applied:

"hanning"

win type = 1

"hamming"

win type = 2

"rectangle"

win type = 3

The window names can be passed as strings or by the win type number. The following defaults are used for unspecified arguments: win size = 80, inc = 24, num coef = 64, and win type = 1. y = stft (x, ...) returns the absolute values of the Fourier coefficients according to the num coef positive frequencies. [y, c] = stft (x, ...) returns the entire STFT-matrix y and a 3-element vector c containing the window size, increment, and window type, which is needed by the

synthesis function.

See also: [synthesis], page 649.

[Function File] x = synthesis (y, c)

Compute a signal from its short-time Fourier transform y and a 3-element vector c specifying window size, increment, and window type. The values y and c can be derived by

     [y, c] = stft (x , ...)

See also: [stft], page 649.

[Function File] [a, v] = yulewalker (c)

Fit an AR (p)-model with Yule-Walker estimates given a vector c of autocovariances

[gamma_0, ..., gamma_p].

Returns the AR coefficients, a, and the variance of white noise, v.

32 Image Processing

Since an image is basically a matrix, Octave is a very powerful environment for processing and analyzing images. To illustrate how easy it is to do image processing in Octave, the following example will load an image, smooth it by a 5-by-5 averaging filter, and compute the gradient of the smoothed image.

I = imread ("myimage.jpg");
S = conv2 (I, ones (5, 5) / 25, "same");
[Dx, Dy] = gradient (S);

In this example S contains the smoothed image, and Dx and Dy contains the partial spatial derivatives of the image.

32.1 Loading and Saving Images

The first step in most image processing tasks is to load an image into Octave which is done with the imread function. The imwrite function is the corresponding function for writing images to the disk.

In summary, most image processing code will follow the structure of this code

I = imread ("my_input_image.img");
J = process_my_image (I);
imwrite (J, "my_output_image.img");
[Function File] [img, map, alpha] = imread (filename)
[Function File] [...] = imread (filename, ext)

[Function File] [...] = imread (url) [Function File] [...] = imread (…, idx) [Function File] [...] = imread (…, param1, val1, …)

Read images from various file formats. Reads an image as a matrix from the file filename. If there is no file filename, and ext was specified, it will look for a file named filename and extension ext, i.e., a file named filename.ext. The size and class of the output depends on the format of the image. A color image is returned as an MxNx3 matrix. Gray-level and black-and-white images are of size MxN. Multipage images will have an additional 4th dimension. The bit depth of the image determines the class of the output: "uint8", "uint16" or "single" for gray and color, and "logical" for black and white. Note that indexed images always return the indexes for a colormap, independent if map is a requested output. To obtain the actual RGB image, use ind2rgb. When more than one indexed image is being read, map is obtained from the first. In some rare cases this may be incorrect and imfinfo can be used to obtain the colormap of each image. See the Octave manual for more information in representing images. Some file formats, such as TIFF and GIF, are able to store multiple images in a single file. idx can be a scalar or vector specifying the index of the images to read. By default, Octave will only read the first page. Depending on the file format, it is possible to configure the reading of images with param, val pairs. The following options are supported:

‘"Frames" or "Index"’

This is an alternative method to specify idx. When specifying it in this way, its value can also be the string "all".

‘"Info"’ This option exists for matlab compatibility and has no effect. For maximum performance while reading multiple images from a single file, use the Index option.

‘"PixelRegion"’

Controls the image region that is read. Takes as value a cell array with two arrays of 3 elements {rows cols}. The elements in the array are the start, increment and end pixel to be read. If the increment value is omitted, defaults to 1.

See also: [imwrite], page 652, [imfinfo], page 653, [imformats], page 656.

[Function File] imwrite (img, filename)
[Function File] imwrite (img, filename, ext)
[Function File] imwrite (img, map, filename)

[Function File] imwrite (…, param1, val1, …)

Write images in various file formats. The image img can be a binary, grayscale, RGB, or multi-dimensional image. The size and class of img should be the same as what should be expected when reading it with imread: the 3rd and 4th dimensions reserved for color space, and multiple pages respectively. If it’s an indexed image, the colormap map must also be specified. If ext is not supplied, the file extension of filename is used to determine the format. The actual supported formats are dependent on options made during the build of Octave. Use imformats to check the support of the different image formats. Depending on the file format, it is possible to configure the writing of images with param, val pairs. The following options are supported:

‘Alpha’ Alpha (transparency) channel for the image. This must be a matrix with same class, and number of rows and columns of img. In case of a multipage image, the size of the 4th dimension must also match and the third dimension must be a singleton. By default, image will be completely opaque.

‘DelayTime’

For formats that accept animations (such as GIF), controls for how long a frame is displayed until it moves to the next one. The value must be scalar (which will applied to all frames in img), or a vector of length equal to the number of frames in im. The value is in seconds, must be between 0 and 655.35, and defaults to 0.5.

‘DisposalMethod’

For formats that accept animations (such as GIF), controls what happens to a frame before drawing the next one. Its value can be one of the following strings: "doNotSpecify" (default); "leaveInPlace"; "restoreBG"; and "restorePrevious", or a cell array of those string with length equal to the number of frames in img.

‘LoopCount’

For formats that accept animations (such as GIF), controls how many times the sequence is repeated. A value of Inf means an infinite loop (default), a value of 0 or 1 that the sequence is played only once (loops zero times), while a value of 2 or above loops that number of times (looping twice means it plays the complete sequence 3 times). This option is ignored when there is only a single image at the end of writing the file.

‘Quality’ Set the quality of the compression. The value should be an integer between 0 and 100, with larger values indicating higher visual quality and lower compression. Defaults to 75.

‘WriteMode’

Some file formats, such as TIFF and GIF, are able to store multiple images in a single file. This option specifies if img should be appended to the file (if it exists) or if a new file should be created for it (possibly overwriting an existing file). The value should be the string "Overwrite" (default), or "Append". Despite this option, the most efficient method of writing a multipage image is to pass a 4 dimensional img to imwrite, the same matrix that could be expected when using imread with the option "Index" set to

"all".

See also: [imread], page 651, [imfinfo], page 653, [imformats], page 656.

[Built-in Function] val = IMAGE_PATH ()

[Built-in Function] old_val = IMAGE_PATH (new_val)

[Built-in Function] IMAGE_PATH (new_val, "local")

Query or set the internal variable that specifies a colon separated list of directories in which to search for image files.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [EXEC PATH], page 724, [OCTAVE HOME], page 733.

It is possible to get information about an image file on disk, without actually reading it into Octave. This is done using the imfinfo function which provides read access to many of the parameters stored in the header of the image file.

[Function File] info = imfinfo (filename)
[Function File] info = imfinfo (filename, ext)

[Function File] info = imfinfo (url)

Read image information from a file.

imfinfo returns a structure containing information about the image stored in the file filename. If there is no file filename, and ext was specified, it will look for a file named filename and extension ext, i.e., a file named filename.ext.

The output structure info contains the following fields:

‘Filename’

The full name of the image file.

‘FileModDate’

Date of last modification to the file.

‘FileSize’

Number of bytes of the image on disk

‘Format’ Image format (e.g., "jpeg").

‘Height’ Image height in pixels.

‘Width’ Image Width in pixels.

‘BitDepth’

Number of bits per channel per pixel.

‘ColorType’
Image type. Value is "grayscale", "indexed", "truecolor", "CMYK",
or "undefined".
‘XResolution’

X resolution of the image.

‘YResolution’

Y resolution of the image.

‘ResolutionUnit’

Units of image resolution. Value is "Inch", "Centimeter", or

"undefined".
‘DelayTime’

Time in 1/100ths of a second (0 to 65535) which must expire before displaying the next image in an animated sequence.

‘LoopCount’

Number of iterations to loop an animation.

‘ByteOrder’

Endian option for formats that support it. Value is "little-endian",

"big-endian", or "undefined".

‘Gamma’ Gamma level of the image. The same color image displayed on two different workstations may look different due to differences in the display monitor.

‘Quality’ JPEG/MIFF/PNG compression level. Value is an integer in the range [0 100].

‘DisposalMethod’

Only valid for GIF images, control how successive frames are rendered (how the preceding frame is disposed of) when creating a GIF anima-

tion. Values can be "doNotSpecify", "leaveInPlace", "restoreBG",

or "restorePrevious". For non-GIF files, value is an empty string.

‘Chromaticities’

Value is a 1x8 Matrix with the x,y chromaticity values for white, red, green, and blue points, in that order.

‘Comment’ Image comment.

‘Compression’

Compression type. Value can be "none", "bzip", "fax3", "fax4",

"jpeg", "lzw", "rle", "deflate", "lzma", "jpeg2000", "jbig2",
"jbig2", or "undefined".
‘Colormap’

Colormap for each image.

‘Orientation’

The orientation of the image with respect to the rows and columns. Value is an integer between 1 and 8 as defined in the TIFF 6 specifications, and for matlab compatibility.

‘Software’

Name and version of the software or firmware of the camera or image input device used to generate the image.

‘Make’ The manufacturer of the recording equipment. This is the manufacture of the DSC, scanner, video digitizer or other equipment that generated the image.

‘Model’ The model name or model number of the recording equipment as mentioned on the field "Make".

‘DateTime’

The date and time of image creation as defined by the Exif standard, i.e., it is the date and time the file was changed.

‘ImageDescription’

The title of the image as defined by the Exif standard.

‘Artist’ Name of the camera owner, photographer or image creator.

‘Copyright’

Copyright notice of the person or organization claiming rights to the image.

‘DigitalCamera’

A struct with information retrieved from the Exif tag.

‘GPSInfo’ A struct with geotagging information retrieved from the Exif tag.

See also: [imread], page 651, [imwrite], page 652, [imshow], page 657, [imformats], page 656.

By default, Octave’s image IO functions (imread, imwrite, and imfinfo) use the GraphicsMagick library for their operations. This means a vast number of image formats is supported but considering the large amount of image formats in science and its commonly closed nature, it is impossible to have a library capable of reading them all. Because of this, the function imformats keeps a configurable list of available formats, their extensions, and what functions should the image IO functions use. This allows to expand Octave’s image IO capabilities by creating functions aimed at acting on specific file formats.

While it would be possible to call the extra functions directly, properly configuring Octave with imformats allows to keep a consistent code that is abstracted from file formats.

It is important to note that a file format is not actually defined by its file extension and that GraphicsMagick is capable to read and write more file formats than the ones listed by imformats. What this means is that even with an incorrect or missing extension the image may still be read correctly, and that even unlisted formats are not necessarily unsupported.

[Function File] imformats ()

[Function File] formats = imformats (ext)
[Function File] formats = imformats (format)
[Function File] formats = imformats ("add", format)

[Function File] formats = imformats ("remove", ext)

[Function File] formats = imformats ("update", ext, format)

[Function File] formats = imformats ("factory")

Manage supported image formats.

formats is a structure with information about each supported file format, or from a specific format ext, the value displayed on the field ext. It contains the following fields:

ext The name of the file format. This may match the file extension but Octave will automatically detect the file format.

description A long description of the file format.

isa A function handle to confirm if a file is of the specified format.

write A function handle to write if a file is of the specified format.

read A function handle to open files the specified format.

info A function handle to obtain image information of the specified format.

alpha Logical value if format supports alpha channel (transparency or matte).

multipage Logical value if format supports multipage (multiple images per file).

It is possible to change the way Octave manages file formats with the options "add", "remove", and "update", and supplying a structure format with the required fields. The option "factory" resets the configuration to the default.

This can be used by Octave packages to extend the image reading capabilities Octave, through use of the PKG ADD and PKG DEL commands.

See also: [imfinfo], page 653, [imread], page 651, [imwrite], page 652.

32.2 Displaying Images

A natural part of image processing is visualization of an image. The most basic function for this is the imshow function that shows the image given in the first input argument.

[Function File] imshow (im) [Function File] imshow (im, limits) [Function File] imshow (im, map) [Function File] imshow (rgb, …) [Function File] imshow (filename)

[Function File] imshow (. . ., string_param1, value1, . . .)

[Function File] h = imshow (…)

Display the image im, where im can be a 2-dimensional (grayscale image) or a 3dimensional (RGB image) matrix. If limits is a 2-element vector [low, high], the image is shown using a display range between low and high. If an empty matrix is passed for limits, the display range is computed as the range between the minimal and the maximal value in the image. If map is a valid color map, the image will be shown as an indexed image using the supplied color map. If a file name is given instead of an image, the file will be read and shown. If given, the parameter string param1 has value value1. string param1 can be any of the following:

"displayrange"

value1 is the display range as described above.

"xdata" If value1 is a two element vector, it must contain horizontal axis limits in the form [xmin xmax]; Otherwise value1 must be a vector and only the first and last elements will be used for xmin and xmax respectively.

"ydata" If value1 is a two element vector, it must contain vertical axis limits in the form [ymin ymax]; Otherwise value1 must be a vector and only the first and last elements will be used for ymin and ymax respectively.

The optional return value h is a graphics handle to the image.

See also: [image], page 657, [imagesc], page 658, [colormap], page 660, [gray2ind], page 659, [rgb2ind], page 659.

[Function File] image (img) [Function File] image (x, y, img) [Function File] image (…, "prop", val, …) [Function File] image ("prop1", val1, …) [Function File] h = image (…)

Display a matrix as an indexed color image. The elements of img are indices into the current colormap. x and y are optional 2-element vectors, [min, max], which specify the range for the axis labels. If a range is specified as [max, min] then the image will be reversed along that axis. For convenience, x and y may be specified as N-element vectors matching the length of the data in img. However, only the first and last elements will be used to determine the axis limits. Warning: x and y are ignored when using gnuplot 4.0 or earlier. Multiple property/value pairs may be specified for the image object, but they must appear in pairs. The optional return value h is a graphics handle to the image. Implementation Note: The origin (0, 0) for images is located in the upper left. For ordinary plots, the origin is located in the lower left. Octave handles this inversion by plotting the data normally, and then reversing the direction of the y-axis by setting the ydir property to "reverse". This has implications whenever an image and an ordinary plot need to be overlaid. The recommended solution is to display the image and then plot the reversed ydata using, for example, flipud (ydata). Calling Forms: The image function can be called in two forms: High-Level and Low-Level. When invoked with normal options, the High-Level form is used which first calls newplot to prepare the graphic figure and axes. When the only inputs to image are property/value pairs the Low-Level form is used which creates a new instance of an image object and inserts it in the current axes.

See also: [imshow], page 657, [imagesc], page 658, [colormap], page 660.

[Function File] imagesc (img) [Function File] imagesc (x, y, img) [Function File] imagesc (…, climits) [Function File] imagesc (…, "prop", val, …) [Function File] imagesc ("prop1", val1, …) [Function File] imagesc (hax, …) [Function File] h = imagesc (…)

Display a scaled version of the matrix img as a color image. The colormap is scaled so that the entries of the matrix occupy the entire colormap. If climits = [lo, hi] is given, then that range is set to the "clim" of the current axes. The axis values corresponding to the matrix elements are specified in x and y, either as pairs giving the minimum and maximum values for the respective axes, or as values for each row and column of the matrix img. The optional return value h is a graphics handle to the image. Calling Forms: The imagesc function can be called in two forms: High-Level and Low-Level. When invoked with normal options, the High-Level form is used which first calls newplot to prepare the graphic figure and axes. When the only inputs to image are property/value pairs the Low-Level form is used which creates a new instance of an image object and inserts it in the current axes.

See also: [image], page 657, [imshow], page 657, [caxis], page 287.

32.3 Representing Images

In general Octave supports four different kinds of images, grayscale images, RGB images, binary images, and indexed images. A grayscale image is represented with an M-by-N matrix in which each element corresponds to the intensity of a pixel. An RGB image is represented with an M-by-N-by-3 array where each 3-vector corresponds to the red, green, and blue intensities of each pixel.

The actual meaning of the value of a pixel in a grayscale or RGB image depends on the class of the matrix. If the matrix is of class double pixel intensities are between 0 and 1, if it is of class uint8 intensities are between 0 and 255, and if it is of class uint16 intensities are between 0 and 65535.

A binary image is an M-by-N matrix of class logical. A pixel in a binary image is black if it is false and white if it is true.

An indexed image consists of an M-by-N matrix of integers and a C-by-3 color map. Each integer corresponds to an index in the color map, and each row in the color map corresponds to an RGB color. The color map must be of class double with values between 0 and 1.

[Function File] iscolormap (cmap)

Return true if cmap is a colormap. A colormap is a real matrix with n rows and 3 columns. Each row represents a single color. The columns contain red, green, and blue intensities respectively. All entries must be between 0 and 1 inclusive.

See also: [colormap], page 660, [rgbplot], page 661.

[Function File] img = gray2ind (I) [Function File] img = gray2ind (I, n) [Function File] img = gray2ind (BW) [Function File] img = gray2ind (BW, n) [Function File] [img, map] = gray2ind (…)

Convert a grayscale or binary intensity image to an indexed image. The indexed image will consist of n different intensity values. If not given n defaults to 64 for grayscale images or 2 for binary black and white images. The output img is of class uint8 if n is less than or equal to 256; Otherwise the return class is uint16.

See also: [ind2gray], page 659, [rgb2ind], page 659.

[Function File] I = ind2gray (x, map)

Convert a color indexed image to a grayscale intensity image. The image x must be an indexed image which will be converted using the colormap cmap. If cmap does not contain enough colors for the image, pixels in x outside the range are mapped to the last color in the map before conversion to grayscale. The output I is of the same class as the input x and may be one of uint8, uint16,

single, or double.

Implementation Note: There are several ways of converting colors to grayscale intensities. This functions uses the luminance value obtained from rgb2ntsc which is I = 0.299*R + 0.587*G + 0.114*B. Other possibilities include the value component from rgb2hsv or using a single color channel from ind2rgb.

See also: [gray2ind], page 659, [ind2rgb], page 660.

[Function File] [x, map] = rgb2ind (rgb)
[Function File] [x, map] = rgb2ind (R, G, B)

Convert an image in red-green-blue (RGB) color space to an indexed image.

The input image rgb can be specified as a single matrix of size MxNx3, or as three separate variables, R, G, and B, its three colour channels, red, green, and blue. It outputs an indexed image x and a colormap map to interpret an image exactly the same as the input. No dithering or other form of color quantization is performed. The output class of the indexed image x can be uint8, uint16 or double, whichever is required to specify the number of unique colors in the image (which will be equal to the number of rows in map) in order Multi-dimensional indexed images (of size MxNx3xK) are also supported, both via a single input (rgb) or its three colour channels as separate variables.

See also: [ind2rgb], page 660, [rgb2hsv], page 666, [rgb2ntsc], page 667.

[Function File] rgb = ind2rgb (x, map)

[Function File] [R, G, B] = ind2rgb (x, map)

Convert an indexed image to red, green, and blue color components. The image x must be an indexed image which will be converted using the colormap map. If map does not contain enough colors for the image, pixels in x outside the range are mapped to the last color in the map. The output may be a single RGB image (MxNx3 matrix where M and N are the original image x dimensions, one for each of the red, green and blue channels). Alternatively, the individual red, green, and blue color matrices of size MxN may be returned. Multi-dimensional indexed images (of size MxNx1xK) are also supported.

See also: [rgb2ind], page 659, [ind2gray], page 659, [hsv2rgb], page 667, [ntsc2rgb], page 667.

[Function File] cmap = colormap ()

[Function File] cmap = colormap (map)

[Function File] cmap = colormap ("default")

[Function File] cmap = colormap ("map_name")

[Function File] cmap = colormap (hax, …)

[Command] colormap map_name

[Function File] cmaps = colormap ("list") [Function File] colormap ("register", "name") [Function File] colormap ("unregister", "name")

Query or set the current colormap. With no input arguments, colormap returns the current color map. colormap (map) sets the current colormap to map. The colormap should be an n row by 3 column matrix. The columns contain red, green, and blue intensities respectively. All entries must be between 0 and 1 inclusive. The new colormap is returned. colormap ("default") restores the default colormap (the jet map with 64 entries). The default colormap is returned. The map may also be specified by a string, "map_name", where map name is the name of a function that returns a colormap. If the first argument hax is an axes handle, then the colormap for the parent figure of hax is queried or set.

For convenience, it is also possible to use this function with the command form,

colormap map_name.

colormap ("list") returns a cell array with all of the available colormaps. The options "register" and "unregister" add or remove the colormap name from this list.

See also: [jet], page 663.

[Function File] rgbplot (cmap) [Function File] rgbplot (cmap, style) [Function File] h = rgbplot (…)

Plot the components of a colormap. Two different styles are available for displaying the cmap:

profile (default)

Plot the RGB line profile of the colormap for each of the channels (red, green and blue) with the plot lines colored appropriately. Each line represents the intensity of each RGB components across the colormap.

composite Draw the colormap across the X-axis so that the actual index colors are visible rather than the individual color components.

The optional return value h is a graphics handle to the created plot. Run demo rgbplot to see an example of rgbplot and each style option.

See also: [colormap], page 660.

[Function File] map = autumn () [Function File] map = autumn (n)

Create color colormap. This colormap ranges from red through orange to yellow. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = bone () [Function File] map = bone (n)

Create color colormap. This colormap varies from black to white with gray-blue shades. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = colorcube () [Function File] map = colorcube (n)

Create color colormap. This colormap is composed of as many equally spaced colors (not grays) in the RGB color space as possible. If there are not a perfect number n of regularly spaced colors then the remaining entries in the colormap are gradients of pure red, green, blue, and gray. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = cool () [Function File] map = cool (n)

Create color colormap. The colormap varies from cyan to magenta. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = copper () [Function File] map = copper (n)

Create color colormap. This colormap varies from black to a light copper tone. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = flag () [Function File] map = flag (n)

Create color colormap. This colormap cycles through red, white, blue, and black with each index change. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = gray () [Function File] map = gray (n)

Create gray colormap. This colormap varies from black to white with shades of gray. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = hot () [Function File] map = hot (n)

Create color colormap. This colormap ranges from black through dark red, red, orange, yellow, to white. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] hsv (n)

Create color colormap. This colormap begins with red, changes through yellow, green, cyan, blue, and magenta, before returning to red. It is useful for displaying periodic functions. The map is obtained by linearly varying the hue through all possible values while keeping constant maximum saturation and value. The equivalent code

is hsv2rgb ([(0:N-1)'/N, ones(N,2)]).

The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = jet () [Function File] map = jet (n)

Create color colormap. This colormap ranges from dark blue through blue, cyan, green, yellow, red, to dark red. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = lines () [Function File] map = lines (n)

Create color colormap. This colormap is composed of the list of colors in the current axes "ColorOrder" property. The default is blue, green, red, cyan, pink, yellow, and gray. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = ocean () [Function File] map = ocean (n)

Create color colormap. This colormap varies from black to white with shades of blue. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = pink () [Function File] map = pink (n)

Create color colormap. This colormap varies from black to white with shades of graypink. It gives a sepia tone when used on grayscale images. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = prism () [Function File] map = prism (n)

Create color colormap. This colormap cycles through red, orange, yellow, green, blue and violet with each index change. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = rainbow () [Function File] map = rainbow (n)

Create color colormap. This colormap ranges from red through orange, yellow, green, blue, to violet. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = spring () [Function File] map = spring (n)

Create color colormap. This colormap varies from magenta to yellow. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = summer () [Function File] map = summer (n)

Create color colormap. This colormap varies from green to yellow. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] map = white () [Function File] map = white (n)

Create color colormap. This colormap is completely white. The argument n should be a scalar. If it is omitted, the length of the current colormap or 64 is assumed.

See also: [colormap], page 660.

[Function File] map = winter () [Function File] map = winter (n)

Create color colormap. This colormap varies from blue to green. The argument n must be a scalar. If unspecified, the length of the current colormap, or 64, is used.

See also: [colormap], page 660.

[Function File] cmap = contrast (x) [Function File] cmap = contrast (x, n)

Return a gray colormap that maximizes the contrast in an image. The returned colormap will have n rows. If n is not defined then the size of the current colormap is used.

See also: [colormap], page 660, [brighten], page 664.

An additional colormap is gmap40. This code map contains only colors with integer values of the red, green and blue components. This is a workaround for a limitation of gnuplot 4.0, that does not allow the color of line or patch objects to be set. gmap40 is chiefly useful to gnuplot 4.0 users, and particularly in conjunction with the bar, surf, and contour functions.

[Function File] map = gmap40 () [Function File] map = gmap40 (n)

Create color colormap. The colormap consists of red, green, blue, yellow, magenta and cyan. This colormap is specifically designed for users of gnuplot 4.0 where these 6 colors are the allowable ones for patch objects. The argument n must be a scalar. If unspecified, a length of 6 is assumed. Larger values of n result in a repetition of the above colors.

See also: [colormap], page 660.

The following three functions modify the existing colormap rather than replace it.

[Function File] map_out = brighten (map, beta)
[Function File] map_out = brighten (beta)
[Function File] map_out = brighten (h, beta)

Brighten or darken a colormap. If the map argument is omitted, the function is applied to the current colormap. The first argument can also be a valid graphics handle h, in which case brighten is applied to the colormap associated with this handle. The argument beta must be a scalar between -1 and 1, where a negative value darkens and a positive value brightens the colormap. If no output is specified then the result is written to the current colormap.

See also: [colormap], page 660, [contrast], page 664.

[Function File] spinmap () [Function File] spinmap (t) [Function File] spinmap (t, inc) [Function File] spinmap ("inf")

Cycle the colormap for t seconds with a color increment of inc. Both parameters are optional. The default cycle time is 5 seconds and the default increment is 2. If the option "inf" is given then cycle continuously until Control-C is pressed. When rotating the original color 1 becomes color 2, color 2 becomes color 3, etc. A positive or negative increment is allowed and a higher value of inc will cause faster cycling through the colormap.

See also: [colormap], page 660.

[Function File] whitebg () [Function File] whitebg (color) [Function File] whitebg ("none") [Function File] whitebg (hfig, …)

Invert the colors in the current color scheme. The root properties are also inverted such that all subsequent plot use the new color scheme. If the optional argument color is present then the background color is set to color rather than inverted. color may be a string representing one of the eight known colors or an RGB triplet. The special string argument "none" restores the plot to the default colors. If the first argument hfig is a figure handle, then operate on this figure rather than the current figure returned by gcf. The root properties will not be changed.

See also: [reset], page 361, [get], page 341, [set], page 342.

The following functions can be used to manipulate colormaps.

[Function File] [Y, newmap] = cmunique (X, map)
[Function File] [Y, newmap] = cmunique (RGB)
[Function File] [Y, newmap] = cmunique (I)

Convert an input image X to an ouput indexed image Y which uses the smallest colormap possible newmap. When the input is an indexed image (X with colormap map) the output is a colormap newmap from which any repeated rows have been eliminated. The output image, Y, is the original input image with the indices adjusted to match the new, possibly smaller, colormap.

When the input is an RGB image (an MxNx3 array), the output colormap will contain one entry for every unique color in the original image. In the worst case the new map could have as many rows as the number of pixels in the original image.

When the input is a grayscale image I, the output colormap will contain one entry for every unique intensity value in the original image. In the worst case the new map could have as many rows as the number of pixels in the original image.

Implementation Details:

newmap is always an Mx3 matrix, even if the input image is an intensity grayscale image I (all three RGB planes are assigned the same value).

The output image is of class uint8 if the size of the new colormap is less than or equal to 256. Otherwise, the output image is of class double.

See also: [rgb2ind], page 659, [gray2ind], page 659.

[Function File] [Y, newmap] = cmpermute (X, map)
[Function File] [Y, newmap] = cmpermute (X, map, index)

Reorder colors in a colormap.

When called with only two arguments, cmpermute randomly rearranges the colormap map and returns a new colormap newmap. It also returns the indexed image Y which is the equivalent of the original input image X when displayed using newmap.

When called with an optional third argument the order of colors in the new colormap is defined by index.

Caution: index should not have repeated elements or the function will fail.

32.4 Plotting on top of Images

If gnuplot is being used to display images it is possible to plot on top of images. Since an image is a matrix it is indexed by row and column values. The plotting system is, however, based on the traditional (x, y) system. To minimize the difference between the two systems Octave places the origin of the coordinate system in the point corresponding to the pixel at (1, 1). So, to plot points given by row and column values on top of an image, one should simply call plot with the column values as the first argument and the row values as the second. As an example the following code generates an image with random intensities between 0 and 1, and shows the image with red circles over pixels with an intensity above 0.99.

I = rand (100, 100);
[row, col] = find (I > 0.99);
hold ("on");
imshow (I);
plot (col, row, "ro");
hold ("off");

32.5 Color Conversion

Octave supports conversion from the RGB color system to NTSC and HSV and vice versa.

[Function File] hsv_map = rgb2hsv (rgb)
[Function File] hsv_map = rgb2hsv (rgb)

Transform a colormap or image from red-green-blue (RGB) space to hue-saturation-value (HSV) space. A color in the RGB space consists of red, green, and blue intensities. A color in HSV space is represented by hue, saturation, and value (brightness) levels. Value gives the amount of light in the color. Hue describes the dominant wavelength. Saturation is the amount of hue mixed into the color.

See also: [hsv2rgb], page 667, [rgb2ind], page 659, [rgb2ntsc], page 667.

[Function File] rgb_map = hsv2rgb (hsv_map)
[Function File] rgb_img = hsv2rgb (hsv_img)

Transform a colormap or image from hue-saturation-value (HSV) space to red-green-blue (RGB) space. A color in HSV space is represented by hue, saturation and value (brightness) levels. Value gives the amount of light in the color. Hue describes the dominant wavelength. Saturation is the amount of hue mixed into the color. A color in the RGB space consists of red, green, and blue intensities.

See also: [rgb2hsv], page 666, [ind2rgb], page 660, [ntsc2rgb], page 667.

[Function File] yiq_map = rgb2ntsc (rgb_map)
[Function File] yiq_img = rgb2ntsc (rgb_img)

Transform a colormap or image from red-green-blue (RGB) color space to luminancechrominance (NTSC) space. The input may be of class uint8, uint16, single, or double. The output is of class double. Implementation Note: The reference matrix for the transformation is

     /Y\     0.299  0.587  0.114  /R\
     |I|  =  0.596 -0.274 -0.322  |G|
     \Q/     0.211 -0.523  0.312  \B/

as documented in http://en.wikipedia.org/wiki/YIQ and truncated to 3 significant figures. Note: The FCC version of NTSC uses only 2 significant digits and is slightly different.

See also: [ntsc2rgb], page 667, [rgb2hsv], page 666, [rgb2ind], page 659.

[Function File] rgb_map = ntsc2rgb (yiq_map)
[Function File] rgb_img = ntsc2rgb (yiq_img)

Transform a colormap or image from luminance-chrominance (NTSC) space to redgreen-blue (RGB) color space. Implementation Note: The conversion matrix is chosen to be the inverse of the matrix used for rgb2ntsc such that

     x == ntsc2rgb (rgb2ntsc (x))

matlab uses a slightly different matrix where rounding means the equality above does not hold.

See also: [rgb2ntsc], page 667, [hsv2rgb], page 667, [ind2rgb], page 660.

33 Audio Processing

Octave provides a few functions for dealing with audio data. An audio ‘sample’ is a single output value from an A/D converter, i.e., a small integer number (usually 8 or 16 bits), and audio data is just a series of such samples. It can be characterized by three parameters: the sampling rate (measured in samples per second or Hz, e.g., 8000 or 44100), the number of bits per sample (e.g., 8 or 16), and the number of channels (1 for mono, 2 for stereo, etc.).

There are many different formats for representing such data. Currently, only the two most popular, linear encoding and mu-law encoding, are supported by Octave. There is an excellent FAQ on audio formats by Guido van Rossum guido@cwi.nl which can be found at any FAQ ftp site, in particular in the directory ‘/pub/usenet/news.answers/audio-fmts’ of the archive site rtfm.mit.edu.

Octave simply treats audio data as vectors of samples (non-mono data are not supported yet). It is assumed that audio files using linear encoding have one of the extensions ‘lin’ or ‘raw’, and that files holding data in mu-law encoding end in ‘au’, ‘mu’, or ‘snd’.

[Function File] lin2mu (x, n)

Convert audio data from linear to mu-law. Mu-law values use 8-bit unsigned integers. Linear values use n-bit signed integers or floating point values in the range -1 ≤ x ≤ 1 if n is 0. If n is not specified it defaults to 0, 8, or 16 depending on the range of values in x.

See also: [mu2lin], page 669, [loadaudio], page 669, [saveaudio], page 669.

[Function File] mu2lin (x, n)

Convert audio data from mu-law to linear. Mu-law values are 8-bit unsigned integers. Linear values use n-bit signed integers or floating point values in the range -1≤y≤1 if n is 0. If n is not specified it defaults to 0.

See also: [lin2mu], page 669, [loadaudio], page 669, [saveaudio], page 669.

[Function File] loadaudio (name, ext, bps)

Load audio data from the file ‘name.ext’ into the vector x. The extension ext determines how the data in the audio file is interpreted; the extensions ‘lin’ (default) and ‘raw’ correspond to linear, the extensions ‘au’, ‘mu’, or ‘snd’ to mu-law encoding. The argument bps can be either 8 (default) or 16, and specifies the number of bits per sample used in the audio file.

See also: [lin2mu], page 669, [mu2lin], page 669, [saveaudio], page 669, [playaudio], page 670, [setaudio], page 670, [record], page 670.

[Function File] saveaudio (name, x, ext, bps)

Save a vector x of audio data to the file ‘name.ext’. The optional parameters ext and bps determine the encoding and the number of bits per sample used in the audio file (see loadaudio); defaults are ‘lin’ and 8, respectively.

See also: [lin2mu], page 669, [mu2lin], page 669, [loadaudio], page 669, [playaudio], page 670, [setaudio], page 670, [record], page 670.

The following functions for audio I/O require special A/D hardware and operating system support. It is assumed that audio data in linear encoding can be played and recorded by reading from and writing to ‘/dev/dsp’, and that similarly ‘/dev/audio’ is used for mu-law encoding. These file names are system-dependent. Improvements so that these functions will work without modification on a wide variety of hardware are welcome.

[Function File] playaudio (name, ext) [Function File] playaudio (x)

Play the audio file ‘name.ext’ or the audio data stored in the vector x.

See also: [lin2mu], page 669, [mu2lin], page 669, [loadaudio], page 669, [saveaudio], page 669, [setaudio], page 670, [record], page 670.

[Function File] record (sec, sampling_rate)

Record sec seconds of audio input into the vector x. The default value for sampling rate is 8000 samples per second, or 8kHz. The program waits until the user types RET and then immediately starts to record.

See also: [lin2mu], page 669, [mu2lin], page 669, [loadaudio], page 669, [saveaudio], page 669, [playaudio], page 670, [setaudio], page 670.

[Function File] setaudio () [Function File] setaudio (w_type)

[Function File] setaudio (w_type, value)

Execute the shell command ‘mixer’, possibly with optional arguments w type and value.

[Function File] y = wavread (filename)
[Function File] [y, Fs, bps] = wavread (filename)
[Function File] [...] = wavread (filename, n)
[Function File] [...] = wavread (filename, [n1 n2])
[Function File] [samples, channels] = wavread (filename, "size")

Load the RIFF/WAVE sound file filename, and return the samples in vector y. If the file contains multichannel data, then y is a matrix with the channels represented as columns.

[y, Fs, bps] = wavread (filename)

Additionally return the sample rate (fs) in Hz and the number of bits per sample (bps).

[...] = wavread (filename, n)

Read only the first n samples from each channel.

wavread (filename, [n1 n2])

Read only samples n1 through n2 from each channel.

[samples, channels] = wavread (filename, "size")

Return the number of samples (n) and number of channels (ch) instead of the audio data.

See also: [wavwrite], page 671.

[Function File] wavwrite (y, filename)
[Function File] wavwrite (y, Fs, filename)
[Function File] wavwrite (y, Fs, bps, filename)

Write y to the canonical RIFF/WAVE sound file filename with sample rate Fs and bits per sample bps. The default sample rate is 8000 Hz with 16-bits per sample. Each column of the data represents a separate channel. If y is either a row vector or a column vector, it is written as a single channel.

See also: [wavread], page 670.

34 Object Oriented Programming

Octave includes the capability to include user classes, including the features of operator and function overloading. Equally a user class can be used to encapsulate certain properties of the class so that they cannot be altered accidentally and can be set up to address the issue of class precedence in mixed class operations.

This chapter discussions the means of constructing a user class with the example of a polynomial class, how to query and set the properties of this class, together with the means to overload operators and functions.

34.1 Creating a Class

We use in the following text a polynomial class to demonstrate the use of object oriented programming within Octave. This class was chosen as it is simple, and so doesn’t distract unnecessarily from the discussion of the programming features of Octave. However, even still a small understand of the polynomial class itself is necessary to fully grasp the techniques described.

The polynomial class is used to represent polynomials of the form

a_{0} + a_{1}x + a_{2}x^{2} + … a_{n}x^{n}

where a_{0}, a_{1}, etc. are elements of ℜ. Thus the polynomial can be represented by a vector

     a = [a0, a1, a2, ..., an];

We therefore now have sufficient information about the requirements of the class constructor for our polynomial class to write it. All object oriented classes in Octave, must be contained with a directory taking the name of the class, prepended with the @ symbol. For example, with our polynomial class, we would place the methods defining the class in the @polynomial directory.

The constructor of the class, must have the name of the class itself and so in our example the constructor with have the name ‘@polynomial/polynomial.m’. Also ideally when the constructor is called with no arguments to should return a value object. So for example our polynomial might look like

     ## -*- texinfo -*-
     ## @deftypefn  {Function File} {} polynomial ()
     ## @deftypefnx {Function File} {} polynomial (@var{a})
     ## Create a polynomial object representing the polynomial
     ##
     ## @example
     ## a0 + a1 * x + a2 * x^2 + @dots{} + an * x^n
     ## @end example
     ##
     ## @noindent
     ## from a vector of coefficients [a0 a1 a2 @dots{} an].
     ## @end deftypefn
     function p = polynomial (a)

       if (nargin == 0)
         p.poly = [0];
         p = class (p, "polynomial");
       elseif (nargin == 1)
         if (strcmp (class (a), "polynomial"))
           p = a;
         elseif (isvector (a) && isreal (a))
           p.poly = a(:).';
           p = class (p, "polynomial");
         else
           error ("polynomial: expecting real vector");
         endif
       else
         print_usage ();
       endif
     endfunction

Note that the return value of the constructor must be the output of the class function called with the first argument being a structure and the second argument being the class name. An example of the call to this constructor function is then

     p = polynomial ([1, 0, 1]);

Note that methods of a class can be documented. The help for the constructor itself can be obtained with the constructor name, that is for the polynomial constructor help polynomial will return the help string. Also the help can be obtained by restricting the search for the help to a particular class, for example help @polynomial/polynomial. This second method is the only means of getting help for the overloaded methods and functions of the class.

The same is true for other Octave functions that take a function name as an argument. For example type @polynomial/display will print the code of the display method of the polynomial class to the screen, and dbstop @polynomial/display will set a breakpoint at the first executable line of the display method of the polynomial class.

To check where a variable is a user class, the isobject and isa functions can be used. For example:

     p = polynomial ([1, 0, 1]);
     isobject (p)
       ⇒ 1
     isa (p, "polynomial")
       ⇒ 1

[Built-in Function] isobject (x)

Return true if x is a class object.

See also: [class], page 39, [typeinfo], page 39, [isa], page 39, [ismethod], page 675.

The available methods of a class can be displayed with the methods function.

[Function File] methods (obj)

[Function File] methods ("classname")

[Function File] mtds = methods (…)

Return a cell array containing the names of the methods for the object obj or the named class classname. obj may be an Octave class object or a Java object.

See also: [fieldnames], page 105.

To inquire whether a particular method is available to a user class, the ismethod function can be used.

[Built-in Function] ismethod (x, method)

Return true if x is a class object and the string method is a method of this class.

See also: [isprop], page 336, [isobject], page 674.

For example:

p = polynomial ([1, 0, 1]);
ismethod (p, "roots")
  ⇒ 1

34.2 Manipulating Classes

There are a number of basic classes methods that can be defined to allow the contents of the classes to be queried and set. The most basic of these is the display method. The display method is used by Octave when displaying a class on the screen, due to an expression that is not terminated with a semicolon. If this method is not defined, then Octave will printed nothing when displaying the contents of a class.

[Function File] display (a)

Display the contents of an object. If a is an object of the class "myclass", then display is called in a case like

     myclass (...)

where Octave is required to display the contents of a variable of the type "myclass".

See also: [class], page 39, [subsref], page 678, [subsasgn], page 680.

An example of a display method for the polynomial class might be

function display (p)
  a = p.poly;
  first = true;
  fprintf ("%s =", inputname (1));
  for i = 1 : length (a);
    if (a(i) != 0)
      if (first)
        first = false;
      elseif (a(i) > 0)
        fprintf (" +");
      endif
      if (a(i) < 0)
        fprintf (" -");
      endif

if (i == 1)
  fprintf (" %g", abs (a(i)));
elseif (abs(a(i)) != 1)
  fprintf (" %g *", abs (a(i)));
endif
if (i > 1)
  fprintf (" X");
endif
if (i > 2)
  fprintf (" ^%d", i - 1);
endif
endif
endfor
if (first)
fprintf (" 0");
endif
fprintf ("\n");
endfunction

Note that in the display method, it makes sense to start the method with the line fprintf ("%s =", inputname (1)) to be consistent with the rest of Octave and print the variable name to be displayed when displaying the class.

To be consistent with the Octave graphic handle classes, a class should also define the get and set methods. The get method should accept one or two arguments, and given one argument of the appropriate class it should return a structure with all of the properties of the class. For example:

function s = get (p, f)
if (nargin == 1)
s.poly = p.poly;
elseif (nargin == 2)
if (ischar (f))
switch (f)
  case "poly"
    s = p.poly;
  otherwise
    error ("get: invalid property %s", f);
endswitch
else
error ("get: expecting the property to be a string");
endif
else
print_usage ();
endif
endfunction

Similarly, the set method should taken as its first argument an object to modify, and then take property/value pairs to be modified.

function s = set (p, varargin)
  s = p;
  if (length (varargin) < 2 || rem (length (varargin), 2) != 0)
    error ("set: expecting property/value pairs");
  endif
  while (length (varargin) > 1)
    prop = varargin{1};
    val = varargin{2};
    varargin(1:2) = [];
    if (ischar (prop) && strcmp (prop, "poly"))
      if (isvector (val) && isreal (val))
        s.poly = val(:).';
      else
        error ("set: expecting the value to be a real vector");
      endif
    else
      error ("set: invalid property of polynomial class");
    endif
  endwhile
endfunction

Note that as Octave does not implement pass by reference, than the modified object is the return value of the set method and it must be called like

p = set (p, "a", [1, 0, 0, 0, 1]);

Also the set method makes use of the subsasgn method of the class, and this method must be defined. The subsasgn method is discussed in the next section.

Finally, user classes can be considered as a special type of a structure, and so they can be saved to a file in the same manner as a structure. For example:

p = polynomial ([1, 0, 1]);
save userclass.mat p
clear p
load userclass.mat

All of the file formats supported by save and load are supported. In certain circumstances, a user class might either contain a field that it makes no sense to save or a field that needs to be initialized before it is saved. This can be done with the saveobj method of the class

[Function File] b = saveobj (a)

Method of a class to manipulate an object prior to saving it to a file. The function saveobj is called when the object a is saved using the save function. An example of the use of saveobj might be to remove fields of the object that don’t make sense to be saved or it might be used to ensure that certain fields of the object are initialized before the object is saved. For example:

     function b = saveobj (a)
       b = a;
       if (isempty (b.field))
          b.field = initfield (b);
       endif
     endfunction

See also: [loadobj], page 678, [class], page 39.

saveobj is called just prior to saving the class to a file. Likely, the loadobj method is called just after a class is loaded from a file, and can be used to ensure that any removed fields are reinserted into the user object.

[Function File] b = loadobj (a)

Method of a class to manipulate an object after loading it from a file. The function loadobj is called when the object a is loaded using the load function. An example of the use of saveobj might be to add fields to an object that don’t make sense to be saved. For example:

     function b = loadobj (a)
       b = a;
       b.addmissingfield = addfield (b);
     endfunction

See also: [saveobj], page 677, [class], page 39.

34.3 Indexing Objects

34.3.1 Defining Indexing And Indexed Assignment

Objects can be indexed with parentheses, either like a (idx) or like a {idx}, or even like a (idx).field. However, it is up to the user to decide what this indexing actually means. In the case of our polynomial class p (n) might mean either the coefficient of the n-th power of the polynomial, or it might be the evaluation of the polynomial at n. The meaning of this subscripted referencing is determined by the subsref method.

[Built-in Function] subsref (val, idx)

Perform the subscripted element selection operation according to the subscript specified by idx.

The subscript idx is expected to be a structure array with fields ‘type’ and ‘subs’. Valid values for ‘type’ are ‘"()"’, ‘"{}"’, and ‘"."’. The ‘subs’ field may be either ‘":"’ or a cell array of index values.

The following example shows how to extract the first two columns of a matrix

val = magic (3)
   ⇒ val = [ 8   1   6
              3   5   7
              4   9   2 ]
idx.type = "()";
idx.subs = {":", 1:2};
subsref (val, idx)
    ⇒ [ 8   1
         3   5
         4   9 ]

Note that this is the same as writing val(:,1:2). If idx is an empty structure array with fields ‘type’ and ‘subs’, return val.

See also: [subsasgn], page 680, [substruct], page 108.

For example we might decide that indexing with "()" evaluates the polynomial and indexing with "{}" returns the n-th coefficient (of n-th power). In this case the subsref method of our polynomial class might look like

function b = subsref (a, s)
if (isempty (s))
error ("polynomial: missing index");
endif
switch (s(1).type)
case "()"
ind = s(1).subs;
if (numel (ind) != 1)
  error ("polynomial: need exactly one index");
else
  b = polyval (fliplr (a.poly), ind{1});
endif
case "{}"
ind = s(1).subs;
if (numel (ind) != 1)
  error ("polynomial: need exactly one index");
else
  if (isnumeric (ind{1}))
    b = a.poly(ind{1}+1);
  else
    b = a.poly(ind{1});
  endif
endif
case "."
fld = s.subs;
if (strcmp (fld, "poly"))
  b = a.poly;
else
  error ("@polynomial/subsref: invalid property \"%s\"", fld);

      endif
    otherwise
      error ("invalid subscript type");
  endswitch
  if (numel (s) > 1)
    b = subsref (b, s(2:end));
  endif
endfunction

The equivalent functionality for subscripted assignments uses the subsasgn method.

[Built-in Function] subsasgn (val, idx, rhs)

Perform the subscripted assignment operation according to the subscript specified by idx.

The subscript idx is expected to be a structure array with fields ‘type’ and ‘subs’. Valid values for ‘type’ are ‘"()"’, ‘"{}"’, and ‘"."’. The ‘subs’ field may be either ‘":"’ or a cell array of index values.

The following example shows how to set the two first columns of a 3-by-3 matrix to zero.

     val = magic (3);
     idx.type = "()";
     idx.subs = {":", 1:2};
     subsasgn (val, idx, 0)
          ⇒ [ 0   0   6
                0   0   7
                0   0   2 ]

Note that this is the same as writing val(:,1:2) = 0.

If idx is an empty structure array with fields ‘type’ and ‘subs’, return rhs.

See also: [subsref], page 678, [substruct], page 108.

[Built-in Function] val = optimize_subsasgn_calls ()
[Built-in Function] old_val = optimize_subsasgn_calls (new_val)
[Built-in Function] optimize_subsasgn_calls (new_val, "local")

Query or set the internal flag for subsasgn method call optimizations.

If true, Octave will attempt to eliminate the redundant copying when calling the subsasgn method of a user-defined class.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

Note that the subsref and subsasgn methods always receive the whole index chain, while they usually handle only the first element. It is the responsibility of these methods to handle the rest of the chain (if needed), usually by forwarding it again to subsref or

subsasgn.

If you wish to use the end keyword in subscripted expressions of an object, then the user needs to define the end method for the class. For example, the end method for our polynomial class might look like

function r = end (obj, index_pos, num_indices)
  if (num_indices != 1)
    error ("polynomial object may only have one index")
  endif
  r = length (obj.poly) - 1;
endfunction

which is a fairly generic end method that has a behavior similar to the end keyword for Octave Array classes. It can then be used as follows:

p = polynomial ([1,2,3,4]);
p(end-1)
  ⇒ 3

Objects can also be used as the index in a subscripted expression themselves and this is controlled with the subsindex function.

[Function File] idx = subsindex (a)

Convert an object to an index vector. When a is a class object defined with a class constructor, then subsindex is the overloading method that allows the conversion of this class object to a valid indexing vector. It is important to note that subsindex must return a zero-based real integer vector of the class "double". For example, if the class constructor

     function b = myclass (a)
       b = class (struct ("a", a), "myclass");
     endfunction

then the subsindex function

     function idx = subsindex (a)
       idx = double (a.a) - 1.0;
     endfunction

can then be used as follows

     a = myclass (1:4);
     b = 1:10;
     b(a)
     ⇒ 1  2  3  4

See also: [class], page 39, [subsref], page 678, [subsasgn], page 680.

Finally, objects can equally be used like ranges, using the colon method

[Function File] r = colon (a, b) [Function File] r = colon (a, b, c)

Method of a class to construct a range with the : operator. For example:

          a = myclass (...);
          b = myclass (...);
          c = a : b

See also: [class], page 39, [subsref], page 678, [subsasgn], page 680.

34.3.2 Indexed Assignment Optimization

Octave’s ubiquitous lazily-copied pass-by-value semantics implies a problem for performance of user-defined subsasgn methods. Imagine a call to subsasgn:

       ss = substruct ("()",{1});
       x = subsasgn (x, ss, 1);

and the corresponding method looking like this:

       function x = subsasgn (x, ss, val)
         ...
         x.myfield (ss.subs{1}) = val;
       endfunction

The problem is that on entry to the subsasgn method, x is still referenced from the caller’s scope, which means that the method will first need to unshare (copy) x and x.myfield before performing the assignment. Upon completing the call, unless an error occurs, the result is immediately assigned to x in the caller’s scope, so that the previous value of x.myfield is forgotten. Hence, the Octave language implies a copy of N elements (N being the size of x.myfield), where modifying just a single element would actually suffice, i.e., degrades a constant-time operation to linear-time one. This may be a real problem for user classes that intrinsically store large arrays.

To partially solve the problem, Octave uses a special optimization for user-defined subsasgn methods coded as m-files. When the method gets called as a result of the built-in assignment syntax (not direct subsasgn call as shown above), i.e.

       x(1) = 1;

AND if the subsasgn method is declared with identical input and output argument, like in the example above, then Octave will ignore the copy of x inside the caller’s scope; therefore, any changes made to x during the method execution will directly affect the caller’s copy as well. This allows, for instance, defining a polynomial class where modifying a single element takes constant time.

It is important to understand the implications that this optimization brings. Since no extra copy of x in the caller’s scope will exist, it is solely the callee’s responsibility to not leave x in an invalid state if an error occurs throughout the execution. Also, if the method partially changes x and then errors out, the changes will affect x in the caller’s scope. Deleting or completely replacing x inside subsasgn will not do anything, however, only indexed assignments matter.

Since this optimization may change the way code works (especially if badly written), a built-in variable optimize_subsasgn_calls is provided to control it. It is on by default. Another option to avoid the effect is to declare subsasgn methods with different output and input arguments, like this:

       function y = subsasgn (x, ss, val)
         ...
       endfunction

34.4 Overloading Objects

34.4.1 Function Overloading

Any Octave function can be overloaded, and allows an object specific version of this function to be called as needed. A pertinent example for our polynomial class might be to overload the polyval function like

     function [y, dy] = polyval (p, varargin)
       if (nargout == 2)
         [y, dy] = polyval (fliplr (p.poly), varargin{:});
       else
         y = polyval (fliplr (p.poly), varargin{:});
       endif
     endfunction

This function just hands off the work to the normal Octave polyval function. Another interesting example for an overloaded function for our polynomial class is the plot function.

     function h = plot (p, varargin)
       n = 128;
       rmax = max (abs (roots (p.poly)));
       x = [0 : (n - 1)] / (n - 1) * 2.2 * rmax - 1.1 * rmax;
       if (nargout > 0)
         h = plot (x, p(x), varargin{:});
       else
         plot (x, p(x), varargin{:});
       endif
     endfunction

which allows polynomials to be plotted in the domain near the region of the roots of the polynomial.

Functions that are of particular interest to be overloaded are the class conversion functions such as double. Overloading these functions allows the cast function to work with the user class and can aid in the use of methods of other classes with the user class. An example double function for our polynomial class might look like.

     function b = double (a)
       b = a.poly;
     endfunction

34.4.2 Operator Overloading

The following table shows, for each built-in numerical operation, the corresponding function name to use when providing an overloaded method for a user class.

Operation  Method  Description
a + b  plus (a, b)  Binary addition operator
a − b  minus (a, b)  Binary subtraction operator
+a  uplus (a)  Unary addition operator
−a  uminus (a)  Unary subtraction operator
a. ∗ b  times (a, b)  Element-wise multiplication operator
a ∗ b  mtimes (a, b)  Matrix multiplication operator
a./b  rdivide (a, b)  Element-wise right division operator
a/b  mrdivide (a, b)  Matrix right division operator
a.\b  ldivide (a, b)  Element-wise left division operator
a\b  mldivide (a, b)  Matrix left division operator
a.ˆ b  power (a, b)  Element-wise power operator
aˆ b  mpower (a, b)  Matrix power operator
a < b  lt (a, b)  Less than operator
a <= b  le (a, b)  Less than or equal to operator
a > b  gt (a, b)  Greater than operator
a >= b  ge (a, b)  Greater than or equal to operator
a == b  eq (a, b)  Equal to operator
a! = b  ne (a, b)  Not equal to operator
a&b  and (a, b)  Logical and operator
a|b  or (a, b)  Logical or operator
!b  not (a)  Logical not operator
a′  ctranspose (a)  Complex conjugate transpose operator
a.′  transpose (a)  Transpose operator
a : b  colon (a, b)  Two element range operator
a : b : c  colon (a, b, c)  Three element range operator
[a, b]  horzcat (a, b)  Horizontal concatenation operator
[a; b]  vertcat (a, b)  Vertical concatenation operator
a(s_{1}, …, s_{n})  subsref (a, s)  Subscripted reference
a(s_{1}, …, s_{n}) = b subsasgn (a, s, b) Subscripted assignment
b(a)  subsindex (a)  Convert to zero-based index
display  display (a)  Commandline display function

Table 34.1: Available overloaded operators and their corresponding class method An example mtimes method for our polynomial class might look like

 function y = mtimes (a, b)
   y = polynomial (conv (double (a), double (b)));
 endfunction

34.4.3 Precedence of Objects

Many functions and operators take two or more arguments and so the case can easily arise that these functions are called with objects of different classes. It is therefore necessary to determine the precedence of which method of which class to call when there are mixed objects given to a function or operator. To do this the superiorto and inferiorto functions can be used

[Built-in Function] superiorto (class_name, …)

When called from a class constructor, mark the object currently constructed as having a higher precedence than class name. More that one such class can be specified in a single call. This function may only be called from a class constructor.

See also: [inferiorto], page 685.

[Built-in Function] inferiorto (class_name, …)

When called from a class constructor, mark the object currently constructed as having a lower precedence than class name. More that one such class can be specified in a single call. This function may only be called from a class constructor.

See also: [superiorto], page 685.

For example with our polynomial class consider the case

2 * polynomial ([1, 0, 1]);

That mixes an object of the class "double" with an object of the class "polynomial". In this case we like to ensure that the return type of the above is of the type "polynomial" and so we use the superiorto function in the class constructor. In particular our polynomial class constructor would be modified to be

## -*- texinfo -*-
## @deftypefn  {Function File} {} polynomial ()
## @deftypefnx {Function File} {} polynomial (@var{a})
## Create a polynomial object representing the polynomial
##
## @example
## a0 + a1 * x + a2 * x^2 + @dots{} + an * x^n
## @end example
##
## @noindent
## from a vector of coefficients [a0 a1 a2 @dots{} an].
## @end deftypefn
function p = polynomial (a)
  if (nargin == 0)
    p.poly = [0];
    p = class (p, "polynomial");
  elseif (nargin == 1)
    if (strcmp (class (a), "polynomial"))
      p = a;
    elseif (isvector (a) && isreal (a))
      p.poly = a(:).';
      p = class (p, "polynomial");
    else

           error ("polynomial: expecting real vector");
         endif
       else
         print_usage ();
       endif
       superiorto ("double");
     endfunction

Note that user classes always have higher precedence than built-in Octave types. So in fact marking our polynomial class higher than the "double" class is in fact not necessary.

When faced with two objects that have the same precedence, Octave will use the method of the object that appears first on the list of arguments.

34.5 Inheritance and Aggregation

Using classes to build new classes is supported by octave through the use of both inheritance and aggregation.

Class inheritance is provided by octave using the class function in the class constructor. As in the case of the polynomial class, the octave programmer will create a struct that contains the data fields required by the class, and then call the class function to indicate that an object is to be created from the struct. Creating a child of an existing object is done by creating an object of the parent class and providing that object as the third argument of the class function.

This is easily demonstrated by example. Suppose the programmer needs an FIR filter, i.e., a filter with a numerator polynomial but a unity denominator polynomial. In traditional octave programming, this would be performed as follows.

     octave:1> x = [some data vector];
     octave:2> n = [some coefficient vector];
     octave:3> y = filter (n, 1, x);

The equivalent class could be implemented in a class directory @FIRfilter that is on the octave path. The constructor is a file FIRfilter.m in the class directory.

     ## -*- texinfo -*-
     ## @deftypefn  {Function File} {} FIRfilter ()
     ## @deftypefnx {Function File} {} FIRfilter (@var{p})
     ## Create a FIR filter with polynomial @var{p} as coefficient vector.
     ## @end deftypefn
     function f = FIRfilter (p)
       f.polynomial = [];
       if (nargin == 0)
         p = @polynomial ([1]);
       elseif (nargin == 1)
         if (!isa (p, "polynomial"))
           error ("FIRfilter: expecting polynomial as input argument");
         endif

  else
    print_usage ();
  endif
  f = class (f, "FIRfilter", p);
endfunction

As before, the leading comments provide command-line documentation for the class constructor. This constructor is very similar to the polynomial class constructor, except that we pass a polynomial object as the third argument to the class function, telling octave that the FIRfilter class will be derived from the polynomial class. Our FIR filter does not have any data fields, but we must provide a struct to the class function. The class function will add an element named polynomial to the object struct, so we simply add a dummy element named polynomial as the first line of the constructor. This dummy element will be overwritten by the class function.

Note further that all our examples provide for the case in which no arguments are supplied. This is important since octave will call the constructor with no arguments when loading objects from save files to determine the inheritance structure.

A class may be a child of more than one class (see the documentation for the class function), and inheritance may be nested. There is no limitation to the number of parents or the level of nesting other than memory or other physical issues.

As before, we need a display method. A simple example might be

function display (f)
  display (f.polynomial);
endfunction

Note that we have used the polynomial field of the struct to display the filter coefficients. Once we have the class constructor and display method, we may create an object by

calling the class constructor. We may also check the class type and examine the underlying structure.

octave:1> f = FIRfilter (polynomial ([1 1 1]/3))
f.polynomial = 0.333333 + 0.333333 * X + 0.333333 * X ^2
octave:2> class (f)
ans = FIRfilter
octave:3> isa (f,"FIRfilter")
ans =  1
octave:4> isa (f,"polynomial")
ans =  1
octave:5> struct (f)
ans =
{
polynomial = 0.333333 + 0.333333 * X + 0.333333 * X ^2
}

We only need to define a method to actually process data with our filter and our class is usable. It is also useful to provide a means of changing the data stored in the class. Since the fields in the underlying struct are private by default, we could provide a mechanism to access the fields. The subsref method may be used for both.

function out = subsref (f, x)
  switch (x.type)
    case "()"
      n = f.polynomial;
      out = filter (n.poly, 1, x.subs{1});
    case "."
      fld = x.subs;
      if (strcmp (fld, "polynomial"))
        out = f.polynomial;
      else
        error ("@FIRfilter/subsref: invalid property \"%s\"", fld);
      endif
    otherwise
      error ("@FIRfilter/subsref: invalid subscript type for FIR filter");
  endswitch
endfunction

The "()" case allows us to filter data using the polynomial provided to the constructor.

octave:2> f = FIRfilter (polynomial ([1 1 1]/3));
octave:3> x = ones (5,1);
octave:4> y = f(x)
y =
   0.33333
   0.66667
   1.00000
   1.00000
   1.00000

The "." case allows us to view the contents of the polynomial field.

octave:1> f = FIRfilter (polynomial ([1 1 1]/3));
octave:2> f.polynomial
ans = 0.333333 + 0.333333 * X + 0.333333 * X ^2

In order to change the contents of the object, we need to define a subsasgn method. For example, we may make the polynomial field publicly writable.

function out = subsasgn (f, index, val)
  switch (index.type)
    case "."
      fld = index.subs;
      if (strcmp (fld, "polynomial"))
        out = f;
        out.polynomial = val;
      else
        error ("@FIRfilter/subsref: invalid property \"%s\"", fld);
      endif
    otherwise
      error ("FIRfilter/subsagn: Invalid index type")
  endswitch
endfunction

So that

octave:6> f = FIRfilter ();
octave:7> f.polynomial = polynomial ([1 2 3]);
f.polynomial = 1 + 2 * X + 3 * X ^2

Defining the FIRfilter class as a child of the polynomial class implies that and FIRfilter object may be used any place that a polynomial may be used. This is not a normal use of a filter, so that aggregation may be a more sensible design approach. In this case, the polynomial is simply a field in the class structure. A class constructor for this case might be

## -*- texinfo -*-
## @deftypefn  {Function File} {} FIRfilter ()
## @deftypefnx {Function File} {} FIRfilter (@var{p})
## Create a FIR filter with polynomial @var{p} as coefficient vector.
## @end deftypefn
function f = FIRfilter (p)
  if (nargin == 0)
    f.polynomial = @polynomial ([1]);
  elseif (nargin == 1)
    if (isa (p, "polynomial"))
      f.polynomial = p;
    else
      error ("FIRfilter: expecting polynomial as input argument");
    endif
  else
    print_usage ();
  endif
  f = class (f, "FIRfilter");
endfunction

690 GNU Octave For our example, the remaining class methods remain unchanged.

35 GUI Development

Octave is principally a batch or command-line language. However, it does offer some limited features for constructing graphical interfaces for interacting with users.

The GUI elements available are I/O dialogs and a progress bar. For example, rather than hardcoding a filename for output results a script can open a dialog box and allow the user to choose a file. Similarly, if a calculation is expected to take a long time a script can display a progress bar.

Several utility functions make it possible to store private data for use with a GUI which will not pollute the user’s variable space.

Finally, a program written in Octave might want to have long term storage of preferences or state variables. This can be done with user-defined preferences.

35.1 I/O Dialogs

Simple dialog menus are available for choosing directories or files. They return a string variable which can then be used with any command requiring a file name.

[Function File] dirname = uigetdir ()
[Function File] dirname = uigetdir (init_path)
[Function File] dirname = uigetdir (init_path, dialog_name)

Open a GUI dialog for selecting a directory. If init path is not given the current working directory is used. dialog name may be used to customize the dialog title.

See also: [uigetfile], page 691, [uiputfile], page 692.

[Function File] [fname, fpath, fltidx] = uigetfile ()
[Function File] [...] = uigetfile (flt)
[Function File] [...] = uigetfile (flt, dialog_name)
[Function File] [...] = uigetfile (flt, dialog_name, default_file)

[Function File] [...] = uigetfile (…, "Position", [px py]) [Function File] [...] = uigetfile (…, "MultiSelect", mode)

Open a GUI dialog for selecting a file and return the filename fname, the path to this file fpath, and the filter index fltidx. flt contains a (list of) file filter string(s) in one of the following formats:

     "/path/to/filename.ext"

If a filename is given then the file extension is extracted and used as filter. In addition, the path is selected as current path and the filename is selected as default file. Example: uigetfile ("myfun.m")

A single file extension "*.ext"

               Example: uigetfile ("*.ext")

A 2-column cell array containing a file extension in the first column and a brief description in the second column. Example: uigetfile ({"*.ext", "My

               Description";"*.xyz", "XYZ-Format"})

The filter string can also contain a semicolon separated list of filter ex-

          tensions. Example: uigetfile ({"*.gif;*.png;*.jpg", "Supported
          Picture Formats"})

dialog name can be used to customize the dialog title. If default file is given then it will be selected in the GUI dialog. If, in addition, a path is given it is also used as current path. The screen position of the GUI dialog can be set using the "Position" key and a 2-element vector containing the pixel coordinates. Two or more files can be selected when setting the "MultiSelect" key to "on". In that case fname is a cell array containing the files.

See also: [uiputfile], page 692, [uigetdir], page 691.

[Function File] [fname, fpath, fltidx] = uiputfile ()
[Function File] [fname, fpath, fltidx] = uiputfile (flt)
[Function File] [fname, fpath, fltidx] = uiputfile (flt, dialog_name)
[Function File] [fname, fpath, fltidx] = uiputfile (flt, dialog_name,
   default_file)

Open a GUI dialog for selecting a file. flt contains a (list of) file filter string(s) in one of the following formats:

"/path/to/filename.ext"

If a filename is given the file extension is extracted and used as filter. In addition the path is selected as current path and the filename is selected as default file. Example: uiputfile ("myfun.m")

"*.ext" A single file extension. Example: uiputfile ("*.ext")

{"*.ext", "My Description"}

A 2-column cell array containing the file extension in the 1st column and a brief description in the 2nd column. Example: uiputfile

          ({"*.ext","My Description";"*.xyz", "XYZ-Format"})

The filter string can also contain a semicolon separated list of filter extensions. Ex-

ample: uiputfile ({"*.gif;*.png;*.jpg", "Supported Picture Formats"})

dialog name can be used to customize the dialog title. If default file is given it is preselected in the GUI dialog. If, in addition, a path is given it is also used as current path.

See also: [uigetfile], page 691, [uigetdir], page 691.

35.2 Progress Bar

[Function File] h = waitbar (frac) [Function File] h = waitbar (frac, msg)

[Function File] h = waitbar (…, "FigureProperty", "Value", …)

[Function File] waitbar (frac) [Function File] waitbar (frac, hwbar)

[Function File] waitbar (frac, hwbar, msg)

Return a handle h to a new waitbar object.

The waitbar is filled to fraction frac which must be in the range [0, 1]. The optional message msg is centered and displayed above the waitbar. The appearance of the waitbar figure window can be configured by passing property/value pairs to the function. When called with a single input the current waitbar, if it exists, is updated to the new value frac. If there are multiple outstanding waitbars they can be updated individually by passing the handle hwbar of the specific waitbar to modify.

35.3 GUI Utility Functions

These functions do not implement a GUI element but are useful when developing programs that do. Warning: The functions uiwait, uiresume, and waitfor are only available for the FLTK tooolkit.

[Function File] used = desktop ("-inuse")

Return true if the desktop (GUI) is currently in use.

See also: [isguirunning], page 693.

[Function File] data = guidata (h) [Function File] guidata (h, data)

Query or set user-custom GUI data. The GUI data is stored in the figure handle h. If h is not a figure handle then it’s parent figure will be used for storage. data must be a single object which means it is usually preferable for it to be a data container such as a cell array or struct so that additional data items can be added easily.

See also: [getappdata], page 363, [setappdata], page 363, [get], page 341, [set], page 342, [getpref], page 695, [setpref], page 695.

[Function File] hdata = guihandles (h)
[Function File] hdata = guihandles

Return a structure of object handles for the figure associated with handle h. If no handle is specified the current figure returned by gcf is used. The fieldname for each entry of hdata is taken from the "tag" property of the graphic object. If the tag is empty then the handle is not returned. If there are multiple graphic objects with the same tag then the entry in hdata will be a vector of handles. guihandles includes all possible handles, including those for which

"HandleVisibility" is "off".

See also: [guidata], page 693, [findobj], page 359, [findall], page 359, [allchild], page 342.

[Built-in Function] isguirunning ()

Return true if Octave is running in GUI mode and false otherwise.

[Function File] uiwait [Function File] uiwait (h)

[Function File] uiwait (h, timeout)

Suspend program execution until the figure with handle h is deleted or uiresume is called. When no figure handle is specified, this function uses the current figure. If the figure handle is invalid or there is no current figure, this functions returns immediately. When specified, timeout defines the number of seconds to wait for the figure deletion or the uiresume call. The timeout value must be at least 1. If a smaller value is specified, a warning is issued and a timeout value of 1 is used instead. If a noninteger value is specified, it is truncated towards 0. If timeout is not specified, the program execution is suspended indefinitely.

See also: [uiresume], page 694, [waitfor], page 694.

[Function File] uiresume (h)

Resume program execution suspended with uiwait. The handle h must be the same as the on specified in uiwait. If the handle is invalid or there is no uiwait call pending for the figure with handle h, this function does nothing.

See also: [uiwait], page 693.

[Built-in Function] waitfor (h) [Built-in Function] waitfor (h, prop) [Built-in Function] waitfor (h, prop, value) [Built-in Function] waitfor (…, "timeout", timeout)

Suspend the execution of the current program until a condition is satisfied on the graphics handle h. While the program is suspended graphics events are still being processed normally, allowing callbacks to modify the state of graphics objects. This function is reentrant and can be called from a callback, while another waitfor call is pending at the top-level. In the first form, program execution is suspended until the graphics object h is destroyed. If the graphics handle is invalid, the function returns immediately. In the second form, execution is suspended until the graphics object is destroyed or the property named prop is modified. If the graphics handle is invalid or the property does not exist, the function returns immediately. In the third form, execution is suspended until the graphics object is destroyed or the property named prop is set to value. The function isequal is used to compare property values. If the graphics handle is invalid, the property does not exist or the property is already set to value, the function returns immediately. An optional timeout can be specified using the property timeout. This timeout value is the number of seconds to wait for the condition to be true. timeout must be at least 1. If a smaller value is specified, a warning is issued and a value of 1 is used instead. If the timeout value is not an integer, it is truncated towards 0. To define a condition on a property named timeout, use the string \timeout instead. In all cases, typing CTRL-C stops program execution immediately.

See also: [waitforbuttonpress], page 332, [isequal], page 141.

35.4 User-Defined Preferences

[Function File] getpref (group, pref, default)

Return the preference value corresponding to the named preference pref in the preference group group. The named preference group must be a character string. If pref does not exist in group and default is specified, return default. The preference pref may be a character string or a cell array of character strings. The corresponding default value default may be any value, or, if pref is a cell array of strings, default must be a cell array of values with the same size as pref. If neither pref nor default are specified, return a structure of preferences for the preference group group. If no arguments are specified, return a structure containing all groups of preferences and their values.

See also: [addpref], page 695, [setpref], page 695, [ispref], page 696, [rmpref], page 695.

[Function File] setpref (group, pref, val)

Set a preference pref to the given val in the named preference group group. The named preference group must be a character string. The preference pref may be a character string or a cell array of character strings. The corresponding value val may be any value, or, if pref is a cell array of strings, val must be a cell array of values with the same size as pref. If the named preference or group does not exist, it is added.

See also: [addpref], page 695, [getpref], page 695, [ispref], page 696, [rmpref], page 695.

[Function File] addpref (group, pref, val)

Add a preference pref and associated value val to the named preference group group. The named preference group must be a character string. The preference pref may be a character string or a cell array of character strings. The corresponding value val may be any value, or, if pref is a cell array of strings, val must be a cell array of values with the same size as pref.

See also: [setpref], page 695, [getpref], page 695, [ispref], page 696, [rmpref], page 695.

[Function File] rmpref (group) [Function File] rmpref (group, pref)

Remove the named preference pref from the preference group group. The named preference group must be a character string. The preference pref may be a character string or cell array of strings. If pref is not specified, remove the preference group group. It is an error to remove a nonexistent preference or group.

See also: [addpref], page 695, [ispref], page 696, [setpref], page 695, [getpref], page 695.

[Function File] ispref (group, pref)

Return true if the named preference pref exists in the preference group group. The named preference group must be a character string. The preference pref may be a character string or a cell array of character strings. If pref is not specified, return true if the preference group group exists. See also: [getpref], page 695, [addpref], page 695, [setpref], page 695, [rmpref], page 695.

[Command] prefdir

[Command] dir = prefdir

Return the directory that contains the preferences for Octave. Examples: Display the preferences directory

     prefdir

Change to the preferences folder

     cd (prefdir)

See also: [getpref], page 695, [setpref], page 695, [addpref], page 695, [rmpref], page 695, [ispref], page 696.

[Command] preferences

Display the GUI preferences dialog window for Octave.

36 System Utilities

This chapter describes the functions that are available to allow you to get information about what is happening outside of Octave, while it is still running, and use this information in your program. For example, you can get information about environment variables, the current time, and even start other programs from the Octave prompt.

36.1 Timing Utilities

Octave’s core set of functions for manipulating time values are patterned after the corresponding functions from the standard C library. Several of these functions use a data structure for time that includes the following elements:

usec Microseconds after the second (0-999999).

sec Seconds after the minute (0-60). This number can be 60 to account for leap seconds.

min Minutes after the hour (0-59).

hour Hours since midnight (0-23).

mday Day of the month (1-31).

mon Months since January (0-11).

year Years since 1900.

wday Days since Sunday (0-6).

yday Days since January 1 (0-365).

isdst Daylight Savings Time flag.

zone Time zone.

In the descriptions of the following functions, this structure is referred to as a tm struct.

[Built-in Function] seconds = time ()

Return the current time as the number of seconds since the epoch. The epoch is referenced to 00:00:00 CUT (Coordinated Universal Time) 1 Jan 1970. For example, on Monday February 17, 1997 at 07:15:06 CUT, the value returned by time was 856163706.

See also: [strftime], page 699, [strptime], page 701, [localtime], page 698, [gmtime], page 698, [mktime], page 699, [now], page 697, [date], page 701, [clock], page 701, [datenum], page 703, [datestr], page 704, [datevec], page 705, [calendar], page 706, [weekday], page 706.

[Function File] t = now ()

Return the current local date/time as a serial day number (see datenum).

The integral part, floor (now) corresponds to the number of days between today and Jan 1, 0000. The fractional part, rem (now, 1) corresponds to the current time.

See also: [clock], page 701, [date], page 701, [datenum], page 703.

[Function File] ctime (t)

Convert a value returned from time (or any other non-negative integer), to the local time and return a string of the same form as asctime. The function ctime (time) is equivalent to asctime (localtime (time)). For example:

ctime (time ())
⇒ "Mon Feb 17 01:15:06 1997"

See also: [asctime], page 699, [time], page 697, [localtime], page 698.

[Built-in Function] tm_struct = gmtime (t)

Given a value returned from time, or any non-negative integer, return a time structure corresponding to CUT (Coordinated Universal Time). For example:

gmtime (time ())
⇒ {
usec = 0
sec = 6
min = 15
hour = 7
mday = 17
mon = 1
year = 97
wday = 1
yday = 47
isdst = 0
zone = CST
}

See also: [strftime], page 699, [strptime], page 701, [localtime], page 698, [mktime], page 699, [time], page 697, [now], page 697, [date], page 701, [clock], page 701, [datenum], page 703, [datestr], page 704, [datevec], page 705, [calendar], page 706, [weekday], page 706.

[Built-in Function] tm_struct = localtime (t)

Given a value returned from time, or any non-negative integer, return a time structure corresponding to the local time zone.

localtime (time ())
⇒ {
usec = 0
sec = 6
min = 15
hour = 1
mday = 17
mon = 1
year = 97
wday = 1
yday = 47
isdst = 0
zone = CST
}

See also: [strftime], page 699, [strptime], page 701, [gmtime], page 698, [mktime], page 699, [time], page 697, [now], page 697, [date], page 701, [clock], page 701, [datenum], page 703, [datestr], page 704, [datevec], page 705, [calendar], page 706, [weekday], page 706.

[Built-in Function] seconds = mktime (tm_struct)

Convert a time structure corresponding to the local time to the number of seconds since the epoch. For example:

     mktime (localtime (time ()))
          ⇒ 856163706

See also: [strftime], page 699, [strptime], page 701, [localtime], page 698, [gmtime], page 698, [time], page 697, [now], page 697, [date], page 701, [clock], page 701, [datenum], page 703, [datestr], page 704, [datevec], page 705, [calendar], page 706, [weekday], page 706.

[Function File] asctime (tm_struct)

Convert a time structure to a string using the following format: "ddd mmm mm

HH:MM:SS yyyy". For example:
     asctime (localtime (time ()))
          ⇒ "Mon Feb 17 01:15:06 1997"

This is equivalent to ctime (time ()).

See also: [ctime], page 698, [localtime], page 698, [time], page 697.

[Built-in Function] strftime (fmt, tm_struct)

Format the time structure tm struct in a flexible way using the format string fmt that contains ‘%’ substitutions similar to those in printf. Except where noted, substituted fields have a fixed size; numeric fields are padded if necessary. Padding is with zeros by default; for fields that display a single number, padding can be changed or inhibited by following the ‘%’ with one of the modifiers described below. Unknown field specifiers are copied as normal characters. All other characters are copied to the output without change. For example:

     strftime ("%r (%Z) %A %e %B %Y", localtime (time ()))
           ⇒ "01:15:06 AM (CST) Monday 17 February 1997"

Octave’s strftime function supports a superset of the ANSI C field specifiers. Literal character fields:

%% % character.

%n Newline character.

%t Tab character.

Numeric modifiers (a nonstandard extension):

- (dash) Do not pad the field.

_(underscore)

Pad the field with spaces.

Time fields:

%H Hour (00-23). %I Hour (01-12). %k Hour (0-23). %l Hour (1-12). %M Minute (00-59). %p Locale’s AM or PM. %r Time, 12-hour (hh:mm:ss [AP]M). %R Time, 24-hour (hh:mm). %s Time in seconds since 00:00:00, Jan 1, 1970 (a nonstandard extension). %S Second (00-61). %T Time, 24-hour (hh:mm:ss). %X Locale’s time representation (%H:%M:%S). %Z Time zone (EDT), or nothing if no time zone is determinable. Date fields: %a Locale’s abbreviated weekday name (Sun-Sat). %A Locale’s full weekday name, variable length (Sunday-Saturday). %b Locale’s abbreviated month name (Jan-Dec). %B Locale’s full month name, variable length (January-December). %c Locale’s date and time (Sat Nov 04 12:02:33 EST 1989). %C Century (00-99). %d Day of month (01-31). %e Day of month ( 1-31). %D Date (mm/dd/yy). %h Same as %b. %j Day of year (001-366). %m Month (01-12). %U Week number of year with Sunday as first day of week (00-53). %w Day of week (0-6). %W Week number of year with Monday as first day of week (00-53). %x Locale’s date representation (mm/dd/yy). %y Last two digits of year (00-99). %Y Year (1970-). See also: [strptime], page 701, [localtime], page 698, [gmtime], page 698, [mktime], page 699, [time], page 697, [now], page 697, [date], page 701, [clock], page 701, [datenum], page 703, [datestr], page 704, [datevec], page 705, [calendar], page 706, [weekday], page 706.

[Built-in Function] [tm_struct, nchars] = strptime (str, fmt)

Convert the string str to the time structure tm struct under the control of the format string fmt.

If fmt fails to match, nchars is 0; otherwise, it is set to the position of last matched character plus 1. Always check for this unless you’re absolutely sure the date string will be parsed correctly.

See also: [strftime], page 699, [localtime], page 698, [gmtime], page 698, [mktime], page 699, [time], page 697, [now], page 697, [date], page 701, [clock], page 701, [datenum], page 703, [datestr], page 704, [datevec], page 705, [calendar], page 706, [weekday], page 706.

Most of the remaining functions described in this section are not patterned after the standard C library. Some are available for compatibility with matlab and others are provided because they are useful.

[Function File] clock ()

Return the current local date and time as a date vector. The date vector contains the following fields: current year, month (1-12), day (1-31), hour (0-23), minute (0-59), and second (0-61). The seconds field has a fractional part after the decimal point for extended accuracy.

For example:

     fix (clock ())
          ⇒ [ 1993, 8, 20, 4, 56, 1 ]

The function clock is more accurate on systems that have the gettimeofday function.

See also: [now], page 697, [date], page 701, [datevec], page 705.

[Function File] date ()

Return the current date as a character string in the form DD-MMM-YYYY.

For example:

     date () ⇒ "20-Aug-1993"

See also: [now], page 697, [clock], page 701, [datestr], page 704, [localtime], page 698.

[Function File] etime (t2, t1)

Return the difference in seconds between two time values returned from clock (t2 − t1). For example:

     t0 = clock ();
     # many computations later...
     elapsed_time = etime (clock (), t0);

will set the variable elapsed_time to the number of seconds since the variable t0 was set.

See also: [tic], page 702, [toc], page 702, [clock], page 701, [cputime], page 702, [addtodate], page 706.

[Built-in Function] [total, user, system] = cputime ();

Return the CPU time used by your Octave session. The first output is the total time spent executing your process and is equal to the sum of second and third outputs, which are the number of CPU seconds spent executing in user mode and the number of CPU seconds spent executing in system mode, respectively. If your system does not have a way to report CPU time usage, cputime returns 0 for each of its output values. Note that because Octave used some CPU time to start, it is reasonable to check to see if cputime works by checking to see if the total CPU time used is nonzero.

See also: [tic], page 702, [toc], page 702.

[Function File] is_leap_year ()

[Function File] is_leap_year (year)

Return true if year is a leap year and false otherwise. If no year is specified, is_ leap_year uses the current year. For example:

     is_leap_year (2000)
        ⇒ 1

See also: [weekday], page 706, [eomday], page 707, [calendar], page 706.

[Built-in Function] tic () [Built-in Function] id = tic () [Built-in Function] toc () [Built-in Function] toc (id) [Built-in Function] val = toc (…)

Set or check a wall-clock timer. Calling tic without an output argument sets the internal timer state. Subsequent calls to toc return the number of seconds since the timer was set. For example,

     tic ();
     # many computations later...
     elapsed_time = toc ();

will set the variable elapsed_time to the number of seconds since the most recent call to the function tic. If called with one output argument, tic returns a scalar of type uint64 that may be later passed to toc.

     id = tic; sleep (5); toc (id)
           ⇒ 5.0010

Calling tic and toc this way allows nested timing calls. If you are more interested in the CPU time that your process used, you should use the cputime function instead. The tic and toc functions report the actual wall clock time that elapsed between the calls. This may include time spent processing other jobs or doing nothing at all.

See also: [toc], page 702, [cputime], page 702.

[Built-in Function] pause () [Built-in Function] pause (n)

Suspend the execution of the program for n seconds. n is a positive real value and may be a fraction of a second. If invoked without an input arguments then the program is suspended until a character is typed. The following example prints a message and then waits 5 seconds before clearing the screen.

     fprintf (stderr, "wait please...\n");
     pause (5);
     clc;

See also: [kbhit], page 229, [sleep], page 703.

[Built-in Function] sleep (seconds)

Suspend the execution of the program for the given number of seconds.

See also: [usleep], page 703, [pause], page 702.

[Built-in Function] usleep (microseconds)

Suspend the execution of the program for the given number of microseconds. On systems where it is not possible to sleep for periods of time less than one second, usleep will pause the execution for round (microseconds / 1e6) seconds.

See also: [sleep], page 703, [pause], page 702.

[Function File] days = datenum (datevec)
[Function File] days = datenum (year, month, day)
[Function File] days = datenum (year, month, day, hour)
[Function File] days = datenum (year, month, day, hour, minute)
[Function File] days = datenum (year, month, day, hour, minute, second)

[Function File] days = datenum ("datestr") [Function File] days = datenum ("datestr", p)

[Function File] [days, secs] = datenum (. . .)

Return the date/time input as a serial day number, with Jan 1, 0000 defined as day 1. The integer part, floor (days) counts the number of complete days in the date input. The fractional part, rem (days, 1) corresponds to the time on the given day. The input may be a date vector (see datevec), datestr (see datestr), or directly specified as input. When processing input datestrings, p is the year at the start of the century to which two-digit years will be referenced. If not specified, it defaults to the current year minus 50. The optional output secs holds the time on the specified day with greater precision than days. Notes:

• Years can be negative and/or fractional.

• Months below 1 are considered to be January.

• Days of the month start at 1.

• Days beyond the end of the month go into subsequent months.

• Days before the beginning of the month go to the previous month.

• Days can be fractional.

Caution: this function does not attempt to handle Julian calendars so dates before October 15, 1582 are wrong by as much as eleven days. Also, be aware that only Roman Catholic countries adopted the calendar in 1582. It took until 1924 for it to be adopted everywhere. See the Wikipedia entry on the Gregorian calendar for more details. Warning: leap seconds are ignored. A table of leap seconds is available on the Wikipedia entry for leap seconds.

See also: [datestr], page 704, [datevec], page 705, [now], page 697, [clock], page 701, [date], page 701.

[Function File] str = datestr (date) [Function File] str = datestr (date, f) [Function File] str = datestr (date, f, p)

Format the given date/time according to the format f and return the result in str. date is a serial date number (see datenum) or a date vector (see datevec). The value of date may also be a string or cell array of strings. f can be an integer which corresponds to one of the codes in the table below, or a date format string. p is the year at the start of the century in which two-digit years are to be interpreted in. If not specified, it defaults to the current year minus 50. For example, the date 730736.65149 (2000-09-07 15:38:09.0934) would be formatted as follows:

Code  Format  Example
0  dd-mmm-yyyy HH:MM:SS  07-Sep-2000 15:38:09
1  dd-mmm-yyyy  07-Sep-2000
2  mm/dd/yy  09/07/00
3  mmm  Sep
4  m  S
5  mm  09
6  mm/dd  09/07
7  dd  07
8  ddd  Thu
9  d  T
10  yyyy  2000
11  yy  00
12  mmmyy  Sep00
13  HH:MM:SS  15:38:09
14  HH:MM:SS PM  03:38:09 PM
15  HH:MM  15:38
16  HH:MM PM  03:38 PM
17  QQ-YY  Q3-00
18  QQ  Q3

19  dd/mm  07/09
20  dd/mm/yy  07/09/00
21  mmm.dd,yyyy HH:MM:SS  Sep.07,2000 15:38:08
22  mmm.dd,yyyy  Sep.07,2000
23  mm/dd/yyyy  09/07/2000
24  dd/mm/yyyy  07/09/2000
25  yy/mm/dd  00/09/07
26  yyyy/mm/dd  2000/09/07
27  QQ-YYYY  Q3-2000
28  mmmyyyy  Sep2000
29  yyyy-mm-dd  2000-09-07
30  yyyymmddTHHMMSS  20000907T153808
31  yyyy-mm-dd HH:MM:SS  2000-09-07 15:38:08

If f is a format string, the following symbols are recognized:

Symbol Meaning  Example
yyyy  Full year  2005
yy  Two-digit year  05
mmmm Full month name  December
mmm  Abbreviated month name  Dec
mm  Numeric month number (padded with zeros)  01, 08, 12
m  First letter of month name (capitalized)  D
dddd  Full weekday name  Sunday
ddd  Abbreviated weekday name  Sun
dd  Numeric day of month (padded with zeros)  11
d  First letter of weekday name (capitalized)  S
HH  Hour of day, padded with zeros if PM is set  09:00
and not padded with zeros otherwise  9:00 AM
MM  Minute of hour (padded with zeros)  10:05
SS  Second of minute (padded with zeros)  10:05:03
FFF  Milliseconds of second (padded with zeros)  10:05:03.012
AM  Use 12-hour time format  11:30 AM
PM  Use 12-hour time format  11:30 PM

If f is not specified or is -1, then use 0, 1 or 16, depending on whether the date portion or the time portion of date is empty.

If p is nor specified, it defaults to the current year minus 50.

If a matrix or cell array of dates is given, a column vector of date strings is returned.

See also: [datenum], page 703, [datevec], page 705, [date], page 701, [now], page 697, [clock], page 701.

[Function File] v = datevec (date) [Function File] v = datevec (date, f) [Function File] v = datevec (date, p) [Function File] v = datevec (date, f, p)

[Function File] [y, m, d, h, mi, s] = datevec (. . .)

Convert a serial date number (see datenum) or date string (see datestr) into a date vector. A date vector is a row vector with six members, representing the year, month, day, hour, minute, and seconds respectively. f is the format string used to interpret date strings (see datestr). If date is a string, but no format is specified, then a relatively slow search is performed through various formats. It is always preferable to specify the format string f if it is known. Formats which do not specify a particular time component will have the value set to zero. Formats which do not specify a date will default to January 1st of the current year. p is the year at the start of the century to which two-digit years will be referenced. If not specified, it defaults to the current year minus 50.

See also: [datenum], page 703, [datestr], page 704, [clock], page 701, [now], page 697, [date], page 701.

[Function File] d = addtodate (d, q, f)

Add q amount of time (with units f ) to the serial datenum, d.

f must be one of "year", "month", "day", "hour", "minute", "second", or
"millisecond".

See also: [datenum], page 703, [datevec], page 705, [etime], page 701.

[Function File] c = calendar () [Function File] c = calendar (d) [Function File] c = calendar (y, m) [Function File] calendar (…)

Return the current monthly calendar in a 6x7 matrix. If d is specified, return the calendar for the month containing the date d, which must be a serial date number or a date string. If y and m are specified, return the calendar for year y and month m. If no output arguments are specified, print the calendar on the screen instead of returning a matrix.

See also: [datenum], page 703, [datestr], page 704.

[Function File] [n, s] = weekday (d)

[Function File] [n, s] = weekday (d, format)

Return the day of the week as a number in n and as a string in s. The days of the week are numbered 1–7 with the first day being Sunday. d is a serial date number or a date string. If the string format is not present or is equal to "short" then s will contain the abbreviated name of the weekday. If format is "long" then s will contain the full name. Table of return values based on format:

n  "short"   "long"

1  Sun  Sunday
2  Mon  Monday
3  Tue  Tuesday
4  Wed  Wednesday
5  Thu  Thursday
6  Fri  Friday
7  Sat  Saturday

See also: [eomday], page 707, [is leap year], page 702, [calendar], page 706, [datenum], page 703, [datevec], page 705.

[Function File] e = eomday (y, m)

Return the last day of the month m for the year y.

See also: [weekday], page 706, [datenum], page 703, [datevec], page 705, [is leap year], page 702, [calendar], page 706.

[Function File] datetick () [Function File] datetick (form) [Function File] datetick (axis, form)

[Function File] datetick (…, "keeplimits") [Function File] datetick (…, "keepticks")

[Function File] datetick (hax, …)

Add date formatted tick labels to an axis. The axis to apply the ticks to is determined by axis which can take the values "x", "y", or "z". The default value is "x". The formatting of the labels is determined by the variable form, which can either be a string or positive integer that datestr accepts.

See also: [datenum], page 703, [datestr], page 704.

36.2 Filesystem Utilities

Octave includes many utility functions for copying, moving, renaming, and deleting files; for creating, reading, and deleting directories; for retrieving status information on files; and for manipulating file and path names.

[Function File] movefile (f1) [Function File] movefile (f1, f2) [Function File] movefile (f1, f2, ’f’)

[Function File] [status, msg, msgid] = movefile (. . .)

Move the file f1 to the destination f2. The name f1 may contain globbing patterns. If f1 expands to multiple file names, f2 must be a directory. If no destination f2 is specified then the destination is the present working directory. If f2 is a file name then f1 is renamed to f2. When the force flag 'f' is given any existing files will be overwritten without prompting. If successful, status is 1, and msg, msgid are empty character strings (""). Otherwise, status is 0, msg contains a system-dependent error message, and msgid contains a unique message identifier. Note that the status code is exactly opposite that of the system command.

See also: [rename], page 708, [copyfile], page 708, [unlink], page 708, [delete], page 323, [glob], page 713.

[Built-in Function] rename old new

[Built-in Function] [err, msg] = rename (old, new)

Change the name of file old to new. If successful, err is 0 and msg is an empty string. Otherwise, err is nonzero and msg contains a system-dependent error message.

See also: [movefile], page 707, [copyfile], page 708, [ls], page 729, [dir], page 729.

[Function File] [status, msg, msgid] = copyfile (f1, f2)
[Function File] [status, msg, msgid] = copyfile (f1, f2, ’f’)

Copy the file f1 to the destination f2. The name f1 may contain globbing patterns. If f1 expands to multiple file names, f2 must be a directory. when the force flag 'f' is given any existing files will be overwritten without prompting. If successful, status is 1, and msg, msgid are empty character strings (""). Otherwise, status is 0, msg contains a system-dependent error message, and msgid contains a unique message identifier. Note that the status code is exactly opposite that of the system command.

See also: [movefile], page 707, [rename], page 708, [unlink], page 708, [delete], page 323, [glob], page 713.

[Built-in Function] [err, msg] = unlink (file)

Delete the file named file. If successful, err is 0 and msg is an empty string. Otherwise, err is nonzero and msg contains a system-dependent error message.

[Built-in Function] link old new [Built-in Function] [err, msg] = link (old, new)

Create a new link (also known as a hard link) to an existing file. If successful, err is 0 and msg is an empty string. Otherwise, err is nonzero and msg contains a system-dependent error message.

See also: [symlink], page 708, [unlink], page 708, [readlink], page 708, [lstat], page 710.

[Built-in Function] symlink old new

[Built-in Function] [err, msg] = symlink (old, new)

Create a symbolic link new which contains the string old. If successful, err is 0 and msg is an empty string. Otherwise, err is nonzero and msg contains a system-dependent error message.

See also: [link], page 708, [unlink], page 708, [readlink], page 708, [lstat], page 710.

[Built-in Function] readlink symlink

[Built-in Function] [result, err, msg] = readlink (symlink)

Read the value of the symbolic link symlink.

If successful, result contains the contents of the symbolic link symlink, err is 0, and msg is an empty string. Otherwise, err is nonzero and msg contains a systemdependent error message.

See also: [lstat], page 710, [symlink], page 708, [link], page 708, [unlink], page 708, [delete], page 323.

[Built-in Function] mkdir dir [Built-in Function] mkdir (parent, dir)

[Built-in Function] [status, msg, msgid] = mkdir (. . .)

Create a directory named dir in the directory parent.

If no parent directory is specified the present working directory is used.

If successful, status is 1, and msg, msgid are empty character strings (""). Otherwise, status is 0, msg contains a system-dependent error message, and msgid contains a unique message identifier.

See also: [rmdir], page 709, [pwd], page 730, [cd], page 729.

[Built-in Function] rmdir dir [Built-in Function] rmdir (dir, "s")

[Built-in Function] [status, msg, msgid] = rmdir (. . .)

Remove the directory named dir.

If successful, status is 1, and msg, msgid are empty character strings (""). Otherwise, status is 0, msg contains a system-dependent error message, and msgid contains a unique message identifier.

If the optional second parameter is supplied with value "s", recursively remove all subdirectories as well.

See also: [mkdir], page 709, [confirm recursive rmdir], page 709, [pwd], page 730.

[Built-in Function] val = confirm_recursive_rmdir ()
[Built-in Function] old_val = confirm_recursive_rmdir (new_val)
[Built-in Function] confirm_recursive_rmdir (new_val, "local")

Query or set the internal variable that controls whether Octave will ask for confirmation before recursively removing a directory tree.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [rmdir], page 709.

[Built-in Function] mkfifo (name, mode)

[Built-in Function] [err, msg] = mkfifo (name, mode)

Create a FIFO special file named name with file mode mode

If successful, err is 0 and msg is an empty string. Otherwise, err is nonzero and msg contains a system-dependent error message.

See also: [pipe], page 724.

[Built-in Function] umask (mask)

Set the permission mask for file creation. The parameter mask is an integer, interpreted as an octal number. If successful, returns the previous value of the mask (as an integer to be interpreted as an octal number); otherwise an error message is printed.

[Built-in Function] [info, err, msg] = stat (file)
[Built-in Function] [info, err, msg] = stat (fid)
[Built-in Function] [info, err, msg] = lstat (file)
[Built-in Function] [info, err, msg] = lstat (fid)

Return a structure info containing the following information about file or file identifier fid.

dev ID of device containing a directory entry for this file.

ino File number of the file.

mode File mode, as an integer. Use the functions S_ISREG, S_ISDIR, S_ISCHR,

          S_ISBLK, S_ISFIFO, S_ISLNK, or S_ISSOCK to extract information from

this value.

modestr File mode, as a string of ten letters or dashes as would be returned by ls

          -l.

nlink Number of links.

uid User ID of file’s owner.

gid Group ID of file’s group.

rdev ID of device for block or character special files.

size Size in bytes.

atime Time of last access in the same form as time values returned from time. See Section 36.1 [Timing Utilities], page 697.

mtime Time of last modification in the same form as time values returned from time. See Section 36.1 [Timing Utilities], page 697.

ctime Time of last file status change in the same form as time values returned from time. See Section 36.1 [Timing Utilities], page 697.

blksize Size of blocks in the file.

blocks Number of blocks allocated for file.

If the call is successful err is 0 and msg is an empty string. If the file does not exist, or some other error occurs, info is an empty matrix, err is −1, and msg contains the corresponding system error message.

If file is a symbolic link, stat will return information about the actual file that is referenced by the link. Use lstat if you want information about the symbolic link itself.

For example:

     [info, err, msg] = stat ("/vmlinuz")
       ⇒ info =
          {
            atime = 855399756
            rdev = 0
            ctime = 847219094
            uid = 0
            size = 389218
            blksize = 4096
            mtime = 847219094
            gid = 6
            nlink = 1
            blocks = 768
            mode = -rw-r--r--
            modestr = -rw-r--r--
            ino = 9316
            dev = 2049
          }
       ⇒ err = 0
       ⇒ msg =

See also: [lstat], page 710, [ls], page 729, [dir], page 729.

[Built-in Function] S_ISBLK (mode)

Return true if mode corresponds to a block device. The value of mode is assumed to be returned from a call to stat.

See also: [stat], page 710, [lstat], page 710.

[Built-in Function] S_ISCHR (mode)

Return true if mode corresponds to a character device. The value of mode is assumed to be returned from a call to stat.

See also: [stat], page 710, [lstat], page 710.

[Built-in Function] S_ISDIR (mode)

Return true if mode corresponds to a directory. The value of mode is assumed to be returned from a call to stat.

See also: [stat], page 710, [lstat], page 710.

[Built-in Function] S_ISFIFO (mode)

Return true if mode corresponds to a fifo. The value of mode is assumed to be returned from a call to stat.

See also: [stat], page 710, [lstat], page 710.

[Built-in Function] S_ISLNK (mode)

Return true if mode corresponds to a symbolic link. The value of mode is assumed to be returned from a call to stat.

See also: [stat], page 710, [lstat], page 710.

[Built-in Function] S_ISREG (mode)

Return true if mode corresponds to a regular file. The value of mode is assumed to be returned from a call to stat.

See also: [stat], page 710, [lstat], page 710.

[Built-in Function] S_ISSOCK (mode)

Return true if mode corresponds to a socket. The value of mode is assumed to be returned from a call to stat.

See also: [stat], page 710, [lstat], page 710.

[Function File] [status, result, msgid] = fileattrib (file)

Return information about file. If successful, status is 1, with result containing a structure with the following fields:

Name Full name of file.

archive True if file is an archive (Windows).

system True if file is a system file (Windows).

hidden True if file is a hidden file (Windows).

directory

True if file is a directory.

UserRead
GroupRead
OtherRead

True if the user (group; other users) has read permission for file.

UserWrite
GroupWrite
OtherWrite

True if the user (group; other users) has write permission for file.

UserExecute
GroupExecute
OtherExecute

True if the user (group; other users) has execute permission for file.

If an attribute does not apply (i.e., archive on a Unix system) then the field is set to NaN. With no input arguments, return information about the current directory. If file contains globbing characters, return information about all the matching files.

See also: [glob], page 713.

[Function File] isdir (f)

Return true if f is a directory.

See also: [exist], page 128, [stat], page 710, [is absolute filename], page 715, [is rooted relative filename], page 715.

[Built-in Function] files = readdir (dir)

[Built-in Function] [files, err, msg] = readdir (dir)

Return the names of files in the directory dir as a cell array of strings. If an error occurs, return an empty cell array in files. If successful, err is 0 and msg is an empty string. Otherwise, err is nonzero and msg contains a system-dependent error message.

See also: [ls], page 729, [dir], page 729, [glob], page 713, [what], page 130.

[Built-in Function] glob (pattern)

Given an array of pattern strings (as a char array or a cell array) in pattern, return a cell array of file names that match any of them, or an empty cell array if no patterns match. The pattern strings are interpreted as filename globbing patterns (as they are used by Unix shells). Within a pattern

* matches any string, including the null string,

? matches any single character, and

[...] matches any of the enclosed characters.

Tilde expansion is performed on each of the patterns before looking for matching file names. For example:

     ls

⇒

           file1  file2  file3  myfile1 myfile1b
     glob ("*file1")

⇒

           {
             [1,1] = file1
             [2,1] = myfile1
           }
     glob ("myfile?")

⇒

           {
             [1,1] = myfile1
           }
     glob ("file[12]")

⇒

           {
             [1,1] = file1
             [2,1] = file2
           }

See also: [ls], page 729, [dir], page 729, [readdir], page 713, [what], page 130, [fnmatch], page 713.

[Built-in Function] fnmatch (pattern, string)

Return true or false for each element of string that matches any of the elements of the string array pattern, using the rules of filename pattern matching. For example:

     fnmatch ("a*b", {"ab"; "axyzb"; "xyzab"})
          ⇒ [ 1; 1; 0 ]

See also: [glob], page 713, [regexp], page 84.

[Built-in Function] file_in_path (path, file) [Built-in Function] file_in_path (path, file, "all")

Return the absolute name of file if it can be found in path. The value of path should be a colon-separated list of directories in the format described for path. If no file is found, return an empty character string. For example:

     file_in_path (EXEC_PATH, "sh")
          ⇒ "/bin/sh"

If the second argument is a cell array of strings, search each directory of the path for element of the cell array and return the first that matches. If the third optional argument "all" is supplied, return a cell array containing the list of all files that have the same name in the path. If no files are found, return an empty cell array.

See also: [file in loadpath], page 182, [find dir in path], page 182, [path], page 181.

[Built-in Function] filesep () [Built-in Function] filesep ("all")

Return the system-dependent character used to separate directory names. If "all" is given, the function returns all valid file separators in the form of a string. The list of file separators is system-dependent. It is ‘/’ (forward slash) under UNIX or Mac OS X, ‘/’ and ‘\’ (forward and backward slashes) under Windows.

See also: [pathsep], page 182.

[Built-in Function] val = filemarker () [Built-in Function] filemarker (new_val) [Built-in Function] filemarker (new_val, "local")

Query or set the character used to separate filename from the the subfunction names contained within the file. This can be used in a generic manner to interact with subfunctions. For example,

     help (["myfunc", filemarker, "mysubfunc"])

returns the help string associated with the subfunction mysubfunc of the function myfunc. Another use of filemarker is when debugging it allows easier placement of breakpoints within subfunctions. For example,

     dbstop (["myfunc", filemarker, "mysubfunc"])

will set a breakpoint at the first line of the subfunction mysubfunc. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

[Function File] [dir, name, ext, ver] = fileparts (filename)

Return the directory, name, extension, and version components of filename.

See also: [fullfile], page 715.

[Function File] filename = fullfile (dir1, dir2, . . ., file)

Return a complete filename constructed from the given components.

See also: [fileparts], page 714.

[Built-in Function] tilde_expand (string)

Perform tilde expansion on string. If string begins with a tilde character, (‘~’), all of the characters preceding the first slash (or all characters, if there is no slash) are treated as a possible user name, and the tilde and the following characters up to the slash are replaced by the home directory of the named user. If the tilde is followed immediately by a slash, the tilde is replaced by the home directory of the user running Octave. For example:

     tilde_expand ("~joeuser/bin")
          ⇒ "/home/joeuser/bin"
     tilde_expand ("~/bin")
          ⇒ "/home/jwe/bin"
[Built-in Function] [cname, status, msg] = canonicalize_file_name
   (fname)

Return the canonical name of file fname. If the file does not exist the empty string ("") is returned.

See also: [make absolute filename], page 715, [is absolute filename], page 715, [is rooted relative filename], page 715.

[Built-in Function] make_absolute_filename (file)

Return the full name of file beginning from the root of the file system. No check is done for the existence of file.

See also: [canonicalize file name], page 715, [is absolute filename], page 715, [is rooted relative filename], page 715, [isdir], page 712.

[Built-in Function] is_absolute_filename (file)

Return true if file is an absolute filename.

See also: [is rooted relative filename], page 715, [make absolute filename], page 715, [isdir], page 712.

[Built-in Function] is_rooted_relative_filename (file)

Return true if file is a rooted-relative filename.

See also: [is absolute filename], page 715, [make absolute filename], page 715, [isdir], page 712.

[Built-in Function] P_tmpdir ()

Return the default name of the directory for temporary files on this system. The name of this directory is system dependent.

[Function File] dir = tempdir ()

Return the name of the system’s directory for temporary files.

[Function File] tempname () [Function File] tempname (dir)

[Function File] tempname (dir, prefix)

This function is an alias for tmpnam.

See also: [tmpnam], page 255.

[Function File] current_state = recycle ()
[Function File] old_state = recycle (new_state)

Query or set the preference for recycling deleted files. Recycling files, instead of permanently deleting them, is not currently implemented in Octave. To help avoid accidental data loss an error will be raised if an attempt is made to enable file recycling.

See also: [delete], page 323.

36.3 File Archiving Utilities

[Function File] bunzip2 (bzfile) [Function File] bunzip2 (bzfile, dir)

Unpack the bzip2 archive bzfile to the directory dir. If dir is not specified, it defaults to the current directory.

See also: [bzip2], page 717, [unpack], page 717, [gunzip], page 716, [unzip], page 717, [untar], page 717.

[Function File] entries = gzip (files)
[Function File] entries = gzip (files, outdir)

Compress the list of files and/or directories specified in files. Each file is compressed separately and a new file with a ‘".gz"’ extension is created. The original files are not modified. Existing compressed files are silently overwritten. If outdir is defined the compressed files are placed in this directory.

See also: [gunzip], page 716, [bzip2], page 717, [zip], page 717, [tar], page 716.

[Function File] gunzip (gzfile, dir)

Unpack the gzip archive gzfile to the directory dir. If dir is not specified, it defaults to the current directory. If gzfile is a directory, all gzfiles in the directory will be recursively gunzipped.

See also: [gzip], page 716, [unpack], page 717, [bunzip2], page 716, [unzip], page 717, [untar], page 717.

[Function File] entries = tar (tarfile, files)
[Function File] entries = tar (tarfile, files, root)

Pack files files into the TAR archive tarfile. The list of files must be a string or a cell array of strings. The optional argument root changes the relative path of files from the current directory. If an output argument is requested the entries in the archive are returned in a cell array.

See also: [untar], page 717, [bzip2], page 717, [gzip], page 716, [zip], page 717.

[Function File] untar (tarfile) [Function File] untar (tarfile, dir)

Unpack the TAR archive tarfile to the directory dir. If dir is not specified, it defaults to the current directory.

See also: [tar], page 716, [unpack], page 717, [bunzip2], page 716, [gunzip], page 716, [unzip], page 717.

[Function File] entries = zip (zipfile, files)
[Function File] entries = zip (zipfile, files, rootdir)

Compress the list of files and/or directories specified in files into the archive zipfile in the same directory. If rootdir is defined the files are located relative to rootdir rather than the current directory.

See also: [unzip], page 717, [bzip2], page 717, [gzip], page 716, [tar], page 716.

[Function File] unzip (zipfile) [Function File] unzip (zipfile, dir)

Unpack the ZIP archive zipfile to the directory dir. If dir is not specified, it defaults to the current directory.

See also: [zip], page 717, [unpack], page 717, [bunzip2], page 716, [gunzip], page 716, [untar], page 717.

[Function File] files = unpack (file)
[Function File] files = unpack (file, dir)
[Function File] files = unpack (file, dir, filetype)

Unpack the archive file based on its extension to the directory dir. If file is a list of strings, then each file is unpacked individually. If dir is not specified, it defaults to the current directory. If a directory is in the file list, then the filetype must also be specified.

The optional return value is a list of files unpacked.

See also: [bzip2], page 717, [gzip], page 716, [zip], page 717, [tar], page 716.

[Function File] entries = bzip2 (files)
[Function File] entries = bzip2 (files, outdir)

Compress the list of files specified in files. Each file is compressed separately and a new file with a ‘".bz2"’ extension is created. The original files are not modified. Existing compressed files are silently overwritten. If outdir is defined the compressed files are placed in this directory.

See also: [bunzip2], page 716, [gzip], page 716, [zip], page 717, [tar], page 716.

36.4 Networking Utilities

[Built-in Function] gethostname ()

Return the hostname of the system where Octave is running.

36.4.1 FTP Objects

Octave supports the FTP protocol through an object-oriented interface. Use the function ftp to create an FTP object which represents the connection. All FTP functions take an FTP object as the first argument.

[Function File] f = ftp (host)

[Function File] f = ftp (host, username, password)

Connect to the FTP server host with username and password. If username and password are not specified, user "anonymous" with no password is used. The returned FTP object f represents the established FTP connection.

The list of actions for an FTP object are shown below. All functions require an FTP object as the first argument.

Method  Description
ascii  Set transfer type to ascii
binary  Set transfer type to binary
cd  Change remote working directory
close  Close FTP connection
delete  Delete remote file
dir  List remote directory contents
mget  Download remote files
mkdir  Create remote directory
mput  Upload local files
rename  Rename remote file or directory
rmdir  Remove remote directory

[Function File] close (f)

Close the FTP connection represented by the FTP object f.

f is an FTP object returned by the ftp function.

[Function File] mget (f, file) [Function File] mget (f, dir)

[Function File] mget (f, remote_name, target)

Download a remote file file or directory dir to the local directory on the FTP connection f. f is an FTP object returned by the ftp function.

The arguments file and dir can include wildcards and any files or directories on the remote server that match will be downloaded.

If a third argument target is given, then a single file or directory will be downloaded to the local directory and the local name will be changed to target.

[Function File] mput (f, file)

Upload the local file file into the current remote directory on the FTP connection f. f is an FTP object returned by the ftp function.

The argument file is passed through the glob function and any files that match the wildcards in file will be uploaded.

[Function File] cd (f) [Function File] cd (f, path)

Get or set the remote directory on the FTP connection f. f is an FTP object returned by the ftp function. If path is not specified, return the remote current working directory. Otherwise, set the remote directory to path and return the new remote working directory. If the directory does not exist, an error message is printed and the working directory is not changed.

[Function File] lst = dir (f)

List the current directory in verbose form for the FTP connection f. f is an FTP object returned by the ftp function.

[Function File] ascii (f)

Set the FTP connection f to use ASCII mode for transfers. ASCII mode is only appropriate for text files as it will convert the remote host’s newline representation to the local host’s newline representation. f is an FTP object returned by the ftp function.

[Function File] binary (f)

Set the FTP connection f to use binary mode for transfers. In binary mode there is no conversion of newlines from the remote representation to the local representation. f is an FTP object returned by the ftp function.

[Function File] delete (f, file)

Delete the remote file file over the FTP connection f. f is an FTP object returned by the ftp function.

[Function File] rename (f, oldname, newname)

Rename or move the remote file or directory oldname to newname, over the FTP connection f. f is an FTP object returned by the ftp function.

[Function File] mkdir (f, path)

Create the remote directory path, over the FTP connection f. f is an FTP object returned by the ftp function.

[Function File] rmdir (f, path)

Remove the remote directory path, over the FTP connection f. f is an FTP object returned by the ftp function.

36.4.2 URL Manipulation

[Loadable Function] s = urlread (url)

[Loadable Function] [s, success] = urlread (url)
[Loadable Function] [s, success, message] = urlread (url)
[Loadable Function] [...] = urlread (url, method, param)

Download a remote file specified by its url and return its content in string s. For example:

     s = urlread ("ftp://ftp.octave.org/pub/octave/README");

The variable success is 1 if the download was successful, otherwise it is 0 in which case message contains an error message. If no output argument is specified and an error occurs, then the error is signaled through Octave’s error handling mechanism. This function uses libcurl. Curl supports, among others, the HTTP, FTP and FILE protocols. Username and password may be specified in the URL. For example:

     s = urlread ("http://user:password@example.com/file.txt");

GET and POST requests can be specified by method and param. The parameter method is either ‘get’ or ‘post’ and param is a cell array of parameter and value pairs. For example:

     s = urlread ("http://www.google.com/search", "get",
                 {"query", "octave"});

See also: [urlwrite], page 720.

[Loadable Function] urlwrite (url, localfile)

[Loadable Function] f = urlwrite (url, localfile)
[Loadable Function] [f, success] = urlwrite (url, localfile)
[Loadable Function] [f, success, message] = urlwrite (url, localfile)

Download a remote file specified by its url and save it as localfile. For example:

     urlwrite ("ftp://ftp.octave.org/pub/octave/README",
               "README.txt");

The full path of the downloaded file is returned in f. The variable success is 1 if the download was successful, otherwise it is 0 in which case message contains an error message. If no output argument is specified and an error occurs, then the error is signaled through Octave’s error handling mechanism. This function uses libcurl. Curl supports, among others, the HTTP, FTP and FILE protocols. Username and password may be specified in the URL, for example:

     urlwrite ("http://username:password@example.com/file.txt",
               "file.txt");

GET and POST requests can be specified by method and param. The parameter method is either ‘get’ or ‘post’ and param is a cell array of parameter and value pairs. For example:

     urlwrite ("http://www.google.com/search", "search.html",
               "get", {"query", "octave"});

See also: [urlread], page 719.

36.4.3 Base64 and Binary Data Transmission

Some transmission channels can not accept binary data. It is customary to encode binary data in Base64 for transmission and to decode the data upon reception.

[Built-in Function] s = base64_encode (x)

Encode a double matrix or array x into the base64 format string s.

See also: [base64 decode], page 721.

[Built-in Function] x = base64_decode (s) [Built-in Function] x = base64_decode (s, dims)

Decode the double matrix or array x from the base64 encoded string s. The optional input parameter dims should be a vector containing the dimensions of the decoded array.

See also: [base64 encode], page 720.

36.5 Controlling Subprocesses

Octave includes some high-level commands like system and popen for starting subprocesses. If you want to run another program to perform some task and then look at its output, you will probably want to use these functions.

Octave also provides several very low-level Unix-like functions which can also be used for starting subprocesses, but you should probably only use them if you can’t find any way to do what you need with the higher-level functions.

[Built-in Function] system ("string")

[Built-in Function] system ("string", return_output)
[Built-in Function] system ("string", return_output, type)

[Built-in Function] [status, output] = system (…)

Execute a shell command specified by string. If the optional argument type is "async", the process is started in the background and the process ID of the child process is returned immediately. Otherwise, the child process is started and Octave waits until it exits. If the type argument is omitted, it defaults to the value "sync". If system is called with one or more output arguments, or if the optional argument return output is true and the subprocess is started synchronously, then the output from the command is returned as a variable. Otherwise, if the subprocess is executed synchronously, its output is sent to the standard output. To send the output of a command executed with system through the pager, use a command like

     [output, text] = system ("cmd");
     disp (text);

or

     printf ("%s\n", nthargout (2, "system", "cmd"));

The system function can return two values. The first is the exit status of the command and the second is any output from the command that was written to the standard output stream. For example,

     [status, output] = system ("echo foo; exit 2");

will set the variable output to the string ‘foo’, and the variable status to the integer ‘2’. For commands run asynchronously, status is the process id of the command shell that is started to run the command.

See also: [unix], page 721, [dos], page 722.

[Function File] unix ("command")

[Function File] status = unix ("command")

[Function File] [status, text] = unix ("command")

[Function File] [...] = unix ("command", "-echo")

Execute a system command if running under a Unix-like operating system, otherwise do nothing. Return the exit status of the program in status and any output from the command in text. When called with no output argument, or the "-echo" argument is given, then text is also sent to standard output.

See also: [dos], page 722, [system], page 721, [isunix], page 733, [ispc], page 732.

[Function File] dos ("command")

[Function File] status = dos ("command")
[Function File] [status, text] = dos ("command")

[Function File] [...] = dos ("command", "-echo")

Execute a system command if running under a Windows-like operating system, otherwise do nothing. Return the exit status of the program in status and any output from the command in text. When called with no output argument, or the "-echo" argument is given, then text is also sent to standard output.

See also: [unix], page 721, [system], page 721, [isunix], page 733, [ispc], page 732.

[Function File] output = perl (scriptfile)
[Function File] output = perl (scriptfile, argument1, argument2, . . .)
[Function File] [output, status] = perl (. . .)

Invoke Perl script scriptfile, possibly with a list of command line arguments. Return output in output and optional status in status. If scriptfile is not an absolute file name it is is searched for in the current directory and then in the Octave loadpath.

See also: [system], page 721, [python], page 722.

[Function File] output = python (scriptfile)
[Function File] output = python (scriptfile, argument1, argument2, . . .)
[Function File] [output, status] = python (. . .)

Invoke Python script scriptfile, possibly with a list of command line arguments. Return output in output and optional status in status. If scriptfile is not an absolute file name it is is searched for in the current directory and then in the Octave loadpath.

See also: [system], page 721, [perl], page 722.

[Built-in Function] fid = popen (command, mode)

Start a process and create a pipe. The name of the command to run is given by command. The file identifier corresponding to the input or output stream of the process is returned in fid. The argument mode may be

"r" The pipe will be connected to the standard output of the process, and open for reading.

"w" The pipe will be connected to the standard input of the process, and open for writing.

For example:

fid = popen ("ls -ltr / | tail -3", "r");
while (ischar (s = fgets (fid)))
  fputs (stdout, s);
endwhile
   ⊣ drwxr-xr-x  33 root  root  3072 Feb 15 13:28 etc
   ⊣ drwxr-xr-x   3 root  root  1024 Feb 15 13:28 lib
   ⊣ drwxrwxrwt  15 root  root  2048 Feb 17 14:53 tmp

[Built-in Function] pclose (fid)

Close a file identifier that was opened by popen. You may also use fclose for the same purpose.

[Built-in Function] [in, out, pid] = popen2 (command, args)

Start a subprocess with two-way communication. The name of the process is given by command, and args is an array of strings containing options for the command. The file identifiers for the input and output streams of the subprocess are returned in in and out. If execution of the command is successful, pid contains the process ID of the subprocess. Otherwise, pid is −1. For example:

[in, out, pid] = popen2 ("sort", "-r");
fputs (in, "these\nare\nsome\nstrings\n");
fclose (in);
EAGAIN = errno ("EAGAIN");
done = false;
do
  s = fgets (out);
  if (ischar (s))
    fputs (stdout, s);
  elseif (errno () == EAGAIN)
    sleep (0.1);
    fclear (out);
  else
    done = true;
  endif
until (done)
fclose (out);
waitpid (pid);

   ⊣ these
   ⊣ strings
   ⊣ some
   ⊣ are

Note that popen2, unlike popen, will not "reap" the child process. If you don’t use waitpid to check the child’s exit status, it will linger until Octave exits.

See also: [popen], page 722, [waitpid], page 725.

[Built-in Function] val = EXEC_PATH ()

[Built-in Function] old_val = EXEC_PATH (new_val)

[Built-in Function] EXEC_PATH (new_val, "local")

Query or set the internal variable that specifies a colon separated list of directories to append to the shell PATH when executing external programs. The initial value of is taken from the environment variable OCTAVE_EXEC_PATH, but that value can be overridden by the command line argument ‘--exec-path PATH’. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [IMAGE PATH], page 653, [OCTAVE HOME], page 733.

In most cases, the following functions simply decode their arguments and make the corresponding Unix system calls. For a complete example of how they can be used, look at the definition of the function popen2.

[Built-in Function] [pid, msg] = fork ()

Create a copy of the current process. Fork can return one of the following values:

> 0 You are in the parent process. The value returned from fork is the process id of the child process. You should probably arrange to wait for any child processes to exit.

0 You are in the child process. You can call exec to start another process. If that fails, you should probably call exit.

< 0 The call to fork failed for some reason. You must take evasive action. A system dependent error message will be waiting in msg.

[Built-in Function] [err, msg] = exec (file, args)

Replace current process with a new process. Calling exec without first calling fork will terminate your current Octave process and replace it with the program named by file. For example,

     exec ("ls" "-l")

will run ls and return you to your shell prompt. If successful, exec does not return. If exec does return, err will be nonzero, and msg will contain a system-dependent error message.

[Built-in Function] [read_fd, write_fd, err, msg] = pipe ()

Create a pipe and return the reading and writing ends of the pipe into read fd and write fd respectively. If successful, err is 0 and msg is an empty string. Otherwise, err is nonzero and msg contains a system-dependent error message.

See also: [mkfifo], page 709.

[Built-in Function] [fid, msg] = dup2 (old, new)

Duplicate a file descriptor. If successful, fid is greater than zero and contains the new file ID. Otherwise, fid is negative and msg contains a system-dependent error message.

[Built-in Function] [pid, status, msg] = waitpid (pid, options)

Wait for process pid to terminate. The pid argument can be:

−1 Wait for any child process.

0 Wait for any child process whose process group ID is equal to that of the Octave interpreter process.

> 0 Wait for termination of the child process with ID pid.

The options argument can be a bitwise OR of zero or more of the following constants:

0 Wait until signal is received or a child process exits (this is the default if the options argument is missing).

WNOHANG Do not hang if status is not immediately available.

WUNTRACED

Report the status of any child processes that are stopped, and whose status has not yet been reported since they stopped.

WCONTINUE

Return if a stopped child has been resumed by delivery of SIGCONT. This value may not be meaningful on all systems.

If the returned value of pid is greater than 0, it is the process ID of the child process that exited. If an error occurs, pid will be less than zero and msg will contain a system-dependent error message. The value of status contains additional systemdependent information about the subprocess that exited.

See also: [WCONTINUE], page 725, [WCOREDUMP], page 725, [WEXITSTA-TUS], page 725, [WIFCONTINUED], page 726, [WIFSIGNALED], page 726, [WIF-STOPPED], page 726, [WNOHANG], page 726, [WSTOPSIG], page 726, [WTERM-SIG], page 727, [WUNTRACED], page 727.

[Built-in Function] WCONTINUE ()

Return the numerical value of the option argument that may be passed to waitpid to indicate that it should also return if a stopped child has been resumed by delivery of a SIGCONT signal.

See also: [waitpid], page 725, [WNOHANG], page 726, [WUNTRACED], page 727.

[Built-in Function] WCOREDUMP (status)

Given status from a call to waitpid, return true if the child produced a core dump. This function should only be employed if WIFSIGNALED returned true. The macro used to implement this function is not specified in POSIX.1-2001 and is not available on some Unix implementations (e.g., AIX, SunOS).

See also: [waitpid], page 725, [WIFEXITED], page 726, [WEXITSTATUS], page 725, [WIFSIGNALED], page 726, [WTERMSIG], page 727, [WIFSTOPPED], page 726, [WSTOPSIG], page 726, [WIFCONTINUED], page 726.

[Built-in Function] WEXITSTATUS (status)

Given status from a call to waitpid, return the exit status of the child. This function should only be employed if WIFEXITED returned true.

See also: [waitpid], page 725, [WIFEXITED], page 726, [WIFSIGNALED], page 726, [WTERMSIG], page 727, [WCOREDUMP], page 725, [WIFSTOPPED], page 726, [WSTOPSIG], page 726, [WIFCONTINUED], page 726.

[Built-in Function] WIFCONTINUED (status)

Given status from a call to waitpid, return true if the child process was resumed by delivery of SIGCONT.

See also: [waitpid], page 725, [WIFEXITED], page 726, [WEXITSTATUS], page 725, [WIFSIGNALED], page 726, [WTERMSIG], page 727, [WCOREDUMP], page 725, [WIFSTOPPED], page 726, [WSTOPSIG], page 726.

[Built-in Function] WIFSIGNALED (status)

Given status from a call to waitpid, return true if the child process was terminated by a signal.

See also: [waitpid], page 725, [WIFEXITED], page 726, [WEXITSTATUS], page 725, [WTERMSIG], page 727, [WCOREDUMP], page 725, [WIFSTOPPED], page 726, [WSTOPSIG], page 726, [WIFCONTINUED], page 726.

[Built-in Function] WIFSTOPPED (status)

Given status from a call to waitpid, return true if the child process was stopped by delivery of a signal; this is only possible if the call was done using WUNTRACED or when the child is being traced (see ptrace(2)).

See also: [waitpid], page 725, [WIFEXITED], page 726, [WEXITSTATUS], page 725, [WIFSIGNALED], page 726, [WTERMSIG], page 727, [WCOREDUMP], page 725, [WSTOPSIG], page 726, [WIFCONTINUED], page 726.

[Built-in Function] WIFEXITED (status)

Given status from a call to waitpid, return true if the child terminated normally.

See also: [waitpid], page 725, [WEXITSTATUS], page 725, [WIFSIGNALED], page 726, [WTERMSIG], page 727, [WCOREDUMP], page 725, [WIFSTOPPED], page 726, [WSTOPSIG], page 726, [WIFCONTINUED], page 726.

[Built-in Function] WNOHANG ()

Return the numerical value of the option argument that may be passed to waitpid to indicate that it should return its status immediately instead of waiting for a process to exit.

See also: [waitpid], page 725, [WUNTRACED], page 727, [WCONTINUE], page 725.

[Built-in Function] WSTOPSIG (status)

Given status from a call to waitpid, return the number of the signal which caused the child to stop. This function should only be employed if WIFSTOPPED returned true.

See also: [waitpid], page 725, [WIFEXITED], page 726, [WEXITSTATUS], page 725, [WIFSIGNALED], page 726, [WTERMSIG], page 727, [WCOREDUMP], page 725, [WIFSTOPPED], page 726, [WIFCONTINUED], page 726.

[Built-in Function] WTERMSIG (status)

Given status from a call to waitpid, return the number of the signal that caused the child process to terminate. This function should only be employed if WIFSIGNALED returned true.

See also: [waitpid], page 725, [WIFEXITED], page 726, [WEXITSTATUS], page 725, [WIFSIGNALED], page 726, [WCOREDUMP], page 725, [WIFSTOPPED], page 726, [WSTOPSIG], page 726, [WIFCONTINUED], page 726.

[Built-in Function] WUNTRACED ()

Return the numerical value of the option argument that may be passed to waitpid to indicate that it should also return if the child process has stopped but is not traced via the ptrace system call

See also: [waitpid], page 725, [WNOHANG], page 726, [WCONTINUE], page 725.

[Built-in Function] [err, msg] = fcntl (fid, request, arg)

Change the properties of the open file fid. The following values may be passed as request:

F_DUPFD Return a duplicate file descriptor.

F_GETFD Return the file descriptor flags for fid.

F_SETFD Set the file descriptor flags for fid.

F_GETFL Return the file status flags for fid. The following codes may be returned (some of the flags may be undefined on some systems).

O_RDONLY Open for reading only.

O_WRONLY Open for writing only.

O_RDWR Open for reading and writing.

O_APPEND Append on each write.

O_CREAT Create the file if it does not exist.

          O_NONBLOCK

Non-blocking mode.

O_SYNC Wait for writes to complete.

O_ASYNC Asynchronous I/O.

F_SETFL Set the file status flags for fid to the value specified by arg. The only flags that can be changed are O_APPEND and O_NONBLOCK.

If successful, err is 0 and msg is an empty string. Otherwise, err is nonzero and msg contains a system-dependent error message.

[Built-in Function] [err, msg] = kill (pid, sig)

Send signal sig to process pid. If pid is positive, then signal sig is sent to pid. If pid is 0, then signal sig is sent to every process in the process group of the current process.

If pid is -1, then signal sig is sent to every process except process 1.

If pid is less than -1, then signal sig is sent to every process in the process group -pid.

If sig is 0, then no signal is sent, but error checking is still performed.

Return 0 if successful, otherwise return -1.

[Built-in Function] SIG ()

Return a structure containing Unix signal names and their defined values.

36.6 Process, Group, and User IDs

[Built-in Function] pgid = getpgrp ()

Return the process group id of the current process.

[Built-in Function] pid = getpid ()

Return the process id of the current process.

[Built-in Function] pid = getppid ()

Return the process id of the parent process.

[Built-in Function] euid = geteuid ()

Return the effective user id of the current process.

[Built-in Function] uid = getuid ()

Return the real user id of the current process.

[Built-in Function] egid = getegid ()

Return the effective group id of the current process.

[Built-in Function] gid = getgid ()

Return the real group id of the current process.

36.7 Environment Variables

[Built-in Function] getenv (var)

Return the value of the environment variable var. For example,

     getenv ("PATH")

returns a string containing the value of your path.

[Built-in Function] putenv (var, value) [Built-in Function] setenv (var, value)

Set the value of the environment variable var to value.

36.8 Current Working Directory

[Command] cd dir

[Command] cd [Built-in Function] old_dir = cd dir

[Command] chdir …

Change the current working directory to dir. If dir is omitted, the current directory is changed to the user’s home directory ("~"). For example,

     cd ~/octave

changes the current working directory to ‘~/octave’. If the directory does not exist, an error message is printed and the working directory is not changed. chdir is an alias for cd and can be used in all of the same calling formats. Compatibility Note: When called with no arguments, matlab prints the present working directory rather than changing to the user’s home directory.

See also: [pwd], page 730, [mkdir], page 709, [rmdir], page 709, [dir], page 729, [ls], page 729.

[Command] ls

[Command] ls filenames

[Command] ls options

[Command] ls options filenames

List directory contents. For example:

     ls -l
          ⊣ total 12
          ⊣ -rw-r--r--   1 jwe  users  4488 Aug 19 04:02 foo.m
          ⊣ -rw-r--r--   1 jwe  users  1315 Aug 17 23:14 bar.m

The dir and ls commands are implemented by calling your system’s directory listing command, so the available options will vary from system to system. Filenames are subject to shell expansion if they contain any wildcard characters ‘*’, ‘?’, ‘[]’. If you want to find a literal example of a wildcard character you must escape it using the backslash operator ‘\’.

See also: [dir], page 729, [readdir], page 713, [glob], page 713, [what], page 130, [stat], page 710, [filesep], page 714, [ls command], page 729.

[Function File] val = ls_command ()

[Function File] old_val = ls_command (new_val)

Query or set the shell command used by Octave’s ls command.

See also: [ls], page 729.

[Function File] dir [Function File] dir (directory)

[Function File] [list] = dir (directory)

Display file listing for directory directory. If directory is not specified then list the present working directory.

If a return value is requested, return a structure array with the fields name File or directory name. date Timestamp of file modification (string value). bytes File size in bytes. isdir True if name is a directory. datenum Timestamp of file modification as serial date number (double). statinfo Information structure returned from stat. If directory is a filename, rather than a directory, then return information about the named file. directory may also be a list rather than a single directory or file. directory is subject to shell expansion if it contains any wildcard characters ‘*’, ‘?’, ‘[]’. If you want to find a literal example of a wildcard character you must escape it using the backslash operator ‘\’. Note that for symbolic links, dir returns information about the file that the symbolic link points to rather than the link itself. However, if the link points to a nonexistent file, dir returns information about the link. See also: [ls], page 729, [readdir], page 713, [glob], page 713, [what], page 130, [stat], page 710.

[Built-in Function] pwd () [Built-in Function] dir = pwd ()

Return the current working directory. See also: [cd], page 729, [dir], page 729, [ls], page 729, [mkdir], page 709, [rmdir], page 709.

36.9 Password Database Functions

Octave’s password database functions return information in a structure with the following fields. name The user name. passwd The encrypted password, if available. uid The numeric user id. gid The numeric group id. gecos The GECOS field. dir The home directory. shell The initial shell.

In the descriptions of the following functions, this data structure is referred to as a pw struct.

[Built-in Function] pw_struct = getpwent ()

Return a structure containing an entry from the password database, opening it if necessary. Once the end of the data has been reached, getpwent returns 0.

[Built-in Function] pw_struct = getpwuid (uid).

Return a structure containing the first entry from the password database with the user ID uid. If the user ID does not exist in the database, getpwuid returns 0.

[Built-in Function] pw_struct = getpwnam (name)

Return a structure containing the first entry from the password database with the user name name. If the user name does not exist in the database, getpwname returns 0.

[Built-in Function] setpwent ()

Return the internal pointer to the beginning of the password database.

[Built-in Function] endpwent ()

Close the password database.

36.10 Group Database Functions

Octave’s group database functions return information in a structure with the following fields.

name The user name.

passwd The encrypted password, if available.

gid The numeric group id.

mem The members of the group.

In the descriptions of the following functions, this data structure is referred to as a grp struct.

[Built-in Function] grp_struct = getgrent ()

Return an entry from the group database, opening it if necessary. Once the end of data has been reached, getgrent returns 0.

[Built-in Function] grp_struct = getgrgid (gid).

Return the first entry from the group database with the group ID gid. If the group ID does not exist in the database, getgrgid returns 0.

[Built-in Function] grp_struct = getgrnam (name)

Return the first entry from the group database with the group name name. If the group name does not exist in the database, getgrnam returns 0.

[Built-in Function] setgrent ()

Return the internal pointer to the beginning of the group database.

[Built-in Function] endgrent ()

Close the group database.

36.11 System Information

[Function File] [c, maxsize, endian] = computer ()

[Function File] arch = computer ("arch")

Print or return a string of the form cpu-vendor-os that identifies the kind of computer Octave is running on. If invoked with an output argument, the value is returned instead of printed. For example:

     computer ()
        ⊣ i586-pc-linux-gnu
     x = computer ()
        ⇒ x = "i586-pc-linux-gnu"

If two output arguments are requested, also return the maximum number of elements for an array. If three output arguments are requested, also return the byte order of the current system as a character ("B" for big-endian or "L" for little-endian). If the argument "arch" is specified, return a string indicating the architecture of the computer on which Octave is running.

[Built-in Function] [uts, err, msg] = uname ()

Return system information in the structure. For example:

     uname ()
        ⇒ {
              sysname = x86_64
              nodename = segfault
              release = 2.6.15-1-amd64-k8-smp
              version = Linux
              machine = #2 SMP Thu Feb 23 04:57:49 UTC 2006
           }

If successful, err is 0 and msg is an empty string. Otherwise, err is nonzero and msg contains a system-dependent error message.

[Built-in Function] nproc () [Built-in Function] nproc (query)

Return the current number of available processors. If called with the optional argument query, modify how processors are counted as follows:

all total number of processors.

current processors available to the current process.

overridable

likewise, but overridable through the OMP_NUM_THREADS environment variable.

[Function File] ispc ()

Return true if Octave is running on a Windows system and false otherwise.

See also: [isunix], page 733, [ismac], page 733.

[Function File] isunix ()

Return true if Octave is running on a Unix-like system and false otherwise.

See also: [ismac], page 733, [ispc], page 732.

[Function File] ismac ()

Return true if Octave is running on a Mac OS X system and false otherwise.

See also: [isunix], page 733, [ispc], page 732.

[Built-in Function] isieee ()

Return true if your computer claims to conform to the IEEE standard for floating point calculations. No actual tests are performed.

[Function File] isdeployed ()

Return true if the current program has been compiled and is running separately from the Octave interpreter and false if it is running in the Octave interpreter. Currently, this function always returns false in Octave.

[Built-in Function] OCTAVE_HOME ()

Return the name of the top-level Octave installation directory.

See also: [EXEC PATH], page 724, [IMAGE PATH], page 653.

[Function File] matlabroot ()

Return the name of the top-level Octave installation directory. This is an alias for the function OCTAVE_HOME provided for compatibility.

See also: [OCTAVE HOME], page 733.

[Built-in Function] OCTAVE_VERSION ()

Return the version number of Octave, as a string.

[Function File] version ()

Return the version number of Octave, as a string. This is an alias for the function OCTAVE_VERSION provided for compatibility.

See also: [OCTAVE VERSION], page 733.

[Function File] ver () [Function File] v = ver ()

[Function File] v = ver ("Octave")

[Function File] v = ver (package)

Display a header containing the current Octave version number, license string, and operating system followed by a list of installed packages, versions, and installation directories.

v = ver ()

Return a vector of structures describing Octave and each installed package. The structure includes the following fields.

Name Package name.

Version Version of the package.

Revision Revision of the package. Date Date of the version/revision.

v = ver ("Octave")

Return version information for Octave only.

v = ver (package)

Return version information for package. See also: [version], page 733, [octave config info], page 735.

[Function File] compare_versions (v1, v2, operator)

Compare two version strings using the given operator. This function assumes that versions v1 and v2 are arbitrarily long strings made of numeric and period characters possibly followed by an arbitrary string (e.g., "1.2.3",

"0.3", "0.1.2+", or "1.2.3.4-test1").

The version is first split into numeric and character portions and then the parts are padded to be the same length (i.e., "1.1" would be padded to be "1.1.0" when being compared with "1.1.1", and separately, the character parts of the strings are padded with nulls). The operator can be any logical operator from the set

• "==" equal

• "<" less than

• "<=" less than or equal to

• ">" greater than

• ">=" greater than or equal to

• "!=" not equal

• "~=" not equal Note that version "1.1-test2" will compare as greater than "1.1-test10". Also, since the numeric part is compared first, "a" compares less than "1a" because the second string starts with a numeric part even though double ("a") is greater than

double ("1").

[Command] license [Function File] license ("inuse") [Function File] retval = license ("inuse")

[Function File] retval = license ("test", feature)
[Function File] license ("test", feature, toggle)

[Function File] retval = license ("checkout", feature)

Display the license of Octave.

license ("inuse")

Display a list of packages currently being used.

retval = license ("inuse")

Return a structure containing the fields feature and user.

retval = license ("test", feature)

Return 1 if a license exists for the product identified by the string feature and 0 otherwise. The argument feature is case insensitive and only the first 27 characters are checked.

license ("test", feature, toggle)

Enable or disable license testing for feature, depending on toggle, which may be one of: "enable" Future tests for the specified license of feature are conducted as usual.

"disable"

Future tests for the specified license of feature return 0.

retval = license ("checkout", feature)

Check out a license for feature, returning 1 on success and 0 on failure. This function is provided for compatibility with matlab. See also: [ver], page 733, [version], page 733.

[Built-in Function] octave_config_info ()

[Built-in Function] octave_config_info (option)

Return a structure containing configuration and installation information for Octave. If option is a string, return the configuration information for the specified option.

[Built-in Function] getrusage ()

Return a structure containing a number of statistics about the current Octave process. Not all fields are available on all systems. If it is not possible to get CPU time statistics, the CPU time slots are set to zero. Other missing data are replaced by NaN. The list of possible fields is: idrss Unshared data size. inblock Number of block input operations. isrss Unshared stack size. ixrss Shared memory size. majflt Number of major page faults. maxrss Maximum data size. minflt Number of minor page faults. msgrcv Number of messages received. msgsnd Number of messages sent. nivcsw Number of involuntary context switches. nsignals Number of signals received. nswap Number of swaps. nvcsw Number of voluntary context switches. oublock Number of block output operations. stime A structure containing the system CPU time used. The structure has the

elements sec (seconds) usec (microseconds).

utime A structure containing the user CPU time used. The structure has the elements sec (seconds) usec (microseconds).

36.12 Hashing Functions

It is often necessary to find if two strings or files are identical. This might be done by comparing them character by character and looking for differences. However, this can be slow, and so comparing a hash of the string or file can be a rapid way of finding if the files differ.

Another use of the hashing function is to check for file integrity. The user can check the hash of the file against a known value and find if the file they have is the same as the one that the original hash was produced with.

Octave supplies the md5sum function to perform MD5 hashes on strings and files. An example of the use of md5sum function might be

     if exist (file, "file")
       hash = md5sum (file);
     else
       # Treat the variable "file" as a string
       hash = md5sum (file, true);
     endif

[Built-in Function] md5sum (file) [Built-in Function] md5sum (str, opt)

Calculate the MD5 sum of the file file. If the second parameter opt exists and is true, then calculate the MD5 sum of the string str.

37 Java Interface

The Java Interface is designed for calling Java functions from within Octave. If you want to do the reverse, and call Octave from within Java, try a li-

brary like javaOctave (http://kenai.com/projects/javaOctave) or joPas
(http://jopas.sourceforge.net).

37.1 Java Interface Functions

The following functions are the core of the Java Interface. They provide a way to create a Java object, get and set its data fields, and call Java methods which return results to Octave.

[Built-in Function] jobj = javaObject (classname)
[Built-in Function] jobj = javaObject (classname, arg1, . . .)

Create a Java object of class classsname, by calling the class constructor with the arguments arg1, … The first example below creates an uninitialized object, while the second example supplies an initial argument to the constructor.

     x = javaObject ("java.lang.StringBuffer")
     x = javaObject ("java.lang.StringBuffer", "Initial string")

See also: [javaMethod], page 739, [javaArray], page 737.

[Function File] jary = javaArray (classname, sz)
[Function File] jary = javaArray (classname, m, n, . . .)

Create a Java array of size sz with elements of class classname. classname may be a Java object representing a class or a string containing the fully qualified class name. The size of the object may also be specified with individual integer arguments m, n, etc. The generated array is uninitialized. All elements are set to null if classname is a reference type, or to a default value (usually 0) if classname is a primitive type. Sample code:

     jary = javaArray ("java.lang.String", 2, 2);
     jary(1,1) = "Hello";

See also: [javaObject], page 737.

There are many different variable types in Octave but only ones created through javaObject can use Java functions. Before using Java with an unknown object the type can be checked with isjava.

[Built-in Function] isjava (x)

Return true if x is a Java object.

See also: [class], page 39, [typeinfo], page 39, [isa], page 39, [javaObject], page 737.

Once an object has been created it is natural to find out what fields the object has and to read (get) and write (set) them.

In Octave the fieldnames function for structures has been overloaded to return the fields of a Java object. For example:

dobj = javaObject ("java.lang.Double", pi);
fieldnames (dobj)

⇒

{
  [1,1] = public static final double java.lang.Double.POSITIVE_INFINITY
  [1,2] = public static final double java.lang.Double.NEGATIVE_INFINITY
  [1,3] = public static final double java.lang.Double.NaN
  [1,4] = public static final double java.lang.Double.MAX_VALUE
  [1,5] = public static final double java.lang.Double.MIN_NORMAL
  [1,6] = public static final double java.lang.Double.MIN_VALUE
  [1,7] = public static final int java.lang.Double.MAX_EXPONENT
  [1,8] = public static final int java.lang.Double.MIN_EXPONENT
  [1,9] = public static final int java.lang.Double.SIZE
  [1,10] = public static final java.lang.Class java.lang.Double.TYPE
}

The analogy of objects with structures is carried over into reading and writing object fields. To read a field the object is indexed with the ‘.’ operator from structures. This is the preferred method for reading fields, but Octave also provides a function interface to read fields with java_get. An example of both styles is shown below.

dobj = javaObject ("java.lang.Double", pi);
dobj.MAX_VALUE
⇒ 1.7977e+308
java_get ("java.lang.Float", "MAX_VALUE")
⇒ 3.4028e+38
[Function File] val = java_get (obj, name)

Get the value of the field name of the Java object obj. For static fields, obj can be a string representing the fully qualified name of the corresponding class. When obj is a regular Java object, structure-like indexing can be used as a shortcut syntax. For instance, the two following statements are equivalent

       java_get (x, "field1")
       x.field1

See also: [java set], page 738, [javaMethod], page 739, [javaObject], page 737.

[Function File] obj = java_set (obj, name, val)

Set the value of the field name of the Java object obj to val. For static fields, obj can be a string representing the fully qualified named of the corresponding Java class. When obj is a regular Java object, structure-like indexing can be used as a shortcut syntax. For instance, the two following statements are equivalent

       java_set (x, "field1", val)
       x.field1 = val

See also: [java get], page 738, [javaMethod], page 739, [javaObject], page 737.

To see what functions can be called with an object use methods. For example, using the previously created dobj:

methods (dobj)

⇒

Methods for class java.lang.Double:
boolean equals(java.lang.Object)
java.lang.String toString(double)
java.lang.String toString()
...

To call a method of an object the same structure indexing operator ‘.’ is used. Octave also provides a functional interface to calling the methods of an object through javaMethod. An example showing both styles is shown below.

dobj = javaObject ("java.lang.Double", pi);
dobj.equals (3)
⇒ 0
javaMethod ("equals", dobj, pi)
⇒ 1
[Built-in Function] ret = javaMethod (methodname, obj)
[Built-in Function] ret = javaMethod (methodname, obj, arg1, . . .)

Invoke the method methodname on the Java object obj with the arguments arg1, … For static methods, obj can be a string representing the fully qualified name of the corresponding class. The function returns the result of the method invocation. When obj is a regular Java object, structure-like indexing can be used as a shortcut syntax. For instance, the two following statements are equivalent

       ret = javaMethod ("method1", x, 1.0, "a string")
       ret = x.method1 (1.0, "a string")

See also: [methods], page 674, [javaObject], page 737.

The following three functions are used to display and modify the class path used by the Java Virtual Machine. This is entirely separate from Octave’s PATH variable and is used by the JVM to find the correct code to execute.

[Function File] javaclasspath ()

[Function File] dpath = javaclasspath ()
[Function File] [dpath, spath] = javaclasspath ()
[Function File] clspath = javaclasspath (what)

Return the class path of the Java virtual machine in the form of a cell array of strings. If called with no inputs:

• If no output is requested, the dynamic and static classpaths are printed to the standard output.

• If one output value dpath is requested, the result is the dynamic classpath.

• If two output valuesdpath and spath are requested, the first variable will contain the dynamic classpath and the second will be contain the static classpath.

If called with a single input parameter what:

"-dynamic"

Return the dynamic classpath.

"-static"

Return the static classpath.

"-all" Return both the static and dynamic classpath in a single cellstr.

See also: [javaaddpath], page 740, [javarmpath], page 740.

[Function File] javaaddpath (clspath)

[Function File] javaaddpath (clspath1, …)

Add clspath to the dynamic class path of the Java virtual machine. clspath may either be a directory where ‘.class’ files are found, or a ‘.jar’ file containing Java classes. Multiple paths may be added at once by specifying additional arguments.

See also: [javarmpath], page 740, [javaclasspath], page 739.

[Function File] javarmpath (clspath)

[Function File] javarmpath (clspath1, …)

Remove clspath from the dynamic class path of the Java virtual machine. clspath may either be a directory where ‘.class’ files are found, or a ‘.jar’ file containing Java classes. Multiple paths may be removed at once by specifying additional arguments.

See also: [javaaddpath], page 740, [javaclasspath], page 739.

The following four functions provide information and control over the interface between Octave and the Java Virtual Machine.

[Function File] usejava (feature)

Return true if the Java element feature is available. Possible features are:

"awt" Abstract Window Toolkit for GUIs.

"desktop"

Interactive desktop is running.

"jvm" Java Virtual Machine.

"swing" Swing components for lightweight GUIs.

usejava determines if specific Java features are available in an Octave session. This function is provided for scripts which may alter their behavior based on the availability of Java. The feature "desktop" always returns false as Octave has no Java-based desktop. Other features may be available if Octave was compiled with the Java Interface and Java is installed.

[Function File] javamem () [Function File] jmem = javamem ()

Show the current memory usage of the Java virtual machine (JVM) and run the garbage collector. When no return argument is given the info is printed to the screen. Otherwise, the output cell array jmem contains Maximum, Total, and Free memory (in bytes). All Java-based routines are run in the JVM’s shared memory pool, a dedicated and separate part of memory claimed by the JVM from your computer’s total memory (which comprises physical RAM and virtual memory / swap space on hard disk).

The maximum allowable memory usage can be configured using the file ‘java.opts’. The directory where this file resides is determined by the environment variable OCTAVE_JAVA_DIR. If unset, the directory where ‘javaaddpath.m’ resides is used

instead (typically ‘OCTAVE_HOME/share/octave/OCTAVE_VERSION/m/java/’

‘java.opts’ is a plain text file with one option per line. The default initial memory size and default maximum memory size (which are both system dependent) can be overridden like so:

-Xms64m

-Xmx512m

(in megabytes in this example). You can adapt these values to your own requirements if your system has limited available physical memory or if you get Java memory errors.

"Total memory" is what the operating system has currently assigned to the JVM and depends on actual and active memory usage. "Free memory" is self-explanatory. During operation of Java-based Octave functions the amount of Total and Free memory will vary, due to Java’s own cleaning up and your operating system’s memory management.

[Built-in Function] val = java_matrix_autoconversion ()
[Built-in Function] old_val = java_matrix_autoconversion (new_val)
[Built-in Function] java_matrix_autoconversion (new_val, "local")

Query or set the internal variable that controls whether Java arrays are automatically converted to Octave matrices. The default value is false.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [java unsigned autoconversion], page 741, [debug java], page 741.

[Built-in Function] val = java_unsigned_autoconversion ()
[Built-in Function] old_val = java_unsigned_autoconversion (new_val)
[Built-in Function] java_unsigned_autoconversion (new_val, "local")

Query or set the internal variable that controls how integer classes are converted when java_matrix_autoconversion is enabled. When enabled, Java arrays of class Byte or Integer are converted to matrices of class uint8 or uint32 respectively. The default value is true.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [java matrix autoconversion], page 741, [debug java], page 741.

[Built-in Function] val = debug_java ()

[Built-in Function] old_val = debug_java (new_val)

[Built-in Function] debug_java (new_val, "local")

Query or set the internal variable that determines whether extra debugging information regarding the initialization of the JVM and any Java exceptions is printed.

When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [java matrix autoconversion], page 741, [java unsigned autoconversion], page 741.

37.2 Dialog Box Functions

The following functions all use the Java Interface to provide some form of dialog box.

[Function File] h = msgbox (msg) [Function File] h = msgbox (msg, title)

[Function File] h = msgbox (msg, title, icon)

Display msg using a message dialog box. The message may have multiple lines separated by newline characters ("\n"), or it may be a cellstr array with one element for each line. The optional input title (character string) can be used to decorate the dialog caption. The optional argument icon selects a dialog icon. It can be one of "none" (default),

"error", "help", or "warn".

The return value is always 1.

See also: [errordlg], page 742, [helpdlg], page 742, [inputdlg], page 743, [listdlg], page 743, [questdlg], page 744, [warndlg], page 744.

[Function File] h = errordlg (msg)

[Function File] h = errordlg (msg, title)

Display msg using an error dialog box. The message may have multiple lines separated by newline characters ("\n"), or it may be a cellstr array with one element for each line. The optional input title (character string) can be used to set the dialog caption. The default title is "Error

Dialog".

The return value is always 1.

See also: [helpdlg], page 742, [inputdlg], page 743, [listdlg], page 743, [msgbox], page 742, [questdlg], page 744, [warndlg], page 744.

[Function File] h = helpdlg (msg)

[Function File] h = helpdlg (msg, title)

Display msg in a help dialog box. The message may have multiple lines separated by newline characters ("\n"), or it may be a cellstr array with one element for each line. The optional input title (character string) can be used to set the dialog caption. The default title is "Help

Dialog".

The return value is always 1.

See also: [errordlg], page 742, [inputdlg], page 743, [listdlg], page 743, [msgbox], page 742, [questdlg], page 744, [warndlg], page 744.

[Function File] cstr = inputdlg (prompt)
[Function File] cstr = inputdlg (prompt, title)
[Function File] cstr = inputdlg (prompt, title, rowscols)
[Function File] cstr = inputdlg (prompt, title, rowscols, defaults)

Return user input from a multi-textfield dialog box in a cell array of strings, or an empty cell array if the dialog is closed by the Cancel button. Inputs:

prompt A cell array with strings labeling each text field. This input is required.

title String to use for the caption of the dialog. The default is "Input Dialog".

rowscols Specifies the size of the text fields and can take three forms:

1. a scalar value which defines the number of rows used for each text field.

2. a vector which defines the individual number of rows used for each text field.

3. a matrix which defines the individual number of rows and columns used for each text field. In the matrix each row describes a single text field. The first column specifies the number of input rows to use and the second column specifies the text field width.

defaults A list of default values to place in each text fields. It must be a cell array of strings with the same size as prompt.

See also: [errordlg], page 742, [helpdlg], page 742, [listdlg], page 743, [msgbox], page 742, [questdlg], page 744, [warndlg], page 744.

[Function File] [sel, ok] = listdlg (key, value, . . .)

Return user inputs from a list dialog box in a vector of selection indices sel and a flag ok indicating how the user closed the dialog box. The value of ok is 1 if the user closed the box with the OK button, otherwise it is 0 and sel is empty. The indices in sel are 1-based. The arguments are specified in form of key, value pairs. The "ListString" argument pair must be specified. Valid key and value pairs are:

"ListString"

a cell array of strings comprising the content of the list.

"SelectionMode"

can be either "Single" or "Multiple" (default).

"ListSize"

a vector with two elements width and height defining the size of the list field in pixels. Default is [160 300].

"InitialValue"

a vector containing 1-based indices of preselected elements. Default is 1 (first item).

"Name" a string to be used as the dialog caption. Default is "".

"PromptString"

a cell array of strings to be displayed above the list field. Default is {}.

"OKString"

a string used to label the OK button. Default is "OK".

"CancelString"

a string used to label the Cancel button. Default is "Cancel".

Example:

     [sel, ok] = listdlg ("ListString", {"An item", "another", "yet another"},
                          "SelectionMode", "Multiple");
     if (ok == 1)
       for i = 1:numel (sel)
         disp (sel(i));
       endfor
     endif

See also: [menu], page 229, [errordlg], page 742, [helpdlg], page 742, [inputdlg], page 743, [msgbox], page 742, [questdlg], page 744, [warndlg], page 744.

[Function File] btn = questdlg (msg)

[Function File] btn = questdlg (msg, title)
[Function File] btn = questdlg (msg, title, default)
[Function File] btn = questdlg (msg, title, btn1, btn2, default)
[Function File] btn = questdlg (msg, title, btn1, btn2, btn3, default)

Display msg using a question dialog box and return the caption of the activated button. The dialog may contain two or three buttons which will all close the dialog. The message may have multiple lines separated by newline characters ("\n"), or it may be a cellstr array with one element for each line. The optional title (character string) can be used to decorate the dialog caption. The string default identifies the default button, which is activated by pressing the ENTER key. It must match one of the strings given in btn1, btn2, or btn3. If only msg and title are specified, three buttons with the default captions "Yes",

"No", and "Cancel" are used.

If only two button captions, btn1 and btn2, are specified the dialog will have only these two buttons.

See also: [errordlg], page 742, [helpdlg], page 742, [inputdlg], page 743, [listdlg], page 743, [warndlg], page 744.

[Function File] h = warndlg (msg)

[Function File] h = warndlg (msg, title)

Display msg using a warning dialog box. The message may have multiple lines separated by newline characters ("\n"), or it may be a cellstr array with one element for each line. The optional input title (character string) can be used to set the dialog caption. The default title is "Warning

     Dialog".

See also: [helpdlg], page 742, [inputdlg], page 743, [listdlg], page 743, [questdlg], page 744.

37.3 FAQ - Frequently asked Questions

37.3.1 How to distinguish between Octave and Matlab?

Octave and matlab are very similar, but handle Java slightly different. Therefore it may be necessary to detect the environment and use the appropriate functions. The following function can be used to detect the environment. Due to the persistent variable it can be called repeatedly without a heavy performance hit.

Example:

     %%
     %% Return: true if the environment is Octave.
     %%
     function retval = isOctave
       persistent cacheval;  % speeds up repeated calls
       if isempty (cacheval)
         cacheval = (exist ("OCTAVE_VERSION", "builtin") > 0);
       end
       retval = cacheval;
     end

37.3.2 How to make Java classes available to Octave?

Java finds classes by searching a classpath. This is a list of Java archive files and/or directories containing class files. In Octave the classpath is composed of two parts:

• the static classpath is initialized once at startup of the JVM, and

• the dynamic classpath which can be modified at runtime.

Octave searches the static classpath first, then the dynamic classpath. Classes appearing in the static as well as in the dynamic classpath will therefore be found in the static classpath and loaded from this location. Classes which will be used frequently or must be available to all users should be added to the static classpath. The static classpath is populated once from the contents of a plain text file named ‘javaclasspath.txt’ (or ‘classpath.txt’ historically) when the Java Virtual Machine starts. This file contains one line for each individual classpath to be added to the static classpath. These lines can identify single class files, directories containing class files, or Java archives with complete class file hierarchies. Comment lines starting with a ‘#’ or a ‘%’ character are ignored.

The search rules for the file ‘javaclasspath.txt’ (or ‘classpath.txt’) are:

• First, Octave tries to locate it in the current directory (where Octave was started from). If such a file is found, it is read and defines the initial static classpath. Thus, it is possible to define a static classpath on a ’per Octave invocation’ basis.

• Next, Octave searches in the user’s home directory. If a file ‘javaclasspath.txt’ exists here, its contents are appended to the static classpath (if any). Thus, it is possible to build an initial static classpath on a ’per user’ basis.

• Finally, Octave looks for a next occurrence of file ‘javaclasspath.txt’ in the m-files directory where Octave Java functions live. This is where ‘javaclasspath.m’ resides,

usually something like ‘OCTAVE_HOME/share/octave/OCTAVE_VERSION/m/java/’. You

can find this directory by executing the command

    which javaclasspath

If this file exists here, its contents are also appended to the static classpath. Note that the archives and class directories defined in this last step will affect all users.

Classes which are used only by a specific script should be placed in the dynamic classpath. This portion of the classpath can be modified at runtime using the javaaddpath and

javarmpath functions.

Example:

octave> base_path = "C:/Octave/java_files";
octave> % add two JARchives to the dynamic classpath
octave> javaaddpath ([base_path, "/someclasses.jar"]);
octave> javaaddpath ([base_path, "/moreclasses.jar"]);
octave> % check the dynamic classpath
octave> p = javaclasspath;
octave> disp (p{1});
C:/Octave/java_files/someclasses.jar
octave> disp (p{2});
C:/Octave/java_files/moreclasses.jar
octave> % remove the first element from the classpath
octave> javarmpath ([base_path, "/someclasses.jar"]);
octave> p = javaclasspath;
octave> disp (p{1});
C:/Octave/java_files/moreclasses.jar
octave> % provoke an error
octave> disp (p{2});
error: A(I): Index exceeds matrix dimension.

Another way to add files to the dynamic classpath exclusively for your user account is to use the file ‘.octaverc’ which is stored in your home directory. All Octave commands in this file are executed each time you start a new instance of Octave. The following example adds the directory ‘octave’ to Octave’s search path and the archive ‘myclasses.jar’ in this directory to the Java search path.

% contents of .octaverc:
addpath ("~/octave");
javaaddpath ("~/octave/myclasses.jar");

37.3.3 How to create an instance of a Java class?

The function javaObject can be used to create Java objects..

Example:

     Passenger = javaObject ("package.FirstClass", row, seat);

37.3.4 How can I handle memory limitations?

In order to execute Java code Octave creates a Java Virtual Machine (JVM). Such a JVM allocates a fixed amount of initial memory and may expand this pool up to a fixed maximum memory limit. The default values depend on the Java version (see [javamem], page 740). The memory pool is shared by all Java objects running in the JVM. This strict memory limit is intended mainly to avoid that runaway applications inside web browsers or in enterprise servers can consume all memory and crash the system. When the maximum memory limit is hit, Java code will throw exceptions so that applications will fail or behave unexpectedly.

You can specify options for the creation of the JVM inside a file named ‘java.opts’. This is a text file where you can enter lines containing ‘-X’ and ‘-D’ options handed to the JVM during initialization.

The directory where the Java options file is located is specified by the environment variable OCTAVE_JAVA_DIR. If unset the directory where ‘javaclasspath.m’ resides is used

instead (typically ‘OCTAVE_HOME/share/octave/OCTAVE_VERSION/m/java/’). You can find

this directory by executing

     which javaclasspath

The ‘-X’ options allow you to increase the maximum amount of memory available to the JVM. The following example allows up to 256 Megabytes to be used by adding the following line to the ‘java.opts’ file:

     -Xmx256m

The maximum possible amount of memory depends on your system. On a Windows system with 2 Gigabytes main memory you should be able to set this maximum to about 1 Gigabyte.

If your application requires a large amount of memory from the beginning, you can also specify the initial amount of memory allocated to the JVM. Adding the following line to the ‘java.opts’ file starts the JVM with 64 Megabytes of initial memory:

     -Xms64m

For more details on the available ‘-X’ options of your Java Virtual Machine issue the command ‘java -X’ at the operating system command prompt and consult the Java documentation.

The ‘-D’ options can be used to define system properties which can then be used by Java classes inside Octave. System properties can be retrieved by using the getProperty() methods of the java.lang.System class. The following example line defines the property MyProperty and assigns it the string 12.34.

     -DMyProperty=12.34

The value of this property can then be retrieved as a string by a Java object or in Octave:

     octave> javaMethod ("getProperty", "java.lang.System", "MyProperty");
     ans = 12.34

See also: javamem.

37.3.5 Which TEX symbols are implemented in dialog functions?

The dialog functions contain a translation table for TEX like symbol codes. Thus messages and labels can be tailored to show some common mathematical symbols or Greek characters. No further TEX formatting codes are supported. The characters are translated to their Unicode equivalent. However, not all characters may be displayable on your system. This depends on the font used by the Java system on your computer.

Each TEX symbol code must be terminated by a space character to make it distinguishable from the surrounding text. Therefore the string ‘\alpha =12.0’ will produce the desired result, whereas ‘\alpha=12.0’ would produce the literal text ’\alpha=12.0’.

See also: errordlg, helpdlg, inputdlg, listdlg, msgbox, questdlg, warndlg.

The table below shows each TEX character code and the corresponding Unicode character:

TEX code  Symbol  TEX code  Symbol  TEX code  Symbol
\alpha  ’α’  \beta  ’β’  \gamma  ’γ’
\delta  ’δ’  \epsilon  ’ϵ’  \zeta  ’ζ’
\eta  ’η’  \theta  ’θ’  \vartheta  ’ϑ’
\iota  ’ι’  \kappa  ’κ’  \lambda  ’λ’
\mu  ’μ’  \nu  ’ν’  \xi  ’ξ’
\pi  ’π’  \rho  ’ρ’  \sigma  ’σ’
\varsigma  ’ς’  \tau  ’τ’  \phi  ’ϕ’
\chi  ’χ’  \psi  ’ψ’  \omega  ’ω’
\upsilon  ’υ’  \Gamma  ’Γ’  \Delta  ’Δ’
\Theta  ’Θ’  \Lambda  ’Λ’  \Pi  ’Π’
\Xi  ’Ξ’  \Sigma  ’Σ’  \Upsilon  ’Υ’
\Phi  ’Φ’  \Psi  ’Ψ’  \Omega  ’Ω’
\Im  ’ℑ’  \Re  ’ℜ’  \leq  ’≤’
\geq  ’≥’  \neq  ’≠’  \pm  ’±’
\infty  ’∞’  \partial  ’∂’  \approx  ’≈’
\circ  ’∘’  \bullet  ’•’  \times  ’×’
\sim  ’∼’  \nabla  ’∇’  \ldots  ’…’
\exists  ’∃’  \neg  ’¬’  \aleph  ’ℵ’
\forall  ’∀’  \cong  ’≅’  \wp  ’℘’
\propto  ’∝’  \otimes  ’⊗’  \oplus  ’⊕’
\oslash  ’⊘’  \cap  ’∩’  \cup  ’∪’
\ni  ’∋’  \in  ’∈’  \div  ’÷’
\equiv  ’≡’  \int  ’∫ ’  \perp  ’⊥’
\wedge  ’∧’  \vee  ’∨’  \supseteq  ’⊇’
\supset  ’⊃’  \subseteq  ’⊆’  \subset  ’⊂’
\clubsuit  ’♣’  \spadesuit  ’♠’  \heartsuit  ’♡’
\diamondsuit  ’♢’  \copyright  ’\begin{matrix}c \\ ◯ \end{matrix}’  \leftarrow  ’←’
\uparrow  ’↑’  \rightarrow  ’→’  \downarrow ’↓’
\leftrightarrow ’↔’  \updownarrow ’↕’

Table: TEX character codes and the resulting symbols.

38 Packages

Since Octave is Free Software users are encouraged to share their programs amongst each other. To aid this sharing Octave supports the installation of extra packages. The ‘Octave-Forge’ project is a community-maintained set of packages that can be downloaded and installed in Octave. At the time of writing the ‘Octave-Forge’ project can be found online at http://octave.sourceforge.net, but since the Internet is an ever-changing place this may not be true at the time of reading. Therefore it is recommended to see the Octave website for an updated reference.

38.1 Installing and Removing Packages

Assuming a package is available in the file ‘image-1.0.0.tar.gz’ it can be installed from the Octave prompt with the command

     pkg install image-1.0.0.tar.gz

If the package is installed successfully nothing will be printed on the prompt, but if an error occurred during installation it will be reported. It is possible to install several packages at once by writing several package files after the pkg install command. If a different version of the package is already installed it will be removed prior to installing the new package. This makes it easy to upgrade and downgrade the version of a package, but makes it impossible to have several versions of the same package installed at once.

To see which packages are installed type

     pkg list
     ⊣ Package Name  | Version | Installation directory
     ⊣ --------------+---------+-----------------------
     ⊣  image *|   1.0.0 | /home/jwe/octave/image-1.0.0

In this case only version 1.0.0 of the image package is installed. The '*' character next to the package name shows that the image package is loaded and ready for use.

It is possible to remove a package from the system using the pkg uninstall command like this

     pkg uninstall image

If the package is removed successfully nothing will be printed in the prompt, but if an error occurred it will be reported. It should be noted that the package file used for installation is not needed for removal, and that only the package name as reported by pkg list should be used when removing a package. It is possible to remove several packages at once by writing several package names after the pkg uninstall command.

To minimize the amount of code duplication between packages it is possible that one package depends on another one. If a package depends on another, it will check if that package is installed during installation. If it is not, an error will be reported and the package will not be installed. This behavior can be disabled by passing the ‘-nodeps’ flag to the pkg install command

     pkg install -nodeps my_package_with_dependencies.tar.gz

Since the installed package expects its dependencies to be installed it may not function correctly. Because of this it is not recommended to disable dependency checking.

[Command] pkg command pkg_name
[Command] pkg command option pkg_name

Manage packages (groups of add-on functions) for Octave. Different actions are available depending on the value of command. Available commands:

‘install’ Install named packages. For example,

     pkg install image-1.0.0.tar.gz

installs the package found in the file ‘image-1.0.0.tar.gz’. The option variable can contain options that affect the manner in which a package is installed. These options can be one or more of

-nodeps The package manager will disable dependency checking. With this option it is possible to install a package even when it depends on another package which is not installed on the system. Use this option with care.

-noauto The package manager will not automatically load the installed package when starting Octave. This overrides any setting within the package.

-auto The package manager will automatically load the installed package when starting Octave. This overrides any setting within the package.

-local A local installation (package available only to current user) is forced, even if the user has system privileges.

-global A global installation (package available to all users) is forced, even if the user doesn’t normally have system privileges.

-forge Install a package directly from the Octave-Forge repository. This requires an internet connection and the cURL library.

-verbose The package manager will print the output of all commands as they are performed.

‘update’ Check installed Octave-Forge packages against repository and update any outdated items. This requires an internet connection and the cURL library. Usage:

     pkg update
‘uninstall’

Uninstall named packages. For example,

     pkg uninstall image

removes the image package from the system. If another installed package depends on the image package an error will be issued. The package can be uninstalled anyway by using the ‘-nodeps’ option.

‘load’ Add named packages to the path. After loading a package it is possible to use the functions provided by the package. For example,

     pkg load image

adds the image package to the path. It is possible to load all installed packages at once with the keyword ‘all’. Usage:

     pkg load all

‘unload’ Remove named packages from the path. After unloading a package it is no longer possible to use the functions provided by the package. It is possible to unload all installed packages at once with the keyword ‘all’. Usage:

     pkg unload all

‘list’ Show the list of currently installed packages. For example,

     installed_packages = pkg ("list")

returns a cell array containing a structure for each installed package. If two output arguments are requested pkg splits the list of installed packages into those which were installed by the current user, and those which were installed by the system administrator.

     [user_packages, system_packages] = pkg ("list")

The option "-forge" lists packages available at the Octave-Forge repository. This requires an internet connection and the cURL library. For example:

     oct_forge_pkgs = pkg ("list", "-forge")
‘describe’

Show a short description of the named installed packages, with the option "-verbose" also list functions provided by the package. For example,

     pkg describe -verbose all

will describe all installed packages and the functions they provide. If one output is requested a cell of structure containing the description and list of functions of each package is returned as output rather than printed on screen:

     desc = pkg ("describe", "secs1d", "image")

If any of the requested packages is not installed, pkg returns an error, unless a second output is requested:

     [desc, flag] = pkg ("describe", "secs1d", "image")

flag will take one of the values "Not installed", "Loaded", or "Not loaded" for each of the named packages.

‘prefix’ Set the installation prefix directory. For example,

     pkg prefix ~/my_octave_packages

sets the installation prefix to ‘~/my_octave_packages’. Packages will be installed in this directory. It is possible to get the current installation prefix by requesting an output argument. For example:

     pfx = pkg ("prefix")

The location in which to install the architecture dependent files can be independently specified with an addition argument. For example:

     pkg prefix ~/my_octave_packages ~/my_arch_dep_pkgs
‘local_list’

Set the file in which to look for information on locally installed packages. Locally installed packages are those that are available only to the current user. For example:

     pkg local_list ~/.octave_packages

It is possible to get the current value of local list with the following

     pkg local_list
‘global_list’

Set the file in which to look for information on globally installed packages. Globally installed packages are those that are available to all users. For example:

     pkg global_list /usr/share/octave/octave_packages

It is possible to get the current value of global list with the following

     pkg global_list

‘build’ Build a binary form of a package or packages. The binary file produced will itself be an Octave package that can be installed normally with pkg. The form of the command to build a binary package is

     pkg build builddir image-1.0.0.tar.gz ...

where builddir is the name of a directory where the temporary installation will be produced and the binary packages will be found. The options ‘-verbose’ and ‘-nodeps’ are respected, while all other options are ignored.

‘rebuild’ Rebuild the package database from the installed directories. This can be used in cases where the package database has been corrupted. It can also take the ‘-auto’ and ‘-noauto’ options to allow the autoloading state of a package to be changed. For example,

     pkg rebuild -noauto image

will remove the autoloading status of the image package.

38.2 Using Packages

By default installed packages are not available from the Octave prompt, but it is possible to control this using the pkg load and pkg unload commands. The functions from a package can be added to the Octave path by typing

pkg load package_name

where package_name is the name of the package to be added to the path.

In much the same way a package can be removed from the Octave path by typing

pkg unload package_name

38.3 Administrating Packages

On UNIX-like systems it is possible to make both per-user and system-wide installations of a package. If the user performing the installation is root the packages will be installed in a system-wide directory that defaults to ‘OCTAVE_HOME/share/octave/packages/’. If the user is not root the default installation directory is ‘~/octave/’. Packages will be installed in a subdirectory of the installation directory that will be named after the package. It is possible to change the installation directory by using the pkg prefix command

     pkg prefix new_installation_directory

The current installation directory can be retrieved by typing

     current_installation_directory = pkg prefix

To function properly the package manager needs to keep some information about the installed packages. For per-user packages this information is by default stored in the file ‘~/.octave_packages’ and for system-wide installations it is stored in

‘OCTAVE_HOME/share/octave/octave_packages’. The path to the per-user file can be

changed with the pkg local_list command

     pkg local_list /path/to/new_file

For system-wide installations this can be changed in the same way using the pkg global_ list command. If these commands are called without a new path, the current path will be returned.

38.4 Creating Packages

Internally a package is simply a gzipped tar file that contains a top level directory of any given name. This directory will in the following be referred to as package and may contain the following files:

package/CITATION

This is am optional file describing instructions on how to cite the package for publication. It will be displayed verbatim by the function citation.

package/COPYING

This is a required file containing the license of the package. No restrictions is made on the license in general. If however the package contains dynamically linked functions the license must be compatible with the GNU General Public License.

package/DESCRIPTION

This is a required file containing information about the package. See Section 38.4.1 [The DESCRIPTION File], page 755, for details on this file.

package/ChangeLog

This is an optional file describing all the changes made to the package source files.

package/INDEX

This is an optional file describing the functions provided by the package. If this file is not given then one with be created automatically from the functions in the package and the Categories keyword in the ‘DESCRIPTION’ file. See Section 38.4.2 [The INDEX File], page 757, for details on this file.

package/NEWS

This is an optional file describing all user-visible changes worth mentioning. As this file increases on size, old entries can be moved into ‘package/ONEWS’.

package/ONEWS

This is an optional file describing old entries from the ‘NEWS’ file.

package/PKG_ADD

An optional file that includes commands that are run when the package is added to the users path. Note that PKG_ADD directives in the source code of the package will also be added to this file by the Octave package manager. Note that symbolic links are to be avoided in packages, as symbolic links do not exist on some file systems, and so a typical use for this file is the replacement of the symbolic link

     ln -s foo.oct bar.oct

with an autoload directive like

     autoload ('bar', which ('foo'));

See Section 38.4.3 [PKG ADD and PKG DEL Directives], page 758, for details on PKG_ADD directives.

package/PKG_DEL

An optional file that includes commands that are run when the package is removed from the users path. Note that PKG_DEL directives in the source code of the package will also be added to this file by the Octave package manager. See Section 38.4.3 [PKG ADD and PKG DEL Directives], page 758, for details on PKG_DEL directives.

package/pre_install.m

This is an optional function that is run prior to the installation of a package. This function is called with a single argument, a struct with fields names after the data in the ‘DESCRIPTION’, and the paths where the package functions will be installed.

package/post_install.m

This is an optional function that is run after the installation of a package. This function is called with a single argument, a struct with fields names after the data in the ‘DESCRIPTION’, and the paths where the package functions were installed.

package/on_uninstall.m

This is an optional function that is run prior to the removal of a package. This function is called with a single argument, a struct with fields names after the data in the ‘DESCRIPTION’, the paths where the package functions are installed, and whether the package is currently loaded.

Besides the above mentioned files, a package can also contain one or more of the following directories:

package/inst

An optional directory containing any files that are directly installed by the package. Typically this will include any m-files.

package/src

An optional directory containing code that must be built prior to the packages installation. The Octave package manager will execute ‘./configure’ in this directory if this script exists, and will then call make if a file ‘Makefile’ exists in this directory. make install will however not be called. The environment variables MKOCTFILE, OCTAVE_CONFIG, and OCTAVE will be set to the full paths of the programs mkoctfile, octave-config, and octave, respectively, of the correct version when configure and make are called. If a file called FILES exists all files listed there will be copied to the inst directory, so they also will be installed. If the FILES file doesn’t exist, ‘src/*.m’ and ‘src/*.oct’ will be copied to the inst directory.

package/doc

An optional directory containing documentation for the package. The files in this directory will be directly installed in a sub-directory of the installed package for future reference.

package/bin

An optional directory containing files that will be added to the Octave EXEC_PATH when the package is loaded. This might contain external scripts, etc., called by functions within the package.

38.4.1 The DESCRIPTION File

The ‘DESCRIPTION’ file contains various information about the package, such as its name, author, and version. This file has a very simple format

• Lines starting with ‘#’ are comments.

• Lines starting with a blank character are continuations from the previous line.

• Everything else is of the form NameOfOption: ValueOfOption.

The following is a simple example of a ‘DESCRIPTION’ file

Name: The name of my package
Version: 1.0.0
Date: 2007-18-04
Author: The name (and possibly email) of the package author.
Maintainer: The name (and possibly email) of the current
package maintainer.
Title: The title of the package
Description: A short description of the package.  If this
description gets too long for one line it can continue
on the next by adding a space to the beginning of the
following lines.
License: GPLv3+

The package manager currently recognizes the following keywords

Name Name of the package.

Version Version of the package. A package version must be 3 numbers separated by dots.

Date Date of last update.

Author Original author of the package.

Maintainer

Maintainer of the package.

Title A one line description of the package.

Description

A one paragraph description of the package.

Categories

Optional keyword describing the package (if no ‘INDEX’ file is given this is mandatory).

Problems Optional list of known problems.

Url Optional list of homepages related to the package.

Autoload Optional field that sets the default loading behavior for the package. If set to yes, true or on, then Octave will automatically load the package when starting. Otherwise the package must be manually loaded with the pkg load command. This default behavior can be overridden when the package is installed.

Depends A list of other Octave packages that this package depends on. This can include

dependencies on particular versions, with a format

     Depends: package (>= 1.0.0)

Possible operators are <, <=, ==, >= or >. If the part of the dependency in () is missing, any version of the package is acceptable. Multiple dependencies can be defined either as a comma separated list or on separate Depends lines.

License An optional short description of the used license (e.g., GPL version 3 or newer). This is optional since the file ‘COPYING’ is mandatory.

SystemRequirements

These are the external install dependencies of the package and are not checked by the package manager. This is here as a hint to the distribution packager. They follow the same conventions as the Depends keyword.

BuildRequires

These are the external build dependencies of the package and are not checked by the package manager. This is here as a hint to the distribution packager. They follow the same conventions as the Depends keyword. Note that in general, packaging systems such as rpm or deb and autoprobe the install dependencies from the build dependencies, and therefore the often a BuildRequires dependency removes the need for a SystemRequirements dependency.

The developer is free to add additional arguments to the ‘DESCRIPTION’ file for their own purposes. One further detail to aid the packager is that the SystemRequirements and BuildRequires keywords can have a distribution dependent section, and the automatic build process will use these. An example of the format of this is

BuildRequires: libtermcap-devel [Mandriva] libtermcap2-devel

where the first package name will be used as a default and if the RPMs are built on a Mandriva distribution, then the second package name will be used instead.

38.4.2 The INDEX File

The optional ‘INDEX’ file provides a categorical view of the functions in the package. This file has a very simple format

• Lines beginning with ‘#’ are comments.

• The first non-comment line should look like this

         toolbox >> Toolbox name

• Lines beginning with an alphabetical character indicates a new category of functions.

• Lines starting with a white space character indicate that the function names on the line belong to the last mentioned category.

The format can be summarized with the following example:

     # A comment
     toolbox >> Toolbox name
     Category Name 1
      function1 function2 function3
      function4
     Category Name 2
      function2 function5

If you wish to refer to a function that users might expect to find in your package but is not there, providing a work around or pointing out that the function is available elsewhere, you can use:

     fn = workaround description

This workaround description will not appear when listing functions in the package with pkg describe but they will be published in the HTML documentation online. Workaround descriptions can use any HTML markup, but keep in mind that it will be enclosed in a bold-italic environment. For the special case of:

     fn = use <code>alternate expression</code>

the bold-italic is automatically suppressed. You will need to use <code> even in references:

     fn = use <a href="someothersite.html"><code>fn</code></a>

Sometimes functions are only partially compatible, in which case you can list the noncompatible cases separately. To refer to another function in the package, use <f>fn</f>. For example:

     eig (a, b) = use <f>qz</f>

Since sites may have many missing functions, you can define a macro rather than typing the same link over and again.

     $id = expansion

defines the macro id. You can use $id anywhere in the description and it will be expanded. For example:

     $TSA = see <a href="link_to_spctools">SPC Tools</a>
     arcov = $TSA <code>armcv</code>

id is any string of letters, numbers and _.

38.4.3 PKG ADD and PKG DEL Directives

If the package contains files called PKG_ADD or PKG_DEL the commands in these files will be executed when the package is added or removed from the users path. In some situations such files are a bit cumbersome to maintain, so the package manager supports automatic creation of such files. If a source file in the package contains a PKG_ADD or PKG_DEL directive they will be added to either the PKG_ADD or PKG_DEL files.

In m-files a PKG_ADD directive looks like this

     ## PKG_ADD: some_octave_command

Such lines should be added before the function keyword. In C++ files a PKG_ADD directive looks like this

     // PKG_ADD: some_octave_command

In both cases some_octave_command should be replaced by the command that should be placed in the PKG_ADD file. PKG_DEL directives work in the same way, except the PKG_ADD keyword is replaced with PKG_DEL and the commands get added to the PKG_DEL file.

38.4.4 Missing Components

If a package relies on a component, such as another Octave package, that may not be present it may be useful to install a function which informs users what to do when a particular component is missing. The function must be written by the package maintainer and registered with Octave using missing_component_hook.

[Built-in Function] val = missing_component_hook ()
[Built-in Function] old_val = missing_component_hook (new_val)
[Built-in Function] missing_component_hook (new_val, "local")

Query or set the internal variable that specifies the function to call when a component of Octave is missing. This can be useful for packagers that may split the Octave installation into multiple sub-packages, for example, to provide a hint to users for how to install the missing components. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function. The hook function is expected to be of the form

          fcn (component)

Octave will call fcn with the name of the function that requires the component and a string describing the missing component. The hook function should return an error message to be displayed.

See also: [missing function hook], page 856.

Appendix A External Code Interface

"The sum of human wisdom is not contained in any one language" —Ezra Pound

Octave is a fantastic language for solving many problems in science and engineering. However, it is not the only computer language and there are times when you may want to use code written in other languages. Good reasons for doing so include: 1) not re-inventing the wheel; existing function libraries which have been thoroughly tested and debugged or large scale simulation codebases are a good example, 2) accessing unique capabilities of a different language; for example the well-known regular expression functions of Perl (but don’t do that because regexp already exists in Octave).

Performance should generally not be a reason for using compiled extensions. Although compiled extensions can run faster, particularly if they replace a loop in Octave code, this is almost never the best path to take. First, there are many techniques to speed up Octave performance while remaining within the language. Second, Octave is a highlevel language that makes it easy to perform common mathematical tasks. Giving that up means shifting the focus from solving the real problem to solving a computer programming problem. It means returning to low-level constructs such as pointers, memory management, mathematical overflow/underflow, etc. Because of the low level nature, and the fact that the compiled code is executed outside of Octave, there is the very real possibility of crashing the interpreter and losing work.

Before going further, you should first determine if you really need to bother writing code outside of Octave.

• Can I get the same functionality using the Octave scripting language alone? Even when a function already exists outside the language, it may be better to simply reproduce the behavior in an m-file rather than attempt to interface to the outside code.

• Is the code thoroughly optimized for Octave?

If performance is an issue you should always start with the in-language techniques for getting better performance. Chief among these is vectorization (see Chapter 19 [Vectorization and Faster Code Execution], page 457) which not only makes the code concise and more understandable but improves performance (10X-100X). If loops must be used, make sure that the allocation of space for variables takes place outside the loops using an assignment to a matrix of the right size, or zeros.

• Does the code make as much use as possible of existing built-in library routines? These routines are highly optimized and many do not carry the overhead of being interpreted.

• Does writing a dynamically linked function represent a useful investment of your time, relative to staying in Octave? It will take time to learn Octave’s interface for external code and there will inevitably be issues with tools such as compilers.

With that said, Octave offers a versatile interface for including chunks of compiled code as dynamically linked extensions. These dynamically linked functions can be called from the interpreter in the same manner as any ordinary function. The interface is bi-directional and external code can call Octave functions (like plot) which otherwise might be very difficult to develop.

The interface is centered around supporting the languages C++, C, and Fortran. Octave itself is written in C++ and can call external C++/C code through its native oct-file interface. The C language is also supported through the mex-file interface for compatibility with matlab. Fortran code is easiest to reach through the oct-file interface.

Because many other languages provide C or C++ APIs it is relatively simple to build bridges between Octave and other languages. This is also a way to bridge to hardware resources which often have device drivers written in C.

A.1 Oct-Files

A.1.1 Getting Started with Oct-Files

Oct-files are pieces of C++ code that have been compiled with the Octave API into a dynamically loadable object. They take their name from the file which contains the object which has the extension ‘.oct’.

Finding a C++ compiler, using the correct switches, adding the right include paths for header files, etc. is a difficult task. Octave automates this by providing the mkoctfile command with which to build oct-files. The command is available from within Octave or at the shell command line.

[Command] mkoctfile [-options] file …

[Function File] [output, status] = mkoctfile (. . .)

The mkoctfile function compiles source code written in C, C++, or Fortran. Depending on the options used with mkoctfile, the compiled code can be called within Octave or can be used as a stand-alone application. mkoctfile can be called from the shell prompt or from the Octave prompt. Calling it from the Octave prompt simply delegates the call to the shell prompt. The output is stored in the output variable and the exit status in the status variable. mkoctfile accepts the following options, all of which are optional except for the file name of the code you wish to compile:

‘-I DIR’ Add the include directory DIR to compile commands.

‘-D DEF’ Add the definition DEF to the compiler call.

‘-l LIB’ Add the library LIB to the link command.

‘-L DIR’ Add the library directory DIR to the link command.

‘-M’
‘--depend’

Generate dependency files (.d) for C and C++ source files.

‘-R DIR’ Add the run-time path to the link command.

‘-Wl,...’ Pass flags though the linker like "-Wl,-rpath=…". The quotes are needed since commas are interpreted as command separators.

‘-W...’ Pass flags though the compiler like "-Wa,OPTION".

‘-c’ Compile but do not link.

‘-g’ Enable debugging options for compilers.

‘-o FILE’
‘--output FILE’

Output file name. Default extension is .oct (or .mex if ‘--mex’ is specified) unless linking a stand-alone executable.

‘-p VAR’
‘--print VAR’

Print the configuration variable VAR. Recognized variables are:

ALL_CFLAGS                  INCFLAGS
ALL_CXXFLAGS                INCLUDEDIR
ALL_FFLAGS                  LAPACK_LIBS
ALL_LDFLAGS                 LD_CXX
AR                          LDFLAGS
BLAS_LIBS                   LD_STATIC_FLAG
CC                          LFLAGS
CFLAGS                      LIBDIR
CPICFLAG                    LIBOCTAVE
CPPFLAGS                    LIBOCTINTERP
CXX                         LIBS
CXXFLAGS                    OCTAVE_HOME
CXXPICFLAG                  OCTAVE_LIBS
DEPEND_EXTRA_SED_PATTERN    OCTAVE_LINK_DEPS
DEPEND_FLAGS                OCTAVE_LINK_OPTS
DL_LD                       OCTAVE_PREFIX
DL_LDFLAGS                  OCTINCLUDEDIR
F77                         OCTLIBDIR
F77_INTEGER8_FLAG           OCT_LINK_DEPS
FFLAGS                      OCT_LINK_OPTS
FFTW3F_LDFLAGS              RANLIB
FFTW3F_LIBS                 RDYNAMIC_FLAG
FFTW3_LDFLAGS               READLINE_LIBS
FFTW3_LIBS                  SED
FFTW_LIBS                   SPECIAL_MATH_LIB
FLIBS                       XTRA_CFLAGS
FPICFLAG                    XTRA_CXXFLAGS
‘--link-stand-alone’

Link a stand-alone executable file.

‘--mex’ Assume we are creating a MEX file. Set the default output extension to ".mex".

‘-s’

‘--strip’ Strip the output file.

‘-v’
‘--verbose’

Echo commands as they are executed.

‘file’ The file to compile or link. Recognized file types are

.c    C source
.cc   C++ source
.C    C++ source
.cpp  C++ source
.f    Fortran source (fixed form)
.F    Fortran source (fixed form)
.f90  Fortran source (free form)
.F90  Fortran source (free form)
.o    object file
.a    library file

Consider the following short example which introduces the basics of writing a C++ function that can be linked to Octave.

#include <octave/oct.h>
DEFUN_DLD (helloworld, args, nargout,
"Hello World Help String")
{
int nargin = args.length ();
octave_stdout << "Hello World has "
<< nargin << " input arguments and "
<< nargout << " output arguments.\n";
return octave_value_list ();
}

The first critical line is #include <octave/oct.h> which makes available most of the definitions necessary for a C++ oct-file. Note that ‘octave/oct.h’ is a C++ header and cannot be directly #include’ed in a C source file, nor any other language.

Included by ‘oct.h’ is a definition for the macro DEFUN_DLD which creates a dynamically loaded function. This macro takes four arguments:

1. The function name as it will be seen in Octave,

2. The list of arguments to the function of type octave_value_list,

3. The number of output arguments, which can and often is omitted if not used, and 4. The string to use for the help text of the function.

The return type of functions defined with DEFUN_DLD is always octave_value_list. There are a couple of important considerations in the choice of function name. First,

it must be a valid Octave function name and so must be a sequence of letters, digits, and underscores not starting with a digit. Second, as Octave uses the function name to define the filename it attempts to find the function in, the function name in the DEFUN_DLD macro must match the filename of the oct-file. Therefore, the above function should be in a file ‘helloworld.cc’, and would be compiled to an oct-file using the command

     mkoctfile helloworld.cc

This will create a file called ‘helloworld.oct’ that is the compiled version of the function. It should be noted that it is perfectly acceptable to have more than one DEFUN_DLD function in a source file. However, there must either be a symbolic link to the oct-file for each of the functions defined in the source code with the DEFUN_DLD macro or the autoload (Section 11.9 [Function Files], page 178) function should be used.

The rest of the function shows how to find the number of input arguments, how to print through the Octave pager, and return from the function. After compiling this function as above, an example of its use is

     helloworld (1, 2, 3)
     ⊣ Hello World has 3 input arguments and 0 output arguments.

Subsequent sections show how to use specific classes from Octave’s core internals. Base classes like dMatrix (a matrix of double values) are found in the directory ‘liboctave/array’. The definitive reference for how to use a particular class is the header file itself. However, it is often enough just to study the examples in the manual in order to be able to use the class.

A.1.2 Matrices and Arrays in Oct-Files

Octave supports a number of different array and matrix classes, the majority of which are based on the Array class. The exception is the sparse matrix types discussed separately below. There are three basic matrix types

Matrix A double precision matrix class defined in ‘dMatrix.h’,

ComplexMatrix

A complex matrix class defined in ‘CMatrix.h’, and

BoolMatrix

A boolean matrix class defined in ‘boolMatrix.h’.

These are the basic two-dimensional matrix types of Octave. In addition there are a number of multi-dimensional array types including

NDArray A double precision array class defined in ‘dNDArray.h’

ComplexNDarray

A complex array class defined in ‘CNDArray.h’

boolNDArray

A boolean array class defined in ‘boolNDArray.h’

int8NDArray
int16NDArray
int32NDArray
int64NDArray

8, 16, 32, and 64-bit signed array classes defined in ‘int8NDArray.h’,

          ‘int16NDArray.h’, etc.

uint8NDArray
uint16NDArray
uint32NDArray
uint64NDArray

8, 16, 32, and 64-bit unsigned array classes defined in ‘uint8NDArray.h’,

          ‘uint16NDArray.h’, etc.

There are several basic ways of constructing matrices or multi-dimensional arrays. Using the class Matrix as an example one can

• Create an empty matrix or array with the empty constructor. For example:

         Matrix a;

This can be used for all matrix and array types.

• Define the dimensions of the matrix or array with a dim vector which has the same

characteristics as the vector returned from size. For example:

         dim_vector dv (2);
         dv(0) = 2; dv(1) = 3;  // 2 rows, 3 columns
         Matrix a (dv);

This can be used on all matrix and array types.

• Define the number of rows and columns in the matrix. For example:

         Matrix a (2, 2)

However, this constructor can only be used with matrix types.

These types all share a number of basic methods and operators. Many bear a resemblance to functions that exist in the interpreter. A selection of useful methods include

[Method] T& operator () (octave idx type)

[Method] T& elem (octave idx type)

The () operator or elem method allow the values of the matrix or array to be read or set. These can take a single argument, which is of type octave_idx_type, that is the index into the matrix or array. Additionally, the matrix type allows two argument versions of the () operator and elem method, giving the row and column index of the value to obtain or set.

Note that these functions do significant error checking and so in some circumstances the user might prefer to access the data of the array or matrix directly through the fortran vec method discussed below.

[Method] octave_idx_type numel (void) const The total number of elements in the matrix or array.

[Method] size_t byte_size (void) const

The number of bytes used to store the matrix or array.

[Method] dim_vector dims (void) const

The dimensions of the matrix or array in value of type dim vector.

[Method] int ndims (void) const The number of dimensions of the matrix or array. Matrices are 2-D, but arrays can be N-dimensional.

[Method] void resize (const dim vector&)

A method taking either an argument of type dim_vector, or in the case of a matrix two arguments of type octave_idx_type defining the number of rows and columns in the matrix.

[Method] T* fortran_vec (void)

This method returns a pointer to the underlying data of the matrix or array so that it can be manipulated directly, either within Octave or by an external library.

Operators such an +, -, or * can be used on the majority of the matrix and array types. In addition there are a number of methods that are of interest only for matrices such as

transpose, hermitian, solve, etc.

The typical way to extract a matrix or array from the input arguments of DEFUN_DLD function is as follows

     #include <octave/oct.h>
     DEFUN_DLD (addtwomatrices, args, , "Add A to B")
     {
       int nargin = args.length ();
       if (nargin != 2)
         print_usage ();
       else
         {
           NDArray A = args(0).array_value ();
           NDArray B = args(1).array_value ();
           if (! error_state)
             return octave_value (A + B);
         }
       return octave_value_list ();
     }

To avoid segmentation faults causing Octave to abort this function explicitly checks that there are sufficient arguments available before accessing these arguments. It then obtains two multi-dimensional arrays of type NDArray and adds these together. Note that the array value method is called without using the is_matrix_type type, and instead the error state is checked before returning A + B. The reason to prefer this is that the arguments might be a type that is not an NDArray, but it would make sense to convert it to one. The array_value method allows this conversion to be performed transparently if possible, and sets error_state if it is not.

A + B, operating on two NDArray’s returns an NDArray, which is cast to an octave_value on the return from the function. An example of the use of this demonstration function is

     addtwomatrices (ones (2, 2), eye (2, 2))
           ⇒ 2  1
               1  2

A list of the basic Matrix and Array types, the methods to extract these from an octave_value, and the associated header file is listed below.

Type  Function  Source Code

RowVector                row_vector_value  ‘dRowVector.h’
ComplexRowVector         complex_row_vector_value  ‘CRowVector.h’
ColumnVector             column_vector_value  ‘dColVector.h’
ComplexColumnVector      complex_column_vector_value ‘CColVector.h’
Matrix                   matrix_value  ‘dMatrix.h’
ComplexMatrix            complex_matrix_value  ‘CMatrix.h’
boolMatrix               bool_matrix_value  ‘boolMatrix.h’
charMatrix               char_matrix_value  ‘chMatrix.h’
NDArray                  array_value  ‘dNDArray.h’
ComplexNDArray           complex_array_value  ‘CNDArray.h’
boolNDArray              bool_array_value  ‘boolNDArray.h’
charNDArray              char_array_value  ‘charNDArray.h’
int8NDArray              int8_array_value  ‘int8NDArray.h’
int16NDArray             int16_array_value  ‘int16NDArray.h’
int32NDArray             int32_array_value  ‘int32NDArray.h’
int64NDArray             int64_array_value  ‘int64NDArray.h’
uint8NDArray             uint8_array_value  ‘uint8NDArray.h’
uint16NDArray            uint16_array_value  ‘uint16NDArray.h’
uint32NDArray            uint32_array_value  ‘uint32NDArray.h’
uint64NDArray            uint64_array_value  ‘uint64NDArray.h’

A.1.3 Character Strings in Oct-Files

A character string in Octave is just a special Array class. Consider the example:

     #include <octave/oct.h>
     DEFUN_DLD (stringdemo, args, , "String Demo")
     {
       octave_value_list retval;
       int nargin = args.length ();
       if (nargin != 1)
         print_usage ();
       else
         {
           charMatrix ch = args(0).char_matrix_value ();
           if (! error_state)
             {
               retval(1) = octave_value (ch, '\'');  // Single Quote String
               octave_idx_type nr = ch.rows ();
               for (octave_idx_type i = 0; i < nr / 2; i++)

            {
              std::string tmp = ch.row_as_string (i);
              ch.insert (ch.row_as_string (nr-i-1).c_str (), i, 0);
              ch.insert (tmp.c_str (), nr-i-1, 0);
            }
          retval(0) = octave_value (ch, '"');  // Double Quote String
        }
    }
  return retval;
}

An example of the use of this function is

s0 = ["First String"; "Second String"];
[s1,s2] = stringdemo (s0)
⇒ s1 = Second String
        First String
⇒ s2 = First String
        Second String
typeinfo (s2)
⇒ sq_string
typeinfo (s1)
⇒ string

One additional complication of strings in Octave is the difference between single quoted and double quoted strings. To find out if an octave_value contains a single or double quoted string use one of the predicate tests shown below.

if (args(0).is_sq_string ())
  octave_stdout << "First argument is a single quoted string\n";
else if (args(0).is_dq_string ())
  octave_stdout << "First argument is a double quoted string\n";

Note, however, that both types of strings are represented by the charNDArray type, and so when assigning to an octave_value, the type of string should be specified. For example:

octave_value_list retval;
charNDArray ch;
...
// Create single quoted string
retval(1) = octave_value (ch);        // default constructor is sq_string
           OR
retval(1) = octave_value (ch, '\'');  // explicitly create sq_string
// Create a double quoted string
retval(0) = octave_value (ch, '"');

A.1.4 Cell Arrays in Oct-Files

Octave’s cell type is also available from within oct-files. A cell array is just an array of octave_values, and thus each element of the cell array can be treated just like any other octave_value. A simple example is

     #include <octave/oct.h>
     #include <octave/Cell.h>
     DEFUN_DLD (celldemo, args, , "Cell Demo")
     {
       octave_value_list retval;
       int nargin = args.length ();
       if (nargin != 1)
         print_usage ();
       else
         {
           Cell c = args(0).cell_value ();
           if (! error_state)
             for (octave_idx_type i = 0; i < c.numel (); i++)
               {
                 retval(i) = c(i);          // using operator syntax
                 //retval(i) = c.elem (i);  // using method syntax
               }
         }
       return retval;
     }

Note that cell arrays are used less often in standard oct-files and so the ‘Cell.h’ header file must be explicitly included. The rest of the example extracts the octave_values one by one from the cell array and returns them as individual return arguments. For example:

     [b1, b2, b3] = celldemo ({1, [1, 2], "test"})

⇒

     b1 =  1
     b2 =
        1   2
     b3 = test

A.1.5 Structures in Oct-Files

A structure in Octave is a map between a number of fields represented and their values. The Standard Template Library map class is used, with the pair consisting of a std::string and an Octave Cell variable.

A simple example demonstrating the use of structures within oct-files is

#include <octave/oct.h>
#include <octave/ov-struct.h>
DEFUN_DLD (structdemo, args, , "Struct Demo")
{
  octave_value retval;
  int nargin = args.length ();
  if (args.length () == 2)
    {
      octave_scalar_map arg0 = args(0).scalar_map_value ();
      //octave_map arg0 = args(0).map_value ();
      if (! error_state)
        {
          std::string arg1 = args(1).string_value ();
          if (! error_state)
            {
              octave_value tmp = arg0.contents (arg1);
              //octave_value tmp = arg0.contents (arg1)(0);
              if (tmp.is_defined ())
                {
                  octave_scalar_map st;
                  st.assign ("selected", tmp);
                  retval = octave_value (st);
                }
              else
                error ("structdemo: struct does not have a field named '%s'\n",
                       arg1.c_str ());
            }
          else
            error ("structdemo: ARG2 must be a character string");
        }
      else
        error ("structdemo: ARG1 must be a struct");
    }
  else
    print_usage ();
  return retval;
}

An example of its use is

     x.a = 1; x.b = "test"; x.c = [1, 2];
     structdemo (x, "b")
     ⇒ selected = test

The example above specifically uses the octave_scalar_map class which is for representing a single struct. For structure arrays the octave_map class is used instead. The commented code shows how the demo could be modified to handle a structure array. In that case the contents method returns a Cell which may have more than one element. Therefore, to obtain the underlying octave_value in this single-struct example we write

     octave_value tmp = arg0.contents (arg1)(0);

where the trailing (0) is the () operator on the Cell object. If this were a true structure array with multiple elements we could iterate over the elements using the () operator.

Structures are a relatively complex data container and there are more functions available in ‘oct-map.h’ which make coding with them easier than relying on just contents.

A.1.6 Sparse Matrices in Oct-Files

There are three classes of sparse objects that are of interest to the user.

SparseMatrix

A double precision sparse matrix class

SparseComplexMatrix

A complex sparse matrix class

SparseBoolMatrix

A boolean sparse matrix class

All of these classes inherit from the Sparse<T> template class, and so all have similar capabilities and usage. The Sparse<T> class was based on Octave’s Array<T> class, and so users familiar with Octave’s Array classes will be comfortable with the use of the sparse classes.

The sparse classes will not be entirely described in this section, due to their similarity with the existing Array classes. However, there are a few differences due the different nature of sparse objects, and these will be described. First, although it is fundamentally possible to have N-dimensional sparse objects, the Octave sparse classes do not allow them at this time; All instances of the sparse classes must be 2-dimensional. This means that SparseMatrix is actually more similar to Octave’s Matrix class than its NDArray class.

A.1.6.1 Array and Sparse Class Differences

The number of elements in a sparse matrix is considered to be the number of non-zero elements rather than the product of the dimensions. Therefore

     SparseMatrix sm;
     ...
     int nel = sm.nelem ();

returns the number of non-zero elements. If the user really requires the number of elements in the matrix, including the non-zero elements, they should use numel rather than nelem. Note that for very large matrices, where the product of the two dimensions is larger than the representation of an unsigned int, then numel can overflow. An example is speye (1e6) which will create a matrix with a million rows and columns, but only a million non-zero elements. Therefore the number of rows by the number of columns in this case is more than two hundred times the maximum value that can be represented by an unsigned int. The use of numel should therefore be avoided useless it is known it won’t overflow.

Extreme care must be take with the elem method and the "()" operator, which perform basically the same function. The reason is that if a sparse object is non-const, then Octave will assume that a request for a zero element in a sparse matrix is in fact a request to create this element so it can be filled. Therefore a piece of code like

     SparseMatrix sm;
     ...
     for (int j = 0; j < nc; j++)
       for (int i = 0; i < nr; i++)
         std::cerr << " (" << i << "," << j << "): " << sm(i,j) << std::endl;

is a great way of turning the sparse matrix into a dense one, and a very slow way at that since it reallocates the sparse object at each zero element in the matrix.

An easy way of preventing the above from happening is to create a temporary constant version of the sparse matrix. Note that only the container for the sparse matrix will be copied, while the actual representation of the data will be shared between the two versions of the sparse matrix. So this is not a costly operation. For example, the above would become

     SparseMatrix sm;
     ...
     const SparseMatrix tmp (sm);
     for (int j = 0; j < nc; j++)
       for (int i = 0; i < nr; i++)
         std::cerr << " (" << i << "," << j << "): " << tmp(i,j) << std::endl;

Finally, as the sparse types aren’t represented by a contiguous block of memory, the fortran_vec method of the Array<T> is not available. It is, however, replaced by three separate methods ridx, cidx and data, that access the raw compressed column format that Octave sparse matrices are stored in. These methods can be used in a manner similar to elem to allow the matrix to be accessed or filled. However, in that case it is up to the user to respect the sparse matrix compressed column format.

A.1.6.2 Creating Sparse Matrices in Oct-Files

There are several useful alternatives for creating a sparse matrix. The first is to create three vectors representing the row index, column index, and data values, and from these create the matrix. The second alternative is to create a sparse matrix with the appropriate amount of space and then fill in the values. Both techniques have their advantages and disadvantages.

Below is an example of creating a small sparse matrix using the first technique

     int nz, nr, nc;
     nz = 4, nr = 3, nc = 4;
     ColumnVector ridx (nz);
     ColumnVector cidx (nz);
     ColumnVector data (nz);
     ridx(0) = 1; cidx(0) = 1; data(0) = 1;
     ridx(1) = 2; cidx(1) = 2; data(1) = 2;
     ridx(2) = 2; cidx(2) = 4; data(2) = 3;
     ridx(3) = 3; cidx(3) = 4; data(3) = 4;
     SparseMatrix sm (data, ridx, cidx, nr, nc);

which creates the matrix given in section Section 22.1.1 [Storage of Sparse Matrices], page 487. Note that the compressed matrix format is not used at the time of the creation of the matrix itself, but is used internally.

As discussed in the chapter on Sparse Matrices, the values of the sparse matrix are stored in increasing column-major ordering. Although the data passed by the user need not respect this requirement, pre-sorting the data will significantly speed up creation of the sparse matrix.

The disadvantage of this technique for creating a sparse matrix is that there is a brief time when two copies of the data exist. For extremely memory constrained problems this may not be the best technique for creating a sparse matrix.

The alternative is to first create a sparse matrix with the desired number of non-zero elements and then later fill those elements in. Sample code:

     int nz, nr, nc;
     nz = 4, nr = 3, nc = 4;
     SparseMatrix sm (nr, nc, nz);
     sm(0,0) = 1; sm(0,1) = 2; sm(1,3) = 3; sm(2,3) = 4;

This creates the same matrix as previously. Again, although not strictly necessary, it is significantly faster if the sparse matrix is created and the elements are added in columnmajor ordering. The reason for this is that when elements are inserted at the end of the current list of known elements then no element in the matrix needs to be moved to allow the new element to be inserted; Only the column indexes need to be updated.

There are a few further points to note about this method of creating a sparse matrix. First, it is possible to create a sparse matrix with fewer elements than are actually inserted in the matrix. Therefore,

     int nr, nc;
     nr = 3, nc = 4;
     SparseMatrix sm (nr, nc, 0);
     sm(0,0) = 1; sm(0,1) = 2; sm(1,3) = 3; sm(2,3) = 4;

is perfectly valid. However, it is a very bad idea because as each new element is added to the sparse matrix the matrix needs to request more space and reallocate memory. This is an expensive operation, that will significantly slow this means of creating a sparse matrix. Furthermore, it is possible to create a sparse matrix with too much storage, so having nz greater than 4 is also valid. The disadvantage is that the matrix occupies more memory than strictly needed.

It is not always possible to know the number of non-zero elements prior to filling a matrix. For this reason the additional unused storage of a sparse matrix can be removed after its creation with the maybe_compress function. In addition, maybe_compress can deallocate the unused storage, but it can also remove zero elements from the matrix. The removal of zero elements from the matrix is controlled by setting the argument of the maybe_compress function to be true. However, the cost of removing the zeros is high because it implies re-sorting the elements. If possible, it is better if the user does not add the unnecessary zeros in the first place. An example of the use of maybe_compress is

int nz, nr, nc;
nz = 6, nr = 3, nc = 4;
SparseMatrix sm1 (nr, nc, nz);
sm1(0,0) = 1; sm1(0,1) = 2; sm1(1,3) = 3; sm1(2,3) = 4;
sm1.maybe_compress ();  // No zero elements were added
SparseMatrix sm2 (nr, nc, nz);
sm2(0,0) = 1; sm2(0,1) = 2; sm(0,2) = 0; sm(1,2) = 0;
sm1(1,3) = 3; sm1(2,3) = 4;
sm2.maybe_compress (true);  // Zero elements were added

The use of the maybe_compress function should be avoided if possible as it will slow the creation of the matrix.

A third means of creating a sparse matrix is to work directly with the data in compressed row format. An example of this technique might be

octave_value arg;
...
int nz, nr, nc;
nz = 6, nr = 3, nc = 4;   // Assume we know the max # nz
SparseMatrix sm (nr, nc, nz);
Matrix m = arg.matrix_value ();
int ii = 0;
sm.cidx (0) = 0;
for (int j = 1; j < nc; j++)
  {
    for (int i = 0; i < nr; i++)
      {
        double tmp = foo (m(i,j));
        if (tmp != 0.)
          {
            sm.data(ii) = tmp;
            sm.ridx(ii) = i;
            ii++;
          }
      }

         sm.cidx(j+1) = ii;
      }
     sm.maybe_compress ();  // If don't know a priori the final # of nz.

which is probably the most efficient means of creating a sparse matrix.

Finally, it might sometimes arise that the amount of storage initially created is insufficient to completely store the sparse matrix. Therefore, the method change_capacity exists to reallocate the sparse memory. The above example would then be modified as

     octave_value arg;
     ...
     int nz, nr, nc;
     nz = 6, nr = 3, nc = 4;   // Assume we know the max # nz
     SparseMatrix sm (nr, nc, nz);
     Matrix m = arg.matrix_value ();
     int ii = 0;
     sm.cidx (0) = 0;
     for (int j = 1; j < nc; j++)
       {
         for (int i = 0; i < nr; i++)
           {
             double tmp = foo (m(i,j));
             if (tmp != 0.)
               {
                 if (ii == nz)
                   {
                     nz += 2;   // Add 2 more elements
                     sm.change_capacity (nz);
                   }
                 sm.data(ii) = tmp;
                 sm.ridx(ii) = i;
                 ii++;
               }
           }
         sm.cidx(j+1) = ii;
      }
     sm.maybe_mutate ();  // If don't know a priori the final # of nz.

Note that both increasing and decreasing the number of non-zero elements in a sparse matrix is expensive as it involves memory reallocation. Also as parts of the matrix, though not its entirety, exist as old and new copies at the same time, additional memory is needed. Therefore, if possible this should be avoided.

A.1.6.3 Using Sparse Matrices in Oct-Files

Most of the same operators and functions on sparse matrices that are available from the Octave command line are also available within oct-files. The basic means of extracting a sparse matrix from an octave_value and returning it as an octave_value, can be seen in the following example.

octave_value_list retval;
SparseMatrix sm = args(0).sparse_matrix_value ();
SparseComplexMatrix scm =
    args(1).sparse_complex_matrix_value ();
SparseBoolMatrix sbm = args(2).sparse_bool_matrix_value ();
...
retval(2) = sbm;
retval(1) = scm;
retval(0) = sm;

The conversion to an octave_value is handled by the sparse octave_value constructors, and so no special care is needed.

A.1.7 Accessing Global Variables in Oct-Files

Global variables allow variables in the global scope to be accessed. Global variables can be accessed within oct-files by using the support functions get_global_value and set_ global_value. get_global_value takes two arguments, the first is a string representing the variable name to obtain. The second argument is a boolean argument specifying what to do if no global variable of the desired name is found. An example of the use of these two functions is

#include <octave/oct.h>
DEFUN_DLD (globaldemo, args, , "Global Demo")
{
  octave_value retval;
  int nargin = args.length ();
  if (nargin != 1)
    print_usage ();
  else
    {
      std::string s = args(0).string_value ();
      if (! error_state)
        {
          octave_value tmp = get_global_value (s, true);
          if (tmp.is_defined ())
            retval = tmp;
          else
            retval = "Global variable not found";
          set_global_value ("a", 42.0);
        }
    }
  return retval;
}

An example of its use is

global a b
b = 10;
globaldemo ("b")
⇒ 10
globaldemo ("c")
⇒ "Global variable not found"
num2str (a)
⇒ 42

A.1.8 Calling Octave Functions from Oct-Files

There is often a need to be able to call another Octave function from within an oct-file, and there are many examples of such within Octave itself. For example, the quad function is an oct-file that calculates the definite integral by quadrature over a user supplied function.

There are also many ways in which a function might be passed. It might be passed as one of

1. Function Handle

2. Anonymous Function Handle

3. Inline Function

4. String

The example below demonstrates an example that accepts all four means of passing a function to an oct-file.

#include <octave/oct.h>
#include <octave/parse.h>
DEFUN_DLD (funcdemo, args, nargout, "Function Demo")
{
  octave_value_list retval;
  int nargin = args.length ();
  if (nargin < 2)
    print_usage ();
  else
    {
      octave_value_list newargs;
      for (octave_idx_type i = nargin - 1; i > 0; i--)
        newargs(i-1) = args(i);
      if (args(0).is_function_handle () || args(0).is_inline_function ())
        {
          octave_function *fcn = args(0).function_value ();
          if (! error_state)
            retval = feval (fcn, newargs, nargout);
        }
      else if (args(0).is_string ())
        {

          std::string fcn = args(0).string_value ();
          if (! error_state)
            retval = feval (fcn, newargs, nargout);
        }
      else
        error ("funcdemo: INPUT must be string, inline, or function handle");
    }
  return retval;
}

The first argument to this demonstration is the user-supplied function and the remaining arguments are all passed to the user function.

funcdemo (@sin,1)
⇒ 0.84147
funcdemo (@(x) sin (x), 1)
⇒ 0.84147
funcdemo (inline ("sin (x)"), 1)
⇒ 0.84147
funcdemo ("sin",1)
⇒ 0.84147
funcdemo (@atan2, 1, 1)
⇒ 0.78540

When the user function is passed as a string the treatment of the function is different. In some cases it is necessary to have the user supplied function as an octave_function object. In that case the string argument can be used to create a temporary function as demonstrated below.

std::octave fcn_name = unique_symbol_name ("__fcn__");
std::string fcode = "function y = ";
fcode.append (fcn_name);
fcode.append ("(x) y = ");
fcn = extract_function (args(0), "funcdemo", fcn_name,
                        fcode, "; endfunction");
...
if (fcn_name.length ())
  clear_function (fcn_name);

There are two important things to know in this case. First, the number of input arguments to the user function is fixed, and in the above example is a single argument. Second, to avoid leaving the temporary function in the Octave symbol table it should be cleared after use. Also, by convention internal function names begin and end with the character sequence ‘__’.

A.1.9 Calling External Code from Oct-Files

Linking external C code to Octave is relatively simple, as the C functions can easily be called directly from C++. One possible issue is that the declarations of the external C functions may need to be explicitly defined as C functions to the compiler. If the declarations of the external C functions are in the header ‘foo.h’, then the tactic to ensure that the C++ compiler treats these declarations as C code is

     #ifdef __cplusplus
     extern "C"
     {
     #endif
     #include "foo.h"
     #ifdef __cplusplus
     }  /* end extern "C" */
     #endif

Calling Fortran code, however, can pose more difficulties. This is due to differences in the manner in which compilers treat the linking of Fortran code with C or C++ code. Octave supplies a number of macros that allow consistent behavior across a number of compilers.

The underlying Fortran code should use the XSTOPX function to replace the Fortran STOP function. XSTOPX uses the Octave exception handler to treat failing cases in the Fortran code explicitly. Note that Octave supplies its own replacement blas XERBLA function, which

uses XSTOPX.

If the code calls XSTOPX, then the F77_XFCN macro should be used to call the underlying Fortran function. The Fortran exception state can then be checked with the global variable

f77_exception_encountered. If XSTOPX will not be called, then the F77_FCN macro should

be used instead to call the Fortran code.

There is no great harm in using F77_XFCN in all cases, except that for Fortran code that is short running and executes a large number of times, there is potentially an overhead in doing so. However, if F77_FCN is used with code that calls XSTOP, Octave can generate a segmentation fault.

An example of the inclusion of a Fortran function in an oct-file is given in the following example, where the C++ wrapper is

     #include <octave/oct.h>
     #include <octave/f77-fcn.h>
     extern "C"
     {
       F77_RET_T
       F77_FUNC (fortransub, FORTSUB)
         (const int&, double*, F77_CHAR_ARG_DECL F77_CHAR_ARG_LEN_DECL);
     }
     DEFUN_DLD (fortrandemo, args, , "Fortran Demo")
     {
       octave_value_list retval;
       int nargin = args.length ();
       if (nargin != 1)
         print_usage ();
       else

{
NDArray a = args(0).array_value ();
if (! error_state)
  {
    double *av = a.fortran_vec ();
    octave_idx_type na = a.numel ();
    OCTAVE_LOCAL_BUFFER (char, ctmp, 128);
    F77_XFCN (fortransub, FORTSUB,
              (na, av, ctmp F77_CHAR_ARG_LEN (128)));
    retval(1) = std::string (ctmp);
    retval(0) = a;
  }
}
return retval;
}

and the Fortran function is

subroutine fortransub (n, a, s)
implicit none
character*(*) s
real*8 a(*)
integer*4 i, n, ioerr
do i = 1, n
  if (a(i) .eq. 0d0) then
    call xstopx ('fortransub: divide by zero')
  else
    a(i) = 1d0 / a(i)
  endif
enddo
write (unit = s, fmt = '(a,i3,a,a)', iostat = ioerr)
$       'There are ', n,
$       ' values in the input vector', char(0)
if (ioerr .ne. 0) then
  call xstopx ('fortransub: error writing string')
endif
return
end

This example demonstrates most of the features needed to link to an external Fortran function, including passing arrays and strings, as well as exception handling. Both the Fortran and C++ files need to be compiled in order for the example to work.

     mkoctfile fortrandemo.cc fortransub.f
     [b, s] = fortrandemo (1:3)

⇒

       b = 1.00000   0.50000   0.33333
       s = There are   3 values in the input vector
     [b, s] = fortrandemo (0:3)
     error: fortrandemo: fortransub: divide by zero

A.1.10 Allocating Local Memory in Oct-Files

Allocating memory within an oct-file might seem easy as the C++ new/delete operators can be used. However, in that case great care must be taken to avoid memory leaks. The preferred manner in which to allocate memory for use locally is to use the OCTAVE_LOCAL_BUFFER macro. An example of its use is

     OCTAVE_LOCAL_BUFFER (double, tmp, len)

that returns a pointer tmp of type double * of length len.

In this case Octave itself will worry about reference counting and variable scope and will properly free memory without programmer intervention.

A.1.11 Input Parameter Checking in Oct-Files

As oct-files are compiled functions they open up the possibility of crashing Octave through careless function calls or memory faults. It is quite important that each and every function have a sufficient level of parameter checking to ensure that Octave behaves well.

The minimum requirement, as previously discussed, is to check the number of input arguments before using them to avoid referencing a non-existent argument. However, in some cases this might not be sufficient as the underlying code imposes further constraints. For example, an external function call might be undefined if the input arguments are not integers, or if one of the arguments is zero, or if the input is complex and a real value was expected. Therefore, oct-files often need additional input parameter checking.

There are several functions within Octave that can be useful for the purposes of parameter checking. These include the methods of the octave value class like is_real_matrix, is_numeric_type, etc. Often, with a knowledge of the Octave m-file language, you can guess at what the corresponding C++ routine will. In addition there are some more specialized input validation functions of which a few are demonstrated below.

     #include <octave/oct.h>
     DEFUN_DLD (paramdemo, args, nargout, "Parameter Check Demo")
     {
       octave_value retval;
       int nargin = args.length ();
       if (nargin != 1)
         print_usage ();
       else if (nargout != 0)
         error ("paramdemo: OUTPUT argument required");
       else

         {
           NDArray m = args(0).array_value ();
           double min_val = -10.0;
           double max_val = 10.0;
           octave_stdout << "Properties of input array:\n";
           if (m.any_element_is_negative ())
             octave_stdout << "  includes negative values\n";
           if (m.any_element_is_inf_or_nan ())
             octave_stdout << "  includes Inf or NaN values\n";
           if (m.any_element_not_one_or_zero ())
             octave_stdout << "  includes other values than 1 and 0\n";
           if (m.all_elements_are_int_or_inf_or_nan ())
             octave_stdout << "  includes only int, Inf or NaN values\n";
           if (m.all_integers (min_val, max_val))
             octave_stdout << "  includes only integers in [-10,10]\n";
         }
       return retval;
     }

An example of its use is:

     paramdemo ([1, 2, NaN, Inf])
     ⇒ Properties of input array:
          includes Inf or NaN values
          includes other values than 1 and 0
          includes only int, Inf or NaN values

A.1.12 Exception and Error Handling in Oct-Files

Another important feature of Octave is its ability to react to the user typing CONTROL-C even during calculations. This ability is based on the C++ exception handler, where memory allocated by the C++ new/delete methods are automatically released when the exception is treated. When writing an oct-file, to allow Octave to treat the user typing CONTROL-C, the OCTAVE_QUIT macro is supplied. For example:

     for (octave_idx_type i = 0; i < a.nelem (); i++)
       {
         OCTAVE_QUIT;
         b.elem (i) = 2. * a.elem (i);
       }

The presence of the OCTAVE_QUIT macro in the inner loop allows Octave to treat the user request with the CONTROL-C. Without this macro, the user must either wait for the function to return before the interrupt is processed, or press CONTROL-C three times to force Octave to exit.

The OCTAVE_QUIT macro does impose a very small speed penalty, and so for loops that are known to be small it might not make sense to include OCTAVE_QUIT.

When creating an oct-file that uses an external libraries, the function might spend a significant portion of its time in the external library. It is not generally possible to use the OCTAVE_QUIT macro in this case. The alternative in this case is

BEGIN_INTERRUPT_IMMEDIATELY_IN_FOREIGN_CODE;
...  some code that calls a "foreign" function ...
END_INTERRUPT_IMMEDIATELY_IN_FOREIGN_CODE;

The disadvantage of this is that if the foreign code allocates any memory internally, then this memory might be lost during an interrupt, without being deallocated. Therefore, ideally Octave itself should allocate any memory that is needed by the foreign code, with either the fortran vec method or the OCTAVE_LOCAL_BUFFER macro.

The Octave unwind protect mechanism (Section 10.8 [The unwind protect Statement], page 164) can also be used in oct-files. In conjunction with the exception handling of Octave, it is important to enforce that certain code is run to allow variables, etc. to be restored even if an exception occurs. An example of the use of this mechanism is

#include <octave/oct.h>
#include <octave/unwind-prot.h>
void
my_err_handler (const char *fmt, ...)
{
  // Do nothing!!
}
DEFUN_DLD (unwinddemo, args, nargout, "Unwind Demo")
{
  octave_value retval;
  int nargin = args.length ();
  if (nargin < 2)
    print_usage ();
  else
    {
      NDArray a = args(0).array_value ();
      NDArray b = args(1).array_value ();
      if (! error_state)
        {
          // Declare unwind_protect frame which lasts as long as
          // the variable frame has scope.
          unwind_protect frame;
          frame.protect_var (current_liboctave_warning_handler);
          set_liboctave_warning_handler (my_err_handler);
          retval = octave_value (quotient (a, b));
        }
    }
  return retval;
}

As can be seen in the example:

     unwinddemo (1, 0)
     ⇒ Inf
     1 / 0
     ⇒ warning: division by zero
        Inf

The warning for division by zero (and in fact all warnings) are disabled in the unwinddemo function.

A.1.13 Documentation and Test of Oct-Files

The documentation of an oct-file is the fourth string parameter of the DEFUN_DLD macro. This string can be formatted in the same manner as the help strings for user functions (see Section C.4 [Documentation Tips], page 811), however there are some issue that are particular to the formatting of help strings within oct-files.

The major issue is that the help string will typically be longer than a single line of text, and so the formatting of long help strings needs to be taken into account. There are several possible solutions, but the most common is illustrated in the following example,

     DEFUN_DLD (do_what_i_want, args, nargout,
       "-*- texinfo -*-\n\
     @deftypefn {Function File} {} do_what_i_say (@var{n})\n\
     A function that does what the user actually wants rather\n\
     than what they requested.\n\
     @end deftypefn")
     {
     ...
     }

where, as can be seen, each line of text is terminated by \n\ which is an embedded new-line in the string together with a C++ string continuation character. Note that the final \ must be the last character on the line.

Octave also includes the ability to embed test and demonstration code for a function within the code itself (see Appendix B [Test and Demo Functions], page 799). This can be used from within oct-files (or in fact any file) with certain provisos. First, the test and demo functions of Octave look for %! as the first two characters of a line to identify test and demonstration code. This is a requirement for oct-files as well. In addition, the test and demonstration code must be wrapped in a comment block to avoid it being interpreted by the compiler. Finally, the Octave test and demonstration code must have access to the original source code of the oct-file and not just the compiled code as the tests are stripped from the compiled code. An example in an oct-file might be

     /*
     %!assert (sin ([1,2]), [sin(1),sin(2)])
     %!error (sin ())
     %!error (sin (1,1))
     */

A.2 Mex-Files

Octave includes an interface to allow legacy mex-files to be compiled and used with Octave. This interface can also be used to share code between Octave and matlab users. However, as mex-files expose matlab’s internal API, and the internal structure of Octave is different, a mex-file can never have the same performance in Octave as the equivalent oct-file. In particular, to support the manner in which variables are passed to mex functions there are a significant number of additional copies of memory blocks when calling or returning from a mex-file function. For this reason, it is recommended that any new code be written with the oct-file interface previously discussed.

A.2.1 Getting Started with Mex-Files

The basic command to build a mex-file is either mkoctfile --mex or mex. The first command can be used either from within Octave or from the command line. However, to avoid issues with matlab’s own mex command, the use of the command mex is limited to within Octave. Compiled mex-files have the extension ‘.mex’.

[Command] mex [options] file … Compile source code written in C, C++, or Fortran, to a MEX file. This is equivalent

     to mkoctfile --mex [options] file.

See also: [mkoctfile], page 760.

[Function File] mexext ()

Return the filename extension used for MEX files.

See also: [mex], page 784.

Consider the following short example:

     #include "mex.h"
     void
     mexFunction (int nlhs, mxArray *plhs[],
                  int nrhs, const mxArray *prhs[])
     {
       mexPrintf ("Hello, World!\n");
       mexPrintf ("I have %d inputs and %d outputs\n", nrhs, nlhs);
     }

The first line #include "mex.h" makes available all of the definitions necessary for a mex-file. One important difference between Octave and matlab is that the header file "matrix.h" is implicitly included through the inclusion of "mex.h". This is necessary to avoid a conflict with the Octave file "Matrix.h" for operating systems and compilers that don’t distinguish between filenames in upper and lower case.

The entry point into the mex-file is defined by mexFunction. The function takes four arguments:

1. The number of return arguments (# of left-hand side args).

2. An array of pointers to return arguments.

3. The number of input arguments (# of right-hand side args).

4. An array of pointers to input arguments.

Note that the function name definition is not explicitly included in mexFunction and so there can only be a single mexFunction entry point per file. Instead, the name of the function as seen in Octave is determined by the name of the mex-file itself minus the extension. Therefore, if the above function is in the file ‘myhello.c’, it can be compiled with

     mkoctfile --mex myhello.c

which creates a file ‘myhello.mex’. The function can then be run from Octave as

     myhello (1,2,3)
     ⇒ Hello, World!
     ⇒ I have 3 inputs and 0 outputs

It should be noted that the mex-file contains no help string for the functions it contains. To document mex-files, there should exist an m-file in the same directory as the mex-file itself. Taking the above as an example, we would therefore have a file ‘myhello.m’ that might contain the text

     %MYHELLO Simple test of the functionality of a mex-file.

In this case, the function that will be executed within Octave will be given by the mexfile, while the help string will come from the m-file. This can also be useful to allow a sample implementation of the mex-file within the Octave language itself for testing purposes.

Although there cannot be multiple entry points in a single mex-file, one can use the mexFunctionName function to determine what name the mex-file was called with. This can be used to alter the behavior of the mex-file based on the function name. For example, if

     #include "mex.h"
     void
     mexFunction (int nlhs, mxArray *plhs[],
                  int nrhs, const mxArray *prhs[])
     {
       const char *nm;
       nm = mexFunctionName ();
       mexPrintf ("You called function: %s\n", nm);
       if (strcmp (nm, "myfunc") == 0)
         mexPrintf ("This is the principal function\n", nm);
       return;
     }

is in file ‘myfunc.c’, and it is compiled with

     mkoctfile --mex myfunc.c
     ln -s myfunc.mex myfunc2.mex

then as can be seen by

     myfunc ()
     ⇒ You called function: myfunc
         This is the principal function
     myfunc2 ()
     ⇒ You called function: myfunc2

the behavior of the mex-file can be altered depending on the functions name.

Although the user should only include ‘mex.h’ in their code, Octave declares additional functions, typedefs, etc., available to the user to write mex-files in the headers ‘mexproto.h’

and ‘mxarray.h’.

A.2.2 Working with Matrices and Arrays in Mex-Files

The basic mex type of all variables is mxArray. Any object, such as a matrix, cell array, or structure is stored in this basic type. As such, mxArray serves basically the same purpose as the octave value class in oct-files in that it acts as a container for the more specialized types.

The mxArray structure contains at a minimum, the name of the variable it represents, its dimensions, its type, and whether the variable is real or complex. It can also contain a number of additional fields depending on the type of the mxArray. There are a number of func-

tions to create mxArray structures, including mxCreateDoubleMatrix, mxCreateCellArray,
mxCreateSparse, and the generic mxCreateNumericArray.

The basic function to access the data contained in an array is mxGetPr. As the mex interface assumes that real and imaginary parts of a complex array are stored separately, there is an equivalent function mxGetPi that gets the imaginary part. Both of these functions are only for use with double precision matrices. The generic functions mxGetData and mxGetImagData perform the same operation on all matrix types. For example:

     mxArray *m;
     mwSize *dims;
     UINT32_T *pr;
     dims = (mwSize *) mxMalloc (2 * sizeof (mwSize));
     dims[0] = 2; dims[1] = 2;
     m = mxCreateNumericArray (2, dims, mxUINT32_CLASS, mxREAL);
     pr = (UINT32_T *) mxGetData (m);

There are also the functions mxSetPr, etc., that perform the inverse, and set the data of an array to use the block of memory pointed to by the argument of mxSetPr.

Note the type mwSize used above, and also mwIndex, are defined as the native precision of the indexing in Octave on the platform on which the mex-file is built. This allows both 32- and 64-bit platforms to support mex-files. mwSize is used to define array dimensions and the maximum number or elements, while mwIndex is used to define indexing into arrays.

An example that demonstrates how to work with arbitrary real or complex double precision arrays is given by the file ‘mypow2.c’ shown below.

     #include "mex.h"
     void
     mexFunction (int nlhs, mxArray* plhs[],

           int nrhs, const mxArray* prhs[])
{
mwSize n;
mwIndex i;
double *vri, *vro;
if (nrhs != 1 || ! mxIsNumeric (prhs[0]))
  mexErrMsgTxt ("ARG1 must be a matrix");
n = mxGetNumberOfElements (prhs[0]);
plhs[0] = mxCreateNumericArray (mxGetNumberOfDimensions (prhs[0]),
                                mxGetDimensions (prhs[0]),
                                mxGetClassID (prhs[0]),
                                mxIsComplex (prhs[0]));
vri = mxGetPr (prhs[0]);
vro = mxGetPr (plhs[0]);
if (mxIsComplex (prhs[0]))
  {
    double *vii, *vio;
    vii = mxGetPi (prhs[0]);
    vio = mxGetPi (plhs[0]);
    for (i = 0; i < n; i++)
      {
        vro[i] = vri[i] * vri[i] - vii[i] * vii[i];
        vio[i] = 2 * vri[i] * vii[i];
      }
  }
else
  {
    for (i = 0; i < n; i++)
      vro[i] = vri[i] * vri[i];
  }
}

with an example of its use

b = randn (4,1) + 1i * randn (4,1);
all (b.^2 == mypow2 (b))
⇒ 1
The example above uses the functions mxGetDimensions, mxGetNumberOfElements, and

mxGetNumberOfDimensions to work with the dimensions of multi-dimensional arrays. The functions mxGetM, and mxGetN are also available to find the number of rows and columns in a 2-D matrix.

A.2.3 Character Strings in Mex-Files

As mex-files do not make the distinction between single and double quoted strings within Octave, there is perhaps less complexity in the use of strings and character matrices in mex-files. An example of their use that parallels the demo in ‘stringdemo.cc’ is given in the file ‘mystring.c’, as shown below.

     #include <string.h>
     #include "mex.h"
     void
     mexFunction (int nlhs, mxArray *plhs[],
                  int nrhs, const mxArray *prhs[])
     {
       mwSize m, n;
       mwIndex i, j;
       mxChar *pi, *po;
       if (nrhs != 1 || ! mxIsChar (prhs[0])
           || mxGetNumberOfDimensions (prhs[0]) > 2)
         mexErrMsgTxt ("ARG1 must be a char matrix");
       m = mxGetM (prhs[0]);
       n = mxGetN (prhs[0]);
       pi = mxGetChars (prhs[0]);
       plhs[0] = mxCreateNumericMatrix (m, n, mxCHAR_CLASS, mxREAL);
       po = mxGetChars (plhs[0]);
       for (j = 0; j < n; j++)
         for (i = 0; i < m; i++)
           po[j*m + m - 1 - i] = pi[j*m + i];
     }

An example of its expected output is

     mystring (["First String"; "Second String"])
     ⇒ Second String
        First String

Other functions in the mex interface for handling character strings are mxCreateString,

mxArrayToString, and mxCreateCharMatrixFromStrings. In a mex-file, a character string

is considered to be a vector rather than a matrix. This is perhaps an arbitrary distinction as the data in the mxArray for the matrix is consecutive in any case.

A.2.4 Cell Arrays with Mex-Files

One can perform exactly the same operations on Cell arrays in mex-files as in oct-files. An example that reduplicates the function of the ‘celldemo.cc’ oct-file in a mex-file is given by ‘mycell.c’ as shown below.

     #include "mex.h"

void
mexFunction (int nlhs, mxArray* plhs[],
             int nrhs, const mxArray* prhs[])
{
  mwSize n;
  mwIndex i;
  if (nrhs != 1 || ! mxIsCell (prhs[0]))
    mexErrMsgTxt ("ARG1 must be a cell");
  n = mxGetNumberOfElements (prhs[0]);
  n = (n > nlhs ? nlhs : n);
  for (i = 0; i < n; i++)
    plhs[i] = mxDuplicateArray (mxGetCell (prhs[0], i));
}

The output is identical to the oct-file version as well.

[b1, b2, b3] = mycell ({1, [1, 2], "test"})

⇒

b1 =  1
b2 =
   1   2
b3 = test

Note in the example the use of the mxDuplicateArray function. This is needed as the mxArray pointer returned by mxGetCell might be deallocated. The inverse function to mxGetCell, used for setting Cell values, is mxSetCell and is defined as

void mxSetCell (mxArray *ptr, int idx, mxArray *val);

Finally, to create a cell array or matrix, the appropriate functions are

mxArray *mxCreateCellArray (int ndims, const int *dims);
mxArray *mxCreateCellMatrix (int m, int n);

A.2.5 Structures with Mex-Files

The basic function to create a structure in a mex-file is mxCreateStructMatrix which creates a structure array with a two dimensional matrix, or mxCreateStructArray.

mxArray *mxCreateStructArray (int ndims, int *dims,
                              int num_keys,
                              const char **keys);
mxArray *mxCreateStructMatrix (int rows, int cols,
                               int num_keys,
                               const char **keys);

Accessing the fields of the structure can then be performed with mxGetField and

mxSetField or alternatively with the mxGetFieldByNumber and mxSetFieldByNumber

functions.

mxArray *mxGetField (const mxArray *ptr, mwIndex index,
                   const char *key);
mxArray *mxGetFieldByNumber (const mxArray *ptr,
                           mwIndex index, int key_num);
void mxSetField (mxArray *ptr, mwIndex index,
               const char *key, mxArray *val);
void mxSetFieldByNumber (mxArray *ptr, mwIndex index,
                       int key_num, mxArray *val);

A difference between the oct-file interface to structures and the mex-file version is that the functions to operate on structures in mex-files directly include an index over the elements of the arrays of elements per field; Whereas, the oct-file structure includes a Cell Array per field of the structure.

An example that demonstrates the use of structures in a mex-file can be found in the file ‘mystruct.c’ shown below.

#include "mex.h"
void
mexFunction (int nlhs, mxArray* plhs[],
           int nrhs, const mxArray* prhs[])
{
int i;
mwIndex j;
mxArray *v;
const char *keys[] = { "this", "that" };
if (nrhs != 1 || ! mxIsStruct (prhs[0]))
  mexErrMsgTxt ("expects struct");
for (i = 0; i < mxGetNumberOfFields (prhs[0]); i++)
  for (j = 0; j < mxGetNumberOfElements (prhs[0]); j++)
    {
      mexPrintf ("field %s(%d) = ", mxGetFieldNameByNumber (prhs[0], i), j);
      v = mxGetFieldByNumber (prhs[0], j, i);
      mexCallMATLAB (0, NULL, 1, &v, "disp");
    }
v = mxCreateStructMatrix (2, 2, 2, keys);
mxSetFieldByNumber (v, 0, 0, mxCreateString ("this1"));
mxSetFieldByNumber (v, 0, 1, mxCreateString ("that1"));
mxSetFieldByNumber (v, 1, 0, mxCreateString ("this2"));
mxSetFieldByNumber (v, 1, 1, mxCreateString ("that2"));
mxSetFieldByNumber (v, 2, 0, mxCreateString ("this3"));

  mxSetFieldByNumber (v, 2, 1, mxCreateString ("that3"));
  mxSetFieldByNumber (v, 3, 0, mxCreateString ("this4"));
  mxSetFieldByNumber (v, 3, 1, mxCreateString ("that4"));
  if (nlhs)
    plhs[0] = v;
}

An example of the behavior of this function within Octave is then

a(1).f1 = "f11"; a(1).f2 = "f12";
a(2).f1 = "f21"; a(2).f2 = "f22";
b = mystruct (a);
⇒ field f1(0) = f11
    field f1(1) = f21
    field f2(0) = f12
    field f2(1) = f22
b
⇒ 2x2 struct array containing the fields:
     this
     that
b(3)
⇒ scalar structure containing the fields:
     this = this3
     that = that3

A.2.6 Sparse Matrices with Mex-Files

The Octave format for sparse matrices is identical to the mex format in that it is a compressed column sparse format. Also in both, sparse matrices are required to be twodimensional. The only difference is that the real and imaginary parts of the matrix are stored separately.

The mex-file interface, in addition to using mxGetM, mxGetN, mxSetM, mxSetN, mxGetPr, mxGetPi, mxSetPr, and mxSetPi, also supplies the following functions.

mwIndex *mxGetIr (const mxArray *ptr);
mwIndex *mxGetJc (const mxArray *ptr);
mwSize mxGetNzmax (const mxArray *ptr);
void mxSetIr (mxArray *ptr, mwIndex *ir);
void mxSetJc (mxArray *ptr, mwIndex *jc);
void mxSetNzmax (mxArray *ptr, mwSize nzmax);

mxGetNzmax gets the maximum number of elements that can be stored in the sparse matrix. This is not necessarily the number of non-zero elements in the sparse matrix. mxGetJc returns an array with one additional value than the number of columns in the sparse matrix.

The difference between consecutive values of the array returned by mxGetJc define the number of non-zero elements in each column of the sparse matrix. Therefore,

mwSize nz, n;
mwIndex *Jc;
mxArray *m;
...
n = mxGetN (m);
Jc = mxGetJc (m);
nz = Jc[n];

returns the actual number of non-zero elements stored in the matrix in nz. As the arrays returned by mxGetPr and mxGetPi only contain the non-zero values of the matrix, we also need a pointer to the rows of the non-zero elements, and this is given by mxGetIr. A complete example of the use of sparse matrices in mex-files is given by the file ‘mysparse.c’ shown below.

#include "mex.h"
void
mexFunction (int nlhs, mxArray *plhs[],
           int nrhs, const mxArray *prhs[])
{
mwSize m, n, nz;
mxArray *v;
mwIndex i;
double *pr, *pi;
double *pr2, *pi2;
mwIndex *ir, *jc;
mwIndex *ir2, *jc2;
if (nrhs != 1 || ! mxIsSparse (prhs[0]))
  mexErrMsgTxt ("ARG1 must be a sparse matrix");
m = mxGetM (prhs[0]);
n = mxGetN (prhs[0]);
nz = mxGetNzmax (prhs[0]);
if (mxIsComplex (prhs[0]))
  {
    mexPrintf ("Matrix is %d-by-%d complex sparse matrix", m, n);
    mexPrintf (" with %d elements\n", nz);
    pr = mxGetPr (prhs[0]);
    pi = mxGetPi (prhs[0]);
    ir = mxGetIr (prhs[0]);
    jc = mxGetJc (prhs[0]);
    i = n;

while (jc[i] == jc[i-1] && i != 0) i--;
mexPrintf ("last non-zero element (%d, %d) = (%g, %g)\n",
           ir[nz-1]+ 1, i, pr[nz-1], pi[nz-1]);
v = mxCreateSparse (m, n, nz, mxCOMPLEX);
pr2 = mxGetPr (v);
pi2 = mxGetPi (v);
ir2 = mxGetIr (v);
jc2 = mxGetJc (v);
for (i = 0; i < nz; i++)
  {
    pr2[i] = 2 * pr[i];
    pi2[i] = 2 * pi[i];
    ir2[i] = ir[i];
  }
for (i = 0; i < n + 1; i++)
  jc2[i] = jc[i];
if (nlhs > 0)
  plhs[0] = v;
}
else if (mxIsLogical (prhs[0]))
{
mxLogical *pbr, *pbr2;
mexPrintf ("Matrix is %d-by-%d logical sparse matrix", m, n);
mexPrintf (" with %d elements\n", nz);
pbr = mxGetLogicals (prhs[0]);
ir = mxGetIr (prhs[0]);
jc = mxGetJc (prhs[0]);
i = n;
while (jc[i] == jc[i-1] && i != 0) i--;
mexPrintf ("last non-zero element (%d, %d) = %d\n",
           ir[nz-1]+ 1, i, pbr[nz-1]);
v = mxCreateSparseLogicalMatrix (m, n, nz);
pbr2 = mxGetLogicals (v);
ir2 = mxGetIr (v);
jc2 = mxGetJc (v);
for (i = 0; i < nz; i++)
  {
    pbr2[i] = pbr[i];
    ir2[i] = ir[i];

  }
for (i = 0; i < n + 1; i++)
  jc2[i] = jc[i];
if (nlhs > 0)
  plhs[0] = v;
}
else
{
mexPrintf ("Matrix is %d-by-%d real sparse matrix", m, n);
mexPrintf (" with %d elements\n", nz);
pr = mxGetPr (prhs[0]);
ir = mxGetIr (prhs[0]);
jc = mxGetJc (prhs[0]);
i = n;
while (jc[i] == jc[i-1] && i != 0) i--;
mexPrintf ("last non-zero element (%d, %d) = %g\n",
           ir[nz-1]+ 1, i, pr[nz-1]);
v = mxCreateSparse (m, n, nz, mxREAL);
pr2 = mxGetPr (v);
ir2 = mxGetIr (v);
jc2 = mxGetJc (v);
for (i = 0; i < nz; i++)
  {
    pr2[i] = 2 * pr[i];
    ir2[i] = ir[i];
  }
for (i = 0; i < n + 1; i++)
  jc2[i] = jc[i];
if (nlhs > 0)
  plhs[0] = v;
}
}

A sample usage of mysparse is

sm = sparse ([1, 0; 0, pi]);
mysparse (sm)

⇒

Matrix is 2-by-2 real sparse matrix with 2 elements
last non-zero element (2, 2) = 3.14159

A.2.7 Calling Other Functions in Mex-Files

It is possible to call other Octave functions from within a mex-file using mexCallMATLAB. An example of the use of mexCallMATLAB can be see in the example below.

#include "mex.h"
void
mexFunction (int nlhs, mxArray* plhs[],
           int nrhs, const mxArray* prhs[])
{
char *str;
mexPrintf ("Starting file myfeval.mex\n");
mexPrintf ("I have %d inputs and %d outputs\n", nrhs, nlhs);
if (nrhs < 1 || ! mxIsString (prhs[0]))
  mexErrMsgTxt ("ARG1 must be a function name");
str = mxArrayToString (prhs[0]);
mexPrintf ("I'm going to call the function %s\n", str);
if (nlhs == 0)
  nlhs = 1;  // Octave's automatic 'ans' variable
/* Cast prhs just to get rid of 'const' qualifier and stop compile warning */
mexCallMATLAB (nlhs, plhs, nrhs-1, (mxArray**)prhs+1, str);
mxFree (str);
}

If this code is in the file ‘myfeval.c’, and is compiled to ‘myfeval.mex’, then an example of its use is

a = myfeval ("sin", 1)
⇒ Starting file myfeval.mex
 I have 2 inputs and 1 outputs
 I'm going to call the interpreter function sin
 a =  0.84147

Note that it is not possible to use function handles or inline functions within a mex-file.

A.3 Standalone Programs

The libraries Octave itself uses can be utilized in standalone applications. These applications then have access, for example, to the array and matrix classes, as well as to all of the Octave algorithms. The following C++ program, uses class Matrix from ‘liboctave.a’ or

‘liboctave.so’.

#include <iostream>
#include <octave/oct.h>
int
main (void)
{
  std::cout << "Hello Octave world!\n";
  int n = 2;
  Matrix a_matrix = Matrix (n, n);
  for (octave_idx_type i = 0; i < n; i++)
    for (octave_idx_type j = 0; j < n; j++)
      a_matrix(i,j) = (i + 1) * 10 + (j + 1);
  std::cout << a_matrix;
  return 0;
}

mkoctfile can be used to build a standalone application with a command like

$ mkoctfile --link-stand-alone standalone.cc -o standalone
$ ./standalone
Hello Octave world!
  11 12
  21 22
$

Note that the application standalone will be dynamically linked against the Octave libraries and any Octave support libraries. The above allows the Octave math libraries to be used by an application. It does not, however, allow the script files, oct-files, or built-in functions of Octave to be used by the application. To do that the Octave interpreter needs to be initialized first. An example of how to do this can then be seen in the code

#include <iostream>
#include <octave/oct.h>
#include <octave/octave.h>
#include <octave/parse.h>
#include <octave/toplev.h>
int
main (void)
{
  string_vector argv (2);
  argv(0) = "embedded";
  argv(1) = "-q";
  octave_main (2, argv.c_str_vec (), 1);

  octave_idx_type n = 2;
  octave_value_list in;
  for (octave_idx_type i = 0; i < n; i++)
    in(i) = octave_value (5 * (i + 2));
  octave_value_list out = feval ("gcd", in, 1);
  if (! error_state && out.length () > 0)
    std::cout << "GCD of ["
              << in(0).int_value ()
              << ", "
              << in(1).int_value ()
              << "] is " << out(0).int_value ()
              << std::endl;
  else
    std::cout << "invalid\n";
  clean_up_and_exit (0);
}

which, as before, is compiled and run as a standalone application with

$ mkoctfile --link-stand-alone embedded.cc -o embedded
$ ./embedded
GCD of [10, 15] is 5
$

It is worth noting that, if only built-in functions are to be called from a C++ standalone program, then it does not need to initialize the interpreter to do so. The general rule is that, for a built-in function named function_name in the interpreter, there will be a C++ function named Ffunction_name (note the prepended capital F) accessible in the C++ API. The declarations for all built-in functions are collected in the header file builtin-defun-decls.h. This feature should be used with care as the list of built-in functions can change. No guarantees can be made that a function that is currently built in won’t be implemented as a .m file or as a dynamically linked function in the future. An example of how to call built-in functions from C++ can be seen in the code

#include <iostream>
#include <octave/oct.h>
#include <octave/builtin-defun-decls.h>
int
main (void)
{
  int n = 2;
  Matrix a_matrix = Matrix (n, n);

for (octave_idx_type i = 0; i < n; i++)
  for (octave_idx_type j = 0; j < n; j++)
    a_matrix(i,j) = (i + 1) * 10 + (j + 1);
std::cout << "This is a matrix:" << std::endl
          << a_matrix            << std::endl;
octave_value_list in;
in(0) = a_matrix;
octave_value_list out = Fnorm (in, 1);
double norm_of_the_matrix = out(0).double_value ();
std::cout << "This is the norm of the matrix:" << std::endl
          << norm_of_the_matrix                << std::endl;
return 0;
}

which, again, is compiled and run as a standalone application with

$ mkoctfile --link-stand-alone standalonebuiltin.cc -o standalonebuiltin
$ ./standalonebuiltin
This is a matrix:
11 12
21 22
This is the norm of the matrix:
34.4952
$

Appendix B Test and Demo Functions

Octave includes a number of functions to allow the integration of testing and demonstration code in the source code of the functions themselves.

B.1 Test Functions

[Command] test name

[Command] test name quiet|normal|verbose [Function File] test ("name", "quiet|normal|verbose", fid)

[Function File] test ([], "explain", fid) [Function File] success = test (…) [Function File] [n, max] = test (…) [Function File] [code, idx] = test ("name", "grabdemo")

Perform tests from the first file in the loadpath matching name. test can be called as a command or as a function. Called with a single argument name, the tests are run interactively and stop after the first error is encountered. With a second argument the tests which are performed and the amount of output is selected.

"quiet" Don’t report all the tests as they happen, just the errors.

"normal" Report all tests as they happen, but don’t do tests which require user interaction.

"verbose"

Do tests which require user interaction.

The argument fid can be used to allow batch processing. Errors can be written to the already open file defined by fid, and hopefully when Octave crashes this file will tell you what was happening when it did. You can use stdout if you want to see the results as they happen. You can also give a file name rather than an fid, in which case the contents of the file will be replaced with the log from the current test. Called with a single output argument success, test returns true if all of the tests were successful. Called with two output arguments n and max, the number of successful tests and the total number of tests in the file name are returned. If the second argument is the string "grabdemo", the contents of the demo blocks are extracted but not executed. Code for all code blocks is concatenated and returned as code with idx being a vector of positions of the ends of the demo blocks. If the second argument is "explain", then name is ignored and an explanation of the line markers used is written to the file fid.

See also: [assert], page 804, [fail], page 804, [error], page 197, [demo], page 805, [example], page 806.

test scans the named script file looking for lines which start with the identifier ‘%!’. The prefix is stripped off and the rest of the line is processed through the Octave interpreter. If the code generates an error, then the test is said to fail.

Since eval() will stop at the first error it encounters, you must divide your tests up into blocks, with anything in a separate block evaluated separately. Blocks are introduced by valid keywords like test, function, or assert immediately following ‘%!’. A block is defined by indentation as in Python. Lines beginning with ‘%!<whitespace>’ are part of the preceeding block.

For example:

%!test error ("this test fails!");
%!test "test doesn't fail. it doesn't generate an error";

When a test fails, you will see something like:

  ***** test error ("this test fails!")
!!!!! test failed
this test fails!

Generally, to test if something works, you want to assert that it produces a correct value. A real test might look something like

%!test
%! a = [1, 2, 3; 4, 5, 6]; B = [1; 2];
%! expect = [ a ; 2*a ];
%! get = kron (b, a);
%! if (any (size (expect) != size (get)))
%!   error ("wrong size: expected %d,%d but got %d,%d",
%!          size (expect), size (get));
%! elseif (any (any (expect != get)))
%!   error ("didn't get what was expected.");
%! endif

To make the process easier, use the assert function. For example, with assert the previous test is reduced to:

%!test
%! a = [1, 2, 3; 4, 5, 6]; b = [1; 2];
%! assert (kron (b, a), [ a; 2*a ]);

assert can accept a tolerance so that you can compare results absolutely or relatively. For example, the following all succeed:

%!test assert (1+eps, 1, 2*eps)          # absolute error
%!test assert (100+100*eps, 100, -2*eps) # relative error

You can also do the comparison yourself, but still have assert generate the error:

%!test assert (isempty ([]))
%!test assert ([1, 2; 3, 4] > 0)

Because assert is so frequently used alone in a test block, there is a shorthand form:

%!assert (...)

which is equivalent to:

%!test assert (...)

Occasionally a block of tests will depend on having optional functionality in Octave. Before testing such blocks the availability of the required functionality must be checked. A %!testif HAVE_XXX block will only be run if Octave was compiled with functionality ‘HAVE_XXX’. For example, the sparse single value decomposition, svds(), depends on having the arpack library. All of the tests for svds begin with

     %!testif HAVE_ARPACK
Review ‘config.h’ or octave_config_info ("features") to see some of the possible val-

ues to check.

Sometimes during development there is a test that should work but is known to fail. You still want to leave the test in because when the final code is ready the test should pass, but you may not be able to fix it immediately. To avoid unnecessary bug reports for these known failures, mark the block with xtest rather than test:

     %!xtest assert (1==0)
     %!xtest fail ("success=1", "error")

In this case, the test will run and any failure will be reported. However, testing is not aborted and subsequent test blocks will be processed normally. Another use of xtest is for statistical tests which should pass most of the time but are known to fail occasionally.

Each block is evaluated in its own function environment, which means that variables defined in one block are not automatically shared with other blocks. If you do want to share variables, then you must declare them as shared before you use them. For example, the following declares the variable a, gives it an initial value (default is empty), and then uses it in several subsequent tests.

     %!shared a
     %! a = [1, 2, 3; 4, 5, 6];
     %!assert (kron ([1; 2], a), [ a; 2*a ]);
     %!assert (kron ([1, 2], a), [ a, 2*a ]);
     %!assert (kron ([1,2; 3,4], a), [ a,2*a; 3*a,4*a ]);

You can share several variables at the same time:

     %!shared a, b

You can also share test functions:

     %!function a = fn (b)
     %!  a = 2*b;
     %!endfunction
     %!assert (fn(2), 4);

Note that all previous variables and values are lost when a new shared block is declared.

Remember that %!function begins a new block and that %!endfunction ends this block. Be aware that until a new block is started, lines starting with ‘%!<space>’ will be discarded as comments. The following is nearly identical to the example above, but does nothing.

     %!function a = fn (b)
     %!  a = 2*b;
     %!endfunction
     %! assert (fn(2), 4);

Because there is a space after ‘%!’ the assert statement does not begin a new block and this line is treated as a comment.

Error and warning blocks are like test blocks, but they only succeed if the code generates an error. You can check the text of the error is correct using an optional regular expression <pattern>. For example:

%!error <passes!> error ("this test passes!");

If the code doesn’t generate an error, the test fails. For example:

%!error "this is an error because it succeeds.";

produces

  ***** error "this is an error because it succeeds.";
!!!!! test failed: no error

It is important to automate the tests as much as possible, however some tests require user interaction. These can be isolated into demo blocks, which if you are in batch mode, are only run when called with demo or the verbose option to test. The code is displayed before it is executed. For example,

%!demo
%! t = [0:0.01:2*pi]; x = sin (t);
%! plot (t, x);
%! # you should now see a sine wave in your figure window

produces

funcname example 1:
 t = [0:0.01:2*pi]; x = sin (t);
 plot (t, x);
 # you should now see a sine wave in your figure window
Press <enter> to continue:

Note that demo blocks cannot use any shared variables. This is so that they can be executed by themselves, ignoring all other tests.

If you want to temporarily disable a test block, put # in place of the block type. This creates a comment block which is echoed in the log file but not executed. For example:

%!#demo
%! t = [0:0.01:2*pi]; x = sin (t);
%! plot (t, x);
%! # you should now see a sine wave in your figure window

The following trivial code snippet provides examples for the use of fail, assert, error and xtest:

     function output = must_be_zero (input)
       if (input != 0)
         error ("Non-zero input!")
       endif
       output = input;
     endfunction
     %!fail ("must_be_zero (1)");
     %!assert (must_be_zero (0), 0);
     %!error <Non-zero> must_be_zero (1);
     %!xtest error ("This code generates an error");

When putting this a file ‘must_be_zero.m’, and running the test, we see

     test must_be_zero verbose

⇒

     >>>>> /path/to/must_be_zero.m
       ***** fail ("must_be_zero (1)");
       ***** assert (must_be_zero (0), 0);
       ***** error <Non-zero> must_be_zero (1);
       ***** xtest error ("This code generates an error");
     !!!!! known failure
     This code generates an error
     PASSES 4 out of 4 tests (1 expected failures)

Block type summary:

%!test check that entire block is correct

%!testif HAVE_XXX

check block only if Octave was compiled with feature HAVE XXX.

%!xtest check block, report a test failure but do not abort testing.

%!error check for correct error message

%!warning

check for correct warning message

%!demo demo only executes in interactive mode

%!# comment: ignore everything within the block

%!shared x,y,z

declare variables for use in multiple tests

%!function

define a function for use in multiple tests

%!endfunction

close a function definition

%!assert (x, y, tol)
          shorthand for %!test assert (x, y, tol)

You can also create test scripts for built-in functions and your own C++ functions. To do so, put a file with the bare function name (no .m extension) in a directory in the load path and it will be discovered by the test function. Alternatively, you can embed tests directly in your C++ code:

/*
%!test disp ("this is a test")
*/

or

#if 0
%!test disp ("this is a test")
#endif

However, in this case the raw source code will need to be on the load path and the user will have to remember to type test ("funcname.cc").

[Function File] assert (cond) [Function File] assert (cond, errmsg, …)

[Function File] assert (cond, msg_id, errmsg, . . .)
[Function File] assert (observed, expected)
[Function File] assert (observed, expected, tol)

Produce an error if the specified condition is not met. assert can be called in three different ways.

assert (cond)
assert (cond, errmsg, ...)
assert (cond, msg_id, errmsg, ...)

Called with a single argument cond, assert produces an error if cond is zero. When called with more than one argument the additional arguments are passed to the error function.

assert (observed, expected)

Produce an error if observed is not the same as expected. Note that observed and expected can be scalars, vectors, matrices, strings, cell arrays, or structures.

assert (observed, expected, tol)

Produce an error if observed is not the same as expected but equality comparison for numeric data uses a tolerance tol. If tol is positive then it is an absolute tolerance which will produce an error if abs (observed - expected) > abs (tol). If tol is negative then it is a relative tolerance

which will produce an error if abs (observed - expected) > abs (tol

* expected). If expected is zero tol will always be interpreted as an absolute tolerance. If tol is not scalar its dimensions must agree with those of observed and expected and tests are performed on an elementwise basis.

See also: [test], page 799, [fail], page 804, [error], page 197.

[Function File] fail (code) [Function File] fail (code, pattern)

[Function File] fail (code, "warning", pattern)

Return true if code fails with an error message matching pattern, otherwise produce an error. Note that code is a string and if code runs successfully, the error produced is:

               expected error <.> but got none

Code must be in the form of a string that may be passed by fail to the Octave interpreter via the evalin function, that is, a (quoted) string constant or a string variable. If called with two arguments, the behavior is similar to fail (code), except the return value will only be true if code fails with an error message containing pattern (case sensitive). If the code fails with a different error to that given in pattern, the message produced is:

               expected <pattern>
               but got <text of actual error>

The angle brackets are not part of the output. Called with three arguments, the behavior is similar to fail (code, pattern), but produces an error if no warning is given during code execution or if the code fails.

See also: [assert], page 804.

B.2 Demonstration Functions

[Command] demo name

[Command] demo name n

[Function File] demo ("name") [Function File] demo ("name", n)

Run example code block n associated with the function name. If n is not specified, all examples are run. Examples are stored in the script file, or in a file with the same name but no extension located on Octave’s load path. To keep examples separate from regular script code, all lines are prefixed by %!. Each example must also be introduced by the keyword "demo" flush left to the prefix with no intervening spaces. The remainder of the example can contain arbitrary Octave code. For example:

     %!demo
     %! t = 0:0.01:2*pi;
     %! x = sin (t);
     %! plot (t, x);
     %! %-------------------------------------------------
     %! % the figure window shows one cycle of a sine wave

Note that the code is displayed before it is executed, so a simple comment at the end suffices for labeling what is being shown. It is generally not necessary to use disp or printf within the demo. Demos are run in a function environment with no access to external variables. This means that every demo must have separate initialization code. Alternatively, all demos can be combined into a single large demo with the code

     %! input("Press <enter> to continue: ","s");

between the sections, but this is discouraged. Other techniques to avoid multiple initialization blocks include using multiple plots with a new figure command between each plot, or using subplot to put multiple plots in the same window. Also, because demo evaluates within a function context, you cannot define new functions inside a demo. If you must have function blocks, rather than just anonymous functions or inline functions, you will have to use eval (example ("function",n)) to see them. Because eval only evaluates one line, or one statement if the statement crosses multiple lines, you must wrap your demo in "if 1 <demo stuff> endif" with the "if" on the same line as "demo". For example:

     %!demo if 1
     %!  function y=f(x)
     %!    y=x;
     %!  endfunction
     %!  f(3)
     %! endif

See also: [test], page 799, [example], page 806.

[Command] example name
[Command] example name n

[Function File] example ("name") [Function File] example ("name", n) [Function File] [s, idx] = example (…)

Display the code for example n associated with the function name, but do not run it. If n is not specified, all examples are displayed. When called with output arguments, the examples are returned in the form of a string s, with idx indicating the ending position of the various examples. See demo for a complete explanation.

See also: [demo], page 805, [test], page 799.

[Function File] rundemos ()

[Function File] rundemos (directory)

Execute built-in demos for all function files in the specified directory. Also executes demos in any C++ source files found in the directory, for use with dynamically linked functions. If no directory is specified, operate on all directories in Octave’s search path for functions.

See also: [runtests], page 806, [path], page 181.

[Function File] runtests ()

[Function File] runtests (directory)

Execute built-in tests for all function files in the specified directory. Also executes tests in any C++ source files found in the directory, for use with dynamically linked functions. If no directory is specified, operate on all directories in Octave’s search path for functions.

See also: [rundemos], page 806, [path], page 181.

[Function File] speed (f, init, max_n, f2, tol)
[Function File] [order, n, T_f, T_f2] = speed (. . .)

Determine the execution time of an expression (f ) for various input values (n). The n are log-spaced from 1 to max n. For each n, an initialization expression (init) is computed to create any data needed for the test. If a second expression (f2) is given then the execution times of the two expressions are compared. When called without output arguments the results are printed to stdout and displayed graphically.

f The code expression to evaluate.

max_n The maximum test length to run. The default value is 100. Alternatively, use [min_n, max_n] or specify the n exactly with [n1, n2, ..., nk].

init Initialization expression for function argument values. Use k for the test number and n for the size of the test. This should compute values for all variables used by f. Note that init will be evaluated first for k = 0, so things which are constant throughout the test series can be computed once. The default value is x = randn (n, 1).

f2 An alternative expression to evaluate, so that the speed of two expressions can be directly compared. The default is [].

tol Tolerance used to compare the results of expression f and expression f2. If tol is positive, the tolerance is an absolute one. If tol is negative, the tolerance is a relative one. The default is eps. If tol is Inf, then no comparison will be made.

order The time complexity of the expression O(a ∗ n^{p}). This is a structure with fields a and p.

n The values n for which the expression was calculated AND the execution time was greater than zero.

T_f The nonzero execution times recorded for the expression f in seconds.

T_f2 The nonzero execution times recorded for the expression f2 in seconds. If required, the mean time ratio is simply mean (T_f ./ T_f2).

The slope of the execution time graph shows the approximate power of the asymptotic running time O(n^{p}). This power is plotted for the region over which it is approximated (the latter half of the graph). The estimated power is not very accurate, but should be sufficient to determine the general order of an algorithm. It should indicate if, for example, the implementation is unexpectedly O(n^{2}) rather than O(n) because it extends a vector each time through the loop rather than pre-allocating storage. In the current version of Octave, the following is not the expected O(n).

     speed ("for i = 1:n, y{i} = x(i); endfor", "", [1000, 10000])

But it is if you preallocate the cell array y:

     speed ("for i = 1:n, y{i} = x(i); endfor", ...
            "x = rand (n, 1); y = cell (size (x));", [1000, 10000])

An attempt is made to approximate the cost of individual operations, but it is wildly inaccurate. You can improve the stability somewhat by doing more work for each n. For example:

     speed ("airy(x)", "x = rand (n, 10)", [10000, 100000])

When comparing two different expressions (f, f2), the slope of the line on the speedup ratio graph should be larger than 1 if the new expression is faster. Better algorithms have a shallow slope. Generally, vectorizing an algorithm will not change the slope of the execution time graph, but will shift it relative to the original. For example:

     speed ("sum (x)", "", [10000, 100000], ...
            "v = 0; for i = 1:length (x), v += x(i); endfor")

The following is a more complex example. If there was an original version of xcorr using for loops and a second version using an FFT, then one could compare the run speed for various lags as follows, or for a fixed lag with varying vector lengths as follows:

     speed ("xcorr (x, n)", "x = rand (128, 1);", 100,
            "xcorr_orig (x, n)", -100*eps)
     speed ("xcorr (x, 15)", "x = rand (20+n, 1);", 100,
            "xcorr_orig (x, n)", -100*eps)

Assuming one of the two versions is in xcorr orig, this would compare their speed and their output values. Note that the FFT version is not exact, so one must specify an acceptable tolerance on the comparison 100*eps. In this case, the comparison should be computed relatively, as abs ((x - y) ./ y) rather than absolutely as abs (x -

y).

Type example ("speed") to see some real examples or demo ("speed") to run them.

Appendix C Tips and Standards

This chapter describes no additional features of Octave. Instead it gives advice on making effective use of the features described in the previous chapters.

C.1 Writing Clean Octave Programs

Here are some tips for avoiding common errors in writing Octave code intended for widespread use:

• Since all global variables share the same name space, and all functions share another name space, you should choose a short word to distinguish your program from other Octave programs. Then take care to begin the names of all global variables, constants, and functions with the chosen prefix. This helps avoid name conflicts. If you write a function that you think ought to be added to Octave under a certain name, such as fiddle_matrix, don’t call it by that name in your program. Call it

mylib_fiddle_matrix in your program, and send mail to maintainers@octave.org

suggesting that it be added to Octave. If and when it is, the name can be changed easily enough. If one prefix is insufficient, your package may use two or three alternative common prefixes, so long as they make sense. Separate the prefix from the rest of the symbol name with an underscore ‘_’. This will be consistent with Octave itself and with most Octave programs.

• When you encounter an error condition, call the function error (or usage). The error and usage functions do not return. See Section 2.5 [Errors], page 34.

• Please put a copyright notice on the file if you give copies to anyone. Use the same lines that appear at the top of the function files distributed with Octave. If you have not signed papers to assign the copyright to anyone else, then place your name in the copyright notice.

C.2 Tips on Writing Comments

Here are the conventions to follow when writing comments.

‘#’ Comments that start with a single sharp-sign, ‘#’, should all be aligned to the same column on the right of the source code. Such comments usually explain how the code on the same line does its job. In the Emacs mode for Octave, the M-; (indent-for-comment) command automatically inserts such a ‘#’ in the right place, or aligns such a comment if it is already present.

‘##’ Comments that start with a double sharp-sign, ‘##’, should be aligned to the same level of indentation as the code. Such comments usually describe the purpose of the following lines or the state of the program at that point.

The indentation commands of the Octave mode in Emacs, such as M-; (indent-for-comment) and TAB (octave-indent-line) automatically indent comments according to these conventions, depending on the number of semicolons. See Section “Manipulating Comments” in The GNU Emacs Manual.

C.3 Conventional Headers for Octave Functions

Octave has conventions for using special comments in function files to give information such as who wrote them. This section explains these conventions.

The top of the file should contain a copyright notice, followed by a block of comments that can be used as the help text for the function. Here is an example:

## Copyright (C) 1996, 1997, 2007 John W. Eaton
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or
## modify it under the terms of the GNU General Public
## License as published by the Free Software Foundation;
## either version 3 of the License, or (at your option) any
## later version.
##
## Octave is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied
## warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
## PURPOSE.  See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public
## License along with Octave; see the file COPYING.  If not,
## see <http://www.gnu.org/licenses/>.
## usage: [IN, OUT, PID] = popen2 (COMMAND, ARGS)
##
## Start a subprocess with two-way communication.  COMMAND
## specifies the name of the command to start.  ARGS is an
## array of strings containing options for COMMAND.  IN and
## OUT are the file ids of the input and streams for the
## subprocess, and PID is the process id of the subprocess,
## or -1 if COMMAND could not be executed.
##
## Example:
##
##  [in, out, pid] = popen2 ("sort", "-nr");
##  fputs (in, "these\nare\nsome\nstrings\n");
##  fclose (in);
##  while (ischar (s = fgets (out)))
##    fputs (stdout, s);
##  endwhile
##  fclose (out);

Octave uses the first block of comments in a function file that do not appear to be a copyright notice as the help text for the file. For Octave to recognize the first comment block as a copyright notice, it must start with the word ‘Copyright’ after stripping the leading comment characters.

After the copyright notice and help text come several header comment lines, each beginning with ‘## header-name:’. For example,

## Author: jwe
## Keywords: subprocesses input-output
## Maintainer: jwe

Here is a table of the conventional possibilities for header-name:

‘Author’ This line states the name and net address of at least the principal author of the

library.

     ## Author: John W. Eaton <jwe@octave.org>
‘Maintainer’

This line should contain a single name/address as in the Author line, or an address only, or the string ‘jwe’. If there is no maintainer line, the person(s) in the Author field are presumed to be the maintainers. The example above is mildly bogus because the maintainer line is redundant. The idea behind the ‘Author’ and ‘Maintainer’ lines is to make possible a function to “send mail to the maintainer” without having to mine the name out by hand. Be sure to surround the network address with ‘<...>’ if you include the person’s full name as well as the network address.

‘Created’ This optional line gives the original creation date of the file. For historical interest only.

‘Version’ If you wish to record version numbers for the individual Octave program, put them in this line.

‘Adapted-By’

In this header line, place the name of the person who adapted the library for installation (to make it fit the style conventions, for example).

‘Keywords’

This line lists keywords. Eventually, it will be used by an apropos command to allow people will find your package when they’re looking for things by topic area. To separate the keywords, you can use spaces, commas, or both.

Just about every Octave function ought to have the ‘Author’ and ‘Keywords’ header comment lines. Use the others if they are appropriate. You can also put in header lines with other header names—they have no standard meanings, so they can’t do any harm.

C.4 Tips for Documentation Strings

As noted above, documentation is typically in a commented header block on an Octave function following the copyright statement. The help string shown above is an unformatted string and will be displayed as is by Octave. Here are some tips for the writing of documentation strings.

• Every command, function, or variable intended for users to know about should have a documentation string.

• An internal variable or subroutine of an Octave program might as well have a documentation string.

• The first line of the documentation string should consist of one or two complete sentences that stand on their own as a summary. The documentation string can have additional lines that expand on the details of how to use the function or variable. The additional lines should also be made up of complete sentences.

• For consistency, phrase the verb in the first sentence of a documentation string as an infinitive with “to” omitted. For instance, use “Return the frob of A and B.” in preference to “Returns the frob of A and B.” Usually it looks good to do likewise for the rest of the first paragraph. Subsequent paragraphs usually look better if they have proper subjects.

• Write documentation strings in the active voice, not the passive, and in the present tense, not the future. For instance, use “Return a list containing A and B.” instead of “A list containing A and B will be returned.”

• Avoid using the word “cause” (or its equivalents) unnecessarily. Instead of, “Cause Octave to display text in boldface,” just write “Display text in boldface.”

• Use two spaces between the period marking the end of a sentence and the word which opens the next sentence. This convention has no effect for typeset formats like TEX, but improves the readability of the documentation in fixed-width environments such as the Info reader.

• Do not start or end a documentation string with whitespace.

• Format the documentation string so that it fits within an 80-column screen. It is a good idea for most lines to be no wider than 60 characters. However, rather than simply filling the entire documentation string, you can make it much more readable by choosing line breaks with care. Use blank lines between topics if the documentation string is long.

• Do not indent subsequent lines of a documentation string so that the text is lined up in the source code with the text of the first line. This looks nice in the source code, but looks bizarre when users view the documentation. Remember that the indentation before the starting double-quote is not part of the string!

• When choosing variable names try to adhere to the following guidelines.

vectors : x,y,z,t,w

matrices : A,B,M

strings : str,s

filenames : fname

cells,cellstrs :

c,cstr

• The documentation string for a variable that is a yes-or-no flag should start with words such as “Nonzero means…”, to make it clear that all nonzero values are equivalent and indicate explicitly what zero and nonzero mean.

• When a function’s documentation string mentions the value of an argument of the function, use the argument name in capital letters as if it were a name for that value. Thus, the documentation string of the operator / refers to its second argument as ‘DIVISOR’, because the actual argument name is divisor. Also use all caps for meta-syntactic variables, such as when you show the decomposition of a list or vector into subunits, some of which may vary.

Octave also allows extensive formatting of the help string of functions using Texinfo. The effect on the online documentation is relatively small, but makes the help string of functions conform to the help of Octave’s own functions. However, the effect on the appearance of printed or online documentation will be greatly improved.

The fundamental building block of Texinfo documentation strings is the Texinfo-macro @deftypefn, which takes three arguments: The class the function is in, its output arguments, and the function’s signature. Typical classes for functions include Function File for standard Octave functions, and Loadable Function for dynamically linked functions. A skeletal Texinfo documentation string therefore looks like this

     -*- texinfo -*-
     @deftypefn {Function File} {@var{ret} =} fn (...)
     @cindex index term
     Help text in Texinfo format.  Code samples should be marked
     like @code{sample of code} and variables should be marked
     as @var{variable}.
     @seealso{fn2, fn3}
     @end deftypefn

This help string must be commented in user functions, or in the help string of the DEFUN_DLD macro for dynamically loadable functions. The important aspects of the documentation string are

-*- texinfo -*- This string signals Octave that the following text is in Texinfo format, and should be the first part of any help string in Texinfo format.

@deftypefn {class} … @end deftypefn The entire help string should be enclosed within the block defined by deftypefn.

@cindex index term This generates an index entry, and can be useful when the function is included as part of a larger piece of documentation. It is ignored within Octave’s help viewer. Only one index term may appear per line but multiple @cindex lines are valid if the function should be filed under different terms.

@var{variable} All variables should be marked with this macro. The markup of variables is then changed appropriately for display.

@code{sample of code} All samples of code should be marked with this macro for the same reasons as the @var macro.

@qcode{"sample code"} @qcode{’sample code’}

All samples of code which are quoted should use this more specialized macro. This happens frequently when discussing graphics properties such as "position" or options such as "on"/"off".

@seealso{function2, function3} This is a comma separated list of function names that allows cross referencing from one function documentation string to another.

Texinfo format has been designed to generate output for online viewing with text terminals as well as generating high-quality printed output. To these ends, Texinfo has commands which control the diversion of parts of the document into a particular output processor. Three formats are of importance: info, HTML, and TEX. These are selected with

@ifinfo
Text area for info only
@end ifinfo
@ifhtml
Text area for HTML only
@end ifhtml
@tex
Text area for TeX only
@end tex

Note that often TEX output can be used in HTML documents and so often the @ifhtml blocks are unnecessary. If no specific output processor is chosen, by default, the text goes into all output processors. It is usual to have the above blocks in pairs to allow the same information to be conveyed in all output formats, but with a different markup. Currently, most Octave documentation only makes a distinction between TEX and all other formats. Therefore, the following construct is seen repeatedly.

@tex
text for TeX only
@end tex
@ifnottex
text for info, HTML, plaintext
@end ifnottex

Another important feature of Texinfo that is often used in Octave help strings is the @example environment. An example of its use is

@example
@group
@code{2 * 2}
@result{} 4
@end group
@end example

which produces

2 * 2 ⇒ 4

The @group block prevents the example from being split across a page boundary, while the @result{} macro produces a right arrow signifying the result of a command. If your example is larger than 20 lines it is better NOT to use grouping so that a reasonable page boundary can be calculated.

In many cases a function has multiple ways in which it can be called, and the @deftypefnx macro can be used to give alternatives. For example

-*- texinfo -*-
@deftypefn  {Function File} {@var{a} =} fn (@var{x}, ...)
@deftypefnx {Function File} {@var{a} =} fn (@var{y}, ...)
Help text in Texinfo format.
@end deftypefn

Many complete examples of Texinfo documentation can be taken from the help strings for the Octave functions themselves. A relatively complete example of which is the nchoosek function. The Texinfo documentation string for nchoosek is

-*- texinfo -*-
@deftypefn  {Function File} {@var{c} =} nchoosek (@var{n}, @var{k})
@deftypefnx {Function File} {@var{c} =} nchoosek (@var{set}, @var{k})
Compute the binomial coefficient or all combinations of a set of items.
If @var{n} is a scalar then calculate the binomial coefficient
of @var{n} and @var{k} which is defined as
@tex
$$
 {n \choose k} = {n (n-1) (n-2) \cdots (n-k+1) \over k!}
               = {n! \over k! (n-k)!}
$$
@end tex
@ifnottex
@example
@group
 /   \
 | n |    n (n-1) (n-2) @dots{} (n-k+1)       n!
 |   |  = ------------------------- =  ---------
 | k |               k!                k! (n-k)!
 \   /
@end group
@end example
@end ifnottex
@noindent
This is the number of combinations of @var{n} items taken in groups of
size @var{k}.
If the first argument is a vector, @var{set}, then generate all

combinations of the elements of @var{set}, taken @var{k} at a time, with
one row per combination.  The result @var{c} has @var{k} columns and
@w{@code{nchoosek (length (@var{set}), @var{k})}} rows.
For example:
How many ways can three items be grouped into pairs?
@example
@group
nchoosek (3, 2)
   @result{} 3
@end group
@end example
What are the possible pairs?
@example
@group
nchoosek (1:3, 2)
   @result{}  1   2
       1   3
       2   3
@end group
@end example
@code{nchoosek} works only for non-negative, integer arguments.  Use
@code{bincoeff} for non-integer and negative scalar arguments, or for
computing many binomial coefficients at once with vector inputs
for @var{n} or @var{k}.
@seealso{bincoeff, perms}
@end deftypefn

which demonstrates most of the concepts discussed above. This documentation string renders as

 -- Function File: C = nchoosek (N, K)
 -- Function File: C = nchoosek (SET, K)
     Compute the binomial coefficient or all combinations of a set of
     items.
     If N is a scalar then calculate the binomial coefficient of N and
     K which is defined as
           /   \
           | n |    n (n-1) (n-2) ... (n-k+1)       n!
           |   |  = ------------------------- =  ---------

      | k |               k!                k! (n-k)!
      \   /
This is the number of combinations of N items taken in groups of
size K.
If the first argument is a vector, SET, then generate all
combinations of the elements of SET, taken K at a time, with one
row per combination.  The result C has K columns and
`nchoosek (length (SET), K)' rows.
For example:
How many ways can three items be grouped into pairs?
     nchoosek (3, 2)
        => 3
What are the possible pairs?
     nchoosek (1:3, 2)
        =>  1   2
            1   3
            2   3
`nchoosek' works only for non-negative, integer arguments.  Use
`bincoeff' for non-integer and negative scalar arguments, or for
computing many binomial coefficients at once with vector inputs
for N or K.
See also: bincoeff, perms

using info, whereas in a printed documentation using TEX it will appear as

[Function File] c = nchoosek (n, k) [Function File] c = nchoosek (set, k)

Compute the binomial coefficient or all combinations of a set of items. If n is a scalar then calculate the binomial coefficient of n and k which is defined as

\binom{n}{k} = \frac{n(n − 1)(n − 2) ⋯ (n − k + 1)}{k!} = \frac{n!}{k!(n − k)!}

This is the number of combinations of n items taken in groups of size k. If the first argument is a vector, set, then generate all combinations of the elements of set, taken k at a time, with one row per combination. The result c has k columns

and nchoosek (length (set), k) rows.

For example: How many ways can three items be grouped into pairs?

     nchoosek (3, 2)
        ⇒ 3

What are the possible pairs?

     nchoosek (1:3, 2)
        ⇒ 1   2
            1   3
            2   3

nchoosek works only for non-negative, integer arguments. Use bincoeff for noninteger and negative scalar arguments, or for computing many binomial coefficients at once with vector inputs for n or k.

See also: bincoeff, perms.

Appendix D Contributing Guidelines

This chapter is dedicated to those who wish to contribute code to Octave.

D.1 How to Contribute

The mailing list for Octave development discussions is maintainers@octave.org. Patches

should be submitted to Octave’s patch tracker (https://savannah.gnu.org/patch/?func=additem&group=octave).

This concerns the development of Octave core, i.e., code that goes in to Octave directly. You may consider developing and publishing a package instead; a great place for this is the allied Octave-Forge project (http://octave.sourceforge.net). Note that the Octave core project is inherently more conservative and follows narrower rules.

D.2 Building the Development Sources

The directions for building from the development sources change from time to time, so you should read the resources for developers on the web or in the development sources archive.

Start here: http://www.octave.org/get-involved.html.

D.3 Basics of Generating a Changeset

The best way to contribute is to create a Mercurial changeset and submit

it to the bug (http://savannah.gnu.org/bugs/?group=octave) or patch
(http://savannah.gnu.org/patch/?func=additem&group=octave)  trackers^{1}.

Mercurial is the source code management system currently used to develop Octave. Other forms of contributions (e.g., simple diff patches) are also acceptable, but they slow down the review process. If you want to make more contributions, you should really get familiar with Mercurial. A good place to start is

http://www.selenic.com/mercurial/wiki/index.cgi/Tutorial. There you will also

find help about how to install Mercurial.

A simple contribution sequence could look like this:

     hg clone http://www.octave.org/hg/octave
                                  # make a local copy of the octave
                                  # source repository
     cd octave
     # change some sources...
     hg commit -m "make Octave the coolest software ever"
                                  # commit the changeset into your
                                  # local repository
     hg export -o ../cool.diff tip
                                  # export the changeset to a diff
                                  # file
     # attach ../cool.diff to your bug report

You may want to get familiar with Mercurial queues to manage your changesets. To work with queues you must activate the extension mq with the following entry in Mercurial’s configuration file ‘.hgrc’ (or ‘Mercurial.ini’ on Windows):

1 Please use the patch tracker only for patches which add new features. If you have a patch to submit

that fixes a bug, you should use the bug tracker instead.

[extensions]
mq=

Here is a slightly more complex example using Mercurial queues, where work on two unrelated changesets is done in parallel and one of the changesets is updated after discussion on the bug tracker:

hg qnew nasty_bug            # create a new patch
# change sources...
hg qref                      # save the changes into the patch
# change even more...
hg qref -m "solution to nasty bug!"
                             # save again with commit message
hg export -o ../nasty.diff tip
                             # export the patch
# attach ../nasty.diff to your bug report
hg qpop                      # undo the application of the patch
                             # and remove the changes from the
                             # source tree
hg qnew doc_improvements     # create an unrelated patch
# change doc sources...
hg qref -m "could not find myfav.m in the doc"
                             # save the changes into the patch
hg export -o ../doc.diff tip
                             # export the second patch
# attach ../doc.diff to your bug report
hg qpop
# discussion in the bug tracker ...
hg qpush nasty_bug           # apply the patch again
# change sources yet again ...
hg qref
hg export -o ../nasty2.diff tip
# attach ../nasty2.diff to your bug report

Mercurial has a few more useful extensions that really should be enabled. They are not enabled by default due to a number of factors (mostly because they don’t work in all terminal types).

The following entries in the ‘.hgrc’ are recommended

[extensions]
graphlog=
color=
progress=
pager=

For the color extension, default color and formatting of hg status can be modified by

[color]
status.modified = magenta bold
status.added = green bold
status.removed = red bold
status.deleted = cyan bold
status.unknown = black  bold
status.ignored = black bold

Sometimes a few further improvements for the pager extension are necessary. The following options should not be enabled unless paging is not working correctly.

[pager]
# Some options for the less pager, see less(1) for their meaning.
pager = LESS='FSRX' less
# Some commands that aren't paged by default; also enable paging
# for them
attend = tags, help, annotate, cat, diff, export, status, \
         outgoing, incoming

Enabling the described extensions should immediately lead to a difference when using the command line version of hg. Of these options, the only one that enables a new command is graphlog. It is recommanded that to use the command hg glog, instead of hg log, for a better feel about what commits are being based on.

D.4 General Guidelines

All Octave’s sources are distributed under the GNU General Public License (GPL). Currently, Octave uses GPL version 3. For details about this license, see http://www.gnu.org/licenses/gpl.html. Therefore, whenever you create a new source file, it should have the following comment header (use appropriate year, name and comment marks):

## Copyright (C) 1996-2013 John W. Eaton <jwe@octave.org>
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING.  If not,
## see <http://www.gnu.org/licenses/>.

Always include commit messages in changesets. After making your source changes, record and briefly describe the changes in your commit message. You should have previously configured your ‘.hgrc’ (or ‘Mercurial.ini’ on Windows) with your name and email, which will be automatically added to your commit message. Your commit message should have a brief one-line explanation of what the commit does. If you are patching a bug, this one-line explanation should mention the bug number at the end. If your change is small and only touches one file then this is typically sufficient. If you are modifying several files, or several parts of one file, you should enumerate your changes roughly following the GNU coding standards for changelogs, as in the following example:

     look for methods before constructors
     * symtab.cc (symbol_table::fcn_info::fcn_info_rep::find):
     Look for class methods before constructors, contrary to matlab
     documentation.
     * test/ctor-vs-method: New directory of test classes.
     * test/test_ctor_vs_method.m: New file.
     * test/Makefile.am: Include ctor-vs-method/module.mk.
     (FCN_FILES): Include test_ctor_vs_method.m in the list.

In this example, the names of the file changed is listed first, and in parentheses the name of the function in that file that was modified. There is no need to mention the function for m-files that only contain one function. The commit message should describe what was changed, not why it was changed. Any explanation for why a change is needed should appear as comments in the code, particularly if there is something that might not be obvious to someone reading it later.

When submitting code which addresses a known bug on the Octave bug tracker (http://bugs.octave.org), please add ’(bug #XXXXX)’ to the first line of the commit messages. For example:

     Fix bug for complex input for gradient (bug #34292).

The preferred comment mark for places that may need further attention is FIXME:.

D.5 Octave Sources (m-files)

Don’t use tabs. Tabs cause trouble. If you are used to them, set up your editor so that it converts tabs to spaces. Indent the bodies of statement blocks. The recommended indent is 2 spaces. When calling functions, put spaces after commas and before the calling parentheses, like this:

       x = max (sin (y+3), 2);

An exception are matrix or cell constructors:

       [sin(x), cos(x)]
       {sin(x), cos(x)}

Here, putting spaces after sin, cos would result in a parse error. For an indexing expression, do not put a space after the identifier (this differentiates indexing and function calls nicely). The space after a comma is not necessary if index expressions are simple, i.e., you may write

       A(:,i,j)

but

       A([1:i-1;i+1:n], XI(:,2:n-1))

Use lowercase names if possible. Uppercase is acceptable for variable names consisting of 1-2 letters. Do not use mixed case names. Function names must be lowercase. Function names are global, so choose them wisely.

Always use a specific end-of-block statement (like endif, endswitch) rather than the generic end.

Enclose the if, while, until, and switch conditions in parentheses, as in C:

     if (isvector (a))
       s = sum (a);
     endif

Do not do this, however, with the iteration counter portion of a for statement. Write:

     for i = 1:n
       b(i) = sum (a(:,i));
     endfor

D.6 C++ Sources

Don’t use tabs. Tabs cause trouble. If you are used to them, set up your editor so that it converts tabs to spaces. Format function headers like this:

     static bool
     matches_patterns (const string_vector& patterns, int pat_idx,
                       int num_pat, const std::string& name)

The function name should start in column 1, and multi-line argument lists should be aligned on the first char after the open parenthesis. You should put a space before the left open parenthesis and after commas, for both function definitions and function calls.

The recommended indent is 2 spaces. When indenting, indent the statement after control structures (like if, while, etc.). If there is a compound statement, indent both the curly braces and the body of the statement (so that the body gets indented by two indents). Example:

     if (have_args)
       {
         idx.push_back (first_args);
         have_args = false;
       }
     else
       idx.push_back (make_value_list (*p_args, *p_arg_nm, &tmp));

If you have nested if statements, use extra braces for extra clarification.

Split long expressions in such a way that a continuation line starts with an operator rather than identifier. If the split occurs inside braces, continuation should be aligned with the first char after the innermost braces enclosing the split. Example:

     SVD::type type = ((nargout == 0 || nargout == 1)
                       ? SVD::sigma_only
                       : (nargin == 2) ? SVD::economy : SVD::std);

Consider putting extra braces around a multi-line expression to make it more readable, even if they are not necessary. Also, do not hesitate to put extra braces anywhere if it improves clarity.

Declare variables just before they are needed. Use local variables of blocks—it helps optimization. Don’t write a multi-line variable declaration with a single type specification and multiple variables. If the variables don’t fit on single line, repeat the type specification. Example:

     octave_value retval;
     octave_idx_type nr = b.rows ();
     octave_idx_type nc = b.cols ();
     double d1, d2;

Use lowercase names if possible. Uppercase is acceptable for variable names consisting of 1-2 letters. Do not use mixed case names.

Use Octave’s types and classes if possible. Otherwise, use the C++ standard library. Use of STL containers and algorithms is encouraged. Use templates wisely to reduce code duplication. Avoid comma expressions, labels and gotos, and explicit typecasts. If you need to typecast, use the modern C++ casting operators. In functions, minimize the number of return statements—use nested if statements if possible.

D.7 Other Sources

Apart from C++ and Octave language (m-files), Octave’s sources include files written in C, Fortran, M4, Perl, Unix shell, AWK, Texinfo, and TEX. There are not many rules to follow when using these other languages; some of them are summarized below. In any case, the golden rule is: if you modify a source file, try to follow any conventions you can detect in the file or other similar files.

For C you should obviously follow all C++ rules that can apply.

If you modify a Fortran file, you should stay within Fortran 77 with common extensions like END DO. Currently, we want all sources to be compilable with the f2c and g77 compilers, without special flags if possible. This usually means that non-legacy compilers also accept the sources.

The M4 macro language is mainly used for Autoconf configuration files. You should follow normal M4 rules when contributing to these files. Some M4 files come from external source, namely the Autoconf archive http://autoconf-archive.cryp.to.

If you give a code example in the documentation written in Texinfo with the @example environment, you should be aware that the text within such an environment will not be wrapped. It is recommended that you keep the lines short enough to fit on pages in the generated pdf or ps documents. Here is a ruler (in an @example environment) for finding the appropriate line width:

     1         2         3         4         5         6 123456789012345678901234567890123456789012345678901234567890

Appendix E Obsolete Functions

After being marked as deprecated for two major releases, the following functions have been removed from Octave. The third column of the table shows the version of Octave in which the function was removed. Prior to removal, each function in the list was marked as deprecated for at least two major releases. All deprecated functions issue warnings explaining that they will be removed in a future version of Octave, and which function should be used instead.

Replacement functions do not always accept precisely the same arguments as the obsolete function, but should provide equivalent functionality.

Obsolete Function  Replacement  Version

beta_cdf                 betacdf  3.4.0
beta_inv                 betainv  3.4.0
beta_pdf                 betapdf  3.4.0
beta_rnd                 betarnd  3.4.0
binomial_cdf             binocdf  3.4.0
binomial_inv             binoinv  3.4.0
binomial_pdf             binopdf  3.4.0
binomial_rnd             binornd  3.4.0
chisquare_cdf            chi2cdf  3.4.0
chisquare_inv            chi2inv  3.4.0
chisquare_pdf            chi2pdf  3.4.0
chisquare_rnd            chi2rnd  3.4.0
clearplot                clf  3.4.0
com2str                  num2str  3.4.0
exponential_cdf          expcdf  3.4.0
exponential_inv          expinv  3.4.0
exponential_pdf          exppdf  3.4.0
exponential_rnd          exprnd  3.4.0
f_cdf                    fcdf  3.4.0
f_inv                    finv  3.4.0
f_pdf                    fpdf  3.4.0
f_rnd                    frnd  3.4.0
gamma_cdf                gamcdf  3.4.0
gamma_inv                gaminv  3.4.0
gamma_pdf                gampdf  3.4.0
gamma_rnd                gamrnd  3.4.0
geometric_cdf            geocdf  3.4.0
geometric_inv            geoinv  3.4.0
geometric_pdf            geopdf  3.4.0
geometric_rnd            geornd  3.4.0
hypergeometric_cdf       hygecdf  3.4.0
hypergeometric_inv       hygeinv  3.4.0
hypergeometric_pdf       hygepdf  3.4.0
hypergeometric_rnd       hygernd  3.4.0

intersection             intersect  3.4.0
is_bool                  isbool  3.4.0
is_complex               iscomplex  3.4.0
is_list                  islist  3.4.0
is_matrix                ismatrix  3.4.0
is_scalar                isscalar  3.4.0
is_square                issquare  3.4.0
is_stream                isstream  3.4.0
is_struct                isstruct  3.4.0
is_symmetric             issymmetric  3.4.0
is_vector                isvector  3.4.0
lognormal_cdf            logncdf  3.4.0
lognormal_inv            logninv  3.4.0
lognormal_pdf            lognpdf  3.4.0
lognormal_rnd            lognrnd  3.4.0
meshdom                  meshgrid  3.4.0
normal_cdf               normcdf  3.4.0
normal_inv               norminv  3.4.0
normal_pdf               normpdf  3.4.0
normal_rnd               normrnd  3.4.0
pascal_cdf               nbincdf  3.4.0
pascal_inv               nbininv  3.4.0
pascal_pdf               nbinpdf  3.4.0
pascal_rnd               nbinrnd  3.4.0
poisson_cdf              poisscdf  3.4.0
poisson_inv              poissinv  3.4.0
poisson_pdf              poisspdf  3.4.0
poisson_rnd              poissrnd  3.4.0
polyinteg                polyint  3.4.0
struct_contains          isfield  3.4.0
struct_elements          fieldnames  3.4.0
t_cdf                    tcdf  3.4.0
t_inv                    tinv  3.4.0
t_pdf                    tpdf  3.4.0
t_rnd                    trnd  3.4.0
uniform_cdf              unifcdf  3.4.0
uniform_inv              unifinv  3.4.0
uniform_pdf              unifpdf  3.4.0
uniform_rnd              unifrnd  3.4.0
weibull_cdf              wblcdf  3.4.0
weibull_inv              wblinv  3.4.0
weibull_pdf              wblpdf  3.4.0
weibull_rnd              wblrnd  3.4.0
wiener_rnd               wienrnd  3.4.0
create_set               unique  3.6.0
dmult                    diag (A) * B  3.6.0
iscommand  None  3.6.0

israwcommand  None  3.6.0
lchol                    chol (..., "lower")  3.6.0
loadimage                load or imread  3.6.0
mark_as_command  None  3.6.0
mark_as_rawcommand  None  3.6.0
spatan2                  atan2  3.6.0
spchol                   chol  3.6.0
spchol2inv               chol2inv  3.6.0
spcholinv                cholinv  3.6.0
spcumprod                cumprod  3.6.0
spcumsum                 cumsum  3.6.0
spdet                    det  3.6.0
spdiag                   sparse (diag (...))  3.6.0
spfind                   find  3.6.0
sphcat                   horzcat  3.6.0
spinv                    inv  3.6.0
spkron                   kron  3.6.0
splchol                  chol (..., "lower")  3.6.0

split

                         char (strsplit (s,
                         t))

3.6.0

splu                     lu  3.6.0
spmax                    max  3.6.0
spmin                    min  3.6.0
spprod                   prod  3.6.0
spqr                     qr  3.6.0
spsum                    sum  3.6.0
spsumsq                  sumsq  3.6.0
spvcat                   vertcat  3.6.0
str2mat                  char  3.6.0
unmark_command  None  3.6.0
unmark_rawcommand  None  3.6.0

Appendix F Known Causes of Trouble

This section describes known problems that affect users of Octave. Most of these are not Octave bugs per se—if they were, we would fix them. But the result for a user may be like the result of a bug.

Some of these problems are due to bugs in other software, some are missing features that are too much work to add, and some are places where people’s opinions differ as to what is best.

F.1 Actual Bugs We Haven’t Fixed Yet

• Output that comes directly from Fortran functions is not sent through the pager and may appear out of sequence with other output that is sent through the pager. One way to avoid this is to force pending output to be flushed before calling a function that will produce output from within Fortran functions. To do this, use the command

         fflush (stdout)

Another possible workaround is to use the command

         page_screen_output (false);

to turn the pager off.

A list of ideas for future enhancements is distributed with Octave. See the file ‘PROJECTS’ in the top level directory in the source distribution.

F.2 Reporting Bugs

Your bug reports play an essential role in making Octave reliable.

When you encounter a problem, the first thing to do is to see if it is already known. See Appendix F [Trouble], page 829. If it isn’t known, then you should report the problem.

Reporting a bug may help you by bringing a solution to your problem, or it may not. In any case, the principal function of a bug report is to help the entire community by making the next version of Octave work better. Bug reports are your contribution to the maintenance of Octave.

In order for a bug report to serve its purpose, you must include the information that makes it possible to fix the bug.

F.2.1 Have You Found a Bug?

If you are not sure whether you have found a bug, here are some guidelines:

• If Octave gets a fatal signal, for any input whatever, that is a bug. Reliable interpreters never crash.

• If Octave produces incorrect results, for any input whatever, that is a bug.

• Some output may appear to be incorrect when it is in fact due to a program whose behavior is undefined, which happened by chance to give the desired results on another system. For example, the range operator may produce different results because of differences in the way floating point arithmetic is handled on various systems.

• If Octave produces an error message for valid input, that is a bug.

• If Octave does not produce an error message for invalid input, that is a bug. However, you should note that your idea of “invalid input” might be my idea of “an extension” or “support for traditional practice”.

• If you are an experienced user of programs like Octave, your suggestions for improvement are welcome in any case.

F.2.2 Where to Report Bugs

To report a bug in Octave, submit a bug report to the Octave bug tracker

http://bugs.octave.org.

Do not send bug reports to ‘help-octave’. Most users of Octave do not want to receive bug reports.

F.2.3 How to Report Bugs

Submit bug reports for Octave to the Octave bug tracker http://bugs.octave.org.

The fundamental principle of reporting bugs usefully is this: report all the facts. If you are not sure whether to state a fact or leave it out, state it!

Often people omit facts because they think they know what causes the problem and they conclude that some details don’t matter. Thus, you might assume that the name of the variable you use in an example does not matter. Well, probably it doesn’t, but one cannot be sure. Perhaps the bug is a stray memory reference which happens to fetch from the location where that name is stored in memory; perhaps, if the name were different, the contents of that location would fool the interpreter into doing the right thing despite the bug. Play it safe and give a specific, complete example.

Keep in mind that the purpose of a bug report is to enable someone to fix the bug if it is not known. Always write your bug reports on the assumption that the bug is not known.

Sometimes people give a few sketchy facts and ask, “Does this ring a bell?” This cannot help us fix a bug. It is better to send a complete bug report to begin with.

Try to make your bug report self-contained. If we have to ask you for more information, it is best if you include all the previous information in your response, as well as the information that was missing.

To enable someone to investigate the bug, you should include all these things:

• The version of Octave. You can get this by noting the version number that is printed when Octave starts, or running it with the ‘-v’ option.

• A complete input file that will reproduce the bug.

A single statement may not be enough of an example—the bug might depend on other details that are missing from the single statement where the error finally occurs.

• The command arguments you gave Octave to execute that example and observe the bug. To guarantee you won’t omit something important, list all the options. If we were to try to guess the arguments, we would probably guess wrong and then we would not encounter the bug.

• The type of machine you are using, and the operating system name and version number.

• The command-line arguments you gave to the configure command when you installed the interpreter.

• A complete list of any modifications you have made to the interpreter source. Be precise about these changes—show a context diff for them.

• Details of any other deviations from the standard procedure for installing Octave.

• A description of what behavior you observe that you believe is incorrect. For example, "The interpreter gets a fatal signal," or, "The output produced at line 208 is incorrect." Of course, if the bug is that the interpreter gets a fatal signal, then one can’t miss it. But if the bug is incorrect output, we might not notice unless it is glaringly wrong. Even if the problem you experience is a fatal signal, you should still say so explicitly. Suppose something strange is going on, such as, your copy of the interpreter is out of sync, or you have encountered a bug in the C library on your system. Your copy might crash and the copy here would not. If you said to expect a crash, then when the interpreter here fails to crash, we would know that the bug was not happening. If you don’t say to expect a crash, then we would not know whether the bug was happening. We would not be able to draw any conclusion from our observations. Often the observed symptom is incorrect output when your program is run. Unfortunately, this is not enough information unless the program is short and simple. It is very helpful if you can include an explanation of the expected output, and why the actual output is incorrect.

• If you wish to suggest changes to the Octave source, send them as context diffs. If you even discuss something in the Octave source, refer to it by context, not by line number, because the line numbers in the development sources probably won’t match those in your sources.

Here are some things that are not necessary:

• A description of the envelope of the bug.

Often people who encounter a bug spend a lot of time investigating which changes to the input file will make the bug go away and which changes will not affect it. Such information is usually not necessary to enable us to fix bugs in Octave, but if you can find a simpler example to report instead of the original one, that is a convenience. Errors in the output will be easier to spot, running under the debugger will take less time, etc. Most Octave bugs involve just one function, so the most straightforward way to simplify an example is to delete all the function definitions except the one in which the bug occurs. However, simplification is not vital; if you don’t want to do this, report the bug anyway and send the entire test case you used.

• A patch for the bug. Patches can be helpful, but if you find a bug, you should report it, even if you cannot send a fix for the problem.

F.2.4 Sending Patches for Octave

If you would like to write bug fixes or improvements for Octave, that is very helpful. When you send your changes, please follow these guidelines to avoid causing extra work for us in studying the patches.

If you don’t follow these guidelines, your information might still be useful, but using it will take extra work. Maintaining Octave is a lot of work in the best of circumstances, and we can’t keep up unless you do your best to help.

• Send an explanation with your changes of what problem they fix or what improvement they bring about. For a bug fix, just include a copy of the bug report, and explain why the change fixes the bug.

• Always include a proper bug report for the problem you think you have fixed. We need to convince ourselves that the change is right before installing it. Even if it is right, we might have trouble judging it if we don’t have a way to reproduce the problem.

• Include all the comments that are appropriate to help people reading the source in the future understand why this change was needed.

• Don’t mix together changes made for different reasons. Send them individually. If you make two changes for separate reasons, then we might not want to install them both. We might want to install just one.

• Use ‘diff -c’ to make your diffs. Diffs without context are hard for us to install reliably. More than that, they make it hard for us to study the diffs to decide whether we want to install them. Unified diff format is better than contextless diffs, but not as easy to read as ‘-c’ format. If you have GNU diff, use ‘diff -cp’, which shows the name of the function that each change occurs in.

• Write the change log entries for your changes.

Read the ‘ChangeLog’ file to see what sorts of information to put in, and to learn the style that we use. The purpose of the change log is to show people where to find what was changed. So you need to be specific about what functions you changed; in large functions, it’s often helpful to indicate where within the function the change was made. On the other hand, once you have shown people where to find the change, you need not explain its purpose. Thus, if you add a new function, all you need to say about it is that it is new. If you feel that the purpose needs explaining, it probably does—but the explanation will be much more useful if you put it in comments in the code. If you would like your name to appear in the header line for who made the change, send us the header line.

F.3 How To Get Help with Octave

The mailing list help@octave.org exists for the discussion of matters related to using and installing Octave. If would like to join the discussion, please send a short note to

help-request@octave.org.

Please do not send requests to be added or removed from the mailing list, or other administrative trivia to the list itself.

If you think you have found a bug in Octave or in the installation procedure, however, you should submit a complete bug report to the Octave bug tracker at http://bugs.octave.org. But before you submit a bug report, please read http://www.octave.org/bugs.html to learn how to submit a useful bug report.

Appendix G Installing Octave

The procedure for installing Octave from source on a Unix-like system is described next. Building on other platforms will follow similar steps. Note that this description applies to Octave releases. Building the development sources from the Mercurial archive requires additional steps as described in Section D.2 [Building the Development Sources], page 819.

G.1 Build Dependencies

Octave is a fairly large program with many build dependencies. You may be able to find pre-packaged versions of the dependencies distributed as part of your system, or you may have to build some or all of them yourself.

G.1.1 Obtaining the Dependencies Automatically

On some systems you can obtain many of Octave’s build dependencies automatically. The commands for doing this vary by system. Similarly, the names of pre-compiled packages vary by system and do not always match exactly the names listed in Section G.1.2 [Build Tools], page 833 and Section G.1.3 [External Packages], page 834.

You will usually need the development version of an external dependency so that you get the libraries and header files for building software, not just for running already compiled programs. These packages typically have names that end with the suffix -dev or -devel.

On systems with apt-get (Debian, Ubuntu, etc.), you may be able to install most of the tools and external packages using a command similar to

     apt-get build-dep octave

The specific package name may be octave3.2 or octave3.4. The set of required tools and external dependencies does not change frequently, so it is not important that the version match exactly, but you should use the most recent one available.

On systems with yum (Fedora, Red Hat, etc.), you may be able to install most of the tools and external packages using a command similar to

     yum-builddep octave

The yum-builddep utility is part of the yum-utils package.

For either type of system, the package name may include a version number. The set of required tools and external dependencies does not change frequently, so it is not important that the version exactly match the version you are installing, but you should use the most recent one available.

G.1.2 Build Tools

The following tools are required:

C++, C, and Fortran compilers The Octave sources are primarily written in C++, but some portions are also written in C and Fortran. The Octave sources are intended to be portable. Recent versions of the GNU compiler collection (GCC) should work (http://gcc.gnu.org). If you use GCC, you should avoid mixing versions. For example, be sure that you are not using the obsolete g77 Fortran compiler with modern versions of gcc and g++.

GNU Make Tool for building software (http://www.gnu.org/software/make). Octave’s build system requires GNU Make. Other versions of Make will not work. Fortunately, GNU Make is highly portable and easy to install.

AWK, sed, and other Unix utilities Basic Unix system utilities are required for building Octave. All will be available with any modern Unix system and also on Windows with either Cygwin or MinGW and MSYS.

Additionally, the following tools may be needed:

Bison Parser generator (http://www.gnu.org/software/bison). You will need Bison if you modify the oct-parse.yy source file or if you delete the files that are generated from it.

Flex Lexer analyzer (http://www.gnu.org/software/flex). You will need Flex if you modify the lex.ll source file or if you delete the files that are generated from it.

Autoconf Package for software configuration (http://www.gnu.org/software/autoconf).

Autoconf is required if you modify Octave’s configure.ac file or other files that it requires.

Automake Package for Makefile generation (http://www.gnu.org/software/automake).

Automake is required if you modify Octave’s Makefile.am files or other files that they depend on.

Libtool Package for building software libraries (http://www.gnu.org/software/libtool). Libtool is required by Automake.

gperf Perfect hash function generator (http://www.gnu.org/software/gperf). You will need gperf if you modify the octave.gperf file or if you delete the file that is generated from it.

Texinfo Package for generating online and print documentation (http://www.gnu.org/software/texinfo). You will need Texinfo to build Octave’s documentation or if you modify the documentation source files or the docstring of any Octave function.

G.1.3 External Packages

The following external packages are required:

BLAS Basic Linear Algebra Subroutine library (http://www.netlib.org/blas). Accelerated blas libraries such as ATLAS (http://math-atlas.sourceforge.net) are recommeded for better performance.

LAPACK Linear Algebra Package (http://www.netlib.org/lapack).

PCRE The Perl Compatible Regular Expression library (http://www.pcre.org).

The following external package is optional but strongly recommended:

GNU Readline Command-line editing library (www.gnu.org/s/readline).

If you wish to build Octave without GNU readline installed, you must use the ‘--disable-readline’ option when running the configure script.

The following external software packages are optional but recommended:

ARPACK Library for the solution of large-scale eigenvalue problems

(http://forge.scilab.org/index.php/p/arpack-ng).  arpack is

required to provide the functions eigs and svds.

cURL Library for transferring data with URL syntax (http://curl.haxx.se). cURL is required to provide the urlread and urlwrite functions and the ftp class.

FFTW3 Library for computing discrete Fourier transforms (http://www.fftw.org).

FFTW3 is used to provide better performance for functions that compute discrete Fourier transforms (fft, ifft, fft2, etc.)

FLTK Portable GUI toolkit (http://www.fltk.org). FLTK is currently used to provide windows for Octave’s OpenGL-based graphics functions.

fontconfig Library for configuring and customizing font access (http://www.freedesktop.org/wiki/Software/fontconfig). Fontconfig is used to manage fonts for Octave’s OpenGL-based graphics functions.

FreeType Portable font engine (http://www.freetype.org). FreeType is used to perform font rendering for Octave’s OpenGL-based graphics functions.

GLPK  GNU Linear Programming Kit (http://www.gnu.org/software/glpk).

GPLK is required for the function glpk.

gl2ps OpenGL to PostScript printing library (http://www.geuz.org/gl2ps/). gl2ps is required for printing when using the FLTK toolkit.

gnuplot Interactive graphics program (http://www.gnuplot.info). gnuplot is currently the default graphics renderer for Octave.

GraphicsMagick++ Image processing library (http://www.graphicsmagick.org). GraphicsMagick++ is used to provide the imread and imwrite functions.

HDF5 Library for manipulating portable data files (http://www.hdfgroup.org/HDF5).

hdf5 is required for Octave’s load and save commands to read and write HDF data files.

Java Development Kit Java programming language compiler and libraries. The OpenJDK free software implementation is recommended (http://openjdk.java.net/), although other JDK implementations may work. Java is required to be able to call Java functions from within Octave.

LLVM Compiler framework, (http://www.llvm.org). LLVM is required for Octave’s experimental just-in-time (JIT) compilation for speeding up the interpreter.

OpenGL API for portable 2-D and 3-D graphics (http://www.opengl.org). An OpenGL implementation is required to provide Octave’s OpenGL-based graphics functions. Octave’s OpenGL-based graphics functions usually outperform the gnuplot-based graphics functions because plot data can be rendered directly instead of sending data and commands to gnuplot for interpretation and rendering.

Qhull Computational geometry library (http://www.qhull.org). Qhull is required

to provide the functions convhull, convhulln, delaunay, delaunay3,
delaunayn, voronoi, and voronoin.

QRUPDATE

QR factorization updating library (http://sourceforge.net/projects/qrupdate).

QRUPDATE is used to provide improved performance for the functions

qrdelete, qrinsert, qrshift, and qrupdate.

QScintilla Source code highlighter and manipulator; a Qt port of Scintilla

(http://www.riverbankcomputing.co.uk/software/qscintilla).

QScintilla is used for syntax highlighting and code completion in the GUI.

Qt GUI and utility libraries (). Qt is required for building the GUI. It is a large framework, but the only components required are the GUI, core, and network modules.

SuiteSparse

Sparse matrix factorization library (http://www.cise.ufl.edu/research/sparse/SuiteSparse).

SuiteSparse is required to provide sparse matrix factorizations and solution of linear equations for sparse systems.

zlib Data compression library (http://zlib.net). The zlib library is required for Octave’s load and save commands to handle compressed data, including matlab v5 MAT files.

G.2 Running Configure and Make

• Run the shell script ‘configure’. This will determine the features your system has (or doesn’t have) and create a file named ‘Makefile’ from each of the files named

‘Makefile.in’.

Here is a summary of the configure options that are most frequently used when building Octave:

--help Print a summary of the options recognized by the configure script.

--prefix=prefix

Install Octave in subdirectories below prefix. The default value of prefix is

    ‘/usr/local’.
--srcdir=dir

Look for Octave sources in the directory dir.

--enable-64

This is an experimental option to enable Octave to use 64-bit integers for array dimensions and indexing on 64-bit platforms. You probably don’t want to use this option unless you know what you are doing. See Section G.3 [Compiling Octave with 64-bit Indexing], page 840, for more details about building Octave with this option.

--enable-bounds-check

Enable bounds checking for indexing operators in the internal array classes. This option is primarily used for debugging Octave. Building Octave with this option has a negative impact on performance and is not recommended for general use.

--disable-docs

Disable building all forms of the documentation (Info, PDF, HTML). The default is to build documentation, but your system will need functioning Texinfo and TEX installs for this to succeed.

--enable-float-truncate

This option allows for truncation of intermediate floating point results in calculations. It is only necessary for certain platforms.

--enable-readline

Use the readline library to provide for editing of the command line in terminal environments. This option is on by default.

--enable-shared

Create shared libraries (this is the default). If you are planning to use the dynamic loading features, you will probably want to use this option. It will make your ‘.oct’ files much smaller and on some systems it may be necessary to build shared libraries in order to use dynamically linked functions. You may also want to build a shared version of libstdc++, if your system doesn’t already have one.

--enable-dl

Use dlopen and friends to make Octave capable of dynamically linking externally compiled functions (this is the default if ‘--enable-shared’ is specified). This option only works on systems that actually have these functions. If you plan on using this feature, you should probably also use ‘--enable-shared’ to reduce the size of your ‘.oct’ files.

--with-blas=<lib>

By default, configure looks for the best blas matrix libraries on your system, including optimized implementations such as the free ATLAS 3.0, as well as vendor-tuned libraries. (The use of an optimized blas will generally result in several-times faster matrix operations.) Use this option to specify a particular blas library that Octave should use.

--with-lapack=<lib>

By default, configure looks for the best lapack matrix libraries on your system, including optimized implementations such as the free ATLAS 3.0, as well as vendor-tuned libraries. (The use of an optimized lapack will generally result in several-times faster matrix operations.) Use this option to specify a particular lapack library that Octave should use.

--with-magick=<lib>

Select the library to use for image I/O. The two possible values are

"GraphicsMagick" (default) or "ImageMagick".

--with-sepchar=<char>

Use <char> as the path separation character. This option can help when running Octave on non-Unix systems.

--without-amd

Don’t use amd, disable some sparse matrix functionality.

--without-camd

Don’t use camd, disable some sparse matrix functionality.

--without-colamd

Don’t use colamd, disable some sparse matrix functionality.

--without-ccolamd

Don’t use ccolamd, disable some sparse matrix functionality.

--without-cholmod

Don’t use cholmod, disable some sparse matrix functionality.

--without-curl

Don’t use the cURL library, disable the ftp objects, urlread and urlwrite functions.

--without-cxsparse

Don’t use cxsparse, disable some sparse matrix functionality.

--without-fftw3

Use the included fftpack library for computing Fast Fourier Transforms instead of the fftw3 library.

--without-fftw3f

Use the included fftpack library for computing Fast Fourier Transforms instead of the fftw3 library when operating on single precision (float) values.

--without-glpk

Don’t use the glpk library for linear programming.

--without-hdf5

Don’t use the hdf5 library, disable reading and writing of hdf5 files.

--without-opengl

Don’t use OpenGL, disable native graphics toolkit for plotting. You will need gnuplot installed in order to make plots.

--without-qhull

Don’t use Qhull, disable delaunay, convhull, and related functions.

--without-qrupdate

Don’t use qrupdate, disable QR and Cholesky update functions.

--without-umfpack

Don’t use umfpack, disable some sparse matrix functionality.

--without-zlib

Don’t use the zlib library, disable data file compression and support for recent MAT file formats.

--without-framework-carbon

Don’t use framework Carbon headers, libraries, or specific source code even if the configure test succeeds (the default is to use Carbon framework if available). This is a platform specific configure option for Mac systems.

--without-framework-opengl

Don’t use framework OpenGL headers, libraries, or specific source code even if the configure test succeeds. If this option is given then OpenGL headers and libraries in standard system locations are tested (the default value is ‘--with-framework-opengl’). This is a platform specific configure option for Mac systems.

See the file ‘INSTALL’ for more general information about the command line options used by configure. That file also contains instructions for compiling in a directory other than the one where the source is located.

• Run make.

You will need a recent version of GNU Make as Octave relies on certain features not generally available in all versions of make. Modifying Octave’s makefiles to work with other make programs is probably not worth your time; instead, we simply recommend installing GNU Make. There are currently two options for plotting in Octave: (1) the external program gnuplot, or (2) the internal graphics engine using OpenGL and FLTK. Gnuplot is a command-driven interactive function plotting program. Gnuplot is copyrighted, but freely distributable. As of Octave release 3.4, gnuplot is the default option for plotting. But, the internal graphics engine is nearly 100% compatible, certainly for most ordinary plots, and users are encouraged to test it. It is anticipated that the internal engine will become the default option at the next major release of Octave. To compile Octave, you will need a recent version of g++ or other ANSI C++ compiler. In addition, you will need a Fortran 77 compiler or f2c. If you use f2c, you will need a script like fort77 that works like a normal Fortran compiler by combining f2c with your C compiler in a single script. If you plan to modify the parser you will also need GNU bison and flex. If you modify the documentation, you will need GNU Texinfo. GNU Make, gcc (and libstdc++), gnuplot, bison, flex, and Texinfo are all available from many anonymous ftp archives. The primary site is ftp.gnu.org, but it is often very busy. A list of sites that mirror the software on ftp.gnu.org is available by

anonymous ftp from ftp://ftp.gnu.org/pub/gnu/GNUinfo/FTP.

Octave requires approximately 1.4 GB of disk storage to unpack and compile from source (significantly less, 400 MB, if you don’t compile with debugging symbols). To compile without debugging symbols try the command

     make CFLAGS=-O CXXFLAGS=-O LDFLAGS=

instead of just make.

• If you encounter errors while compiling Octave, first see Section G.4 [Installation Problems], page 843 for a list of known problems and if there is a workaround or solution for your problem. If not, see Appendix F [Trouble], page 829 for information about how to report bugs.

• Once you have successfully compiled Octave, run make install. This will install a copy of Octave, its libraries, and its documentation in the destination directory. As distributed, Octave is installed in the following directories. In the table below, prefix defaults to ‘/usr/local’, version stands for the current version number of the interpreter, and arch is the type of computer on which Octave is installed (for

example, ‘i586-unknown-gnu’).
‘prefix/bin’

Octave and other binaries that people will want to run directly.

‘prefix/lib/octave-version’

Libraries like liboctave.a and liboctinterp.a.

‘prefix/octave-version/include/octave’

Include files distributed with Octave.

‘prefix/share’

Architecture-independent data files.

‘prefix/share/man/man1’

Unix-style man pages describing Octave.

‘prefix/share/info’

Info files describing Octave.

‘prefix/share/octave/version/m’

Function files distributed with Octave. This includes the Octave version, so that multiple versions of Octave may be installed at the same time.

‘prefix/libexec/octave/version/exec/arch’

Executables to be run by Octave rather than the user.

‘prefix/lib/octave/version/oct/arch’

Object files that will be dynamically loaded.

‘prefix/share/octave/version/imagelib’

Image files that are distributed with Octave.

G.3 Compiling Octave with 64-bit Indexing

Note: the following only applies to systems that have 64-bit pointers. Configuring Octave with ‘--enable-64’ cannot magically make a 32-bit system have a 64-bit address space.

On 64-bit systems, Octave is limited to (approximately) the following array sizes when using the default 32-bit indexing mode:

 double:         16GB
 single:          8GB
 uint64, int64:  16GB
 uint32, int32:   8GB
 uint16, int16:   4GB
 uint8, int8:     2GB

In each case, the limit is really (approximately) 2^{31} elements because of the default type of the value used for indexing arrays (signed 32-bit integer, corresponding to the size of a Fortran INTEGER value).

Trying to create larger arrays will produce the following error:

     octave:1> a = zeros (1024*1024*1024*3, 1, 'int8');
     error: memory exhausted or requested size too large
            for range of Octave's index type --
            trying to return to prompt

You will obtain this error even if your system has enough memory to create this array (4 GB in the above case).

To use arrays larger than 2 GB, Octave has to be configured with the option ‘--enable-64’. This option is experimental and you are encouraged to submit bug reports if you find a problem. With this option, Octave will use 64-bit integers internally for array dimensions and indexing. However, all numerical libraries used by Octave will also need to use 64-bit integers for array dimensions and indexing. In most cases, this means they will need to be compiled from source since most (all?) distributions which package these libraries compile them with the default Fortran integer size, which is normally 32-bits wide.

The following instructions were tested with the development version of Octave and GCC 4.3.4 on an x86 64 Debian system.

The versions listed below are the versions used for testing. If newer versions of these packages are available, you should try to use them, although there may be some differences.

All libraries and header files will be installed in subdirectories of $prefix64 (you must choose the location of this directory).

  • blas and lapack (http://www.netlib.org/lapack)

Reference versions for both libraries are included in the reference lapack 3.2.1 distribution from netlib.org. − Copy the file ‘make.inc.example’ and name it ‘make.inc’. The options ‘-fdefault-integer-8’ and ‘-fPIC’ (on 64-bit CPU) have to be added to the variable OPTS and NOOPT.

− Once you have compiled this library make sure that you use it for compiling Suite Sparse and Octave. In the following we assume that you installed the lapack library as $prefix64/lib/liblapack.a.

  • QRUPDATE (http://sourceforge.net/projects/qrupdate)

In the ‘Makeconf’ file:

     − Add ‘-fdefault-integer-8’ to FFLAGS.

− Adjust the blas and lapack variables as needed if your 64-bit aware blas and

lapack libraries are in a non-standard location. − Set PREFIX to the top-level directory of your install tree. − Run make solib to make a shared library. − Run make install to install the library.

  • SuiteSparse (http://www.cise.ufl.edu/research/sparse/SuiteSparse)

Pass the following options to make to enable 64-bit integers for blas library calls. On 64-bit Windows systems, use -DLONGBLAS="long long" instead.

         CFLAGS='-DLONGBLAS=long'
         CXXFLAGS='-DLONGBLAS=long'

The SuiteSparse makefiles don’t generate shared libraries. On some systems, you can generate them by doing something as simple as

     top=$(pwd)
     for f in *.a; do
       mkdir tmp
       cd tmp
       ar vx ../$f
       gcc -shared -o ../${f%%.a}.so *.o
       cd $top
       rm -rf tmp
     done

Other systems may require a different solution.

• ATLAS instead of reference blas and lapack Suggestions on how to compile ATLAS would be most welcome.

• glpk

• Qhull (http://www.qhull.org)

Both glpk and Qhull use int internally so maximum problem sizes may be limited.

• Octave

Octave’s 64-bit index support is activated with the configure option ‘--enable-64’.

     ./configure \
       LD_LIBRARY_PATH="$prefix64/lib" \
       CPPFLAGS="-I$prefix64/include" LDFLAGS="-L$prefix64/lib" \
       --enable-64

You must ensure that all Fortran sources except those in the ‘liboctave/cruft/ranlib’ directory are compiled such that INTEGERS are 8-bytes wide. If you are using gfortran, the configure script should automatically set the Makefile variable

F77_INTEGER_8_FLAG to ‘-fdefault-integer-8’. If you are using another compiler,

you must set this variable yourself. You should NOT set this flag in FFLAGS, otherwise the files in ‘liboctave/cruft/ranlib’ will be miscompiled.

• Other dependencies Probably nothing special needs to be done for the following dependencies. If you discover that something does need to be done, please submit a bug report. − pcre − zlib − hdf5 − fftw3 − cURL − GraphicsMagick++ − OpenGL − freetype − fontconfig − fltk

G.4 Installation Problems

This section contains a list of problems (and some apparent problems that don’t really mean anything is wrong) that may show up during installation of Octave.

• On some SCO systems, info fails to compile if HAVE_TERMIOS_H is defined in ‘config.h’. Simply removing the definition from ‘info/config.h’ should allow it to compile.

• If configure finds dlopen, dlsym, dlclose, and dlerror, but not the header file

‘dlfcn.h’, you need to find the source for the header file and install it in the directory ‘usr/include’. This is reportedly a problem with Slackware 3.1. For Linux/GNU systems, the source for ‘dlfcn.h’ is in the ldso package.

• Building ‘.oct’ files doesn’t work.

You should probably have a shared version of libstdc++. A patch is needed to build shared versions of version 2.7.2 of libstdc++ on the HP-PA architecture. You can find

the patch at ftp://ftp.cygnus.com/pub/g++/libg++-2.7.2-hppa-gcc-fix.

• On some DEC alpha systems there may be a problem with the libdxml library, resulting in floating point errors and/or segmentation faults in the linear algebra routines called by Octave. If you encounter such problems, then you should modify the configure script so that SPECIAL_MATH_LIB is not set to -ldxml.

• On FreeBSD systems Octave may hang while initializing some internal constants. The

fix appears to be to use

     options      GPL_MATH_EMULATE

rather than

     options      MATH_EMULATE

in the kernel configuration files (typically found in the directory ‘/sys/i386/conf’. After making this change, you’ll need to rebuild the kernel, install it, and reboot.

• If you encounter errors like

     passing `void (*)()' as argument 2 of
       `octave_set_signal_handler(int, void (*)(int))'

or

     warning: ANSI C++ prohibits conversion from `(int)'
              to `(...)'

while compiling ‘sighandlers.cc’, you may need to edit some files in the gcc include subdirectory to add proper prototypes for functions there. For example, Ultrix 4.2 needs proper declarations for the signal function and the SIG_IGN macro in the file

‘signal.h’.

On some systems the SIG_IGN macro is defined to be something like this:

     #define  SIG_IGN  (void (*)())1

when it should really be something like:

     #define  SIG_IGN  (void (*)(int))1

to match the prototype declaration for the signal function. This change should also be made for the SIG_DFL and SIG_ERR symbols. It may be necessary to change the definitions in ‘sys/signal.h’ as well.

The gcc fixincludes and fixproto scripts should probably fix these problems when gcc installs its modified set of header files, but I don’t think that’s been done yet.

You should not change the files in ‘/usr/include’. You can find the gcc include directory tree by running the command

     gcc -print-libgcc-file-name

The directory of gcc include files normally begins in the same directory that contains the file ‘libgcc.a’.

• Some of the Fortran subroutines may fail to compile with older versions of the Sun Fortran compiler. If you get errors like

     zgemm.f:
             zgemm:
     warning: unexpected parent of complex expression subtree
     zgemm.f, line 245: warning: unexpected parent of complex
       expression subtree
     warning: unexpected parent of complex expression subtree
     zgemm.f, line 304: warning: unexpected parent of complex
       expression subtree
     warning: unexpected parent of complex expression subtree
     zgemm.f, line 327: warning: unexpected parent of complex
       expression subtree
     pcc_binval: missing IR_CONV in complex op
     make[2]: *** [zgemm.o] Error 1

when compiling the Fortran subroutines in the ‘liboctave/cruft’ subdirectory, you should either upgrade your compiler or try compiling with optimization turned off.

• On NeXT systems, if you get errors like this:

     /usr/tmp/cc007458.s:unknown:Undefined local
           symbol LBB7656
     /usr/tmp/cc007458.s:unknown:Undefined local
           symbol LBE7656

when compiling ‘Array.cc’ and ‘Matrix.cc’, try recompiling these files without ‘-g’.

• Some people have reported that calls to system() and the pager do not work on SunOS systems. This is apparently due to having G_HAVE_SYS_WAIT defined to be 0 instead of 1 when compiling libg++.

• On NeXT systems, linking to ‘libsys_s.a’ may fail to resolve the following functions

     _tcgetattr _tcsetattr _tcflow

which are part of ‘libposix.a’. Unfortunately, linking Octave with ‘-posix’ results in the following undefined symbols.

     .destructors_used
     .constructors_used
     _objc_msgSend
     _NXGetDefaultValue
     _NXRegisterDefaults
     .objc_class_name_NXStringTable
     .objc_class_name_NXBundle

One kluge around this problem is to extract ‘termios.o’ from ‘libposix.a’, put it in Octave’s ‘src’ directory, and add it to the list of files to link together in the makefile. Suggestions for better ways to solve this problem are welcome!

• If Octave crashes immediately with a floating point exception, it is likely that it is failing to initialize the IEEE floating point values for infinity and NaN. If your system actually does support IEEE arithmetic, you should be able to fix this problem by modifying the function octave_ieee_init in the file ‘lo-ieee.cc’ to correctly initialize Octave’s internal infinity and NaN variables. If your system does not support IEEE arithmetic but Octave’s configure script incorrectly determined that it does, you can work around the problem by editing the file

‘config.h’ to not define HAVE_ISINF, HAVE_FINITE, and HAVE_ISNAN.

In any case, please report this as a bug since it might be possible to modify Octave’s configuration script to automatically determine the proper thing to do.

• If Octave is unable to find a header file because it is installed in a location that is not normally searched by the compiler, you can add the directory to the include search

path by specifying (for example) CPPFLAGS=-I/some/nonstandard/directory as an

argument to configure. Other variables that can be specified this way are CFLAGS, CXXFLAGS, FFLAGS, and LDFLAGS. Passing them as options to the configure script also records them in the ‘config.status’ file. By default, CPPFLAGS and LDFLAGS are

empty, CFLAGS and CXXFLAGS are set to "-g -O" and FFLAGS is set to "-O".

Appendix H Emacs Octave Support

The development of Octave code can greatly be facilitated using Emacs with Octave mode, a major mode for editing Octave files which can e.g. automatically indent the code, do some of the typing (with Abbrev mode) and show keywords, comments, strings, etc. in different faces (with Font-lock mode on devices that support it).

It is also possible to run Octave from within Emacs, either by directly entering commands at the prompt in a buffer in Inferior Octave mode, or by interacting with Octave from within a file with Octave code. This is useful in particular for debugging Octave code.

Finally, you can convince Octave to use the Emacs info reader for help -i.

All functionality is provided by the Emacs Lisp package EOS (for “Emacs Octave Support”). This chapter describes how to set up and use this package.

Please contact Kurt.Hornik@wu-wien.ac.at if you have any questions or suggestions on using EOS.

H.1 Installing EOS

The Emacs package EOS consists of the three files ‘octave-mod.el’, ‘octave-inf.el’, and ‘octave-hlp.el’. These files, or better yet their byte-compiled versions, should be somewhere in your Emacs load-path.

If you have GNU Emacs with a version number at least as high as 19.35, you are all set up, because EOS is respectively will be part of GNU Emacs as of version 19.35.

Otherwise, copy the three files from the ‘emacs’ subdirectory of the Octave distribution to a place where Emacs can find them (this depends on how your Emacs was installed). Byte-compile them for speed if you want.

H.2 Using Octave Mode

If you are lucky, your sysadmins have already arranged everything so that Emacs automatically goes into Octave mode whenever you visit an Octave code file as characterized by its extension ‘.m’. If not, proceed as follows.

1. To begin using Octave mode for all ‘.m’ files you visit, add the following lines to a file

loaded by Emacs at startup time, typically your ‘~/.emacs’ file:

         (autoload 'octave-mode "octave-mod" nil t)
         (setq auto-mode-alist
               (cons '("\\.m$" . octave-mode) auto-mode-alist))

2. Finally, to turn on the abbrevs, auto-fill and font-lock features automatically, also add

the following lines to one of the Emacs startup files:

         (add-hook 'octave-mode-hook
                   (lambda ()
                     (abbrev-mode 1)
                     (auto-fill-mode 1)
                     (if (eq window-system 'x)
                         (font-lock-mode 1))))

See the Emacs manual for more information about how to customize Font-lock mode.

In Octave mode, the following special Emacs commands can be used in addition to the standard Emacs commands.

C-h m Describe the features of Octave mode.

LFD Reindent the current Octave line, insert a newline and indent the new line

(octave-reindent-then-newline-and-indent). An abbrev before point is

expanded if abbrev-mode is non-nil.

TAB Indents current Octave line based on its contents and on previous lines (indent-

according-to-mode).
;  Insert an “electric” semicolon (octave-electric-semi). If octave-auto-

indent is non-nil, reindent the current line. If octave-auto-newline is non-nil, automagically insert a newline and indent the new line.

` Start entering an abbreviation (octave-abbrev-start). If Abbrev mode is turned on, typing `C-h or `? lists all abbrevs. Any other key combination is executed normally. Note that all Octave abbrevs start with a grave accent.

M-LFD Break line at point and insert continuation marker and alignment (octave-

split-line).

M-TAB Perform completion on Octave symbol preceding point, comparing that symbol against Octave’s reserved words and built-in variables (octave-complete-

symbol).

M-C-a Move backward to the beginning of a function (octave-beginning-of-defun). With prefix argument N, do it that many times if N is positive; otherwise, move forward to the N-th following beginning of a function.

M-C-e Move forward to the end of a function (octave-end-of-defun). With prefix argument N, do it that many times if N is positive; otherwise, move back to the N-th preceding end of a function.

M-C-h Puts point at beginning and mark at the end of the current Octave function, i.e., the one containing point or following point (octave-mark-defun).

M-C-q Properly indents the Octave function which contains point (octave-indent-

defun).

M-; If there is no comment already on this line, create a code-level comment (started by two comment characters) if the line is empty, or an in-line comment (started by one comment character) otherwise (octave-indent-for-comment). Point is left after the start of the comment which is properly aligned.

C-c ; Puts the comment character ‘#’ (more precisely, the string value of octavecomment-start) at the beginning of every line in the region (octave-comment-region). With just C-u prefix argument, uncomment each line in the region. A numeric prefix argument N means use N comment characters.

C-c : Uncomments every line in the region (octave-uncomment-region).

C-c C-p Move one line of Octave code backward, skipping empty and comment lines (octave-previous-code-line). With numeric prefix argument N, move that many code lines backward (forward if N is negative).

C-c C-n Move one line of Octave code forward, skipping empty and comment lines (octave-next-code-line). With numeric prefix argument N, move that many code lines forward (backward if N is negative).

C-c C-a Move to the ‘real’ beginning of the current line (octave-beginning-of-line). If point is in an empty or comment line, simply go to its beginning; otherwise, move backwards to the beginning of the first code line which is not inside a continuation statement, i.e., which does not follow a code line ending in ‘...’ or ‘\’, or is inside an open parenthesis list.

C-c C-e Move to the ‘real’ end of the current line (octave-end-of-line). If point is in a code line, move forward to the end of the first Octave code line which does not end in ‘...’ or ‘\’ or is inside an open parenthesis list. Otherwise, simply go to the end of the current line.

C-c M-C-n Move forward across one balanced begin-end block of Octave code (octaveforward-block). With numeric prefix argument N, move forward across n such blocks (backward if N is negative).

C-c M-C-p Move back across one balanced begin-end block of Octave code (octavebackward-block). With numeric prefix argument N, move backward across N such blocks (forward if N is negative).

C-c M-C-d Move forward down one begin-end block level of Octave code (octave-down-block). With numeric prefix argument, do it that many times; a negative argument means move backward, but still go down one level.

C-c M-C-u Move backward out of one begin-end block level of Octave code (octavebackward-up-block). With numeric prefix argument, do it that many times; a negative argument means move forward, but still to a less deep spot.

C-c M-C-h Put point at the beginning of this block, mark at the end (octave-mark-block). The block marked is the one that contains point or follows point.

C-c ] Close the current block on a separate line (octave-close-block). An error is signaled if no block to close is found.

C-c f Insert a function skeleton, prompting for the function’s name, arguments and return values which have to be entered without parentheses (octave-insert-

defun).

C-c C-h Search the function, operator and variable indices of all info files with documentation for Octave for entries (octave-help). If used interactively, the entry is prompted for with completion. If multiple matches are found, one can cycle through them using the standard ‘,’ (Info-index-next) command of the Info reader. The variable octave-help-files is a list of files to search through and defaults to '("octave"). If there is also an Octave Local Guide with corresponding info file, say, ‘octave-LG’, you can have octave-help search both files by

     (setq octave-help-files '("octave" "octave-LG"))

in one of your Emacs startup files.

A common problem is that the RET key does not indent the line to where the new text should go after inserting the newline. This is because the standard Emacs convention is that RET (aka C-m) just adds a newline, whereas LFD (aka C-j) adds a newline and indents it. This is particularly inconvenient for users with keyboards which do not have a special LFD key at all; in such cases, it is typically more convenient to use RET as the LFD key (rather than typing C-j).

You can make RET do this by adding

     (define-key octave-mode-map "\C-m"
       'octave-reindent-then-newline-and-indent)

to one of your Emacs startup files. Another, more generally applicable solution is

     (defun RET-behaves-as-LFD ()
       (let ((x (key-binding "\C-j")))
         (local-set-key "\C-m" x)))
     (add-hook 'octave-mode-hook 'RET-behaves-as-LFD)

(this works for all modes by adding to the startup hooks, without having to know the particular binding of RET in that mode!). Similar considerations apply for using M-RET as M-LFD. As Barry A. Warsaw bwarsaw@cnri.reston.va.us says in the documentation for his cc-mode, “This is a very common question. :-) If you want this to be the default behavior, don’t lobby me, lobby RMS!”

The following variables can be used to customize Octave mode.

octave-auto-indent

Non-nil means auto-indent the current line after a semicolon or space. Default

          is nil.
octave-auto-newline

Non-nil means auto-insert a newline and indent after semicolons are typed. The default value is nil.

octave-blink-matching-block

Non-nil means show matching begin of block when inserting a space, newline or ‘;’ after an else or end keyword. Default is t. This is an extremely useful feature for automatically verifying that the keywords match—if they don’t, an error message is displayed.

octave-block-offset

Extra indentation applied to statements in block structures. Default is 2.

octave-continuation-offset

Extra indentation applied to Octave continuation lines. Default is 4.

octave-continuation-string

String used for Octave continuation lines. Normally ‘\’.

octave-mode-startup-message

If t (default), a startup message is displayed when Octave mode is called.

If Font Lock mode is enabled, Octave mode will display

  • strings in font-lock-string-face
  • comments in font-lock-comment-face

• the Octave reserved words (such as all block keywords) and the text functions (such as ‘cd’ or ‘who’) which are also reserved using font-lock-keyword-face

• the built-in operators (‘&&’, ‘==’, …) using font-lock-reference-face

• and the function names in function declarations in font-lock-function-name-face.

There is also rudimentary support for Imenu (currently, function names can be indexed). You can generate TAGS files for Emacs from Octave ‘.m’ files using the shell script

octave-tags that is installed alongside your copy of Octave.

Customization of Octave mode can be performed by modification of the variable octavemode-hook. If the value of this variable is non-nil, turning on Octave mode calls its value.

If you discover a problem with Octave mode, you can conveniently send a bug report using C-c C-b (octave-submit-bug-report). This automatically sets up a mail buffer with version information already added. You just need to add a description of the problem, including a reproducible test case and send the message.

H.3 Running Octave from Within Emacs

The package ‘octave’ provides commands for running an inferior Octave process in a special Emacs buffer. Use

     M-x run-octave

to directly start an inferior Octave process. If Emacs does not know about this command, add the line

     (autoload 'run-octave "octave-inf" nil t)

to your ‘.emacs’ file.

This will start Octave in a special buffer the name of which is specified by the variable

inferior-octave-buffer and defaults to "*Inferior Octave*". From within this buffer,

you can interact with the inferior Octave process ‘as usual’, i.e., by entering Octave commands at the prompt. The buffer is in Inferior Octave mode, which is derived from the standard Comint mode, a major mode for interacting with an inferior interpreter. See the documentation for comint-mode for more details, and use C-h b to find out about available special keybindings.

You can also communicate with an inferior Octave process from within files with Octave code (i.e., buffers in Octave mode), using the following commands.

C-c i l Send the current line to the inferior Octave process (octave-send-line). With positive prefix argument N, send that many lines. If octave-send-line-auto-forward is non-nil, go to the next unsent code line.

C-c i b Send the current block to the inferior Octave process (octave-send-block).

C-c i f Send the current function to the inferior Octave process (octave-send-defun).

C-c i r Send the region to the inferior Octave process (octave-send-region).

C-c i s Make sure that ‘inferior-octave-buffer’ is displayed (octave-show-process-

          buffer).

C-c i h Delete all windows that display the inferior Octave buffer (octave-hide-

          process-buffer).

C-c i k Kill the inferior Octave process and its buffer (octave-kill-process).

The effect of the commands which send code to the Octave process can be customized by the following variables.

octave-send-echo-input

Non-nil means echo input sent to the inferior Octave process. Default is t.

octave-send-show-buffer

Non-nil means display the buffer running the Octave process after sending a command (but without selecting it). Default is t.

If you send code and there is no inferior Octave process yet, it will be started automatically.

The startup of the inferior Octave process is highly customizable. The variable inferior-octave-startup-args can be used for specifying command lines arguments to be passed to Octave on startup as a list of strings. For example, to suppress the startup message and use ‘traditional’ mode, set this to '("-q" "--traditional"). You can also specify a startup file of Octave commands to be loaded on startup; note that these commands will not produce any visible output in the process buffer. Which file to use is controlled by the variable inferior-octave-startup-file. If this is nil, the file ‘~/.emacs-octave’ is used if it exists.

And finally, inferior-octave-mode-hook is run after starting the process and putting its buffer into Inferior Octave mode. Hence, if you like the up and down arrow keys to behave in the interaction buffer as in the shell, and you want this buffer to use nice colors, add

     (add-hook 'inferior-octave-mode-hook
               (lambda ()
                 (turn-on-font-lock)
                 (define-key inferior-octave-mode-map [up]
                   'comint-previous-input)
                 (define-key inferior-octave-mode-map [down]
                   'comint-next-input)))

to your ‘.emacs’ file. You could also swap the roles of C-a (beginning-of-line) and C-c C-a (comint-bol) using this hook.

Note that if you set your Octave prompts to something different from the defaults, make sure that inferior-octave-prompt matches them. Otherwise, nothing will work, because Emacs will not know when Octave is waiting for input, or done sending output.

H.4 Using the Emacs Info Reader for Octave

You may also use the Emacs Info reader with Octave’s doc function. For this, the package ‘gnuserv’ needs to be installed.

If ‘gnuserv’ is installed, add the lines

     (autoload 'octave-help "octave-hlp" nil t)
     (require 'gnuserv)
     (gnuserv-start)

to your ‘.emacs’ file.

You can use either ‘plain’ Emacs Info or the function octave-help as your Octave info

reader (for ‘help -i’). In the former case, use info_program ("info-emacs-info"). The

latter is perhaps more attractive because it allows to look up keys in the indices of several info files related to Octave (provided that the Emacs variable octave-help-files is set

correctly). In this case, use info_program ("info-emacs-octave-help").

If you use Octave from within Emacs, it is best to add these settings to your ‘~/.emacs-octave’ startup file (or the file pointed to by the Emacs variable

inferior-octave-startup-file).

Appendix I Grammar and Parser

This appendix will eventually contain a semi-formal description of Octave’s language.

I.1 Keywords

The following identifiers are keywords, and may not be used as variable or function names:

__FILE____LINE__break
case                       catch                    classdef
continue                   do                       else
elseif                     end                      end_try_catch
end_unwind_protect         endclassdef              endenumeration
endevents                  endfor                   endfunction
endif                      endmethods               endparfor
endproperties              endswitch                endwhile
enumeration                events                   for
function                   global                   if
methods                    otherwise                parfor
persistent                 properties               return
static                     switch                   try
until                      unwind_protect           unwind_protect_cleanup

while

The function iskeyword can be used to quickly check whether an identifier is reserved by Octave.

[Built-in Function] iskeyword () [Built-in Function] iskeyword (name)

Return true if name is an Octave keyword. If name is omitted, return a list of keywords.

See also: [isvarname], page 121, [exist], page 128.

I.2 Parser

The parser has a number of variables that affect its internal operation. These variables are generally documented in the manual alongside the code that they affect. For example, allow_noninteger_range_as_index is discussed in the section on index expressions.

In addition, there are three non-specific parser customization functions. add_input_ event_hook can be used to schedule a user function for periodic evaluation. remove_ input_event_hook will stop a user function from being evaluated periodically.

[Built-in Function] id = add_input_event_hook (fcn)
[Built-in Function] id = add_input_event_hook (fcn, data)

Add the named function or function handle fcn to the list of functions to call periodically when Octave is waiting for input. The function should have the form

          fcn (data)

If data is omitted, Octave calls the function without any arguments.

The returned identifier may be used to remove the function handle from the list of input hook functions.

See also: [remove input event hook], page 856.

[Built-in Function] remove_input_event_hook (name)
[Built-in Function] remove_input_event_hook (fcn_id)

Remove the named function or function handle with the given identifier from the list of functions to call periodically when Octave is waiting for input.

See also: [add input event hook], page 855.

Finally, when the parser cannot identify an input token it calls a particular function to handle this. By default, this is the internal function "__unimplemented__" which makes suggestions about possible Octave substitutes for matlab functions.

[Built-in Function] val = missing_function_hook ()
[Built-in Function] old_val = missing_function_hook (new_val)
[Built-in Function] missing_function_hook (new_val, "local")

Query or set the internal variable that specifies the function to call when an unknown identifier is requested. When called from inside a function with the "local" option, the variable is changed locally for the function and any subroutines it calls. The original variable value is restored when exiting the function.

See also: [missing component hook], page 758.

Appendix J GNU GENERAL PUBLIC LICENSE

Version 3, 29 June 2007

Copyright \begin{matrix}c \\ ◯ \end{matrix} 2007 Free Software Foundation, Inc. http://fsf.org/

Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

Preamble

The GNU General Public License is a free, copyleft license for software and other kinds of works.

The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program—to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for them if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs, and that you know you can do these things.

To protect your rights, we need to prevent others from denying you these rights or asking you to surrender the rights. Therefore, you have certain responsibilities if you distribute copies of the software, or if you modify it: responsibilities to respect the freedom of others.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must pass on to the recipients the same freedoms that you received. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.

Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.

For the developers’ and authors’ protection, the GPL clearly explains that there is no warranty for this free software. For both users’ and authors’ sake, the GPL requires that modified versions be marked as changed, so that their problems will not be attributed erroneously to authors of previous versions.

Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users’ freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users.

Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free.

The precise terms and conditions for copying, distribution and modification follow.

TERMS AND CONDITIONS

0. Definitions. “This License” refers to version 3 of the GNU General Public License. “Copyright” also means copyright-like laws that apply to other kinds of works, such as semiconductor masks. “The Program” refers to any copyrightable work licensed under this License. Each licensee is addressed as “you”. “Licensees” and “recipients” may be individuals or organizations. To “modify” a work means to copy from or adapt all or part of the work in a fashion requiring copyright permission, other than the making of an exact copy. The resulting work is called a “modified version” of the earlier work or a work “based on” the earlier work. A “covered work” means either the unmodified Program or a work based on the Program. To “propagate” a work means to do anything with it that, without permission, would make you directly or secondarily liable for infringement under applicable copyright law, except executing it on a computer or modifying a private copy. Propagation includes copying, distribution (with or without modification), making available to the public, and in some countries other activities as well. To “convey” a work means any kind of propagation that enables other parties to make or receive copies. Mere interaction with a user through a computer network, with no transfer of a copy, is not conveying. An interactive user interface displays “Appropriate Legal Notices” to the extent that it includes a convenient and prominently visible feature that (1) displays an appropriate copyright notice, and (2) tells the user that there is no warranty for the work (except to the extent that warranties are provided), that licensees may convey the work under this License, and how to view a copy of this License. If the interface presents a list of user commands or options, such as a menu, a prominent item in the list meets this criterion.

1. Source Code. The “source code” for a work means the preferred form of the work for making modifications to it. “Object code” means any non-source form of a work. A “Standard Interface” means an interface that either is an official standard defined by a recognized standards body, or, in the case of interfaces specified for a particular programming language, one that is widely used among developers working in that language.

The “System Libraries” of an executable work include anything, other than the work as a whole, that (a) is included in the normal form of packaging a Major Component, but which is not part of that Major Component, and (b) serves only to enable use of the work with that Major Component, or to implement a Standard Interface for which an implementation is available to the public in source code form. A “Major Component”, in this context, means a major essential component (kernel, window system, and so on) of the specific operating system (if any) on which the executable work runs, or a compiler used to produce the work, or an object code interpreter used to run it.

The “Corresponding Source” for a work in object code form means all the source code needed to generate, install, and (for an executable work) run the object code and to modify the work, including scripts to control those activities. However, it does not include the work’s System Libraries, or general-purpose tools or generally available free programs which are used unmodified in performing those activities but which are not part of the work. For example, Corresponding Source includes interface definition files associated with source files for the work, and the source code for shared libraries and dynamically linked subprograms that the work is specifically designed to require, such as by intimate data communication or control flow between those subprograms and other parts of the work.

The Corresponding Source need not include anything that users can regenerate automatically from other parts of the Corresponding Source.

The Corresponding Source for a work in source code form is that same work.

2. Basic Permissions.

All rights granted under this License are granted for the term of copyright on the Program, and are irrevocable provided the stated conditions are met. This License explicitly affirms your unlimited permission to run the unmodified Program. The output from running a covered work is covered by this License only if the output, given its content, constitutes a covered work. This License acknowledges your rights of fair use or other equivalent, as provided by copyright law.

You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force. You may convey covered works to others for the sole purpose of having them make modifications exclusively for you, or provide you with facilities for running those works, provided that you comply with the terms of this License in conveying all material for which you do not control copyright. Those thus making or running the covered works for you must do so exclusively on your behalf, under your direction and control, on terms that prohibit them from making any copies of your copyrighted material outside their relationship with you.

Conveying under any other circumstances is permitted solely under the conditions stated below. Sublicensing is not allowed; section 10 makes it unnecessary.

3. Protecting Users’ Legal Rights From Anti-Circumvention Law.

No covered work shall be deemed part of an effective technological measure under any applicable law fulfilling obligations under article 11 of the WIPO copyright treaty adopted on 20 December 1996, or similar laws prohibiting or restricting circumvention of such measures.

When you convey a covered work, you waive any legal power to forbid circumvention of technological measures to the extent such circumvention is effected by exercising rights under this License with respect to the covered work, and you disclaim any intention to limit operation or modification of the work as a means of enforcing, against the work’s users, your or third parties’ legal rights to forbid circumvention of technological measures.

4. Conveying Verbatim Copies.

You may convey verbatim copies of the Program’s source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice; keep intact all notices stating that this License and any non-permissive terms added in accord with section 7 apply to the code; keep intact all notices of the absence of any warranty; and give all recipients a copy of this License along with the Program. You may charge any price or no price for each copy that you convey, and you may offer support or warranty protection for a fee.

5. Conveying Modified Source Versions. You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions: a. The work must carry prominent notices stating that you modified it, and giving a

relevant date. b. The work must carry prominent notices stating that it is released under this Li-

cense and any conditions added under section 7. This requirement modifies the requirement in section 4 to “keep intact all notices”.

c. You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.

d. If the work has interactive user interfaces, each must display Appropriate Legal Notices; however, if the Program has interactive interfaces that do not display Appropriate Legal Notices, your work need not make them do so.

A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an “aggregate” if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation’s users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.

6. Conveying Non-Source Forms.

You may convey a covered work in object code form under the terms of sections 4 and 5, provided that you also convey the machine-readable Corresponding Source under the terms of this License, in one of these ways: a. Convey the object code in, or embodied in, a physical product (including a phys-

ical distribution medium), accompanied by the Corresponding Source fixed on a durable physical medium customarily used for software interchange.

b. Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by a written offer, valid for at least three years and valid for as long as you offer spare parts or customer support for that product model, to give anyone who possesses the object code either (1) a copy of the Corresponding Source for all the software in the product that is covered by this License, on a durable physical medium customarily used for software interchange, for a price no more than your reasonable cost of physically performing this conveying of source, or (2) access to copy the Corresponding Source from a network server at no charge.

c. Convey individual copies of the object code with a copy of the written offer to provide the Corresponding Source. This alternative is allowed only occasionally and noncommercially, and only if you received the object code with such an offer, in accord with subsection 6b.

d. Convey the object code by offering access from a designated place (gratis or for a charge), and offer equivalent access to the Corresponding Source in the same way through the same place at no further charge. You need not require recipients to copy the Corresponding Source along with the object code. If the place to copy the object code is a network server, the Corresponding Source may be on a different server (operated by you or a third party) that supports equivalent copying facilities, provided you maintain clear directions next to the object code saying where to find the Corresponding Source. Regardless of what server hosts the Corresponding Source, you remain obligated to ensure that it is available for as long as needed to satisfy these requirements.

e. Convey the object code using peer-to-peer transmission, provided you inform other peers where the object code and Corresponding Source of the work are being offered to the general public at no charge under subsection 6d.

A separable portion of the object code, whose source code is excluded from the Corresponding Source as a System Library, need not be included in conveying the object code work. A “User Product” is either (1) a “consumer product”, which means any tangible personal property which is normally used for personal, family, or household purposes, or (2) anything designed or sold for incorporation into a dwelling. In determining whether a product is a consumer product, doubtful cases shall be resolved in favor of coverage. For a particular product received by a particular user, “normally used” refers to a typical or common use of that class of product, regardless of the status of the particular user or of the way in which the particular user actually uses, or expects or is expected to use, the product. A product is a consumer product regardless of whether the product has substantial commercial, industrial or non-consumer uses, unless such uses represent the only significant mode of use of the product. “Installation Information” for a User Product means any methods, procedures, authorization keys, or other information required to install and execute modified versions of a covered work in that User Product from a modified version of its Corresponding Source.

The information must suffice to ensure that the continued functioning of the modified object code is in no case prevented or interfered with solely because modification has been made. If you convey an object code work under this section in, or with, or specifically for use in, a User Product, and the conveying occurs as part of a transaction in which the right of possession and use of the User Product is transferred to the recipient in perpetuity or for a fixed term (regardless of how the transaction is characterized), the Corresponding Source conveyed under this section must be accompanied by the Installation Information. But this requirement does not apply if neither you nor any third party retains the ability to install modified object code on the User Product (for example, the work has been installed in ROM). The requirement to provide Installation Information does not include a requirement to continue to provide support service, warranty, or updates for a work that has been modified or installed by the recipient, or for the User Product in which it has been modified or installed. Access to a network may be denied when the modification itself materially and adversely affects the operation of the network or violates the rules and protocols for communication across the network. Corresponding Source conveyed, and Installation Information provided, in accord with this section must be in a format that is publicly documented (and with an implementation available to the public in source code form), and must require no special password or key for unpacking, reading or copying.

7. Additional Terms. “Additional permissions” are terms that supplement the terms of this License by making exceptions from one or more of its conditions. Additional permissions that are applicable to the entire Program shall be treated as though they were included in this License, to the extent that they are valid under applicable law. If additional permissions apply only to part of the Program, that part may be used separately under those permissions, but the entire Program remains governed by this License without regard to the additional permissions. When you convey a copy of a covered work, you may at your option remove any additional permissions from that copy, or from any part of it. (Additional permissions may be written to require their own removal in certain cases when you modify the work.) You may place additional permissions on material, added by you to a covered work, for which you have or can give appropriate copyright permission. Notwithstanding any other provision of this License, for material you add to a covered work, you may (if authorized by the copyright holders of that material) supplement the terms of this License with terms: a. Disclaiming warranty or limiting liability differently from the terms of sections 15

and 16 of this License; or b. Requiring preservation of specified reasonable legal notices or author attributions

in that material or in the Appropriate Legal Notices displayed by works containing it; or

c. Prohibiting misrepresentation of the origin of that material, or requiring that modified versions of such material be marked in reasonable ways as different from the original version; or d. Limiting the use for publicity purposes of names of licensors or authors of the

material; or e. Declining to grant rights under trademark law for use of some trade names, trade-

marks, or service marks; or f. Requiring indemnification of licensors and authors of that material by anyone who

conveys the material (or modified versions of it) with contractual assumptions of liability to the recipient, for any liability that these contractual assumptions directly impose on those licensors and authors.

All other non-permissive additional terms are considered “further restrictions” within the meaning of section 10. If the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term. If a license document contains a further restriction but permits relicensing or conveying under this License, you may add to a covered work material governed by the terms of that license document, provided that the further restriction does not survive such relicensing or conveying. If you add terms to a covered work in accord with this section, you must place, in the relevant source files, a statement of the additional terms that apply to those files, or a notice indicating where to find the applicable terms. Additional terms, permissive or non-permissive, may be stated in the form of a separately written license, or stated as exceptions; the above requirements apply either way.

8. Termination.

You may not propagate or modify a covered work except as expressly provided under this License. Any attempt otherwise to propagate or modify it is void, and will automatically terminate your rights under this License (including any patent licenses granted under the third paragraph of section 11). However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation. Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice. Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, you do not qualify to receive new licenses for the same material under section 10.

9. Acceptance Not Required for Having Copies. You are not required to accept this License in order to receive or run a copy of the Program. Ancillary propagation of a covered work occurring solely as a consequence of using peer-to-peer transmission to receive a copy likewise does not require acceptance.

However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so.

10. Automatic Licensing of Downstream Recipients. Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License. An “entity transaction” is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party’s predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts. You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it.

11. Patents.

A “contributor” is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor’s “contributor version”. A contributor’s “essential patent claims” are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, “control” includes the right to grant patent sublicenses in a manner consistent with the requirements of this License. Each contributor grants you a non-exclusive, worldwide, royalty-free patent license under the contributor’s essential patent claims, to make, use, sell, offer for sale, import and otherwise run, modify and propagate the contents of its contributor version. In the following three paragraphs, a “patent license” is any express agreement or commitment, however denominated, not to enforce a patent (such as an express permission to practice a patent or covenant not to sue for patent infringement). To “grant” such a patent license to a party means to make such an agreement or commitment not to enforce a patent against the party. If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. “Knowingly relying” means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient’s use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid.

If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it.

A patent license is “discriminatory” if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007.

Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law.

12. No Surrender of Others’ Freedom.

If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program.

13. Use with the GNU Affero General Public License.

Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such.

14. Revised Versions of this License.

The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License “or any later version” applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation.

If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy’s public statement of acceptance of a version permanently authorizes you to choose that version for the Program.

Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version.

15. Disclaimer of Warranty.

THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PER-MITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM “AS IS” WITHOUT WARRANTY OF ANY KIND, EITHER EX-PRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFEC-TIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

16. Limitation of Liability.

IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, IN-CIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUS-TAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAM-AGES.

17. Interpretation of Sections 15 and 16.

If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee.

END OF TERMS AND CONDITIONS

How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the “copyright” line and a pointer to where the full notice is found.

      one line to give the program's name and a brief idea of what it does.
      Copyright (C) year name of author
      This program is free software: you can redistribute it and/or modify
      it under the terms of the GNU General Public License as published by
      the Free Software Foundation, either version 3 of the License, or (at
      your option) any later version.
      This program is distributed in the hope that it will be useful, but
      WITHOUT ANY WARRANTY; without even the implied warranty of
      MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
      General Public License for more details.
      You should have received a copy of the GNU General Public License
      along with this program.  If not, see http://www.gnu.org/licenses/.

Also add information on how to contact you by electronic and paper mail.

If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode:

      program Copyright (C) year name of author
      This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
      This is free software, and you are welcome to redistribute it
      under certain conditions; type `show c' for details.

The hypothetical commands ‘show w’ and ‘show c’ should show the appropriate parts of the General Public License. Of course, your program’s commands might be different; for a GUI interface, you would use an “about box”.

You should also get your employer (if you work as a programmer) or school, if any, to sign a “copyright disclaimer” for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see http://www.gnu.org/licenses/.

The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please

read http://www.gnu.org/philosophy/why-not-lgpl.html.

Concept Index 869

Concept Index

#

‘#’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ‘#!’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ‘#{’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

%

‘%’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ‘%{’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

-

--braindead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

--built-in-docstrings-file filename . . . . . . . 15

--debug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 --debug-jit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

--doc-cache-file filename . . . . . . . . . . . . . . . . . . 15

--echo-commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 --exec-path path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 --force-gui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --image-path path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --info-file filename . . . . . . . . . . . . . . . . . . . . . . . . 16 --info-program program . . . . . . . . . . . . . . . . . . . . . . 16 --interactive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --jit-compiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --line-editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --no-gui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --no-history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --no-init-file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --no-init-path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --no-line-editing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --no-site-file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 --no-window-system . . . . . . . . . . . . . . . . . . . . . . . . . . 17 --norc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 --path path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 --persist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 --quiet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 --silent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

--texi-macros-file filename . . . . . . . . . . . . . . . . 17

--traditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 --verbose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 --version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 -? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 -d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 -f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 -h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 -H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 -i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 -p path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 -q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 -v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 -V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 -x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

.

... continuation marker . . . . . . . . . . . . . . . . . . . . . . 165 .octaverc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

:

:end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

                              \

\ continuation marker . . . . . . . . . . . . . . . . . . . . . . . . 165

                              ~

~/.inputrc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ~/.octaverc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

A

acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 683 and operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142, 683 anonymous functions . . . . . . . . . . . . . . . . . . . . . . . . . 190 ans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 answers, incorrect. . . . . . . . . . . . . . . . . . . . . . . . 829, 831 application-defined data . . . . . . . . . . . . . . . . . . . . . . 363 apply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 area series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 arguments in function call . . . . . . . . . . . . . . . . . . . . 135 arithmetic operators . . . . . . . . . . . . . . . . . . . . . 137, 683 array, creating a Java array . . . . . . . . . . . . . . . . . . . 737 assignment expressions . . . . . . . . . . . . . . . . . . . . . . . 145 assignment operators . . . . . . . . . . . . . . . . . . . . . . . . . 145 axes graphics object . . . . . . . . . . . . . . . . . . . . . . . . . . 336 axes properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

B

bar series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 batch processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 block comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 body of a loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 boolean expressions . . . . . . . . . . . . . . . . . . . . . . 142, 683 boolean operators. . . . . . . . . . . . . . . . . . . . . . . . 142, 683 break statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 broadcast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 broadcasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 BSX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 bug criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 bug tracker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 bugs, investigating. . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 bugs, known. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 bugs, reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 built-in data types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 built-in function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

C

callbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 calling Java from Octave. . . . . . . . . . . . . . . . . . . . . . 737 calling Octave from Java. . . . . . . . . . . . . . . . . . . . . . 737 case statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 catch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 cell arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43, 110 character strings . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 65 Cholesky factorization . . . . . . . . . . . . . . . . . . . . . . . . 442 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Citing Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 classes, making available to Octave . . . . . . . . . . . 745 classpath, adding new path . . . . . . . . . . . . . . . . . . . 740 classpath, difference between static and dynamic

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 classpath, displaying. . . . . . . . . . . . . . . . . . . . . . . . . . 739 classpath, dynamic . . . . . . . . . . . . . . . . . . . . . . 739, 740 classpath, removing path . . . . . . . . . . . . . . . . . . . . . 740 classpath, setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 classpath, static . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 ‘classpath.txt’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 clearing the screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 code profiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 coding standards . . . . . . . . . . . . . . . . . . . . . . . . 809, 819 colors, graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 comma separated lists . . . . . . . . . . . . . . . . . . . . . . . . 118 command and output logs . . . . . . . . . . . . . . . . . . . . . 33 command completion . . . . . . . . . . . . . . . . . . . . . . . . . . 27 command descriptions . . . . . . . . . . . . . . . . . . . . . . . . . 13 command echoing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 command history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 command options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 command-line editing . . . . . . . . . . . . . . . . . . . . . . . . . . 24 comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 comparison expressions . . . . . . . . . . . . . . . . . . 140, 683 complex-conjugate transpose . . . . . . . . . . . . . 137, 683 containers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 continuation lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 continue statement . . . . . . . . . . . . . . . . . . . . . . . . . . 163 contour series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 contributing to Octave . . . . . . . . . . . . . . . . . . . . . . . . . . 5 contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 conversion specifications (printf). . . . . . . . . . . . . 244 conversion specifications (scanf) . . . . . . . . . . . . . . 250 copy-on-write. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 copyright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 core dump. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 COW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 creating graphics objects . . . . . . . . . . . . . . . . . . . . . 336 cs-lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 customizing readline . . . . . . . . . . . . . . . . . . . . . . . . . 31 customizing the prompt. . . . . . . . . . . . . . . . . . . . . . . . 31

D

DAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 data sources in object groups . . . . . . . . . . . . . . . . . 367 data structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 97 data types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 data types, built-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 data types, user-defined. . . . . . . . . . . . . . . . . . . . . . . . 43 decrement operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 default arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 default graphics properties. . . . . . . . . . . . . . . . . . . . 360 defining functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 deprecated functions. . . . . . . . . . . . . . . . . . . . . . . . . . 825 description format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 diagonal and permutation matrices . . . . . . . . . . . 479 diagonal matrix expressions. . . . . . . . . . . . . . . . . . . 481 dialog, displaying a help dialog . . . . . . . . . . . . . . . 742 dialog, displaying a list dialog. . . . . . . . . . . . . . . . . 743 dialog, displaying a question dialog . . . . . . . . . . . 744 dialog, displaying a warning dialog . . . . . . . 742, 744 dialog, displaying an error dialog. . . . . . . . . . . . . . 742 dialog, displaying an input dialog . . . . . . . . . . . . . 743 diary of commands and output. . . . . . . . . . . . . . . . . 33 differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 535 diffs, submitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 distribution of Octave. . . . . . . . . . . . . . . . . . . . . . . . . . . 5 division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 683 do-until statement . . . . . . . . . . . . . . . . . . . . . . . . . . 160 documentation fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 documentation notation . . . . . . . . . . . . . . . . . . . . . . . 11 documenting functions . . . . . . . . . . . . . . . . . . . . . . . . . 37 documenting Octave programs . . . . . . . . . . . . . . . . . 36 documenting user scripts . . . . . . . . . . . . . . . . . . . . . . 37 Dulmage-Mendelsohn decomposition . . . . . . . . . . 506 dynamic classpath . . . . . . . . . . . . . . . . . . . . . . . 739, 745 dynamic classpath, adding new path . . . . . . . . . . 740 dynamic naming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 dynamic-linking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 Dynamically Linked Functions . . . . . . . . . . . . . . . . 759

E

echoing executing commands. . . . . . . . . . . . . . . . . . . 33 editing the command line . . . . . . . . . . . . . . . . . . . . . . 24 element-by-element evaluation . . . . . . . . . . . . . . . . 142 else statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 elseif statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Emacs TAGS files . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851 end statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 end, indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 end_try_catch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 end_unwind_protect . . . . . . . . . . . . . . . . . . . . . . . . . 164 endfor statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 endfunction statement . . . . . . . . . . . . . . . . . . . . . . . 167 endif statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 endswitch statement . . . . . . . . . . . . . . . . . . . . . . . . . 157 endwhile statement . . . . . . . . . . . . . . . . . . . . . . . . . . 159 equality operator . . . . . . . . . . . . . . . . . . . . . . . . 140, 683 equality, tests for . . . . . . . . . . . . . . . . . . . . . . . . 140, 683 equations, nonlinear . . . . . . . . . . . . . . . . . . . . . . . . . . 473 erroneous messages . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 erroneous results. . . . . . . . . . . . . . . . . . . . . . . . . 829, 831 error bar series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 error ids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 error message notation. . . . . . . . . . . . . . . . . . . . . . . . . 12 error messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 error messages, incorrect. . . . . . . . . . . . . . . . . . . . . . 829 escape sequence notation . . . . . . . . . . . . . . . . . . . . . . 65 evaluation notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 executable scripts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 execution speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 exiting octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 19 exponentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 683 expression, range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 expressions, assignment . . . . . . . . . . . . . . . . . . . . . . . 145 expressions, boolean . . . . . . . . . . . . . . . . . . . . . 142, 683 expressions, comparison. . . . . . . . . . . . . . . . . . 140, 683 expressions, logical. . . . . . . . . . . . . . . . . . . . . . . 142, 683

F

factorial function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 fatal signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 field, returning value of Java object field . . . . . . 738 field, setting value of Java object field. . . . . . . . . 738 fields, displaying available fields of a Java object

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 figure graphics object . . . . . . . . . . . . . . . . . . . . . . . . . 336 figure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 finding minimums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 flag character (printf) . . . . . . . . . . . . . . . . . . . . . . . 245 flag character (scanf) . . . . . . . . . . . . . . . . . . . . . . . . 250 for statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Frobenius norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 function application . . . . . . . . . . . . . . . . . . . . . . . . . . 462 function descriptions. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 function file. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 178 function statement . . . . . . . . . . . . . . . . . . . . . . . . . . 167 functions, deprecated . . . . . . . . . . . . . . . . . . . . . . . . . 825 functions, obsolete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 functions, user-defined . . . . . . . . . . . . . . . . . . . . . . . . 167 funding Octave development . . . . . . . . . . . . . . . . . . . . 5

G

general p-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 global statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 global variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 grammar rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 graphics colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 graphics data structures . . . . . . . . . . . . . . . . . . . . . . 334 graphics line styles . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 graphics marker styles . . . . . . . . . . . . . . . . . . . . . . . . 361 graphics object properties . . . . . . . . . . . . . . . . . . . . 343 graphics object, axes . . . . . . . . . . . . . . . . . . . . . . . . . 336 graphics object, figure . . . . . . . . . . . . . . . . . . . . . . . . 336 graphics object, image . . . . . . . . . . . . . . . . . . . . . . . . 336 graphics object, line . . . . . . . . . . . . . . . . . . . . . . . . . . 336 graphics object, patch . . . . . . . . . . . . . . . . . . . . . . . . 336 graphics object, root figure . . . . . . . . . . . . . . . . . . . 336 graphics object, surface . . . . . . . . . . . . . . . . . . . . . . . 336 graphics object, text. . . . . . . . . . . . . . . . . . . . . . . . . . 336 graphics objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 graphics objects, saving. . . . . . . . . . . . . . . . . . . . . . . 343 graphics properties, default . . . . . . . . . . . . . . . . . . . 360 graphics toolkits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 greater than operator . . . . . . . . . . . . . . . . . . . . 140, 683 group objects . . . . . . . . . . . . . . . . . . . . . . . 372, 373, 375

H

handle functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 handle, function handles . . . . . . . . . . . . . . . . . . . . . . 190 header comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 help, online. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 help, user-defined functions . . . . . . . . . . . . . . . . . . . . 37 help, where to find . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 Hermitian operator . . . . . . . . . . . . . . . . . . . . . . 137, 683 Hessenberg decomposition . . . . . . . . . . . . . . . . . . . . 444 history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 history of commands. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

I

if statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 image graphics object . . . . . . . . . . . . . . . . . . . . . . . . 336 image properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 improving Octave . . . . . . . . . . . . . . . . . . . . . . . . 830, 831 incorrect error messages . . . . . . . . . . . . . . . . . . . . . . 829 incorrect output . . . . . . . . . . . . . . . . . . . . . . . . . 829, 831 incorrect results . . . . . . . . . . . . . . . . . . . . . . . . . 829, 831 increment operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 infinity norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 inline, inline functions . . . . . . . . . . . . . . . . . . . . . . . . 190 input conversions, for scanf . . . . . . . . . . . . . . . . . . 250 input history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 installation trouble . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 installing Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 instance, how to create . . . . . . . . . . . . . . . . . . . . . . . 747 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 introduction to graphics structures. . . . . . . . . . . . 334 invalid input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829

J

Java, calling from Octave . . . . . . . . . . . . . . . . . . . . . 737 Java, using with Octave . . . . . . . . . . . . . . . . . . . . . . 737 ‘javaclasspath.txt’ . . . . . . . . . . . . . . . . . . . . . . . . . 745

K

Kendall’s Tau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 known causes of trouble . . . . . . . . . . . . . . . . . . . . . . 829

L

language definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 less than operator . . . . . . . . . . . . . . . . . . . . . . . 140, 683 line graphics object . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 line properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 line series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 line styles, graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 linear algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 linear algebra, techniques . . . . . . . . . . . . . . . . . . . . . 435 loadable function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 loading data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 local minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 logging commands and output . . . . . . . . . . . . . . . . . 33 logical expressions . . . . . . . . . . . . . . . . . . . . . . . 142, 683 logical operators . . . . . . . . . . . . . . . . . . . . . . . . . 142, 683 loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 looping over structure elements . . . . . . . . . . . . . . . 161 LP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 LU decomposition . . . . . . . . . . . . . . . . . . . . . . . 444, 520 lvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

M

map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 mapping function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 marker styles, graphics . . . . . . . . . . . . . . . . . . . . . . . 361 matching failure, in scanf . . . . . . . . . . . . . . . . . . . . 250 matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 matrices, diagonal and permutation . . . . . . . . . . . 479 matrix factorizations . . . . . . . . . . . . . . . . . . . . . . . . . 442 matrix functions, basic . . . . . . . . . . . . . . . . . . . . . . . 435 matrix multiplication . . . . . . . . . . . . . . . . . . . . 137, 683 matrix, functions of . . . . . . . . . . . . . . . . . . . . . . . . . . 452 matrix, permutation functions . . . . . . . . . . . . . . . . 483 matrix, specialized solvers . . . . . . . . . . . . . . . . . . . . 454 matrix, zero elements . . . . . . . . . . . . . . . . . . . . . . . . . 484 maximum field width (scanf). . . . . . . . . . . . . . . . . 250 memory management . . . . . . . . . . . . . . . . . . . . . . . . . 470 memory, displaying Java memory status . . . . . . 740 memory, limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 747 messages, error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 method, invoking a method of a Java object . . 739 methods, displaying available methods of a Java

object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 mex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 mex-files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 minimum field width (printf) . . . . . . . . . . . . . . . . 245 missing data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 mkoctfile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 multi-line comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 683

N

negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 683 NLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 nonlinear programming . . . . . . . . . . . . . . . . . . . . . . . 547 not operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142, 683 numeric constant. . . . . . . . . . . . . . . . . . . . . . . . . . . 41, 47 numeric value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41, 47

O

object groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 object, creating a Java object . . . . . . . . . . . . . . . . . 737 object, how to create . . . . . . . . . . . . . . . . . . . . . . . . . 747 obsolete functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 oct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 oct-files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 Octave and matlab, how to distinguish between

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Octave API . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 Octave command options . . . . . . . . . . . . . . . . . . . . . . 15 Octave development . . . . . . . . . . . . . . . . . . . . . . . . . . 819 Octave, calling from Java . . . . . . . . . . . . . . . . . . . . . 737 octave-tags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851 ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 online help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 operator precedence . . . . . . . . . . . . . . . . . . . . . . . . . . 148 operators, arithmetic . . . . . . . . . . . . . . . . . . . . 137, 683 operators, assignment . . . . . . . . . . . . . . . . . . . . . . . . 145 operators, boolean . . . . . . . . . . . . . . . . . . . . . . . 142, 683 operators, decrement . . . . . . . . . . . . . . . . . . . . . . . . . 147 operators, increment. . . . . . . . . . . . . . . . . . . . . . . . . . 147 operators, logical . . . . . . . . . . . . . . . . . . . . . . . . 142, 683 operators, relational . . . . . . . . . . . . . . . . . . . . . 140, 683 optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470, 547 options, Octave command . . . . . . . . . . . . . . . . . . . . . 15 or operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142, 683 oregonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 otherwise statement . . . . . . . . . . . . . . . . . . . . . . . . . 157 output conversions, for printf . . . . . . . . . . . . . . . . 246

P

parser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 patch graphics object . . . . . . . . . . . . . . . . . . . . . . . . . 336 patch properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 patches, submitting. . . . . . . . . . . . . . . . . . . . . . . . . . . 831 path, adding to classpath . . . . . . . . . . . . . . . . . . . . . 740 path, removing from classpath . . . . . . . . . . . . . . . . 740 permutation matrix functions . . . . . . . . . . . . . . . . . 483 persistent statement . . . . . . . . . . . . . . . . . . . . . . . . 124 persistent variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 personal startup file . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 PKG ADD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 PKG DEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 plotting, high-level . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 plotting, multiple plot windows . . . . . . . . . . . . . . . 319 plotting, multiple plots per figure . . . . . . . . . . . . . 318 plotting, saving and printing plots . . . . . . . . . . . . 327 plotting, three-dimensional . . . . . . . . . . . . . . . . . . . 291 plotting, two-dimensional functions . . . . . . . . . . . 288 plotting, window manipulation. . . . . . . . . . . . . . . . 319 precision (printf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 printing notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 printing plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 program, self contained . . . . . . . . . . . . . . . . . . . . . . . . 35 Progress Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 project startup file. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 prompt customization . . . . . . . . . . . . . . . . . . . . . . . . . 31 pseudoinverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441, 482

Q

QP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 QR factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 quadratic programming . . . . . . . . . . . . . . . . . . . . . . . 547 quitting octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 19 quiver group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 683

R

range expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 readline customization . . . . . . . . . . . . . . . . . . . . . . . 31 recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 relational operators . . . . . . . . . . . . . . . . . . . . . . 140, 683 reporting bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . 829, 830 results, incorrect . . . . . . . . . . . . . . . . . . . . . . . . . 829, 831 root figure graphics object . . . . . . . . . . . . . . . . . . . . 336 root figure properties . . . . . . . . . . . . . . . . . . . . . . . . . 344

S

saving data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 saving graphics objects . . . . . . . . . . . . . . . . . . . . . . . 343 saving plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 scatter group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Schur decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . 450 script files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 scripts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 self contained programs . . . . . . . . . . . . . . . . . . . . . . . . 35 series objects. . . . . . . . . . . . . . . . . . . 368, 369, 371, 374 short-circuit evaluation . . . . . . . . . . . . . . . . . . . . . . . 143 side effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 SIMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 singular value decomposition. . . . . . . . . . . . . . . . . . 450 site startup file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Spearman’s Rho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 speedups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 stair group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 standards of coding style . . . . . . . . . . . . . . . . . . . . . 809 startup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 startup files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 static classpath . . . . . . . . . . . . . . . . . . . . . . . . . . 739, 745 stem series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 65 structural rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 structure elements, looping over . . . . . . . . . . . . . . 161 structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 97 submitting diffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 submitting patches . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 683 suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 surface graphics object . . . . . . . . . . . . . . . . . . . . . . . 336 surface group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 surface properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 switch statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 symbols, translation table . . . . . . . . . . . . . . . . . . . . 748

T

TAGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851 test functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 tests for equality. . . . . . . . . . . . . . . . . . . . . . . . . 140, 683 TEX symbols, translation table. . . . . . . . . . . . . . . . 748 text graphics object . . . . . . . . . . . . . . . . . . . . . . . . . . 336 text properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 toolkit customization . . . . . . . . . . . . . . . . . . . . . . . . . 376 toolkits, graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 translation table for TEX symbols. . . . . . . . . . . . . 748 transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 683 transpose, complex-conjugate . . . . . . . . . . . . 137, 683 troubleshooting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 try statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

U

unary minus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 683 undefined behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 undefined function value . . . . . . . . . . . . . . . . . . . . . . 829 unwind_protect statement . . . . . . . . . . . . . . . . . . . 164 unwind_protect_cleanup. . . . . . . . . . . . . . . . . . . . . 164 use of comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 user-defined data types . . . . . . . . . . . . . . . . . . . . . . . . 43 user-defined functions . . . . . . . . . . . . . . . . . . . . . . . . 167 user-defined variables . . . . . . . . . . . . . . . . . . . . . . . . . 121 using Octave with Java . . . . . . . . . . . . . . . . . . . . . . . 737

V

varargin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 varargout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 variable-length argument lists. . . . . . . . . . . . . . . . . 174 variable-length return lists . . . . . . . . . . . . . . . . . . . . 176 variables, global . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 variables, persistent . . . . . . . . . . . . . . . . . . . . . . . . . . 124 variables, user-defined . . . . . . . . . . . . . . . . . . . . . . . . 121 vectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 vectorize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 version startup file. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

874 GNU Octave

W

warning ids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 warranty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 while statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 wrong answers . . . . . . . . . . . . . . . . . . . . . . . . . . . 829, 831

Function Index 875

Function Index

A

abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 accumarray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 accumdim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 acos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 acosd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 acosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 acot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 acotd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 acoth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 acsc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 acscd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 acsch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 add_input_event_hook . . . . . . . . . . . . . . . . . . . . . . . 855 addlistener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 addpath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 addpref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 addproperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 addtodate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 airy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 all . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 allchild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

allow_noninteger_range_as_index . . . . . . . . . . 134

amd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 ancestor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 anova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 any . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 arch_fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 arch_rnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 arch_test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 arg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 argnames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 argv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 arma_rnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 arrayfun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 ascii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 asctime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 asec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 asecd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 asech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 asin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 asind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 asinh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 assert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 assignin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 atan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 atan2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 atan2d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 atand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 atanh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 atexit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 autoload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 autoreg_matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 autumn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

                                                available_graphics_toolkits . . . . . . . . . . . . . . . 376

axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285, 286

B

balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264, 265 barh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 bartlett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 bartlett_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 base2dec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 base64_decode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 base64_encode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 beep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 beep_on_error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 besselh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 besseli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 besselj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 besselk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 bessely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 betacdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 betainc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 betaincinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 betainv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 betaln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 betapdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 betarnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 bicg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 bicgstab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 bicubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 bin2dec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 binary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 bincoeff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 binocdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 binoinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 binopdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 binornd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 bitand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 bitcmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 bitget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 bitmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 bitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 bitpack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 bitset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 bitshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 bitunpack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 bitxor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 blackman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 blanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 blkdiag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 blkmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 brighten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 bsxfun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

built_in_docstrings_file . . . . . . . . . . . . . . . . . . . 23

builtin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 bunzip2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 byte_size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 bzip2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

C

calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 canonicalize_file_name. . . . . . . . . . . . . . . . . . . . . 715 cart2pol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 cart2sph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 cast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 cauchy_cdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 cauchy_inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 cauchy_pdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 cauchy_rnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 caxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 cbrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 ccolamd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 cd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 719, 729 ceil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 cell2mat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 cell2struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 celldisp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 cellfun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 cellindexmat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 cellslices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 cellstr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 cgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 char . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 chdir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 729 chi2cdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 chi2inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 chi2pdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 chi2rnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

chisquare_test_homogeneity . . . . . . . . . . . . . . . . 582
chisquare_test_independence . . . . . . . . . . . . . . . 582

chol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 chol2inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 choldelete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 cholinsert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 cholinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 cholshift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 cholupdate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 chop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 circshift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 citation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 cla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 clabel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 clc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 clear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 clf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 cloglog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 close . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323, 718 closereq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 cmpermute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 cmunique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 colamd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 colloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 colon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 colorbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316, 317 colorcube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 colormap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 colperm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 colstyle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 comet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 comet3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 command_line_path . . . . . . . . . . . . . . . . . . . . . . . . . . 182 common_size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 commutation_matrix . . . . . . . . . . . . . . . . . . . . . . . . . 424 compan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 compare_versions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 compass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 completion_append_char . . . . . . . . . . . . . . . . . . . . . . 27 completion_matches . . . . . . . . . . . . . . . . . . . . . . . . . . 27 complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 cond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 condest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

confirm_recursive_rmdir . . . . . . . . . . . . . . . . . . . 709

conj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 contour3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 contourc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 contourf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 conv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 conv2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 convhull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 convhulln. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 convn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 cool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 copyfile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 copyobj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 cor_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 corr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 cos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 cosd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 cosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 cotd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 coth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 cplxpair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 cputime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

crash_dumps_octave_core . . . . . . . . . . . . . . . . . . . 238

cross . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 csc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 cscd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 csch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 cstrcat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 csvread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 csvwrite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 csymamd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 ctime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 ctranspose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 cummax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cummin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 cumprod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cumsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 cumtrapz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

D

daspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 daspk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 daspk_options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 dasrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 dasrt_options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 dassl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 dassl_options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 datenum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 datestr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 datetick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 datevec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 dawson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 dbclear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 dbcont . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 dbdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 dblist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 dblquad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 dbnext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 dbquit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 dbstack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 dbstatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 dbstep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 dbstop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 dbtype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214, 215 dbup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 dbwhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 deal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 deblank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 debug_java . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 debug_jit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 debug_on_error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 debug_on_interrupt . . . . . . . . . . . . . . . . . . . . . . . . . 211 debug_on_warning. . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 dec2base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 dec2bin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 dec2hex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 deconv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 del2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 delaunay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 delaunay3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 delaunayn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 delete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323, 719 dellistener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 demo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 det . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 detrend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 diag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 diary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 diff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 diffpara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 diffuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 dims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 dir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719, 729 discrete_cdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 discrete_inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 discrete_pdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 discrete_rnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 disp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 dlmread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 dlmwrite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 dmperm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

                                                do_braindead_shortcircuit_evaluation. . . . . 144

do_string_escapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 doc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 doc_cache_create. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 doc_cache_file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 dos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 double . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 drawnow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 dsearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 dsearchn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 dump_prefs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 dup2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 duplication_matrix . . . . . . . . . . . . . . . . . . . . . . . . . 424 durbinlevinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646

E

e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

                                                echo_executing_commands. . . . . . . . . . . . . . . . . . . . . 33

edit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 edit_history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 EDITOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 eig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 eigs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 elem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 ellipj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 ellipke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 ellipsoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 empirical_cdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 empirical_inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 empirical_pdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 empirical_rnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 endgrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 endpwent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 eomday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 eps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 eq. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 erf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 erfc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 erfcinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 erfcx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 erfi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 erfinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 errno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 errno_list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 errorbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276, 277 errordlg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 etime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 etree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 etreeplot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 eval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 evalin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 exec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 EXEC_PATH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 exist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 exit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 expcdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 expint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 expinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 expm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 expm1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 exppdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 exprnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 ezcontour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 ezcontourf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 ezmesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 ezmeshc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309, 310 ezplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 ezplot3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 ezpolar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 ezsurf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 ezsurfc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

F

f_test_regression . . . . . . . . . . . . . . . . . . . . . . . . . . 583 factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 factorial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 fail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 false . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 fcdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 fclear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 fclose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 fcntl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 fdisp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 feather . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 feof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 ferror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 feval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 fflush . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 fft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 fft2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 fftconv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 fftfilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 fftn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 fftshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 fftw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 fgetl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 fgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 fieldnames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 file_in_loadpath. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 file_in_path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 fileattrib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 filemarker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 fileparts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 fileread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 filesep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 fill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 filter2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 find . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 find_dir_in_path. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 findall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359, 360 findfigs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 findobj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 findstr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 finv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 fix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 fixed_point_format . . . . . . . . . . . . . . . . . . . . . . . . . . 51 flag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 flintmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 flipdim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 fliplr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 flipud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 fminbnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 fminsearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 fminunc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 fmod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 fnmatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 foo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 fopen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 fork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 fortran_vec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 fpdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 fplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 fprintf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 fputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 fractdiff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 fread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 freport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 freqz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 freqz_plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 frewind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 frnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 fscanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 fseek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 fskipl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 fsolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 ftell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 ftp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 full . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 fullfile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 func2str . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 fwrite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 fzero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

G

gallery . . . . . . . . . . . . . . 399, 400, 401, 402, 403, 404 gamcdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 gaminv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 gammainc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 gammaln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 gampdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 gamrnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 gca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 gcbf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 gcbo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 gcd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 gcf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 gco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 ge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 genpath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 genvarname . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 geocdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 geoinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 geopdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 geornd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

get_first_help_sentence. . . . . . . . . . . . . . . . . . . . . 24

get_help_text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

get_help_text_from_file. . . . . . . . . . . . . . . . . . . . . 24

getappdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 getegid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 getenv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 geteuid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 getfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 getgid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 getgrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 getgrgid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 getgrnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 gethostname . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 getpgrp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 getpid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 getppid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 getpref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 getpwent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 getpwnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 getpwuid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 getrusage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 getuid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 ginput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 givens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 glob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 glpk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 gls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 gmap40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 gmres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 gmtime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 gnuplot_binary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 gplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417, 418 graphics_toolkit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 gray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 gray2ind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 griddata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 griddata3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 griddatan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 gt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 gtext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 gui_mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 guidata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 guihandles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 gunzip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 gzip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

H

hadamard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 hamming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 hankel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 hanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 hdl2struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 helpdlg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 hess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 hex2dec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 hex2num . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 hggroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 hidden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 hilb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 hist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 histc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 history_control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 history_file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 history_save . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 history_size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

history_timestamp_format_string . . . . . . . . . . . 30

hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 home . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 horzcat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 hot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 hotelling_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 hotelling_test_2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 housh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 hsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 hsv2rgb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 hurst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 hygecdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 hygeinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 hygepdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 hygernd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 hypot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

I

i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 idivide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ifelse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 ifft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 ifft2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 ifftn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 ifftshift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

ignore_function_time_stamp . . . . . . . . . . . . . . . . 180

imag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 IMAGE_PATH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 imagesc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 imfinfo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 imformats. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 importdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 imread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 imshow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 imwrite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 ind2gray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 ind2rgb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 ind2sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 inf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Inf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 inferiorto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 info . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 info_file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 info_program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 inline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 inpolygon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 inputdlg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 inputname. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 int16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 int2str . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 int32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 int64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 int8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 interp1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 interp2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 interp3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 interpft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 interpn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 intersect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 intmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 intmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 invhilb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 ipermute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 iqr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 is_absolute_filename . . . . . . . . . . . . . . . . . . . . . . . 715 is_dq_string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 is_function_handle . . . . . . . . . . . . . . . . . . . . . . . . . 191 is_leap_year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

is_rooted_relative_filename . . . . . . . . . . . . . . . 715

is_sq_string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 is_valid_file_id. . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 isa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 isalnum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 isalpha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 isappdata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 isargout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 isascii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 isaxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 isbool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 iscell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 iscellstr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 ischar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 iscntrl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 iscolormap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 iscolumn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 iscomplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 isdebugmode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 isdefinite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 isdeployed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 isdigit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 isdir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 isempty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 isequal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 isequaln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 isfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 isfigure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 isfinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 isfloat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 isglobal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 isgraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 isguirunning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 ishandle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 ishermitian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 ishghandle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 ishold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 isieee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 isindex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 isinf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 isinteger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 isjava . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 iskeyword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 isletter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 islogical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 islower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 ismac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 ismatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ismember . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 ismethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 isna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 isnan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 isnull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 isnumeric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 isobject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 isocolors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 isonormals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 isosurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 ispc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 ispref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 isprime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 isprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 isprop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 ispunct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 isreal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 isrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 isscalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 issorted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 isspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 issparse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 issquare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 isstrprop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 isstruct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 issymmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 isunix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 isupper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 isvarname. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 isvector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 isxdigit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

J

j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 java_get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738

java_matrix_autoconversion . . . . . . . . . . . . . . . . 741

java_set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738

java_unsigned_autoconversion . . . . . . . . . . . . . . 741

javaaddpath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 javaArray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 javaclasspath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 javamem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 javaMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 javaObject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 javarmpath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 jit_enable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 jit_startcnt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

K

kbhit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 kendall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 keyboard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 kill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 kolmogorov_smirnov_cdf. . . . . . . . . . . . . . . . . . . . . 577

                                                kolmogorov_smirnov_test . . . . . . . . . . . . . . . . . . . 583
                                                kolmogorov_smirnov_test_2 . . . . . . . . . . . . . . . . . 584

kron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 kruskal_wallis_test . . . . . . . . . . . . . . . . . . . . . . . . 584 krylov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564

L

laplace_cdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 laplace_inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 laplace_pdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 laplace_rnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 lasterr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 lasterror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 lastwarn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 lcm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 ldivide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 le. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 legendre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 lgamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 lin2mu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 linkprop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 linsolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 linspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 list_in_columns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 list_primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 listdlg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 loadaudio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

                                                loaded_graphics_toolkits . . . . . . . . . . . . . . . . . . 376

loadobj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 localtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 log10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 log1p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 log2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 logical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 logistic_cdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 logistic_inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 logistic_pdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 logistic_regression . . . . . . . . . . . . . . . . . . . . . . . . 572 logistic_rnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 loglog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 loglogerr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 logm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 logncdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 logninv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 lognpdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 lognrnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 logspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 lookfor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 lookup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 lower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 ls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 ls_command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 lsode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 lsode_options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 lsqnonneg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 lstat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 lt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 lu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 luinc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 luupdate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

M

magic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 mahalanobis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 make_absolute_filename. . . . . . . . . . . . . . . . . . . . . 715 makeinfo_program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 manova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 mat2cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 mat2str . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 matlabroot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 matrix_type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 max_recursion_depth . . . . . . . . . . . . . . . . . . . . . . . . 137 mcnemar_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 md5sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 meansq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 merge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 meshc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 meshgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 meshz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 mex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 mexext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 mfilename. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 mget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 mgorth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 minus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 mislocked. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 missing_component_hook. . . . . . . . . . . . . . . . . . . . . 758 missing_function_hook . . . . . . . . . . . . . . . . . . . . . . 856 mkdir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709, 719 mkfifo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 mkoctfile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 mkpp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 mkstemp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 mktime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 mldivide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 mlock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 more . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 mouse_wheel_zoom. . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 movefile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 mpoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 mpower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 mput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 mrdivide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 msgbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 mtimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 mu2lin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 munlock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

N

NA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 namelengthmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 nan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 NaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 nargchk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 nargin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 narginchk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 nargout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 nargoutchk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 native_float_format . . . . . . . . . . . . . . . . . . . . . . . . 234 nbincdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 nbininv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 nbinpdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 nbinrnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 nchoosek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568, 817 ndgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 ndims. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43, 764 ne. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 newplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 news . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 nextpow2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 nfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 nnz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 nonzeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 normcdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 normest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 norminv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 normpdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 normrnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 not . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 nproc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 nth_element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 nthargout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 nthroot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 ntsc2rgb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 null . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 num2cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 num2hex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 num2str . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 numel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 nzmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

O

ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 octave_config_info . . . . . . . . . . . . . . . . . . . . . . . . . 735 octave_core_file_limit. . . . . . . . . . . . . . . . . . . . . 239 octave_core_file_name . . . . . . . . . . . . . . . . . . . . . . 239

octave_core_file_options . . . . . . . . . . . . . . . . . . 239

OCTAVE_HOME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 octave_idx_type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 octave_tmp_file_name . . . . . . . . . . . . . . . . . . . . . . . 255 OCTAVE_VERSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 ols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 onCleanup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 onenormest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 optimget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

optimize_subsasgn_calls . . . . . . . . . . . . . . . . . . . 680

optimset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 or. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 orderfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 orient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 orth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 ostrsplit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 output_max_field_width . . . . . . . . . . . . . . . . . . . . . . 49 output_precision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

P

P_tmpdir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 pack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

page_output_immediately . . . . . . . . . . . . . . . . . . . 227

page_screen_output . . . . . . . . . . . . . . . . . . . . . . . . . 227 PAGER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 PAGER_FLAGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 parseparams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 pascal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 pathdef . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 pathsep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 pause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 pbaspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 pcg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 pchip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 pclose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 pcolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 pcr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 periodogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 perl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 perms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 permute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 pi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 pie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 pie3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 pink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 pinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 pkg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749, 750 planerot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 playaudio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 plot3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 plotmatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 plotyy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 plus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 poisscdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 poissinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 poisspdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 poissrnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 pol2cart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 polar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 poly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 polyaffine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 polyarea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 polyder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 polyeig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 polyfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 polygcd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 polyint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 polyout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 polyreduce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 polyval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 polyvalm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 popen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 popen2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 postpad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 pow2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 powerset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 ppder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 ppint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 ppjumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 ppplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 ppval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 pqpnonneg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 prctile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 prefdir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 prepad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 print_empty_dimensions . . . . . . . . . . . . . . . . . . . . . . 51

print_struct_array_contents . . . . . . . . . . . . . . . . 99

print_usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 printd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 printf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 probit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 prod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 profexplore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 profshow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

program_invocation_name. . . . . . . . . . . . . . . . . . . . . 18

program_name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 prop_test_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 PS1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 PS2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 PS4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 putenv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 puts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 pwd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 python . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722

Q

qp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 qqplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 qr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 qrdelete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 qrinsert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 qrshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 qrupdate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 quad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 quad_options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 quadcc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 quadgk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 quadl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 quadv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 quantile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 questdlg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 quit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 quiver3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 qz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 qzhess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

R

rainbow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 rand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 rande . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 randg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396, 397 randi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 randn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 randp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 randperm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 rat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 rats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 rcond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 rdivide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 readdir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713

readline_re_read_init_file . . . . . . . . . . . . . . . . . 31
readline_read_init_file. . . . . . . . . . . . . . . . . . . . . 31

readlink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 real . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 reallog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 realmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 realmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432, 433 realpow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 realsqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 record . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 rectangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 rectint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 recycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 refresh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 refreshdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 regexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 regexpi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 regexprep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 regexptranslate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

register_graphics_toolkit . . . . . . . . . . . . . . . . . 376

rehash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 rem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

remove_input_event_hook . . . . . . . . . . . . . . . . . . . 856

rename . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708, 719 repelems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 repmat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 reset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 reshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 residue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 resize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385, 765 restoredefaultpath . . . . . . . . . . . . . . . . . . . . . . . . . 182 rethrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 rgb2hsv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 rgb2ind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 rgb2ntsc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 rgbplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 ribbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 rindex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 rmappdata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 rmdir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709, 719 rmfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 rmpath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 rmpref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 rose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 rosser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 rot90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 rotdim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 round . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 roundb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 rows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 rref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 rsf2csf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 run_count. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 run_history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 run_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 rundemos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 runlength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 runtests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806

S

S_ISBLK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 S_ISCHR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 S_ISDIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 S_ISFIFO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 S_ISLNK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 S_ISREG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 S_ISSOCK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 save . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 save_default_options . . . . . . . . . . . . . . . . . . . . . . . 231

save_header_format_string . . . . . . . . . . . . . . . . . 232

save_precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 saveas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 saveaudio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 saveobj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 savepath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 scanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 scatter3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 schur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 secd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 sech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 SEEK_CUR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 SEEK_END . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 SEEK_SET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 semilogx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 semilogxerr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 semilogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 semilogyerr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 setappdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 setaudio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 setdiff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 setenv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 setfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 setgrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 setpref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 setpwent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 setxor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 shading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 shg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 shiftdim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 shrinkfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 SIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728

sighup_dumps_octave_core . . . . . . . . . . . . . . . . . . 238

sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 sign_test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 signbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

sigterm_dumps_octave_core . . . . . . . . . . . . . . . . . 238

silent_functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 sinc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 sind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 sinetone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 sinewave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 single . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 sinh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 size_equal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 sizemax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 sizeof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 sleep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 sombrero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 sortrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 spalloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 sparse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 sparse_auto_mutate . . . . . . . . . . . . . . . . . . . . . . . . . 499 spaugment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 spconvert. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 spdiags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 spearman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 spectral_adf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 spectral_xdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 specular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 spencer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 speye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 spfun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 sph2cart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 spinmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 splinefit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 split_long_rows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 spones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 spparms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 sprand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 sprandn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 sprandsym. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 sprank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 sprintf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 spstats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 spy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 sqp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 sqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 sqrtm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 squeeze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 sscanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 stairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 stat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 std . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 stderr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 stdin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 stdnormal_cdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 stdnormal_inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 stdnormal_pdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 stdnormal_rnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 stdout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 stem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 stem3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 stemleaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 stft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 str2double. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 str2func . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 str2num . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 strcat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 strchr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 strcmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 strcmpi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 strfind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 strftime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 string_fill_char. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 strjoin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 strjust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 strmatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 strncmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 strncmpi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 strptime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 strread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 strrep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 strsplit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 strtok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 strtrim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 strtrunc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 struct_levels_to_print . . . . . . . . . . . . . . . . . . . . . . 99 struct2cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 struct2hdl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 structfun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 strvcat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 sub2ind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 subplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 subsasgn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 subsindex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 subsref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 substr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 substruct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 summer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 sumsq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 superiorto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

suppress_verbose_help_message . . . . . . . . . . . . . . 23

surf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 surfc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 surfl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 surfnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 svd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 svd_driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 svds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 swapbytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 syl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 symamd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 symbfact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 symlink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 symrcm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 symvar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

T

t_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 t_test_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 t_test_regression . . . . . . . . . . . . . . . . . . . . . . . . . . 586 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 tand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 tanh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 tar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 tcdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 tempdir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 tempname . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 terminal_size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 tetramesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 texi_macros_file. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 textread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 textscan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236, 237 tic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 tilde_expand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 tinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 tmpfile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 tmpnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 toascii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 toc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 toeplitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 tolower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 toupper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 tpdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 transpose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 trapz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 treelayout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 treeplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 tril . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 trimesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 triplequad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 triplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 trisurf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 triu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 trnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 true . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 tsearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 tsearchn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 typecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 typeinfo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

U

u_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 uigetdir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 uigetfile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 uimenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 uint16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 uint32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 uint64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 uint8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 uiputfile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 uiresume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 uiwait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 umask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 uminus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 uname . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 undo_string_escapes . . . . . . . . . . . . . . . . . . . . . . . . . 93 unidcdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 unidinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 unidpdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 unidrnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 unifcdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 unifinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 unifpdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 unifrnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 unique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 unix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721, 722 unlink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 unmkpp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 unpack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 untabify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 untar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 unwrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 unzip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 uplus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 upper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 urlread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 urlwrite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 usejava . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 usleep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

V

validatestring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 vander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 var . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 var_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 vec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 vech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 vectorize. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 ver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 vertcat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 voronoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 voronoin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

W

waitbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 waitfor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 waitforbuttonpress . . . . . . . . . . . . . . . . . . . . . . . . . 332 waitpid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 warndlg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 warranty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 waterfall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 wavread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 wavwrite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 wblcdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 wblinv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 wblpdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 wblrnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 WCONTINUE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 WCOREDUMP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 weekday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 welch_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 WEXITSTATUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 what . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 which . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 white . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 whitebg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 who . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 whos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 whos_line_format. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 wienrnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 WIFCONTINUED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 WIFEXITED. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 WIFSIGNALED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 WIFSTOPPED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 wilcoxon_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 wilkinson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 winter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 WNOHANG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 WSTOPSIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 WTERMSIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 WUNTRACED. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

X

xlabel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 xlim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 xor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

Y

yes_or_no. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

888 GNU Octave

ylim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 yulewalker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

Z

z_test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 z_test_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 zip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 zlim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 zscore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Operator Index

! ; ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142, 143, 684 ; != . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141, 684 ; " ; " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 65 ; & ; & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142, 684 ; && . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 ; ’ ; ’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 65, 138, 684 ; ( ; ( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 ; ) ; ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 ; * ; * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 139, 684 ; ** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138, 139 ; *= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 ; + ; + . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 138, 139, 140, 684 ; ++. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 ; += . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 ; , ; , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ; - ; - . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 138, 139, 140, 684 ; -- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147, 148 ; -= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 ; . ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 ; .’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138, 140, 684 ; .*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 140, 684 ; .** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138, 139  .+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 ; ./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 140, 684 ; .^. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138, 139, 684 ; .\. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138, 139, 684 ; / ; / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 139, 684 ; /= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 ; : ; : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52, 131, 684 ; ; ; ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ; < ; < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140, 141, 684 ; <=. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140, 141, 684 ; = ; = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 ; == . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140, 141, 684 ; > ; > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140, 141, 684 ; >=. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140, 141, 684 ; [ ; [ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ; ] ; ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ; ^ ; ^. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138, 139, 684 ; { ; { . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 ; } ; } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

\ ; \ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137, 139, 684 ; | ; | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142, 143, 684  ||. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 ; ~ ; ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142, 143 ; ~= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141, 684