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.

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)
     For a vector argument, return 1 if any element of the vector is nonzero.

     For a matrix argument, return a row vector of ones and zeros with each element indicating whether any of the elements of the corresponding column of the matrix are nonzero.  For example,

          any (eye (2, 4))
               => [ 1, 1, 0, 0 ]

     To see if any of the elements of a matrix are nonzero, you can use a statement like

          any (any (a))

 -- Built-in Function: all (X)
     The function all behaves like the function any, except that it returns true only if all the elements of a vector, or all the elements in a column of a matrix, are nonzero.

Since the comparison operators (see Comparison Ops) 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 it's 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)).

 -- 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, errorcode 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.

 -- Function File: diff (X, K)
     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.

     The second argument is optional.  If supplied, diff (X, K), where K is a nonnegative integer, returns the K-th differences.

 -- Mapping Function: isinf (X)
     Return 1 for elements of X that are infinite and zero otherwise. For example,

          isinf ([13, Inf, NaN])
               => [ 0, 1, 0 ]

 -- Mapping Function: isnan (X)
     Return 1 for elements of X that are NaN values and zero otherwise. For example,

          isnan ([13, Inf, NaN])
               => [ 0, 0, 1 ]

 -- Mapping Function: finite (X)
     Return 1 for elements of X that are NaN values and zero otherwise. For example,

          finite ([13, Inf, NaN])
               => [ 1, 0, 0 ]

 -- Loadable Function: find (X)
     Return a vector of indices of nonzero elements of a matrix.  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 ]

Rearranging Matrices
====================

 -- Function File: fliplr (X)
     Return a copy of X with the order of the columns reversed.  For example,

          fliplr ([1, 2; 3, 4])
               =>  2  1
                   4  3

 -- Function File: flipud (X)
     Return a copy of X with the order of the rows reversed.  For example,

          flipud ([1, 2; 3, 4])
               =>  3  4
                   1  2

 -- Function File: rot90 (X, N)
     Return a copy of X 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 N 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)

 -- Function File: reshape (A, M, N)
     Return a matrix with M rows and N columns whose elements are taken from the matrix A.  To decide how to order the elements, Octave pretends that the elements of a matrix are stored in column-major order (like Fortran arrays are stored).

     For example,

          reshape ([1, 2, 3, 4], 2, 2)
               =>  1  3
                   2  4

     If the variable do_fortran_indexing is nonzero, the reshape function is equivalent to

          retval = zeros (m, n);
          retval (:) = a;

     but it is somewhat less cryptic to use reshape instead of the colon operator.  Note that the total number of elements in the original matrix must match the total number of elements in the new matrix.

 -- Function File: shift (X, B)
     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.

 -- Loadable Function: [S, I] = sort (X)
     Return a copy of X with the elements elements arranged in increasing order.  For matrices, sort orders the elements in each column.

     For example,

          sort ([1, 2; 2, 3; 3, 1])
               =>  1  1
                   2  2
                   3  3

     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

Since the sort function does not allow sort keys to be specified, it can't be used to order the rows of a matrix according to the values of the elements in various columns (For example, to first sort based on the values in column 1, and then, for any values that are repeated in column 1, sort based on the values found in column 2, etc.) in a single call.  Using the second output, however, it is possible to sort all rows based on the values in a given column.  Here's an example that sorts the rows of a matrix based on the values in the second column.

     a = [1, 2; 2, 3; 3, 1];
     [s, i] = sort (a (:, 2));
     a (i, :)
          =>  3  1
              1  2
              2  3

 -- Function File: tril (A, K)
 -- Function File: triu (A, K)
     Return a new matrix formed by extracting extract 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 matrix.

     If the value of K is negative, additional elements above (for tril) or below (for triu) the main diagonal are also selected.

     The absolute value of K must not be greater than the number of sub- 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

 -- Function File: vec (X)
     Return the vector obtained by stacking the columns of the matrix X one above the other.

 -- 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.

Special Utility Matrices
========================

 -- Built-in Function: eye (X)
 -- Built-in Function: eye (N, M)
     Return an identity matrix.  If invoked with a single scalar argument, eye returns a square matrix with the dimension specified.  If you supply two scalar arguments, 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])

     For compatibility with MATLAB, calling eye with no arguments is equivalent to calling it with an argument of 1.

 -- Built-in Function: ones (X)
 -- Built-in Function: ones (N, M)
     Return a matrix whose elements are all 1.  The arguments are handled the same as the arguments for eye.

     If you need to create a matrix whose values are all the same, you should use an expression like

          val_matrix = val * ones (n, m)

 -- Built-in Function: zeros (X)
 -- Built-in Function: zeros (N, M)
     Return a matrix whose elements are all 0.  The arguments are handled the same as the arguments for eye.

 -- Loadable Function: rand (X)
 -- Loadable Function: rand (N, M)
 -- Loadable Function: rand ("seed", X)
     Return a matrix with random elements uniformly distributed on the interval (0, 1).  The arguments are handled the same as the arguments for eye.  In addition, you can set the seed for the random number generator using the form

          rand ("seed", X)

     where X is a scalar value.  If called as

          rand ("seed")

     rand returns the current value of the seed.

 -- Loadable Function: randn (X)
 -- Loadable Function: randn (N, M)
 -- Loadable Function: randn ("seed", X)
     Return a matrix with normally distributed random elements.  The arguments are handled the same as the arguments for eye.  In addition, you can set the seed for the random number generator using the form

          randn ("seed", X)

     where X is a scalar value.  If called as

          randn ("seed")

     randn returns the current value of the seed.

The rand and randn functions use separate generators. This ensures that

     rand ("seed", 13);
     randn ("seed", 13);
     u = rand (100, 1);
     n = randn (100, 1);

and

     rand ("seed", 13);
     randn ("seed", 13);
     u = zeros (100, 1);
     n = zeros (100, 1);
     for i = 1:100
       u(i) = rand ();
       n(i) = randn ();
     end

produce equivalent results.

Normally, rand and randn obtain their initial seeds from the system clock, so that the sequence of random numbers is not the same each time you run Octave.  If you really do need for to reproduce a sequence of numbers exactly, you can set the seed to a specific value.

If it is invoked without arguments, rand and randn return a single element of a random sequence.

The 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: diag (V, 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 functions linspace and logspace make it very easy to create vectors with evenly or logarithmically spaced elements. See Ranges.

 -- Function File: linspace (BASE, LIMIT, N)
     Return a row vector with N linearly spaced elements between BASE and LIMIT.  The number of elements, N, must be greater than 1.  The 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, regardless of the value of prefer_column_vectors.

 -- Function File: logspace (BASE, LIMIT, N)
     Similar to linspace except that the values are logarithmically spaced from 10^base to 10^limit.

     If LIMIT is equal to pi, the points are between 10^base and pi, not 10^base and 10^pi, in order to  be compatible with the corresponding MATLAB function.

 -- Built-in Variable: treat_neg_dim_as_zero
     If the value of treat_neg_dim_as_zero is nonzero, expressions like

          eye (-1)

     produce an empty matrix (i.e., row and column dimensions are zero). Otherwise, an error message is printed and control is returned to the top level.  The default value is 0.

Famous Matrices
===============

The following functions return famous matrix forms.

 -- Function File: hankel (C, R)
     Return the Hankel matrix constructed given 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, the last row is taken to be the same as the first column.

     A Hankel matrix formed from an m-vector C, and an n-vector R, has the elements

          H (i, j) = c (i+j-1),  i+j-1 <= m;
          H (i, j) = r (i+j-m),  otherwise

 -- 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) = 1 / (i + j - 1)

 -- Function File: invhilb (N)
     Return the inverse of a Hilbert matrix of order N.  This is exact. Compare 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.

 -- Function File: sylvester_matrix (K)
     Return the Sylvester matrix of order n = 2^k.

 -- Function File: toeplitz (C, R)
     Return the Toeplitz matrix constructed given the first column C, and (optionally) the first row R.  If the first element of C is not the same as the first element of R, 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

          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)

 -- Function File: vander (C)
     Return the Vandermonde matrix whose next to last column is C.

     A Vandermonde matrix has the form

          c(0)^n ... c(0)^2  c(0)  1
          c(1)^n ... c(1)^2  c(1)  1
           .           .      .    .
           .           .      .    .
           .           .      .    .

          c(n)^n ... c(n)^2  c(n)  1