1 Introduction
    1.1 What is ALGLIB
    1.2 ALGLIB license
    1.3 Documentation license
    1.4 Reference Manual and User Guide
    1.5 Acknowledgements
2 ALGLIB structure
    2.1 Packages
    2.2 Subpackages
    2.3 Open Source and Commercial versions
3 Compatibility
    3.1 CPU
    3.2 OS
    3.3 Compiler
    3.4 Optimization settings
4 Compiling ALGLIB
    4.1 Adding to your project
    4.2 Configuring for your compiler
    4.3 Improving performance (CPU-specific and OS-specific optimizations)
    4.4 Examples (free and commercial editions)
        4.4.1 Introduction
        4.4.2 Compiling under Windows
        4.4.3 Compiling under Linux
5 Using ALGLIB
    5.1 Thread-safety
    5.2 Global definitions
    5.3 Datatypes
    5.4 Constants
    5.5 Functions
    5.6 Working with vectors and matrices
    5.7 Using functions: 'expert' and 'friendly' interfaces
    5.8 Handling errors
    5.9 Working with Level 1 BLAS functions
    5.10 Reading data from CSV files
6 Working with commercial version
    6.1 Benefits of commercial version
    6.2 Working with SIMD support (Intel/AMD users)
    6.3 Using multithreading
        6.3.1 SMT (CMT/hyper-threading) issues
    6.4 Linking with Intel MKL
        6.4.1 Using lightweight Intel MKL supplied by ALGLIB Project
        6.4.2 Using your own installation of Intel MKL
7 Advanced topics
    7.1 Exception-free mode
    7.2 Partial compilation
    7.3 Testing ALGLIB
8 ALGLIB packages and subpackages
    8.1 AlglibMisc package
    8.2 DataAnalysis package
    8.3 DiffEquations package
    8.4 FastTransforms package
    8.5 Integration package
    8.6 Interpolation package
    8.7 LinAlg package
    8.8 Optimization package
    8.9 Solvers package
    8.10 SpecialFunctions package
    8.11 Statistics package

1 Introduction

1.1 What is ALGLIB

ALGLIB is a cross-platform numerical analysis and data mining library. It supports several programming languages (C++, C#, Delphi, VB.NET, Python) and several operating systems (Windows, *nix family).

ALGLIB features include:

Data analysis (classification/regression, including neural networks)
Optimization and nonlinear solvers
Interpolation and linear/nonlinear least-squares fitting
Linear algebra (direct algorithms, EVD/SVD), direct and iterative linear solvers, Fast Fourier Transform and many other algorithms (numerical integration, ODEs, statistics, special functions)

ALGLIB Project (the company behind ALGLIB) delivers to you several editions of ALGLIB:

ALGLIB Free Edition - full functionality but limited performance and license
ALGLIB Commercial Edition - high-performance version of ALGLIB with business-friendly license

Free Edition is a serial version without multithreading support or extensive low-level optimizations (generic C or C# code). Commercial Edition is a heavily optimized version of ALGLIB. It supports multithreading, it was extensively optimized, and (on Intel platforms) - our commercial users may enjoy precompiled version of ALGLIB which internally calls Intel MKL to accelerate low-level tasks. We obtained license from Intel corp., which allows us to integrate Intel MKL into ALGLIB, so you don't have to buy separate license from Intel.

1.2 ALGLIB license

ALGLIB Free Edition is distributed under license which favors non-commmercial usage, but is not well suited for commercial applications:

ALGLIB for C++ and ALGLIB for C# are distributed under GPL 2+, GPL license version 2 or at your option any later version. A copy of the GNU General Public License is available at http://www.fsf.org/licensing/licenses
ALGLIB for Delphi and ALGLIB for CPython are distributed under ALGLIB Personal and Academic Use License Agreement.

ALGLIB Commercial Edition is distributed under license which is friendly to commericial users. A copy of the commercial license can be found at http://www.alglib.net/commercial.php.

1.3 Documentation license

This reference manual is licensed under BSD-like documentation license:

Redistribution and use of this document (ALGLIB Reference Manual) with or without modification, are permitted provided that such redistributions will retain the above copyright notice, this condition and the following disclaimer as the first (or last) lines of this file.

THIS DOCUMENTATION IS PROVIDED BY THE ALGLIB PROJECT "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE ALGLIB PROJECT BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS DOCUMENTATION, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

1.4 Reference Manual and User Guide

ALGLIB Project provides two sources of information: ALGLIB Reference Manual (this document) and ALGLIB User Guide.

ALGLIB Reference Manual contains full description of all publicly accessible ALGLIB units accompanied with examples. Reference Manual is focused on the source code: it documents units, functions, structures and so on. If you want to know what unit YYY can do or what subroutines unit ZZZ contains Reference Manual is a place to go. Free software needs free documentation - that's why ALGLIB Reference Manual is licensed under BSD-like documentation license.

Additionally to the Reference Manual we provide you User Guide. User Guide is focused on more general questions: how fast ALGLIB is? how reliable it is? what are the strong and weak sides of the algorithms used? We aim to make ALGLIB User Guide an important source of information both about ALGLIB and numerical analysis algorithms in general. We want it to be a book about algorithms, not just software documentation. And we want it to be unique - that's why ALGLIB User Guide is distributed under less-permissive personal-use-only license.

1.5 Acknowledgements

ALGLIB was not possible without contribution of the next open source projects:

We also want to thank developers of the Intel's local development center (Nizhny Novgorod branch) for their help during MKL integration.

2 ALGLIB structure

2.1 Packages

ALGLIB is a C++ interface to the computational core written in C. Both C library and C++ wrapper are automatically generated by code generation tools developed within ALGLIB project. Pre-3.0 versions of ALGLIB included more than 100 units, but it was difficult to work with such large number of files. Since ALGLIB 3.0 all units are merged into 11 packages and two support units:

alglibmisc.cpp - contains different algorithms which are hard to classify
dataanalysis.cpp - contains data mining algorithms
diffequations.cpp - contains differential equation solvers
fasttransforms.cpp - contains FFT and other related algorithms
integration.cpp - contains numerical integration algorithms
interpolation.cpp - contains interpolation algorithms
linalg.cpp - contains linear algebra algorithms
optimization.cpp - contains optimization algorithms
solvers.cpp - contains linear and nonlinear solvers
specialfunctions.cpp - contains special functions
statistics.cpp - statistics
alglibinternal.cpp - contains internal functions which are used by other packages, but not exposed to the external world
ap.cpp - contains publicly accessible vector/matrix classes, most important and general functions and other "basic" functionality

One package may rely on other ones, but we have tried to reduce number of dependencies. Every package relies on ap.cpp and many packages rely on alglibinternal.cpp. But many packages require only these two to work, and many other packages need significantly less than 13 packages. For example, statistics.cpp requires two packages mentioned above and only one additional package - specialfunctions.cpp.

2.2 Subpackages

There is one more concept to learn - subpackages. Every package was created from several source files. For example (as of ALGLIB 3.0.0), linalg.cpp was created by merging together 14 .cpp files (C++ interface) and 14 .c files (computational core). These files provide different functionality: one of them calculates triangular factorizations, another generates random matrices, and so on. We've merged source code, but what to do with their documentation?

Of course, we can merge their documentation (as we've merged units) in one big list of functions and data structures, but such list will be hard to read. Instead, we have decided to merge source code, but leave documentation separate.

If you look at the list of ALGLIB packages, you will see that each package includes several subpackages. For example, linalg.cpp includes trfac, svd, evd and other subpackages. These subpackages do no exist as separate files, namespaces or other entities. They are just subsets of one large unit which provide significantly different functionality. They have separate documentation sections, but if you want to use svd subpackage, you have to include linalg.h, not svd.h.

2.3 Open Source and Commercial versions

ALGLIB comes in two versions - open source (GPL-licensed) and commercial (closed source) one. Both versions have same functionality, i.e. may solve same set of problems. However, commercial version differs from open source one in following aspects:

License. Commercial ALGLIB is licensed under non-copyleft license which is friendly to commercial users.
Performance. Many algorithms in commercial ALGLIB are multi-threaded and SIMD-optimized (when used on Intel systems). Open source ALGLIB is single-threaded and can not efficiently use modern multicore CPU's.
You have to study comments on specific functions if you want to know whether they have multithreaded versions or not.

This documentation applies to both versions of ALGLIB. Detailed description of commercial version can be found below.

3 Compatibility

3.1 CPU

ALGLIB is compatible with any CPU which:

supports double precision arithmetics
complies with IEEE 754 floating point standard (especially in its handling of IEEE special values)
either big-endian or little-endian (but not mixed-endian)

Most mainstream CPU's (in particular, x86, x86_64, ARM and SPARC) satisfy these requirements.

As for Intel architecture, ALGLIB works with both FPU-based and SIMD-based implementations of floating point math. 80-bit internal representation used by Intel FPU is not a problem for ALGLIB.

3.2 OS

ALGLIB for C++ (both open source and commercial versions) can be compiled in OS-agnostic mode (no OS-specific preprocessor definitions), when it is compatible with any OS which supports C++98 standard library. In particular, it will work under any POSIX-compatible OS and under Windows.

If you want to use multithreaded capabilities of commercial version of ALGLIB, you should compile it in OS-specific mode by #defining either AE_OS=AE_WINDOWS or AE_OS=AE_POSIX at compile time, depending on OS being used. Former corresponds to any modern OS (32/64-bit Windows XP and later) from Windows family, while latter means almost any POSIX-compatible OS. It applies only to commercial version of ALGLIB. Open source version is always OS-agnostic, even in the presence of OS-specific definitions.

3.3 Compiler

ALGLIB is compatible with any C++ compiler which:

supports 32-bit and 64-bit signed integer datatypes
emits code which handles comparisons with IEEE special values without raising exception. We don't require that x/0 will return INF. But at least we must be able to compare double precision value with infinity or NAN without raising exception.

All modern compilers satisfy these requirements.

However, some very old compilers (ten years old version of Borland C++ Builder, for example) may emit code which does not correctly work with IEEE special values. If you use one of these old compilers, we recommend you to run ALGLIB test suite to ensure that library works.

3.4 Optimization settings

ALGLIB is compatible with any kind of optimizing compiler as long as:

volatile modifier is correctly handled (i.e. compiler does not optimize volatile reads/writes)
optimized code correctly handles IEEE special values

Generally, all kinds of optimization that were marked by compiler vendor as "safe" are possible. For example, ALGLIB can be compiled:

under MSVC: with /O1, /O2, /Og, /Os, /Ox, /Ot, /Oy, /fp:precise, /fp:except, /fp:strict
under GCC: with -O1, -O2, -O3, -Os

From the other side, following "unsafe" optimizations will break ALGLIB:

under MSVC: /fp:fast
under GCC: -Ofast, -ffast-math

4 Compiling ALGLIB

4.1 Adding to your project

Adding ALGLIB to your project is easy - just pick packages you need and... add them to your project! Under most used compilers (GCC, MSVC) it will work without any additional settings. In other cases you will need to define several preprocessor definitions (this topic will be detailed below), but everything will still be simple.

By "adding to your project" we mean that you should a) compile .cpp files with the rest of your project, and b) include .h files you need. Do not include .cpp files - these files must be compiled separately, not as part of some larger source file. The only files you should include are .h files, stored in the /src folder of the ALGLIB distribution.

As you see, ALGLIB has no project files or makefiles. Why? There are several reasons:

first, because many ALGLIB users don't need separate static library (which will be created by invoking makefile) - they prefer to integrate source code in their projects. We have provided script-based build system before, but majority of our users prefer to build ALGLIB themselves.
second, because we want ALGLIB to be usable in any programming environment, whether it is Visual Studio, GNU build system or something else. The best solution is to write package which doesn't depend on any particular programming environment.

In any case, compiling ALGLIB is so simple that even without project file you can do it in several minutes.

4.2 Configuring for your compiler

If you use modern versions of MSVC or GCC, you don't need to configure ALGLIB at all. But if you use outdated versions of these compilers (or something else), then you may need to tune definitions of several data types:

alglib_impl::ae_int32_t - signed integer which is 32 bits wide
alglib_impl::ae_int64_t - signed integer which is 64 bits wide
alglib_impl::ae_uint64_t - unsigned integer which is 64 bits wide
alglib_impl::ae_int_t - signed integer which has same width as pointer

ALGLIB tries to autodetect your compiler and to define these types in compiler-specific manner:

ae_int32_t is defined as int, because this type is 32 bits wide in all modern compilers.
ae_int64_t is defined as _int64 (MSVC) or as signed long long (GCC, Sun Studio).
ae_uint64_t is defined as unsigned _int64 (MSVC) or as unsigned long long (GCC, Sun Studio).
ae_int_t is defined as ptrdiff_t.

In most cases, it is enough. But if anything goes wrong, you have several options:

if your compiler provides stdint.h, you can define AE_HAVE_STDINT conditional symbol
alternatively, you can manually define AE_INT32_T and/or AE_INT64_T and/or AE_UINT64_T and/or AE_INT_T symbols. Just assign datatype name to them, and ALGLIB will automatically use your definition. You may define all or just one/two types (those which are not detected automatically).

4.3 Improving performance (CPU-specific and OS-specific optimizations)

You can improve performance of ALGLIB in a several ways:

by compiling with advanced optimization turned on
by telling ALGLIB about CPU it will run on

ALGLIB has two-layered structure: some set of basic performance-critical primitives is implemented using optimized code, and the rest of the library is built on top of these primitives. By default, ALGLIB uses generic C code to implement these primitives (matrix multiplication, decompositions, etc.). This code works everywhere from Intel to SPARC. However, you can tell ALGLIB that it will work under particular architecture by defining appropriate macro at the global level:

defining AE_CPU=AE_INTEL - to tell ALGLIB that it will work under Intel

When AE_CPU macro is defined and equals to the AE_INTEL, it enables SIMD support. ALGLIB will use cpuid instruction to determine SIMD presence at run-time and use SIMD-capable code. ALGLIB uses SIMD intrinsics which are portable across different compilers and efficient enough for most practical purposes.

If you want to use multithreaded capabilities of commercial version of ALGLIB, you should compile it in OS-specific mode by #defining either AE_OS=AE_WINDOWS, AE_OS=AE_POSIX or AE_OS=AE_LINUX (POSIX with Linux-specific extensions) at compile time, depending on OS being used. Former corresponds to any modern OS (32/64-bit Windows XP and later) from Windows family, while latter means almost any POSIX-compatible OS (or any OS from the Linux family). It applies only to commercial version of ALGLIB. Open source version is always OS-agnostic, even in the presence of OS-specific definitions.

4.4 Examples (free and commercial editions)

4.4.1 Introduction

In this section we'll consider different compilation scenarios for free and commercial versions of ALGLIB - from simple platform-agnostic compilation to compiling/linking with MKL extensions.

We assume that you unpacked ALGLIB distribution in the current directory and saved here demo.cpp file, whose code is given below. Thus, in the current directory you should have exactly one file (demo.cpp) and exactly one subdirectory (cpp folder with ALGLIB distribution).

4.4.2 Compiling under Windows

File listing below contains the very basic program which uses ALGLIB to perform matrix-matrix multiplication. After that program evaluates performance of GEMM (function being called) and prints result to console. We'll show how performance of this program continually increases as we add more and more sophisticated compiler options.

demo.cpp (WINDOWS EXAMPLE)

#include <stdio.h>
#include <windows.h>
#include "LinAlg.h"

double counter()
{
    return 0.001*GetTickCount();
}

int main()
{
    alglib::real_2d_array a, b, c;
    int n = 2000;
    int i, j;
    double timeneeded, flops;
    
    // Initialize arrays
    a.setlength(n, n);
    b.setlength(n, n);
    c.setlength(n, n);
    for(i=0; i<n; i++)
        for(j=0; j<n; j++)
        {
            a[i][j] = alglib::randomreal()-0.5;
            b[i][j] = alglib::randomreal()-0.5;
            c[i][j] = 0.0;
        }
    
    // Set global threading settings (applied to all ALGLIB functions);
    // default is to perform serial computations, unless parallel execution
    // is activated. Parallel execution tries to utilize all cores; this
    // behavior can be changed with alglib::setnworkers() call.
    alglib::setglobalthreading(alglib::parallel);
    
    // Perform matrix-matrix product.
    flops = 2*pow((double)n, (double)3);
    timeneeded = counter();
    alglib::rmatrixgemm(
        n, n, n,
        1.0,
        a, 0, 0, 0,
        b, 0, 0, 1,
        0.0,
        c, 0, 0);
    timeneeded = counter()-timeneeded;
    
    // Evaluate performance
    printf("Performance is %.1f GFLOPS\n", (double)(1.0E-9*flops/timeneeded));
    
    return 0;
}

Examples below cover Windows compilation from command line with MSVC. It is very straightforward to adapt them to compilation from MSVC IDE - or to another compilers. We assume that you already called %VCINSTALLDIR%\bin\amd64\vcvars64.bat batch file which loads 64-bit build environment (or its 32-bit counterpart). We also assume that current directory is clean before example is executed (i.e. it has ONLY demo.cpp file and cpp folder). We used 3.2 GHz 4-core CPU for this test.

First example covers platform-agnostic compilation without optimization settings - the most simple way to compile ALGLIB. This step is same in both open source and commercial editions. However, in platform-agnostic mode ALGLIB is unable to use all performance related features present in commercial edition.

We starts from copying all cpp and h files to current directory, then we will compile them along with demo.cpp. In this and following examples we will omit compiler output for the sake of simplicity.

OS-agnostic mode, no compiler optimizations

> copy cpp\src\*.* .
> cl /I. /EHsc /Fedemo.exe *.cpp
> demo.exe
Performance is 0.7 GFLOPS

Well, 0.7 GFLOPS is not very impressing for a 3.2GHz CPU... Let's add /Ox to compiler parameters.

OS-agnostic mode, /Ox optimization

> cl /I. /EHsc /Fedemo.exe /Ox *.cpp
> demo.exe
Performance is 0.9 GFLOPS

Still not impressed. Let's turn on optimizations for x86 architecture: define AE_CPU=AE_INTEL. This option provides some speed-up in both free and commercial editions of ALGLIB.

OS-agnostic mode, ALGLIB knows it is x86/x64

> cl /I. /EHsc /Fedemo.exe /Ox /DAE_CPU=AE_INTEL *.cpp
> demo.exe
Performance is 4.5 GFLOPS

It is good, but we have 4 cores - and only one of them was used. Defining AE_OS=AE_WINDOWS allows ALGLIB to use Windows threads to parallelize execution of some functions. Starting from this moment, our example applies only to Commercial Edition.

ALGLIB knows it is Windows on x86/x64 CPU (COMMERCIAL EDITION)

> cl /I. /EHsc /Fedemo.exe /Ox /DAE_CPU=AE_INTEL /DAE_OS=AE_WINDOWS *.cpp
> demo.exe
Performance is 16.0 GFLOPS

Not bad. And now we are ready to the final test - linking with MKL extensions.

Linking with MKL extensions differs a bit from standard way of linking with ALGLIB. ALGLIB itself is compiled with one more preprocessor definition: we define AE_MKL symbol. We also link ALGLIB with appropriate (32-bit or 64-bit) alglib???_??mkl.lib static library, which is an import library for special lightweight MKL distribution, shipped with ALGLIB. We also should copy to current directory appropriate alglib???_??mkl.dll binary file which contains Intel MKL.

Linking with MKL extensions (COMMERCIAL EDITION)

> copy cpp\addons-mkl\alglib*64mkl.lib .
> copy cpp\addons-mkl\alglib*64mkl.dll .
> cl /I. /EHsc /Fedemo.exe /Ox /DAE_CPU=AE_INTEL /DAE_OS=AE_WINDOWS /DAE_MKL *.cpp alglib*64mkl.lib
> demo.exe
Performance is 33.1 GFLOPS

From 0.7 GFLOPS to 33.1 GFLOPS - you may see that commercial version of ALGLIB is really worth it!

4.4.3 Compiling under Linux

demo.cpp (LINUX EXAMPLE)

#include <stdio.h>
#include <sys/time.h>
#include "LinAlg.h"

double counter()
{
    struct timeval now;
    alglib_impl::ae_int64_t r, v;
    gettimeofday(&now, NULL);
    v = now.tv_sec;
    r = v*1000;
    v = now.tv_usec/1000;
    r = r+v;
    return 0.001*r;
}

int main()
{
    alglib::real_2d_array a, b, c;
    int n = 2000;
    int i, j;
    double timeneeded, flops;
    
    // Initialize arrays
    a.setlength(n, n);
    b.setlength(n, n);
    c.setlength(n, n);
    for(i=0; i<n; i++)
        for(j=0; j<n; j++)
        {
            a[i][j] = alglib::randomreal()-0.5;
            b[i][j] = alglib::randomreal()-0.5;
            c[i][j] = 0.0;
        }
    
    // Set global threading settings (applied to all ALGLIB functions);
    // default is to perform serial computations, unless parallel execution
    // is activated. Parallel execution tries to utilize all cores; this
    // behavior can be changed with alglib::setnworkers() call.
    alglib::setglobalthreading(alglib::parallel);
    
    // Perform matrix-matrix product.
    flops = 2*pow((double)n, (double)3);
    timeneeded = counter();
    alglib::rmatrixgemm(
        n, n, n,
        1.0,
        a, 0, 0, 0,
        b, 0, 0, 1,
        0.0,
        c, 0, 0);
    timeneeded = counter()-timeneeded;
    
    // Evaluate performance
    printf("Performance is %.1f GFLOPS\n", (double)(1.0E-9*flops/timeneeded));
    
    return 0;
}

Examples below cover x64 Linux compilation from command line with GCC. We assume that current directory is clean before example is executed (i.e. it has ONLY demo.cpp file and cpp folder). We used 2.3 GHz 2-core Skylake CPU with 2x Hyperthreading enabled for this test.

OS-agnostic mode, no compiler optimizations

> cp cpp/src/* .
> g++ -I. -o demo.out *.cpp
> ./demo.out
Performance is 0.9 GFLOPS

Let's add -O3 to compiler parameters.

OS-agnostic mode, -O3 optimization

> g++ -I. -o demo.out -O3 *.cpp
> ./demo.out
Performance is 2.8 GFLOPS

Better, but not impressed. Let's turn on optimizations for x86 architecture: define AE_CPU=AE_INTEL. This option provides some speed-up in both free and commercial editions of ALGLIB.

OS-agnostic mode, ALGLIB knows it is x86/x64

> g++ -I. -o demo.out -O3 -DAE_CPU=AE_INTEL *.cpp
> ./demo.out
Performance is 5.0 GFLOPS

It is good, but we have 4 cores (in fact, 2 cores - it is 2-way hyperthreaded system) and only one of them was used. Defining AE_OS=AE_POSIX allows ALGLIB to use POSIX threads to parallelize execution of some functions. You should also specify -pthread flag to link with pthreads standard library. Starting from this moment, our example applies only to Commercial Edition.

ALGLIB knows it is POSIX OS on x86/x64 CPU (COMMERCIAL EDITION)

> g++ -I. -o demo.out -O3 -DAE_CPU=AE_INTEL -DAE_OS=AE_POSIX -pthread *.cpp
> ./demo.out
Performance is 9.0 GFLOPS

Not bad. You may notice that performance growth was ~2x, not 4x. The reason is that we tested ALGLIB on hyperthreaded system: although we have 4 logical cores, they share computational resources of just 2 physical cores. And now we are ready to the final test - linking with MKL extensions.

We should note that on typical Linux system shared libraries are not loaded from current directory by default. Either you install them into one of the system directories, or use some way to tell linker/loader that you want to load shared library from some specific directory. For our example we choose to update LD_LIBRARY_PATH environment variable.

Linking with MKL extensions (COMMERCIAL EDITION, relevant for ALGLIB 3.13)

> cp cpp/addons-mkl/libalglib*64mkl.so .
> ls *.so
libalglib313_64mkl.so
> g++ -I. -o demo.out -O3 -DAE_CPU=AE_INTEL -DAE_OS=AE_POSIX -pthread -DAE_MKL -L. *.cpp -lalglib313_64mkl
> LD_LIBRARY_PATH=.
> export LD_LIBRARY_PATH
> ./demo.out
Performance is 33.8 GFLOPS

Final result: from 0.9 GFLOPS to 33.8 GFLOPS!

5 Using ALGLIB

5.1 Thread-safety

Both open source and commercial versions of ALGLIB are 100% thread-safe as long as different user threads work with different instances of objects/arrays. Thread-safety is guaranteed by having no global shared variables.

However, any kind of sharing ALGLIB objects/arrays between different threads is potentially hazardous. Even when this object is seemingly used in read-only mode!

Say, you use ALGLIB neural network NET to process two input vectors X0 and X1, and get two output vectors Y0 and Y1. You may decide that neural network is used in read-only mode which does not change state of NET, because output is written to distinct arrays Y. Thus, you may want to process these vectors from parallel threads.

But it is not read-only operation, even if it looks like that! Neural network object NET allocates internal temporary buffers, which are modified by neural processing functions. Thus, sharing one instance of neural network between two threads is thread-unsafe!

5.2 Global definitions

ALGLIB defines several conditional symbols (all start with "AE_" which means "ALGLIB environment") and two namespaces: alglib_impl (contains computational core) and alglib (contains C++ interface).

Although this manual mentions both alglib_impl and alglib namespaces, only alglib namespace should be used by you. It contains user-friendly C++ interface with automatic memory management, exception handling and all other nice features. alglib_impl is less user-friendly, is less documented, and it is too easy to crash your system or cause memory leak if you use it directly.

5.3 Datatypes

ALGLIB (ap.h header) defines several "basic" datatypes (types which are used by all packages) and many package-specific datatypes. "Basic" datatypes are:

alglib::ae_int_t - signed integer type used by library
alglib::complex - double precision complex datatype, safer replacement for std::complex
alglib::ap_error - exception which is thrown by library
boolean_1d_array - 1-dimensional boolean array
integer_1d_array - 1-dimensional integer array
real_1d_array - 1-dimensional real (double precision) array
complex_1d_array - 1-dimensional complex array
boolean_2d_array - 2-dimensional boolean array
integer_2d_array - 2-dimensional integer array
real_2d_array - 2-dimensional real (double precision) array
complex_2d_array - 2-dimensional complex array

Package-specific datatypes are classes which can be divided into two distinct groups:

"struct-like" classes with public fields which you can access directly.
"object-like" classes which have no public fields. You should use ALGLIB functions to work with them.

5.4 Constants

The most important constants (defined in the ap.h header) from ALGLIB namespace are:

alglib::machineepsilon - small number which is close to the double precision &eps;, but is slightly larger
alglib::maxrealnumber - very large number which is close to the maximum real number, but is slightly smaller
alglib::minrealnumber - very small number which is close to the minimum nonzero real number, but is slightly larger
alglib::fp_nan - NAN (non-signalling under most platforms except for PA-RISC, where it is signalling; but when PA-RISC CPU is in its default state, it is silently converted to the quiet NAN)
alglib::fp_posinf - positive infinity
alglib::fp_neginf - negative infinity

5.5 Functions

The most important "basic" functions from ALGLIB namespace (ap.h header) are:

alglib::randomreal() - returns random real number from [0,1)
alglib::randominteger(mx) - returns random integer number from [0,nx); mx must be less than RAND_MAX
alglib::fp_eq(v1,v2) - makes IEEE-compliant comparison of two double precision numbers. If numbers are represented with greater precision than specified by IEEE 754 (as with Intel 80-bit FPU), this functions converts them to 64 bits before comparing.
alglib::fp_neq(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_less(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_less_eq(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_greater(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_greater_eq(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_isnan - checks whether number is NAN
alglib::fp_isposinf - checks whether number is +INF
alglib::fp_isneginf - checks whether number is -INF
alglib::fp_isinf - checks whether number is +INF or -INF
alglib::fp_isfinite - checks whether number is finite value (possibly subnormalized)

5.6 Working with vectors and matrices

ALGLIB (ap.h header) supports matrixes and vectors (one-dimensional and two-dimensional arrays) of variable size, with numeration starting from zero.

Everything starts from array creation. You should distinguish the creation of array class instance and the memory allocation for the array. When creating the class instance, you can use constructor without any parameters, that creates an empty array without any elements. An attempt to address them may cause the program failure.

You can use copy and assignment constructors that copy one array into another. If, during the copy operation, the source array has no memory allocated for the array elements, destination array will contain no elements either. If the source array has memory allocated for its elements, destination array will allocate the same amount of memory and copy the elements there. That is, the copy operation yields into two independent arrays with indentical contents.

You can also create array from formatted string like "[]", "[true,FALSE,tRUe]", "[[]]]" or "[[1,2],[3.2,4],[5.2]]" (note: '.' is used as decimal point independently from locale settings).

alglib::boolean_1d_array b1;
b1 = "[true]";

alglib::real_2d_array r2("[[2,3],[3,4]]");
alglib::real_2d_array r2_1("[[]]");
alglib::real_2d_array r2_2(r2);
r2_1 = r2;

alglib::complex_1d_array c2;
c2 = "[]";
c2 = "[0]";
c2 = "[1,2i]";
c2 = "[+1-2i,-1+5i]";
c2 = "[ 4i-2,  8i+2]";
c2 = "[+4i-2, +8i+2]";
c2 = "[-4i-2, -8i+2]";

After an empty array has been created, you can allocate memory for its elements, using the setlength() method. The content of the created array elements is not defined. If the setlength method is called for the array with already allocated memory, then, after changing its parameters, the newly allocated elements also become undefined and the old content is destroyed.

alglib::boolean_1d_array b1;
b1.setlength(2);

alglib::integer_2d_array r2;
r2.setlength(4,3);

Another way to initialize array is to call setcontent() method. This method accepts pointer to data which are copied into newly allocated array. Vectors are stored in contiguous order, matrices are stored row by row.

alglib::real_1d_array r1;
double _r1[] = {2, 3};
r1.setcontent(2,_r1);

alglib::real_2d_array r2;
double _r2[] = {11, 12, 13, 21, 22, 23};
r2.setcontent(2,3,_r2);

You can also attach real vector/matrix object to already allocated double precision array (attaching to boolean/integer/complex arrays is not supported). In this case, no actual data is copied, and attached vector/matrix object becomes a read/write proxy for external array.

alglib::real_1d_array r1;
double a1[] = {2, 3};
r1.attach_to_ptr(2,a1);

alglib::real_2d_array r2;
double a2[] = {11, 12, 13, 21, 22, 23};
r2.attach_to_ptr(2,3,_r2);

To access the array elements, an overloaded operator() or operator[] can used. That is, the code addressing the element of array a with indexes [i,j] can look like a(i,j) or a[i][j].

alglib::integer_1d_array a("[1,2,3]");
alglib::integer_1d_array b("[3,9,27]");
a[0] = b(0);

alglib::integer_2d_array c("[[1,2,3],[9,9,9]]");
alglib::integer_2d_array d("[[3,9,27],[8,8,8]]");
d[1][1] = c(0,0);

You can access contents of 1-dimensional array by calling getcontent() method which returns pointer to the array memory. For historical reasons 2-dimensional arrays do not provide getcontent() method, but you can use create reference to any element of array. 2-dimensional arrays store data in row-major order with aligned rows (i.e. generally distance between rows is not equal to number of columns). You can get stride (distance between consequtive elements in different rows) with getstride() call.

alglib::integer_1d_array a("[1,2]");
alglib::real_2d_array b("[[0,1],[10,11]]");

alglib::ae_int_t *a_row = a.getcontent();

// all three pointers point to the same location
double *b_row0 = &b[0][0];
double *b_row0_2 = &b(0,0);
double *b_row0_3 = b[0];

// advancing to the next row of 2-dimensional array
double *b_row1 = b_row0 + b.getstride();

Finally, you can get array size with length(), rows() or cols() methods:

alglib::integer_1d_array a("[1,2]");
alglib::real_2d_array b("[[0,1],[10,11]]");

printf("%ld\n", (long)a.length());
printf("%ld\n", (long)b.rows());
printf("%ld\n", (long)b.cols());

5.7 Using functions: 'expert' and 'friendly' interfaces

Most ALGLIB functions provide two interfaces: 'expert' and 'friendly'. What is the difference between two? When you use 'friendly' interface, ALGLIB:

tries to automatically determine size of input arguments
throws an exception when arguments have inconsistent size (for example, square matrix is expected, but non-square is passed; another example - two parameters must have same size, but have different size)
if semantics of input parameter assumes that it is symmetric/Hermitian matrix, checks that lower triangle is equal to upper triangle (conjugate of upper triangle) and throws an exception otherwise
if semantics of output parameter assumes that it is symmetric/Hermitian matrix, returns full matrix (both upper and lower triangles)

When you use 'expert' interface, ALGLIB:

requires caller to explicitly specify size of input arguments. If vector/matrix is larger than size being specified (say, N), only N leading elements are used
if semantics of input parameter assumes that it is symmetric/Hermitian matrix, uses only upper or lower triangle of input matrix and requires caller to specify what triangle to use
if semantics of output parameter assumes that it is symmetric/Hermitian matrix, returns only upper or lower triangle (when you look at specific function, it is clear what triangle is returned)

Here are several examples of 'friendly' and 'expert' interfaces:

#include "interpolation.h"

...

alglib::real_1d_array    x("[0,1,2,3]");
alglib::real_1d_array    y("[1,5,3,9]");
alglib::real_1d_array   y2("[1,5,3,9,0]");
alglib::spline1dinterpolant s;

alglib::spline1dbuildlinear(x, y, 4, s);  // 'expert' interface is used
alglib::spline1dbuildlinear(x, y, s);     // 'friendly' interface - input size is
                                          // automatically determined

alglib::spline1dbuildlinear(x, y2, 4, s); // y2.length() is 5, but it will work

alglib::spline1dbuildlinear(x, y2, s);    // it won't work because sizes of x and y2
                                          // are inconsistent

'Friendly' interface - matrix semantics:

#include "linalg.h"

...

alglib::real_2d_array a;
alglib::matinvreport  rep;
alglib::ae_int_t      info;

// 
// 'Friendly' interface: spdmatrixinverse() accepts and returns symmetric matrix
// 

// symmetric positive definite matrix
a = "[[2,1],[1,2]]";

// after this line A will contain [[0.66,-0.33],[-0.33,0.66]]
// which is symmetric too
alglib::spdmatrixinverse(a, info, rep); 

// you may try to pass nonsymmetric matrix
a = "[[2,1],[0,2]]";

// but exception will be thrown in such case
alglib::spdmatrixinverse(a, info, rep);

Same function but with 'expert' interface:

#include "linalg.h"

...

alglib::real_2d_array a;
alglib::matinvreport  rep;
alglib::ae_int_t      info;

// 
// 'Expert' interface, spdmatrixinverse()
// 

// only upper triangle is used; a[1][0] is initialized by NAN,
// but it can be arbitrary number
a = "[[2,1],[NAN,2]]";

// after this line A will contain [[0.66,-0.33],[NAN,0.66]]
// only upper triangle is modified
alglib::spdmatrixinverse(a, 2 /* N */, true /* upper triangle is used */, info, rep);

5.8 Handling errors

ALGLIB uses two error handling strategies:

returning error code
throwing exception

What is actually done depends on function being used and error being reported:

if function returns some error code and has corresponding value for this kind of error, ALLGIB returns error code
if function does not return error code (or returns error code, but there is no code for error being reported), ALGLIB throws alglib::ap_error exception. Exception object has msg parameter which contains short description of error.

To make things clear we consider several examples of error handling.

Example 1. mincgreate function creates nonlinear CG optimizer. It accepts problem size N and initial point X. Several things can go wrong - you may pass array which is too short, filled by NAN's, or otherwise pass incorrect data. However, this function returns no error code - so it throws an exception in case something goes wrong. There is no other way to tell caller that something went wrong.

Example 2. rmatrixinverse function calculates inverse matrix. It returns error code, which is set to +1 when problem is solved and is set to -3 if singular matrix was passed to the function. However, there is no error code for matrix which is non-square or contains infinities. Well, we could have created corresponding error codes - but we didn't. So if you pass singular matrix to rmatrixinverse, you will get completion code -3. But if you pass matrix which contains INF in one of its elements, alglib::ap_error will be thrown.

First error handling strategy (error codes) is used to report "frequent" errors which can occur during normal execution of user program. Second error handling strategy (exceptions) is used to report "rare" errors which are result of serious flaws in your program (or ALGLIB) - infinities/NAN's in the inputs, inconsistent inputs, etc.

5.9 Working with Level 1 BLAS functions

ALGLIB (ap.h header) includes following Level 1 BLAS functions:

alglib::vdotproduct() family, which allows to calculate dot product of two real or complex vectors
alglib::vmove() family, which allows to copy real/complex vector to another location with optimal multiplication by real/complex value
alglib::vmoveneg() family, which allows to copy real/complex vector to another location with multiplication by -1
alglib::vadd() and alglib::vsub() families, which allows to add or subtract two real/complex vectors with optimal multiplication by real/complex value
alglib::vmul() family, which implements in-place multiplication of real/complex vector by real/complex value

Each Level 1 BLAS function accepts input stride and output stride, which are expected to be positive. Input and output vectors should not overlap. Functions operating with complex vectors accept additional parameter conj_src, which specifies whether input vector is conjugated or not.

For each real/complex function there exists "simple" companion which accepts no stride or conjugation modifier. "Simple" function assumes that input/output stride is +1, and no input conjugation is required.

alglib::real_1d_array    rvec("[0,1,2,3]");
alglib::real_2d_array    rmat("[[1,2],[3,4]]");
alglib::complex_1d_array cvec("[0+1i,1+2i,2-1i,3-2i]");
alglib::complex_2d_array cmat("[[3i,1],[9,2i]]");

alglib::vmove(&rvec[0],  1, &rmat[0][0], rmat.getstride(), 2); // now rvec is [1,3,2,3]

alglib::vmove(&cvec[0],  1, &cmat[0][0], rmat.getstride(), "No conj", 2); // now cvec is [3i, 9, 2-1i, 3-2i]
alglib::vmove(&cvec[2],  1, &cmat[0][0], 1,                "Conj", 2);    // now cvec is [3i, 9, -3i,  1]

Here is full list of Level 1 BLAS functions implemented in ALGLIB:

double vdotproduct(
    const double *v0,
     ae_int_t stride0,
     const double *v1,
     ae_int_t stride1,
     ae_int_t n);
double vdotproduct(
    const double *v1,
     const double *v2,
     ae_int_t N);

alglib::complex vdotproduct(
    const alglib::complex *v0,
     ae_int_t stride0,
     const char *conj0,
     const alglib::complex *v1,
     ae_int_t stride1,
     const char *conj1,
     ae_int_t n);
alglib::complex vdotproduct(
    const alglib::complex *v1,
     const alglib::complex *v2,
     ae_int_t N);

void vmove(
    double *vdst,
      ae_int_t stride_dst,
     const double* vsrc,
      ae_int_t stride_src,
     ae_int_t n);
void vmove(
    double *vdst,
     const double* vsrc,
     ae_int_t N);

void vmove(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex* vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n);
void vmove(
    alglib::complex *vdst,
     const alglib::complex* vsrc,
     ae_int_t N);

void vmoveneg(
    double *vdst,
      ae_int_t stride_dst,
     const double* vsrc,
      ae_int_t stride_src,
     ae_int_t n);
void vmoveneg(
    double *vdst,
     const double *vsrc,
     ae_int_t N);

void vmoveneg(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex* vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n);
void vmoveneg(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N);

void vmove(
    double *vdst,
      ae_int_t stride_dst,
     const double* vsrc,
      ae_int_t stride_src,
     ae_int_t n,
     double alpha);
void vmove(
    double *vdst,
     const double *vsrc,
     ae_int_t N,
     double alpha);

void vmove(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex* vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     double alpha);
void vmove(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     double alpha);

void vmove(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex* vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     alglib::complex alpha);
void vmove(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     alglib::complex alpha);

void vadd(
    double *vdst,
      ae_int_t stride_dst,
     const double *vsrc,
      ae_int_t stride_src,
     ae_int_t n);
void vadd(
    double *vdst,
     const double *vsrc,
     ae_int_t N);

void vadd(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n);
void vadd(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N);

void vadd(
    double *vdst,
      ae_int_t stride_dst,
     const double *vsrc,
      ae_int_t stride_src,
     ae_int_t n,
     double alpha);
void vadd(
    double *vdst,
     const double *vsrc,
     ae_int_t N,
     double alpha);

void vadd(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     double alpha);
void vadd(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     double alpha);

void vadd(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     alglib::complex alpha);
void vadd(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     alglib::complex alpha);

void vsub(
    double *vdst,
      ae_int_t stride_dst,
     const double *vsrc,
      ae_int_t stride_src,
     ae_int_t n);
void vsub(
    double *vdst,
     const double *vsrc,
     ae_int_t N);

void vsub(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n);
void vsub(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N);

void vsub(
    double *vdst,
      ae_int_t stride_dst,
     const double *vsrc,
      ae_int_t stride_src,
     ae_int_t n,
     double alpha);
void vsub(
    double *vdst,
     const double *vsrc,
     ae_int_t N,
     double alpha);

void vsub(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     double alpha);
void vsub(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     double alpha);

void vsub(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     alglib::complex alpha);
void vsub(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     alglib::complex alpha);

void vmul(
    double *vdst,
      ae_int_t stride_dst,
     ae_int_t n,
     double alpha);
void vmul(
    double *vdst,
     ae_int_t N,
     double alpha);

void vmul(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     ae_int_t n,
     double alpha);
void vmul(
    alglib::complex *vdst,
     ae_int_t N,
     double alpha);

void vmul(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     ae_int_t n,
     alglib::complex alpha);
void vmul(
    alglib::complex *vdst,
     ae_int_t N,
     alglib::complex alpha);

5.10 Reading data from CSV files

ALGLIB (ap.h header) has alglib::read_csv() function which allows to read data from CSV file. Entire file is loaded into memory as double precision 2D array (alglib::real_2d_array object). This function provides following features:

support for ASCI encoding and UTF-8 without BOM (in header names)
ability to use any character (comma/tab/space) as field separator (as long as it is distinct from one used for decimal point)
ability to use both comma and full stop as decimal point (as long is decimal point is distinct from field separator).
support for Unix and Windows text files (CR vs CRLF)

See comments on alglib::read_csv() function for more information about its functionality.

6 Working with commercial version

6.1 Benefits of commercial version

Commercial version of ALGLIB for C++ features four important improvements over open source one:

License. Commercial license used by ALGLIB is friendly to closed source applications. Unlike GPL, it does not require you to open source your application. Thus, almost any commercial software developer is interested in obtaining commercial license.
Low-level optimizations. Commercial version of ALGLIB includes SIMD-optimized versions of many computationally intensive functions. It allows to increase performance on Intel/AMD platforms while still being able to use software under non-x86 CPU's.
Multithreading. Commercial version of ALGLIB can utilize multicore capabilities of modern CPU's. Large computational problems can be automatically split between different cores. ALGLIB uses its own multithreading framework which does not depend on vendor/compiler support for technologies like OpenMP/MPI/... It gives ALGLIB unprecedented portability across operating systems and compilers.
Integrated Intel MKL. Commercial version of ALGLIB includes special MKL extensions - special lightweight distribution of Intel MKL, high-performance numerical analysis library, accompanied by ALGLIB-MKL interface. We obtained license from Intel which allows us to integrate MKL into ALGLIB distribution. Linking with MKL accelerates many ALGLIB functions, however due to license restrictions you can not use MKL directly (i.e. bypass ALGLIB interface between your program and MKL).

6.2 Working with SIMD support (Intel/AMD users)

ALGLIB for C++ can utilize SIMD instructions supported by Intel and AMD processors. This feature is optional and must be explicitly turned on during compile-time. If you do not activate it, ALGLIB will use generic C code, without any processor-specific assembly/intrinsics.

Thus, if you turn on this feature, your code will run faster on x86_32 and x86_64 processors, but will be unportable to non-x86 platforms (and Intel MIC platform, which is not exactly x86!). From the other side, if you do not activate this feature, your code will be portable to almost any modern CPU (SPARC, ARM, ...).

In order to turn on x86-specific optimizations, you should define AE_CPU=AE_INTEL preprocessor definition at global level. It will tell ALGLIB to use SIMD intrinsics supported by GCC, MSVC and Intel compilers. Additionally you should tell compiler to generate SIMD-capable code. It can be done in the project settings of your IDE or in the command line:


GCC example:
> g++ -msse2 -I. -DAE_CPU=AE_INTEL *.cpp -lm

MSVC example:
> cl /I. /EHsc /DAE_CPU=AE_INTEL *.cpp

6.3 Using multithreading

Commercial version of ALGLIB includes out-of-the-box support for multithreading. Many (not all) computationally intensive problems can be solved in multithreaded mode. You should read comments on specific ALGLIB functions to determine what can be multithreaded and what can not.

ALGLIB does not depend on vendor/compiler support for technologies like OpenMP/MPI/... Under Windows ALGLIB uses OS threads and custom synchronization framework. Under POSIX-compatible OS (Solaris, Linux, FreeBSD, NetBSD, OpenBSD, ...) ALGLIB uses POSIX Threads (standard *nix library which is shipped with any POSIX system) with its threading and synchronization primitives. It gives ALGLIB unprecedented portability across operating systems and compilers. ALGLIB does not depend on presence of any custom multithreading library or compiler support for any multithreading technology.

If you want to use multithreaded capabilities of ALGLIB, you should:

compile it in OS-specific mode (ALGLIB have to know what OS it is running on)
activate multithreading at global level with alglib::setglobalthreading function call or enable it at per-function basis by passing alglib::parallel to the specific computational function
optionally, tell ALGLIB about number of worker threads to use (default is to utilize all cores)

Let explain it in more details...

1. You should compile ALGLIB in OS-specific mode by #defining either AE_OS=AE_WINDOWS or AE_OS=AE_POSIX (or AE_OS=AE_LINUX, which means "POSIX with Linux extensions") at compile time, depending on OS being used. Former corresponds to any modern OS (32/64-bit Windows XP and later) from Windows family, while latter two mean almost any POSIX-compatible OS or any OS from the Linux family. When compiling on POSIX/Linux, do not forget to link ALGLIB with libpthread library.

2. By default, ALGLIB is configured to perform all calculations in serial manner. Parallel execution can be manually enabled at either global or per-function level. Former means that all ALGLIB functions will be able to parallelize themselves at their discretion. Similarly, you can enable global parallelism, but selectively disable it for specific function calls.

Global parallelism is enabled by alglib::setglobalthreading(alglib::parallel) and disabled by alglib::setglobalthreading(alglib::serial) call. Function-level parallelism can be enabled (or disabled, if global default is to parallelize) by adding alglib::parallel (or alglib::serial) to the end of the parameters list of specific ALGLIB function being called.

Enabling parallelism does not guarantee that ALGLIB will parallelize its computations. Small problems (say, products of 64x64 matrices) can not be efficiently parallelized. ALGLIB will automatically decide whether your problem is large enough or not for efficient parallelization.

3. ALGLIB automatically determines number of cores on application startup. On Windows it is done using GetSystemInfo() call. On POSIX systems ALGLIB performs sysconf(_SC_NPROCESSORS_ONLN) system call. This system call is supported by all modern POSIX-compatible systems: Solaris, Linux, FreeBSD, NetBSD, OpenBSD.

By default, ALGLIB uses all available cores (when told to parallelize calculations). Such behavior may be changed with setnworkers() call:

alglib::setnworkers(0) = use all cores
alglib::setnworkers(-1) = leave one core unused
alglib::setnworkers(-2) = leave two cores unused
alglib::setnworkers(+2) = use 2 cores (even if you have more)

You may want to specify maximum number of worker threads during compile time by means of preprocessor definition AE_NWORKERS=N. You can add this definition to compiler command line or change corresponding project settings in your IDE. Here N can be any positive number. ALGLIB will use exactly N worker threads, unless being told to use less by setnworkers() call.

Some old POSIX-compatible operating systems do not support sysconf(_SC_NPROCESSORS_ONLN) system call which is required in order to automatically determine number of active cores. On these systems you should specify number of cores manually at compile time. Without it ALGLIB will run in single-threaded mode.

6.3.1 SMT (CMT/hyper-threading) issues

Simultaneous multithreading (SMT) also known as Hyper-threading (Intel) and Cluster-based Multithreading (AMD) is a CPU design where several (usually two) logical cores share resources of one physical core. Say, on dual-core system with 2x HT scale factor you will see 4 logical cores. Each pair of these 4 cores, however, share same hardware resources. Thus, you may get only marginal speedup when running highly optimized software which fully utilizes CPU resources.

Say, if one thread occupies floating-point unit, another thread on the same physical core may work with integer numbers at the same time without any performance penalties. In this case you may get some speedup due to having additional cores. But if both threads keep FPU unit 100% busy, they won't get any multithreaded speedup.

So, if 2 math-intensive threads are dispatched by OS scheduler to different physical cores, you will get 2x speedup due to use of multithreading. But if these threads are dispatched to different logical cores - but same physical core - you won't get any speedup at all! One physical core will be 100% busy, and another one will be 100% idle. From the other side, if you start four threads instead of two, your system will be 100% utilized independently of thread scheduling details.

Let we stress it one more time - multithreading speedup on SMT systems is highly dependent on number of threads you are running and decisions made by OS scheduler. It is not 100% deterministic! With "true SMP" when you run 2 threads, you get 2x speedup (or 1.95, or 1.80 - it depends on algorithm, but this factor is always same). With SMT when you run 2 threads you may get your 2x speedup - or no speedup at all. Modern OS schedulers do a good job on single-socket hardware, but even in this "simple" case they give no guarantees of fair distribution of hardware resources. And things become a bit tricky when you work with multi-socket hardware. On SMT systems the only guaranteed way to 100% utilize your CPU is to create as many worker threads as there are logical cores. In this case OS scheduler has no chance to make its work in a wrong way.

6.4 Linking with Intel MKL

6.4.1 Using lightweight Intel MKL supplied by ALGLIB Project

Commercial edition of ALGLIB includes MKL extensions - special lightweight distribution of Intel MKL, highly optimized numerical library from Intel - and precompiled ALGLIB-MKL interface libraries. Linking your programs with MKL extensions allows you to run ALGLIB with maximum performance. MKL binaries are delivered for x86/x64 Windows and x64 Linux platforms.

Unlike the rest of the library, MKL extensions are distributed in binary-only form. ALGLIB itself is still distributed in source code form, but Intel MKL and ALGLIB-MKL interface are distributed as precompiled dynamic/static libraries. We can not distribute them in source because of license restrictions associated with Intel MKL. Also due to license restrictions we can not give you direct access to MKL functionality. You may use MKL to accelerate ALGLIB - without paying for MKL license - but you may not call its functions directly. It is technically possible, but strictly prohibited by both MKL's EULA and ALGLIB License Agreement. If you want to work with MKL, you should obtain separate license from Intel (as of 2018, free licenses are available).

MKL extensions are located in the /cpp/addons-mkl subdirectory of the ALGLIB distribution. This directory includes following files:

alglib???_32mkl.lib - x86 Windows import library (MSVC) for Intel MKL extensions
alglib???_64mkl.lib - x64 Windows import library (MSVC) for Intel MKL extensions
alglib???_32mkl.dll - x86 Windows binary with Intel MKL inside
alglib???_64mkl.dll - x64 Windows binary with Intel MKL inside
libalglib???_64mkl.so - x64 Linux binary with Intel MKL inside

Here ??? stands for specific ALGLIB version: 313 for ALGLIB 3.13, and so on. Files above are just MKL extensions - ALGLIB itself is not included in these binaries, and you still have to compile primary ALGLIB distribution.

In order to activate MKL extensions you should:

#define globally AE_MKL to activate ALGLIB-MKL connection.
additionally, #define globally AE_OS=AE_WINDOWS or AE_OS=AE_POSIX to activate multithreading capabilities
additionally, #define globally AE_CPU=AE_INTEL to use SIMD instructions provided by x86/x64 CPU's in the rest of ALGLIB
Depending on your OS, perform one of the following:
- Windows: choose 32-bit or 64-bit import library alglib???_32/64mkl.lib and link it with your application
- Linux: choose 64-bit SO file alglib???_64mkl.so and link it with your application
place DLL/SO binary into directory where it can be found during application startup (Windows: application dir; Linux: one of the system directories)

Several examples of ALGLIB+MKL usage are given in the 'compiling ALGLIB: examples' section.

6.4.2 Using your own installation of Intel MKL

If you bought separate license for Intel MKL, and want to use your own installation of MKL - and not our lightweight distribution - then you should compile ALGLIB as it was told in the previous section, with all necessary preprocessor definitions (AE_OS=AE_WINDOWS or AE_OS=AE_POSIX, AE_CPU=AE_INTEL and AE_MKL defined). But instead of linking with MKL Extensions binary, you should add to your project alglib2mkl.c file from addons-mkl directory and compile it (as C file) along with the rest of ALGLIB.

This C file implements interface between MKL and ALGLIB. Having this file in your project and defining AE_MKL preprocessor definition results in ALGLIB using MKL functions.

However, this C file is just interface! It is your responsibility to make sure that C/C++ compiler can find MKL headers, and appropriate MKL static/dynamic libraries are linked to your application.

7 Advanced topics

7.1 Exception-free mode

ALGLIB for C++ can be compiled in exception-free mode, with exceptions (throw/try/catch constructs) being disabled at compiler level. Such feature is sometimes used by developers of embedded software.

ALGLIB uses two-level model of errors: "expected" errors (like degeneracy of linear system or inconsistency of linear constraints) are reported with dedicated completion codes, and "critical" errors (like malloc failures, unexpected NANs/INFs in the input data and so on) are reported with exceptions. The idea is that it is hard to put (and handle) completion codes in every ALGLIB function, so we use exceptions to signal errors which should never happen under normal circumstances.

Internally ALGLIB for C++ is implemented as C++ wrapper around computational core written in pure C. Thus, internals of ALGLIB core use C-specific methods of error handling - completion codes and setjmp/longjmp functions. These error handling strategies are combined with sophisticated machinery of C memory management which makes sure that not even a byte of dynamic memory is lost when we make longjmp to the error handler. So, the only point where C++ exceptions are actually used is a boundary between C core and C++ interface.

If you choose to use exceptions (default mode), ALGLIB will throw an exception with short textual description of the situation. And if you choose to work without exceptions, ALGLIB will set global error flag and silently return from the current function/constructor/... instead of throwing an exception. Due to portability issues this error flag is made to be a non-TLS variable, i.e. it is shared between different threads. So, you can use exception-free error handling only in single-threaded programs - although multithreaded programs won't break, there is no way to determine which thread caused an "exception without exceptions".

Exception-free method of reporting critical errors can be activated by #defining two preprocessor symbols at global level:

AE_NO_EXCEPTIONS - to switch from exception-based to exception-free code
AE_THREADING=AE_SERIAL_UNSAFE - to confirm that you are aware of limitations associated with exception-free mode (it does not support multithreading)

We must also note that exception-free mode is incompatible with OS-aware compiling: you can not have AE_OS=??? defined together with AE_NO_EXCEPTIONS.

After you #define all the necessary preprocessor symbols, two functions will appear in alglib namespace:

bool alglib::get_error_flag(const char **p_msg = NULL), which returns current error status (true is returned on error), with optional char** parameter used to get human-readable error message.
void alglib::clear_error_flag(), which clears error flag (ALGLIB functions set flag on failure, but do not clear it on successful calls)

You must check error flag after EVERY operation with ALGLIB objects and functions. In addition to calling computational ALGLIB functions, following kinds of operations may result in "exception":

calling default constructor for ALGLIB object (simply instantiating object may result in "exception"). Due to large size ALGLIB objects are allocated dynamically (they look like value types, but internally everything is stored in the heap memory). Thus every constructor, even default one, makes at least one malloc() call which may fail.
calling copy/assignment constructors for ALGLIB objects (same reason - malloc)
resizing arrays with setlength()/setcontents() and attaching to external memory with attach_to_ptr()

7.2 Partial compilation

Due to ALGLIB modular structure it is possible to selectively enable/disable some of its subpackages along with their dependencies. Deactivation of ALGLIB source code is performed at preprocessor level - compiler does not even see disabled code. Partial compilation can be used for two purposes:

to reduce code size in cases when linker is unable to remove unused code (happens with some compilers when ALGLIB is compiled as part of the shared library)
to reduce build time (disabled code is silently removed by preprocessor, thus no time is spent in the compiler)

You can activate partial compilation by #defining at global level following symbols:

AE_PARTIAL_BUILD - to disable everything not enabled by default
AE_COMPILE_SUBPACKAGE - to selectively enable subpackage SUBPACKAGE, with its name given in upper case (case sensitive). You may combine several definitions like this in order to enable several subpackages. Subpackage names can be found in the list of ALGLIB packages and subpackages.

7.3 Testing ALGLIB

There are three test suites in ALGLIB: computational tests, interface tests, extended tests. Computational tests are located in /tests/test_c.cpp. They are focused on numerical properties of algorithms, stress testing and "deep" tests (large automatically generated problems). They require significant amount of time to finish (tens of minutes).

Interface tests are located in /tests/test_i.cpp. These tests are focused on ability to correctly pass data between computational core and caller, ability to detect simple problems in inputs, and on ability to at least compile ALGLIB with your compiler. They are very fast (about a minute to finish including compilation time).

Extended tests are located in /tests/test_x.cpp. These tests are focused on testing some special properties (say, testing that cloning object indeed results in 100% independent copy being created) and performance of several chosen algorithms.

Running test suite is easy - just

compile one of these files (test_c.cpp, test_i.cpp or test_x.cpp) along with the rest of the library
launch executable you will get. It may take from several seconds (interface tests) to several minutes (computational tests) to get final results

If you want to be sure that ALGLIB will work with some sophisticated optimization settings, set corresponding flags during compile time. If your compiler/system are not in the list of supported ones, we recommend you to run both test suites. But if you are running out of time, run at least test_i.cpp.

8 ALGLIB packages and subpackages

8.1 `AlglibMisc` package
hqrnd		High quality random numbers generator
nearestneighbor		Nearest neighbor search: approximate and exact
xdebug		Debug functions to test ALGLIB interface generator

8.2 `DataAnalysis` package
bdss		Basic dataset functions
clustering		Clustering functions (hierarchical, k-means, k-means++)
datacomp		Backward compatibility functions
dforest		Decision forest classifier (regression model)
filters		Different filters used in data analysis
lda		Linear discriminant analysis
linreg		Linear models
logit		Logit models
mcpd		Markov Chains for Population/proportional Data
mlpbase		Basic functions for neural networks
mlpe		Basic functions for neural ensemble models
mlptrain		Neural network training
pca		Principal component analysis
ssa		Singular Spectrum Analysis

8.3 `DiffEquations` package
odesolver		Ordinary differential equation solver

8.4 `FastTransforms` package
conv		Fast real/complex convolution
corr		Fast real/complex cross-correlation
fft		Real/complex FFT
fht		Real Fast Hartley Transform

8.5 `Integration` package
autogk		Adaptive 1-dimensional integration
gkq		Gauss-Kronrod quadrature generator
gq		Gaussian quadrature generator

8.6 `Interpolation` package
fitsphere		Fitting circle/sphere to data (least squares, minimum circumscribed, maximum inscribed, minimum zone)
idwint		Inverse distance weighting: interpolation/fitting
intcomp		Backward compatibility functions
lsfit		Fitting with least squates target function (linear and nonlinear least-squares)
parametric		Parametric curves
polint		Polynomial interpolation/fitting
ratint		Rational interpolation/fitting
rbf		Scattered N-dimensional interpolation with RBF models
spline1d		1D spline interpolation/fitting
spline2d		2D spline interpolation
spline3d		3D spline interpolation

8.7 `LinAlg` package
ablas		Level 2 and Level 3 BLAS operations
bdsvd		Bidiagonal SVD
evd		Direct and iterative eigensolvers
inverseupdate		Sherman-Morrison update of the inverse matrix
matdet		Determinant calculation
matgen		Random matrix generation
matinv		Matrix inverse
normestimator		Estimates norm of the sparse matrix (from below)
ortfac		Real/complex QR/LQ, bi(tri)diagonal, Hessenberg decompositions
rcond		Condition number estimate
schur		Schur decomposition
sparse		Sparse matrices
spdgevd		Generalized symmetric eigensolver
svd		Singular value decomposition
trfac		LU and Cholesky decompositions (dense and sparse)

8.8 `Optimization` package
minbc		Box constrained optimizer with fast activation of multiple constraints per step
minbleic		Bound constrained optimizer with additional linear equality/inequality constraints
mincg		Conjugate gradient optimizer
mincomp		Backward compatibility functions
minlbfgs		Limited memory BFGS optimizer
minlm		Improved Levenberg-Marquardt optimizer
minnlc		Nonlinearly constrained optimizer
minns		Nonsmooth constrained optimizer
minqp		Quadratic programming with bound and linear equality/inequality constraints

8.9 `Solvers` package
directdensesolvers		Direct dense linear solvers
directsparsesolvers		Direct sparse linear solvers
lincg		Sparse linear CG solver
linlsqr		Sparse linear LSQR solver
nleq		Solvers for nonlinear equations
polynomialsolver		Polynomial solver

8.10 `SpecialFunctions` package
airyf		Airy functions
bessel		Bessel functions
betaf		Beta function
binomialdistr		Binomial distribution
chebyshev		Chebyshev polynomials
chisquaredistr		Chi-Square distribution
dawson		Dawson integral
elliptic		Elliptic integrals
expintegrals		Exponential integrals
fdistr		F-distribution
fresnel		Fresnel integrals
gammafunc		Gamma function
hermite		Hermite polynomials
ibetaf		Incomplete beta function
igammaf		Incomplete gamma function
jacobianelliptic		Jacobian elliptic functions
laguerre		Laguerre polynomials
legendre		Legendre polynomials
normaldistr		Normal distribution
poissondistr		Poisson distribution
psif		Psi function
studenttdistr		Student's t-distribution
trigintegrals		Trigonometric integrals

8.11 `Statistics` package
basestat		Mean, variance, covariance, correlation, etc.
correlationtests		Hypothesis testing: correlation tests
jarquebera		Hypothesis testing: Jarque-Bera test
mannwhitneyu		Hypothesis testing: Mann-Whitney-U test
stest		Hypothesis testing: sign test
studentttests		Hypothesis testing: Student's t-test
variancetests		Hypothesis testing: F-test and one-sample variance test
wsr		Hypothesis testing: Wilcoxon signed rank test

`ablas` subpackage

Functions

cmatrixcopy
cmatrixgemm
cmatrixherk
cmatrixlefttrsm
cmatrixmv
cmatrixrank1
cmatrixrighttrsm
cmatrixsyrk
cmatrixtranspose
rmatrixcopy
rmatrixenforcesymmetricity
rmatrixgemm
rmatrixgemv
rmatrixger
rmatrixlefttrsm
rmatrixmv
rmatrixrank1
rmatrixrighttrsm
rmatrixsymv
rmatrixsyrk
rmatrixsyvmv
rmatrixtranspose
rmatrixtrsv

Examples

ablas_d_gemm		Matrix multiplication (single-threaded)
ablas_d_syrk		Symmetric rank-K update (single-threaded)

`cmatrixcopy` function

/*************************************************************************
Copy

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied and transposed
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void alglib::cmatrixcopy(
    ae_int_t m,
    ae_int_t n,
    complex_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    complex_2d_array& b,
    ae_int_t ib,
    ae_int_t jb,
    const xparams _params = alglib::xdefault);

`cmatrixgemm` function

/*************************************************************************
This subroutine calculates C = alpha*op1(A)*op2(B) +beta*C where:
* C is MxN general matrix
* op1(A) is MxK matrix
* op2(B) is KxN matrix
* "op" may be identity transformation, transposition, conjugate transposition

Additional info:
* cache-oblivious algorithm is used.
* multiplication result replaces C. If Beta=0, C elements are not used in
  calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

IMPORTANT:

This function does NOT preallocate output matrix C, it MUST be preallocated
by caller prior to calling this function. In case C does not have  enough
space to store result, exception will be generated.

INPUT PARAMETERS
    M       -   matrix size, M>0
    N       -   matrix size, N>0
    K       -   matrix size, K>0
    Alpha   -   coefficient
    A       -   matrix
    IA      -   submatrix offset
    JA      -   submatrix offset
    OpTypeA -   transformation type:
                * 0 - no transformation
                * 1 - transposition
                * 2 - conjugate transposition
    B       -   matrix
    IB      -   submatrix offset
    JB      -   submatrix offset
    OpTypeB -   transformation type:
                * 0 - no transformation
                * 1 - transposition
                * 2 - conjugate transposition
    Beta    -   coefficient
    C       -   matrix (PREALLOCATED, large enough to store result)
    IC      -   submatrix offset
    JC      -   submatrix offset

  -- ALGLIB routine --
     2009-2019
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixgemm(
    ae_int_t m,
    ae_int_t n,
    ae_int_t k,
    alglib::complex alpha,
    complex_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    complex_2d_array b,
    ae_int_t ib,
    ae_int_t jb,
    ae_int_t optypeb,
    alglib::complex beta,
    complex_2d_array& c,
    ae_int_t ic,
    ae_int_t jc,
    const xparams _params = alglib::xdefault);

Examples: [1]

`cmatrixherk` function

/*************************************************************************
This subroutine calculates  C=alpha*A*A^H+beta*C  or  C=alpha*A^H*A+beta*C
where:
* C is NxN Hermitian matrix given by its upper/lower triangle
* A is NxK matrix when A*A^H is calculated, KxN matrix otherwise

Additional info:
* multiplication result replaces C. If Beta=0, C elements are not used in
  calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    N       -   matrix size, N>=0
    K       -   matrix size, K>=0
    Alpha   -   coefficient
    A       -   matrix
    IA      -   submatrix offset (row index)
    JA      -   submatrix offset (column index)
    OpTypeA -   multiplication type:
                * 0 - A*A^H is calculated
                * 2 - A^H*A is calculated
    Beta    -   coefficient
    C       -   preallocated input/output matrix
    IC      -   submatrix offset (row index)
    JC      -   submatrix offset (column index)
    IsUpper -   whether upper or lower triangle of C is updated;
                this function updates only one half of C, leaving
                other half unchanged (not referenced at all).

  -- ALGLIB routine --
     16.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixherk(
    ae_int_t n,
    ae_int_t k,
    double alpha,
    complex_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    double beta,
    complex_2d_array& c,
    ae_int_t ic,
    ae_int_t jc,
    bool isupper,
    const xparams _params = alglib::xdefault);

Examples: [1]

`cmatrixlefttrsm` function

/*************************************************************************
This subroutine calculates op(A^-1)*X where:
* X is MxN general matrix
* A is MxM upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition, conjugate transposition
Multiplication result replaces X.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    N   -   matrix size, N>=0
    M   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[I1:I1+M-1,J1:J1+M-1]
    I1      -   submatrix offset
    J1      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
                * 2 - conjugate transposition
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixlefttrsm(
    ae_int_t m,
    ae_int_t n,
    complex_2d_array a,
    ae_int_t i1,
    ae_int_t j1,
    bool isupper,
    bool isunit,
    ae_int_t optype,
    complex_2d_array& x,
    ae_int_t i2,
    ae_int_t j2,
    const xparams _params = alglib::xdefault);

`cmatrixmv` function

/*************************************************************************
Matrix-vector product: y := op(A)*x

INPUT PARAMETERS:
    M   -   number of rows of op(A)
            M>=0
    N   -   number of columns of op(A)
            N>=0
    A   -   target matrix
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    OpA -   operation type:
            * OpA=0     =>  op(A) = A
            * OpA=1     =>  op(A) = A^T
            * OpA=2     =>  op(A) = A^H
    X   -   input vector
    IX  -   subvector offset
    IY  -   subvector offset
    Y   -   preallocated matrix, must be large enough to store result

OUTPUT PARAMETERS:
    Y   -   vector which stores result

if M=0, then subroutine does nothing.
if N=0, Y is filled by zeros.


  -- ALGLIB routine --

     28.01.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixmv(
    ae_int_t m,
    ae_int_t n,
    complex_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t opa,
    complex_1d_array x,
    ae_int_t ix,
    complex_1d_array& y,
    ae_int_t iy,
    const xparams _params = alglib::xdefault);

`cmatrixrank1` function

/*************************************************************************
Rank-1 correction: A := A + u*v'

INPUT PARAMETERS:
    M   -   number of rows
    N   -   number of columns
    A   -   target matrix, MxN submatrix is updated
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    U   -   vector #1
    IU  -   subvector offset
    V   -   vector #2
    IV  -   subvector offset
*************************************************************************/
void alglib::cmatrixrank1(
    ae_int_t m,
    ae_int_t n,
    complex_2d_array& a,
    ae_int_t ia,
    ae_int_t ja,
    complex_1d_array& u,
    ae_int_t iu,
    complex_1d_array& v,
    ae_int_t iv,
    const xparams _params = alglib::xdefault);

`cmatrixrighttrsm` function

/*************************************************************************
This subroutine calculates X*op(A^-1) where:
* X is MxN general matrix
* A is NxN upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition, conjugate transposition
Multiplication result replaces X.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    N   -   matrix size, N>=0
    M   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[I1:I1+N-1,J1:J1+N-1]
    I1      -   submatrix offset
    J1      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
                * 2 - conjugate transposition
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  -- ALGLIB routine --
     20.01.2018
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixrighttrsm(
    ae_int_t m,
    ae_int_t n,
    complex_2d_array a,
    ae_int_t i1,
    ae_int_t j1,
    bool isupper,
    bool isunit,
    ae_int_t optype,
    complex_2d_array& x,
    ae_int_t i2,
    ae_int_t j2,
    const xparams _params = alglib::xdefault);

`cmatrixsyrk` function

/*************************************************************************
This subroutine is an older version of CMatrixHERK(), one with wrong  name
(it is HErmitian update, not SYmmetric). It  is  left  here  for  backward
compatibility.

  -- ALGLIB routine --
     16.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixsyrk(
    ae_int_t n,
    ae_int_t k,
    double alpha,
    complex_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    double beta,
    complex_2d_array& c,
    ae_int_t ic,
    ae_int_t jc,
    bool isupper,
    const xparams _params = alglib::xdefault);

`cmatrixtranspose` function

/*************************************************************************
Cache-oblivous complex "copy-and-transpose"

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied and transposed
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void alglib::cmatrixtranspose(
    ae_int_t m,
    ae_int_t n,
    complex_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    complex_2d_array& b,
    ae_int_t ib,
    ae_int_t jb,
    const xparams _params = alglib::xdefault);

`rmatrixcopy` function

/*************************************************************************
Copy

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied and transposed
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void alglib::rmatrixcopy(
    ae_int_t m,
    ae_int_t n,
    real_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    real_2d_array& b,
    ae_int_t ib,
    ae_int_t jb,
    const xparams _params = alglib::xdefault);

`rmatrixenforcesymmetricity` function

/*************************************************************************
This code enforces symmetricy of the matrix by copying Upper part to lower
one (or vice versa).

INPUT PARAMETERS:
    A   -   matrix
    N   -   number of rows/columns
    IsUpper - whether we want to copy upper triangle to lower one (True)
            or vice versa (False).
*************************************************************************/
void alglib::rmatrixenforcesymmetricity(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    const xparams _params = alglib::xdefault);

`rmatrixgemm` function

/*************************************************************************
This subroutine calculates C = alpha*op1(A)*op2(B) +beta*C where:
* C is MxN general matrix
* op1(A) is MxK matrix
* op2(B) is KxN matrix
* "op" may be identity transformation, transposition

Additional info:
* cache-oblivious algorithm is used.
* multiplication result replaces C. If Beta=0, C elements are not used in
  calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

IMPORTANT:

This function does NOT preallocate output matrix C, it MUST be preallocated
by caller prior to calling this function. In case C does not have  enough
space to store result, exception will be generated.

INPUT PARAMETERS
    M       -   matrix size, M>0
    N       -   matrix size, N>0
    K       -   matrix size, K>0
    Alpha   -   coefficient
    A       -   matrix
    IA      -   submatrix offset
    JA      -   submatrix offset
    OpTypeA -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    B       -   matrix
    IB      -   submatrix offset
    JB      -   submatrix offset
    OpTypeB -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    Beta    -   coefficient
    C       -   PREALLOCATED output matrix, large enough to store result
    IC      -   submatrix offset
    JC      -   submatrix offset

  -- ALGLIB routine --
     2009-2019
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixgemm(
    ae_int_t m,
    ae_int_t n,
    ae_int_t k,
    double alpha,
    real_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    real_2d_array b,
    ae_int_t ib,
    ae_int_t jb,
    ae_int_t optypeb,
    double beta,
    real_2d_array& c,
    ae_int_t ic,
    ae_int_t jc,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rmatrixgemv` function

/*************************************************************************

*************************************************************************/
void alglib::rmatrixgemv(
    ae_int_t m,
    ae_int_t n,
    double alpha,
    real_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t opa,
    real_1d_array x,
    ae_int_t ix,
    double beta,
    real_1d_array& y,
    ae_int_t iy,
    const xparams _params = alglib::xdefault);

`rmatrixger` function

/*************************************************************************
Rank-1 correction: A := A + alpha*u*v'

NOTE: this  function  expects  A  to  be  large enough to store result. No
      automatic preallocation happens for  smaller  arrays.  No  integrity
      checks is performed for sizes of A, u, v.

INPUT PARAMETERS:
    M   -   number of rows
    N   -   number of columns
    A   -   target matrix, MxN submatrix is updated
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    Alpha-  coefficient
    U   -   vector #1
    IU  -   subvector offset
    V   -   vector #2
    IV  -   subvector offset


  -- ALGLIB routine --

     16.10.2017
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixger(
    ae_int_t m,
    ae_int_t n,
    real_2d_array& a,
    ae_int_t ia,
    ae_int_t ja,
    double alpha,
    real_1d_array u,
    ae_int_t iu,
    real_1d_array v,
    ae_int_t iv,
    const xparams _params = alglib::xdefault);

`rmatrixlefttrsm` function

/*************************************************************************
This subroutine calculates op(A^-1)*X where:
* X is MxN general matrix
* A is MxM upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition
Multiplication result replaces X.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    N   -   matrix size, N>=0
    M   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[I1:I1+M-1,J1:J1+M-1]
    I1      -   submatrix offset
    J1      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixlefttrsm(
    ae_int_t m,
    ae_int_t n,
    real_2d_array a,
    ae_int_t i1,
    ae_int_t j1,
    bool isupper,
    bool isunit,
    ae_int_t optype,
    real_2d_array& x,
    ae_int_t i2,
    ae_int_t j2,
    const xparams _params = alglib::xdefault);

`rmatrixmv` function

/*************************************************************************
IMPORTANT: this function is deprecated since ALGLIB 3.13. Use RMatrixGEMV()
           which is more generic version of this function.

Matrix-vector product: y := op(A)*x

INPUT PARAMETERS:
    M   -   number of rows of op(A)
    N   -   number of columns of op(A)
    A   -   target matrix
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    OpA -   operation type:
            * OpA=0     =>  op(A) = A
            * OpA=1     =>  op(A) = A^T
    X   -   input vector
    IX  -   subvector offset
    IY  -   subvector offset
    Y   -   preallocated matrix, must be large enough to store result

OUTPUT PARAMETERS:
    Y   -   vector which stores result

if M=0, then subroutine does nothing.
if N=0, Y is filled by zeros.


  -- ALGLIB routine --

     28.01.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixmv(
    ae_int_t m,
    ae_int_t n,
    real_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t opa,
    real_1d_array x,
    ae_int_t ix,
    real_1d_array& y,
    ae_int_t iy,
    const xparams _params = alglib::xdefault);

`rmatrixrank1` function

/*************************************************************************
IMPORTANT: this function is deprecated since ALGLIB 3.13. Use RMatrixGER()
           which is more generic version of this function.

Rank-1 correction: A := A + u*v'

INPUT PARAMETERS:
    M   -   number of rows
    N   -   number of columns
    A   -   target matrix, MxN submatrix is updated
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    U   -   vector #1
    IU  -   subvector offset
    V   -   vector #2
    IV  -   subvector offset
*************************************************************************/
void alglib::rmatrixrank1(
    ae_int_t m,
    ae_int_t n,
    real_2d_array& a,
    ae_int_t ia,
    ae_int_t ja,
    real_1d_array& u,
    ae_int_t iu,
    real_1d_array& v,
    ae_int_t iv,
    const xparams _params = alglib::xdefault);

`rmatrixrighttrsm` function

/*************************************************************************
This subroutine calculates X*op(A^-1) where:
* X is MxN general matrix
* A is NxN upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition
Multiplication result replaces X.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    N   -   matrix size, N>=0
    M   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[I1:I1+N-1,J1:J1+N-1]
    I1      -   submatrix offset
    J1      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixrighttrsm(
    ae_int_t m,
    ae_int_t n,
    real_2d_array a,
    ae_int_t i1,
    ae_int_t j1,
    bool isupper,
    bool isunit,
    ae_int_t optype,
    real_2d_array& x,
    ae_int_t i2,
    ae_int_t j2,
    const xparams _params = alglib::xdefault);

`rmatrixsymv` function

/*************************************************************************

*************************************************************************/
void alglib::rmatrixsymv(
    ae_int_t n,
    double alpha,
    real_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    bool isupper,
    real_1d_array x,
    ae_int_t ix,
    double beta,
    real_1d_array& y,
    ae_int_t iy,
    const xparams _params = alglib::xdefault);

`rmatrixsyrk` function

/*************************************************************************
This subroutine calculates  C=alpha*A*A^T+beta*C  or  C=alpha*A^T*A+beta*C
where:
* C is NxN symmetric matrix given by its upper/lower triangle
* A is NxK matrix when A*A^T is calculated, KxN matrix otherwise

Additional info:
* multiplication result replaces C. If Beta=0, C elements are not used in
  calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    N       -   matrix size, N>=0
    K       -   matrix size, K>=0
    Alpha   -   coefficient
    A       -   matrix
    IA      -   submatrix offset (row index)
    JA      -   submatrix offset (column index)
    OpTypeA -   multiplication type:
                * 0 - A*A^T is calculated
                * 2 - A^T*A is calculated
    Beta    -   coefficient
    C       -   preallocated input/output matrix
    IC      -   submatrix offset (row index)
    JC      -   submatrix offset (column index)
    IsUpper -   whether C is upper triangular or lower triangular

  -- ALGLIB routine --
     16.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixsyrk(
    ae_int_t n,
    ae_int_t k,
    double alpha,
    real_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    double beta,
    real_2d_array& c,
    ae_int_t ic,
    ae_int_t jc,
    bool isupper,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rmatrixsyvmv` function

/*************************************************************************

*************************************************************************/
double alglib::rmatrixsyvmv(
    ae_int_t n,
    real_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    bool isupper,
    real_1d_array x,
    ae_int_t ix,
    real_1d_array& tmp,
    const xparams _params = alglib::xdefault);

`rmatrixtranspose` function

/*************************************************************************
Cache-oblivous real "copy-and-transpose"

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied and transposed
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void alglib::rmatrixtranspose(
    ae_int_t m,
    ae_int_t n,
    real_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    real_2d_array& b,
    ae_int_t ib,
    ae_int_t jb,
    const xparams _params = alglib::xdefault);

`rmatrixtrsv` function

/*************************************************************************
This subroutine solves linear system op(A)*x=b where:
* A is NxN upper/lower triangular/unitriangular matrix
* X and B are Nx1 vectors
* "op" may be identity transformation, transposition, conjugate transposition

Solution replaces X.

IMPORTANT: * no overflow/underflow/denegeracy tests is performed.
           * no integrity checks for operand sizes, out-of-bounds accesses
             and so on is performed

INPUT PARAMETERS
    N   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[IA:IA+N-1,JA:JA+N-1]
    IA      -   submatrix offset
    JA      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    X       -   right part, actual vector is stored in X[IX:IX+N-1]
    IX      -   offset

OUTPUT PARAMETERS
    X       -   solution replaces elements X[IX:IX+N-1]

  -- ALGLIB routine / remastering of LAPACK's DTRSV --
     (c) 2017 Bochkanov Sergey - converted to ALGLIB
     (c) 2016 Reference BLAS level1 routine (LAPACK version 3.7.0)
     Reference BLAS is a software package provided by Univ. of Tennessee,
     Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.
*************************************************************************/
void alglib::rmatrixtrsv(
    ae_int_t n,
    real_2d_array a,
    ae_int_t ia,
    ae_int_t ja,
    bool isupper,
    bool isunit,
    ae_int_t optype,
    real_1d_array& x,
    ae_int_t ix,
    const xparams _params = alglib::xdefault);

ablas_d_gemm example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_2d_array a = "[[2,1],[1,3]]";
    real_2d_array b = "[[2,1],[0,1]]";
    real_2d_array c = "[[0,0],[0,0]]";

    //
    // rmatrixgemm() function allows us to calculate matrix product C:=A*B or
    // to perform more general operation, C:=alpha*op1(A)*op2(B)+beta*C,
    // where A, B, C are rectangular matrices, op(X) can be X or X^T,
    // alpha and beta are scalars.
    //
    // This function:
    // * can apply transposition and/or multiplication by scalar to operands
    // * can use arbitrary part of matrices A/B (given by submatrix offset)
    // * can store result into arbitrary part of C
    // * for performance reasons requires C to be preallocated
    //
    // Parameters of this function are:
    // * M, N, K            -   sizes of op1(A) (which is MxK), op2(B) (which
    //                          is KxN) and C (which is MxN)
    // * Alpha              -   coefficient before A*B
    // * A, IA, JA          -   matrix A and offset of the submatrix
    // * OpTypeA            -   transformation type:
    //                          0 - no transformation
    //                          1 - transposition
    // * B, IB, JB          -   matrix B and offset of the submatrix
    // * OpTypeB            -   transformation type:
    //                          0 - no transformation
    //                          1 - transposition
    // * Beta               -   coefficient before C
    // * C, IC, JC          -   preallocated matrix C and offset of the submatrix
    //
    // Below we perform simple product C:=A*B (alpha=1, beta=0)
    //
    // IMPORTANT: this function works with preallocated C, which must be large
    //            enough to store multiplication result.
    //
    ae_int_t m = 2;
    ae_int_t n = 2;
    ae_int_t k = 2;
    double alpha = 1.0;
    ae_int_t ia = 0;
    ae_int_t ja = 0;
    ae_int_t optypea = 0;
    ae_int_t ib = 0;
    ae_int_t jb = 0;
    ae_int_t optypeb = 0;
    double beta = 0.0;
    ae_int_t ic = 0;
    ae_int_t jc = 0;
    rmatrixgemm(m, n, k, alpha, a, ia, ja, optypea, b, ib, jb, optypeb, beta, c, ic, jc);
    printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[4,3],[2,4]]

    //
    // Now we try to apply some simple transformation to operands: C:=A*B^T
    //
    optypeb = 1;
    rmatrixgemm(m, n, k, alpha, a, ia, ja, optypea, b, ib, jb, optypeb, beta, c, ic, jc);
    printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[5,1],[5,3]]
    return 0;
}

ablas_d_syrk example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // rmatrixsyrk() function allows us to calculate symmetric rank-K update
    // C := beta*C + alpha*A'*A, where C is square N*N matrix, A is square K*N
    // matrix, alpha and beta are scalars. It is also possible to update by
    // adding A*A' instead of A'*A.
    //
    // Parameters of this function are:
    // * N, K       -   matrix size
    // * Alpha      -   coefficient before A
    // * A, IA, JA  -   matrix and submatrix offsets
    // * OpTypeA    -   multiplication type:
    //                  * 0 - A*A^T is calculated
    //                  * 2 - A^T*A is calculated
    // * Beta       -   coefficient before C
    // * C, IC, JC  -   preallocated input/output matrix and submatrix offsets
    // * IsUpper    -   whether upper or lower triangle of C is updated;
    //                  this function updates only one half of C, leaving
    //                  other half unchanged (not referenced at all).
    //
    // Below we will show how to calculate simple product C:=A'*A
    //
    // NOTE: beta=0 and we do not use previous value of C, but still it
    //       MUST be preallocated.
    //
    ae_int_t n = 2;
    ae_int_t k = 1;
    double alpha = 1.0;
    ae_int_t ia = 0;
    ae_int_t ja = 0;
    ae_int_t optypea = 2;
    double beta = 0.0;
    ae_int_t ic = 0;
    ae_int_t jc = 0;
    bool isupper = true;
    real_2d_array a = "[[1,2]]";

    // preallocate space to store result
    real_2d_array c = "[[0,0],[0,0]]";

    // calculate product, store result into upper part of c
    rmatrixsyrk(n, k, alpha, a, ia, ja, optypea, beta, c, ic, jc, isupper);

    // output result.
    // IMPORTANT: lower triangle of C was NOT updated!
    printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[1,2],[0,4]]
    return 0;
}

`airyf` subpackage

Functions

airy

Examples

`airy` function

/*************************************************************************
Airy function

Solution of the differential equation

y"(x) = xy.

The function returns the two independent solutions Ai, Bi
and their first derivatives Ai'(x), Bi'(x).

Evaluation is by power series summation for small x,
by rational minimax approximations for large x.



ACCURACY:
Error criterion is absolute when function <= 1, relative
when function > 1, except * denotes relative error criterion.
For large negative x, the absolute error increases as x^1.5.
For large positive x, the relative error increases as x^1.5.

Arithmetic  domain   function  # trials      peak         rms
IEEE        -10, 0     Ai        10000       1.6e-15     2.7e-16
IEEE          0, 10    Ai        10000       2.3e-14*    1.8e-15*
IEEE        -10, 0     Ai'       10000       4.6e-15     7.6e-16
IEEE          0, 10    Ai'       10000       1.8e-14*    1.5e-15*
IEEE        -10, 10    Bi        30000       4.2e-15     5.3e-16
IEEE        -10, 10    Bi'       30000       4.9e-15     7.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void alglib::airy(
    double x,
    double& ai,
    double& aip,
    double& bi,
    double& bip,
    const xparams _params = alglib::xdefault);

`autogk` subpackage

Integrating f=exp(x) by adaptive integrator

`autogkreport` class

/*************************************************************************
Integration report:
* TerminationType = completetion code:
    * -5    non-convergence of Gauss-Kronrod nodes
            calculation subroutine.
    * -1    incorrect parameters were specified
    *  1    OK
* Rep.NFEV countains number of function calculations
* Rep.NIntervals contains number of intervals [a,b]
  was partitioned into.
*************************************************************************/
class autogkreport
{
    ae_int_t             terminationtype;
    ae_int_t             nfev;
    ae_int_t             nintervals;
};

`autogkstate` class

/*************************************************************************
This structure stores state of the integration algorithm.

Although this class has public fields,  they are not intended for external
use. You should use ALGLIB functions to work with this class:
* autogksmooth()/AutoGKSmoothW()/... to create objects
* autogkintegrate() to begin integration
* autogkresults() to get results
*************************************************************************/
class autogkstate
{
};

`autogkintegrate` function

/*************************************************************************
This function is used to launcn iterations of ODE solver

It accepts following parameters:
    diff    -   callback which calculates dy/dx for given y and x
    obj     -   optional object which is passed to diff; can be NULL


  -- ALGLIB --
     Copyright 07.05.2009 by Bochkanov Sergey
*************************************************************************/
void autogkintegrate(autogkstate &state,
    void (*func)(double x, double xminusa, double bminusx, double &y, void *ptr),
    void *ptr = NULL, const xparams _xparams = alglib::xdefault);

Examples: [1]

`autogkresults` function

/*************************************************************************
Adaptive integration results

Called after AutoGKIteration returned False.

Input parameters:
    State   -   algorithm state (used by AutoGKIteration).

Output parameters:
    V       -   integral(f(x)dx,a,b)
    Rep     -   optimization report (see AutoGKReport description)

  -- ALGLIB --
     Copyright 14.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::autogkresults(
    autogkstate state,
    double& v,
    autogkreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`autogksingular` function

/*************************************************************************
Integration on a finite interval [A,B].
Integrand have integrable singularities at A/B.

F(X) must diverge as "(x-A)^alpha" at A, as "(B-x)^beta" at B,  with known
alpha/beta (alpha>-1, beta>-1).  If alpha/beta  are  not known,  estimates
from below can be used (but these estimates should be greater than -1 too).

One  of  alpha/beta variables (or even both alpha/beta) may be equal to 0,
which means than function F(x) is non-singular at A/B. Anyway (singular at
bounds or not), function F(x) is supposed to be continuous on (A,B).

Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
is calculated with accuracy close to the machine precision.

INPUT PARAMETERS:
    A, B    -   interval boundaries (A<B, A=B or A>B)
    Alpha   -   power-law coefficient of the F(x) at A,
                Alpha>-1
    Beta    -   power-law coefficient of the F(x) at B,
                Beta>-1

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state

SEE ALSO
    AutoGKSmooth, AutoGKSmoothW, AutoGKResults.


  -- ALGLIB --
     Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::autogksingular(
    double a,
    double b,
    double alpha,
    double beta,
    autogkstate& state,
    const xparams _params = alglib::xdefault);

`autogksmooth` function

/*************************************************************************
Integration of a smooth function F(x) on a finite interval [a,b].

Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
is calculated with accuracy close to the machine precision.

Algorithm works well only with smooth integrands.  It  may  be  used  with
continuous non-smooth integrands, but with  less  performance.

It should never be used with integrands which have integrable singularities
at lower or upper limits - algorithm may crash. Use AutoGKSingular in such
cases.

INPUT PARAMETERS:
    A, B    -   interval boundaries (A<B, A=B or A>B)

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state

SEE ALSO
    AutoGKSmoothW, AutoGKSingular, AutoGKResults.


  -- ALGLIB --
     Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::autogksmooth(
    double a,
    double b,
    autogkstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`autogksmoothw` function

/*************************************************************************
Integration of a smooth function F(x) on a finite interval [a,b].

This subroutine is same as AutoGKSmooth(), but it guarantees that interval
[a,b] is partitioned into subintervals which have width at most XWidth.

Subroutine  can  be  used  when  integrating nearly-constant function with
narrow "bumps" (about XWidth wide). If "bumps" are too narrow, AutoGKSmooth
subroutine can overlook them.

INPUT PARAMETERS:
    A, B    -   interval boundaries (A<B, A=B or A>B)

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state

SEE ALSO
    AutoGKSmooth, AutoGKSingular, AutoGKResults.


  -- ALGLIB --
     Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::autogksmoothw(
    double a,
    double b,
    double xwidth,
    autogkstate& state,
    const xparams _params = alglib::xdefault);

autogk_d1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "integration.h"

using namespace alglib;
void int_function_1_func(double x, double xminusa, double bminusx, double &y, void *ptr) 
{
    // this callback calculates f(x)=exp(x)
    y = exp(x);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates integration of f=exp(x) on [0,1]:
    // * first, autogkstate is initialized
    // * then we call integration function
    // * and finally we obtain results with autogkresults() call
    //
    double a = 0;
    double b = 1;
    autogkstate s;
    double v;
    autogkreport rep;

    autogksmooth(a, b, s);
    alglib::autogkintegrate(s, int_function_1_func);
    autogkresults(s, v, rep);

    printf("%.2f\n", double(v)); // EXPECTED: 1.7182
    return 0;
}

`basestat` subpackage

Functions

cov2
covm
covm2
pearsoncorr2
pearsoncorrelation
pearsoncorrm
pearsoncorrm2
rankdata
rankdatacentered
sampleadev
samplekurtosis
samplemean
samplemedian
samplemoments
samplepercentile
sampleskewness
samplevariance
spearmancorr2
spearmancorrm
spearmancorrm2
spearmanrankcorrelation

Examples

basestat_d_base		Basic functionality (moments, adev, median, percentile)
basestat_d_c2		Correlation (covariance) between two random variables
basestat_d_cm		Correlation (covariance) between components of random vector
basestat_d_cm2		Correlation (covariance) between two random vectors

`cov2` function

/*************************************************************************
2-sample covariance

Input parameters:
    X       -   sample 1 (array indexes: [0..N-1])
    Y       -   sample 2 (array indexes: [0..N-1])
    N       -   N>=0, sample size:
                * if given, only N leading elements of X/Y are processed
                * if not given, automatically determined from input sizes

Result:
    covariance (zero for N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
double alglib::cov2(
    real_1d_array x,
    real_1d_array y,
    const xparams _params = alglib::xdefault);
double alglib::cov2(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

Examples: [1]

`covm` function

/*************************************************************************
Covariance matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X are used
            * if not given, automatically determined from input size
    M   -   M>0, number of variables:
            * if given, only leading M columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M,M], covariance matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::covm(
    real_2d_array x,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);
void alglib::covm(
    real_2d_array x,
    ae_int_t n,
    ae_int_t m,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);

Examples: [1]

`covm2` function

/*************************************************************************
Cross-covariance matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M1], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    Y   -   array[N,M2], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X/Y are used
            * if not given, automatically determined from input sizes
    M1  -   M1>0, number of variables in X:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size
    M2  -   M2>0, number of variables in Y:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M1,M2], cross-covariance matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::covm2(
    real_2d_array x,
    real_2d_array y,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);
void alglib::covm2(
    real_2d_array x,
    real_2d_array y,
    ae_int_t n,
    ae_int_t m1,
    ae_int_t m2,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);

Examples: [1]

`pearsoncorr2` function

/*************************************************************************
Pearson product-moment correlation coefficient

Input parameters:
    X       -   sample 1 (array indexes: [0..N-1])
    Y       -   sample 2 (array indexes: [0..N-1])
    N       -   N>=0, sample size:
                * if given, only N leading elements of X/Y are processed
                * if not given, automatically determined from input sizes

Result:
    Pearson product-moment correlation coefficient
    (zero for N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
double alglib::pearsoncorr2(
    real_1d_array x,
    real_1d_array y,
    const xparams _params = alglib::xdefault);
double alglib::pearsoncorr2(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

Examples: [1]

`pearsoncorrelation` function

/*************************************************************************
Obsolete function, we recommend to use PearsonCorr2().

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::pearsoncorrelation(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`pearsoncorrm` function

/*************************************************************************
Pearson product-moment correlation matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X are used
            * if not given, automatically determined from input size
    M   -   M>0, number of variables:
            * if given, only leading M columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M,M], correlation matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pearsoncorrm(
    real_2d_array x,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);
void alglib::pearsoncorrm(
    real_2d_array x,
    ae_int_t n,
    ae_int_t m,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);

Examples: [1]

`pearsoncorrm2` function

/*************************************************************************
Pearson product-moment cross-correlation matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M1], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    Y   -   array[N,M2], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X/Y are used
            * if not given, automatically determined from input sizes
    M1  -   M1>0, number of variables in X:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size
    M2  -   M2>0, number of variables in Y:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M1,M2], cross-correlation matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pearsoncorrm2(
    real_2d_array x,
    real_2d_array y,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);
void alglib::pearsoncorrm2(
    real_2d_array x,
    real_2d_array y,
    ae_int_t n,
    ae_int_t m1,
    ae_int_t m2,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rankdata` function

/*************************************************************************
This function replaces data in XY by their ranks:
* XY is processed row-by-row
* rows are processed separately
* tied data are correctly handled (tied ranks are calculated)
* ranking starts from 0, ends at NFeatures-1
* sum of within-row values is equal to (NFeatures-1)*NFeatures/2

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    XY      -   array[NPoints,NFeatures], dataset
    NPoints -   number of points
    NFeatures-  number of features

OUTPUT PARAMETERS:
    XY      -   data are replaced by their within-row ranks;
                ranking starts from 0, ends at NFeatures-1

  -- ALGLIB --
     Copyright 18.04.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::rankdata(
    real_2d_array& xy,
    const xparams _params = alglib::xdefault);
void alglib::rankdata(
    real_2d_array& xy,
    ae_int_t npoints,
    ae_int_t nfeatures,
    const xparams _params = alglib::xdefault);

`rankdatacentered` function

/*************************************************************************
This function replaces data in XY by their CENTERED ranks:
* XY is processed row-by-row
* rows are processed separately
* tied data are correctly handled (tied ranks are calculated)
* centered ranks are just usual ranks, but centered in such way  that  sum
  of within-row values is equal to 0.0.
* centering is performed by subtracting mean from each row, i.e it changes
  mean value, but does NOT change higher moments

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    XY      -   array[NPoints,NFeatures], dataset
    NPoints -   number of points
    NFeatures-  number of features

OUTPUT PARAMETERS:
    XY      -   data are replaced by their within-row ranks;
                ranking starts from 0, ends at NFeatures-1

  -- ALGLIB --
     Copyright 18.04.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::rankdatacentered(
    real_2d_array& xy,
    const xparams _params = alglib::xdefault);
void alglib::rankdatacentered(
    real_2d_array& xy,
    ae_int_t npoints,
    ae_int_t nfeatures,
    const xparams _params = alglib::xdefault);

`sampleadev` function

/*************************************************************************
ADev

Input parameters:
    X   -   sample
    N   -   N>=0, sample size:
            * if given, only leading N elements of X are processed
            * if not given, automatically determined from size of X

Output parameters:
    ADev-   ADev

  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::sampleadev(
    real_1d_array x,
    double& adev,
    const xparams _params = alglib::xdefault);
void alglib::sampleadev(
    real_1d_array x,
    ae_int_t n,
    double& adev,
    const xparams _params = alglib::xdefault);

Examples: [1]

`samplekurtosis` function

/*************************************************************************
Calculation of the kurtosis.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

NOTE:

This function return result  which calculated by 'SampleMoments' function
and stored at 'Kurtosis' variable.


  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double alglib::samplekurtosis(
    real_1d_array x,
    const xparams _params = alglib::xdefault);
double alglib::samplekurtosis(
    real_1d_array x,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`samplemean` function

/*************************************************************************
Calculation of the mean.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

NOTE:

This function return result  which calculated by 'SampleMoments' function
and stored at 'Mean' variable.


  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double alglib::samplemean(
    real_1d_array x,
    const xparams _params = alglib::xdefault);
double alglib::samplemean(
    real_1d_array x,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`samplemedian` function

/*************************************************************************
Median calculation.

Input parameters:
    X   -   sample (array indexes: [0..N-1])
    N   -   N>=0, sample size:
            * if given, only leading N elements of X are processed
            * if not given, automatically determined from size of X

Output parameters:
    Median

  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::samplemedian(
    real_1d_array x,
    double& median,
    const xparams _params = alglib::xdefault);
void alglib::samplemedian(
    real_1d_array x,
    ae_int_t n,
    double& median,
    const xparams _params = alglib::xdefault);

Examples: [1]

`samplemoments` function

/*************************************************************************
Calculation of the distribution moments: mean, variance, skewness, kurtosis.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

OUTPUT PARAMETERS
    Mean    -   mean.
    Variance-   variance.
    Skewness-   skewness (if variance<>0; zero otherwise).
    Kurtosis-   kurtosis (if variance<>0; zero otherwise).

NOTE: variance is calculated by dividing sum of squares by N-1, not N.

  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::samplemoments(
    real_1d_array x,
    double& mean,
    double& variance,
    double& skewness,
    double& kurtosis,
    const xparams _params = alglib::xdefault);
void alglib::samplemoments(
    real_1d_array x,
    ae_int_t n,
    double& mean,
    double& variance,
    double& skewness,
    double& kurtosis,
    const xparams _params = alglib::xdefault);

Examples: [1]

`samplepercentile` function

/*************************************************************************
Percentile calculation.

Input parameters:
    X   -   sample (array indexes: [0..N-1])
    N   -   N>=0, sample size:
            * if given, only leading N elements of X are processed
            * if not given, automatically determined from size of X
    P   -   percentile (0<=P<=1)

Output parameters:
    V   -   percentile

  -- ALGLIB --
     Copyright 01.03.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::samplepercentile(
    real_1d_array x,
    double p,
    double& v,
    const xparams _params = alglib::xdefault);
void alglib::samplepercentile(
    real_1d_array x,
    ae_int_t n,
    double p,
    double& v,
    const xparams _params = alglib::xdefault);

Examples: [1]

`sampleskewness` function

/*************************************************************************
Calculation of the skewness.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

NOTE:

This function return result  which calculated by 'SampleMoments' function
and stored at 'Skewness' variable.


  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double alglib::sampleskewness(
    real_1d_array x,
    const xparams _params = alglib::xdefault);
double alglib::sampleskewness(
    real_1d_array x,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`samplevariance` function

/*************************************************************************
Calculation of the variance.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

NOTE:

This function return result  which calculated by 'SampleMoments' function
and stored at 'Variance' variable.


  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double alglib::samplevariance(
    real_1d_array x,
    const xparams _params = alglib::xdefault);
double alglib::samplevariance(
    real_1d_array x,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`spearmancorr2` function

/*************************************************************************
Spearman's rank correlation coefficient

Input parameters:
    X       -   sample 1 (array indexes: [0..N-1])
    Y       -   sample 2 (array indexes: [0..N-1])
    N       -   N>=0, sample size:
                * if given, only N leading elements of X/Y are processed
                * if not given, automatically determined from input sizes

Result:
    Spearman's rank correlation coefficient
    (zero for N=0 or N=1)

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::spearmancorr2(
    real_1d_array x,
    real_1d_array y,
    const xparams _params = alglib::xdefault);
double alglib::spearmancorr2(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spearmancorrm` function

/*************************************************************************
Spearman's rank correlation matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X are used
            * if not given, automatically determined from input size
    M   -   M>0, number of variables:
            * if given, only leading M columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M,M], correlation matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spearmancorrm(
    real_2d_array x,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);
void alglib::spearmancorrm(
    real_2d_array x,
    ae_int_t n,
    ae_int_t m,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spearmancorrm2` function

/*************************************************************************
Spearman's rank cross-correlation matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M1], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    Y   -   array[N,M2], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X/Y are used
            * if not given, automatically determined from input sizes
    M1  -   M1>0, number of variables in X:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size
    M2  -   M2>0, number of variables in Y:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M1,M2], cross-correlation matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spearmancorrm2(
    real_2d_array x,
    real_2d_array y,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);
void alglib::spearmancorrm2(
    real_2d_array x,
    real_2d_array y,
    ae_int_t n,
    ae_int_t m1,
    ae_int_t m2,
    real_2d_array& c,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spearmanrankcorrelation` function

/*************************************************************************
Obsolete function, we recommend to use SpearmanCorr2().

    -- ALGLIB --
    Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::spearmanrankcorrelation(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

basestat_d_base example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_1d_array x = "[0,1,4,9,16,25,36,49,64,81]";
    double mean;
    double variance;
    double skewness;
    double kurtosis;
    double adev;
    double p;
    double v;

    //
    // Here we demonstrate calculation of sample moments
    // (mean, variance, skewness, kurtosis)
    //
    samplemoments(x, mean, variance, skewness, kurtosis);
    printf("%.1f\n", double(mean)); // EXPECTED: 28.5
    printf("%.1f\n", double(variance)); // EXPECTED: 801.1667
    printf("%.1f\n", double(skewness)); // EXPECTED: 0.5751
    printf("%.1f\n", double(kurtosis)); // EXPECTED: -1.2666

    //
    // Average deviation
    //
    sampleadev(x, adev);
    printf("%.1f\n", double(adev)); // EXPECTED: 23.2

    //
    // Median and percentile
    //
    samplemedian(x, v);
    printf("%.1f\n", double(v)); // EXPECTED: 20.5
    p = 0.5;
    samplepercentile(x, p, v);
    printf("%.1f\n", double(v)); // EXPECTED: 20.5
    return 0;
}

basestat_d_c2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We have two samples - x and y, and want to measure dependency between them
    //
    real_1d_array x = "[0,1,4,9,16,25,36,49,64,81]";
    real_1d_array y = "[0,1,2,3,4,5,6,7,8,9]";
    double v;

    //
    // Three dependency measures are calculated:
    // * covariation
    // * Pearson correlation
    // * Spearman rank correlation
    //
    v = cov2(x, y);
    printf("%.2f\n", double(v)); // EXPECTED: 82.5
    v = pearsoncorr2(x, y);
    printf("%.2f\n", double(v)); // EXPECTED: 0.9627
    v = spearmancorr2(x, y);
    printf("%.2f\n", double(v)); // EXPECTED: 1.000
    return 0;
}

basestat_d_cm example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // X is a sample matrix:
    // * I-th row corresponds to I-th observation
    // * J-th column corresponds to J-th variable
    //
    real_2d_array x = "[[1,0,1],[1,1,0],[-1,1,0],[-2,-1,1],[-1,0,9]]";
    real_2d_array c;

    //
    // Three dependency measures are calculated:
    // * covariation
    // * Pearson correlation
    // * Spearman rank correlation
    //
    // Result is stored into C, with C[i,j] equal to correlation
    // (covariance) between I-th and J-th variables of X.
    //
    covm(x, c);
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[1.80,0.60,-1.40],[0.60,0.70,-0.80],[-1.40,-0.80,14.70]]
    pearsoncorrm(x, c);
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[1.000,0.535,-0.272],[0.535,1.000,-0.249],[-0.272,-0.249,1.000]]
    spearmancorrm(x, c);
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[1.000,0.556,-0.306],[0.556,1.000,-0.750],[-0.306,-0.750,1.000]]
    return 0;
}

basestat_d_cm2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // X and Y are sample matrices:
    // * I-th row corresponds to I-th observation
    // * J-th column corresponds to J-th variable
    //
    real_2d_array x = "[[1,0,1],[1,1,0],[-1,1,0],[-2,-1,1],[-1,0,9]]";
    real_2d_array y = "[[2,3],[2,1],[-1,6],[-9,9],[7,1]]";
    real_2d_array c;

    //
    // Three dependency measures are calculated:
    // * covariation
    // * Pearson correlation
    // * Spearman rank correlation
    //
    // Result is stored into C, with C[i,j] equal to correlation
    // (covariance) between I-th variable of X and J-th variable of Y.
    //
    covm2(x, y, c);
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[4.100,-3.250],[2.450,-1.500],[13.450,-5.750]]
    pearsoncorrm2(x, y, c);
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[0.519,-0.699],[0.497,-0.518],[0.596,-0.433]]
    spearmancorrm2(x, y, c);
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[0.541,-0.649],[0.216,-0.433],[0.433,-0.135]]
    return 0;
}

`bdss` subpackage

Functions

dsoptimalsplit2
dsoptimalsplit2fast

Examples

`dsoptimalsplit2` function

/*************************************************************************
Optimal binary classification

Algorithms finds optimal (=with minimal cross-entropy) binary partition.
Internal subroutine.

INPUT PARAMETERS:
    A       -   array[0..N-1], variable
    C       -   array[0..N-1], class numbers (0 or 1).
    N       -   array size

OUTPUT PARAMETERS:
    Info    -   completetion code:
                * -3, all values of A[] are same (partition is impossible)
                * -2, one of C[] is incorrect (<0, >1)
                * -1, incorrect pararemets were passed (N<=0).
                *  1, OK
    Threshold-  partiton boundary. Left part contains values which are
                strictly less than Threshold. Right part contains values
                which are greater than or equal to Threshold.
    PAL, PBL-   probabilities P(0|v<Threshold) and P(1|v<Threshold)
    PAR, PBR-   probabilities P(0|v>=Threshold) and P(1|v>=Threshold)
    CVE     -   cross-validation estimate of cross-entropy

  -- ALGLIB --
     Copyright 22.05.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::dsoptimalsplit2(
    real_1d_array a,
    integer_1d_array c,
    ae_int_t n,
    ae_int_t& info,
    double& threshold,
    double& pal,
    double& pbl,
    double& par,
    double& pbr,
    double& cve,
    const xparams _params = alglib::xdefault);

`dsoptimalsplit2fast` function

/*************************************************************************
Optimal partition, internal subroutine. Fast version.

Accepts:
    A       array[0..N-1]       array of attributes     array[0..N-1]
    C       array[0..N-1]       array of class labels
    TiesBuf array[0..N]         temporaries (ties)
    CntBuf  array[0..2*NC-1]    temporaries (counts)
    Alpha                       centering factor (0<=alpha<=1, recommended value - 0.05)
    BufR    array[0..N-1]       temporaries
    BufI    array[0..N-1]       temporaries

Output:
    Info    error code (">0"=OK, "<0"=bad)
    RMS     training set RMS error
    CVRMS   leave-one-out RMS error

Note:
    content of all arrays is changed by subroutine;
    it doesn't allocate temporaries.

  -- ALGLIB --
     Copyright 11.12.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::dsoptimalsplit2fast(
    real_1d_array& a,
    integer_1d_array& c,
    integer_1d_array& tiesbuf,
    integer_1d_array& cntbuf,
    real_1d_array& bufr,
    integer_1d_array& bufi,
    ae_int_t n,
    ae_int_t nc,
    double alpha,
    ae_int_t& info,
    double& threshold,
    double& rms,
    double& cvrms,
    const xparams _params = alglib::xdefault);

`bdsvd` subpackage

Functions

rmatrixbdsvd

Examples

`rmatrixbdsvd` function

/*************************************************************************
Singular value decomposition of a bidiagonal matrix (extended algorithm)

COMMERCIAL EDITION OF ALGLIB:

  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives approximately constant  (with  respect  to  number  of
  ! worker threads) acceleration factor which depends on CPU  being  used,
  ! problem  size  and  "baseline"  ALGLIB  edition  which  is  used   for
  ! comparison.
  !
  ! Generally, commercial ALGLIB is several times faster than  open-source
  ! generic C edition, and many times faster than open-source C# edition.
  !
  ! Multithreaded acceleration is NOT supported for this function.
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

The algorithm performs the singular value decomposition  of  a  bidiagonal
matrix B (upper or lower) representing it as B = Q*S*P^T, where Q and  P -
orthogonal matrices, S - diagonal matrix with non-negative elements on the
main diagonal, in descending order.

The  algorithm  finds  singular  values.  In  addition,  the algorithm can
calculate  matrices  Q  and P (more precisely, not the matrices, but their
product  with  given  matrices U and VT - U*Q and (P^T)*VT)).  Of  course,
matrices U and VT can be of any type, including identity. Furthermore, the
algorithm can calculate Q'*C (this product is calculated more  effectively
than U*Q,  because  this calculation operates with rows instead  of matrix
columns).

The feature of the algorithm is its ability to find  all  singular  values
including those which are arbitrarily close to 0  with  relative  accuracy
close to  machine precision. If the parameter IsFractionalAccuracyRequired
is set to True, all singular values will have high relative accuracy close
to machine precision. If the parameter is set to False, only  the  biggest
singular value will have relative accuracy  close  to  machine  precision.
The absolute error of other singular values is equal to the absolute error
of the biggest singular value.

Input parameters:
    D       -   main diagonal of matrix B.
                Array whose index ranges within [0..N-1].
    E       -   superdiagonal (or subdiagonal) of matrix B.
                Array whose index ranges within [0..N-2].
    N       -   size of matrix B.
    IsUpper -   True, if the matrix is upper bidiagonal.
    IsFractionalAccuracyRequired -
                THIS PARAMETER IS IGNORED SINCE ALGLIB 3.5.0
                SINGULAR VALUES ARE ALWAYS SEARCHED WITH HIGH ACCURACY.
    U       -   matrix to be multiplied by Q.
                Array whose indexes range within [0..NRU-1, 0..N-1].
                The matrix can be bigger, in that case only the  submatrix
                [0..NRU-1, 0..N-1] will be multiplied by Q.
    NRU     -   number of rows in matrix U.
    C       -   matrix to be multiplied by Q'.
                Array whose indexes range within [0..N-1, 0..NCC-1].
                The matrix can be bigger, in that case only the  submatrix
                [0..N-1, 0..NCC-1] will be multiplied by Q'.
    NCC     -   number of columns in matrix C.
    VT      -   matrix to be multiplied by P^T.
                Array whose indexes range within [0..N-1, 0..NCVT-1].
                The matrix can be bigger, in that case only the  submatrix
                [0..N-1, 0..NCVT-1] will be multiplied by P^T.
    NCVT    -   number of columns in matrix VT.

Output parameters:
    D       -   singular values of matrix B in descending order.
    U       -   if NRU>0, contains matrix U*Q.
    VT      -   if NCVT>0, contains matrix (P^T)*VT.
    C       -   if NCC>0, contains matrix Q'*C.

Result:
    True, if the algorithm has converged.
    False, if the algorithm hasn't converged (rare case).

NOTE: multiplication U*Q is performed by means of transposition to internal
      buffer, multiplication and backward transposition. It helps to avoid
      costly columnwise operations and speed-up algorithm.

Additional information:
    The type of convergence is controlled by the internal  parameter  TOL.
    If the parameter is greater than 0, the singular values will have
    relative accuracy TOL. If TOL<0, the singular values will have
    absolute accuracy ABS(TOL)*norm(B).
    By default, |TOL| falls within the range of 10*Epsilon and 100*Epsilon,
    where Epsilon is the machine precision. It is not  recommended  to  use
    TOL less than 10*Epsilon since this will  considerably  slow  down  the
    algorithm and may not lead to error decreasing.

History:
    * 31 March, 2007.
        changed MAXITR from 6 to 12.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1999.
*************************************************************************/
bool alglib::rmatrixbdsvd(
    real_1d_array& d,
    real_1d_array e,
    ae_int_t n,
    bool isupper,
    bool isfractionalaccuracyrequired,
    real_2d_array& u,
    ae_int_t nru,
    real_2d_array& c,
    ae_int_t ncc,
    real_2d_array& vt,
    ae_int_t ncvt,
    const xparams _params = alglib::xdefault);

`bessel` subpackage

Functions

besseli0
besseli1
besselj0
besselj1
besseljn
besselk0
besselk1
besselkn
bessely0
bessely1
besselyn

Examples

`besseli0` function

/*************************************************************************
Modified Bessel function of order zero

Returns modified Bessel function of order zero of the
argument.

The function is defined as i0(x) = j0( ix ).

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        30000       5.8e-16     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::besseli0(
    double x,
    const xparams _params = alglib::xdefault);

`besseli1` function

/*************************************************************************
Modified Bessel function of order one

Returns modified Bessel function of order one of the
argument.

The function is defined as i1(x) = -i j1( ix ).

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.9e-15     2.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::besseli1(
    double x,
    const xparams _params = alglib::xdefault);

`besselj0` function

/*************************************************************************
Bessel function of order zero

Returns Bessel function of order zero of the argument.

The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval the following rational
approximation is used:


       2         2
(w - r  ) (w - r  ) P (w) / Q (w)
      1         2    3       8

           2
where w = x  and the two r's are zeros of the function.

In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.

ACCURACY:

                     Absolute error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       60000       4.2e-16     1.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::besselj0(
    double x,
    const xparams _params = alglib::xdefault);

`besselj1` function

/*************************************************************************
Bessel function of order one

Returns Bessel function of order one of the argument.

The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 24 term Chebyshev
expansion is used. In the second, the asymptotic
trigonometric representation is employed using two
rational functions of degree 5/5.

ACCURACY:

                     Absolute error:
arithmetic   domain      # trials      peak         rms
   IEEE      0, 30       30000       2.6e-16     1.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::besselj1(
    double x,
    const xparams _params = alglib::xdefault);

`besseljn` function

/*************************************************************************
Bessel function of integer order

Returns Bessel function of order n, where n is a
(possibly negative) integer.

The ratio of jn(x) to j0(x) is computed by backward
recurrence.  First the ratio jn/jn-1 is found by a
continued fraction expansion.  Then the recurrence
relating successive orders is applied until j0 or j1 is
reached.

If n = 0 or 1 the routine for j0 or j1 is called
directly.

ACCURACY:

                     Absolute error:
arithmetic   range      # trials      peak         rms
   IEEE      0, 30        5000       4.4e-16     7.9e-17


Not suitable for large n or x. Use jv() (fractional order) instead.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::besseljn(
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`besselk0` function

/*************************************************************************
Modified Bessel function, second kind, order zero

Returns modified Bessel function of the second kind
of order zero of the argument.

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

Tested at 2000 random points between 0 and 8.  Peak absolute
error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.2e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::besselk0(
    double x,
    const xparams _params = alglib::xdefault);

`besselk1` function

/*************************************************************************
Modified Bessel function, second kind, order one

Computes the modified Bessel function of the second kind
of order one of the argument.

The range is partitioned into the two intervals [0,2] and
(2, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.2e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::besselk1(
    double x,
    const xparams _params = alglib::xdefault);

`besselkn` function

/*************************************************************************
Modified Bessel function, second kind, integer order

Returns modified Bessel function of the second kind
of order n of the argument.

The range is partitioned into the two intervals [0,9.55] and
(9.55, infinity).  An ascending power series is used in the
low range, and an asymptotic expansion in the high range.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        90000       1.8e-8      3.0e-10

Error is high only near the crossover point x = 9.55
between the two expansions used.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::besselkn(
    ae_int_t nn,
    double x,
    const xparams _params = alglib::xdefault);

`bessely0` function

/*************************************************************************
Bessel function of the second kind, order zero

Returns Bessel function of the second kind, of order
zero, of the argument.

The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval a rational approximation
R(x) is employed to compute
  y0(x)  = R(x)  +   2 * log(x) * j0(x) / PI.
Thus a call to j0() is required.

In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.



ACCURACY:

 Absolute error, when y0(x) < 1; else relative error:

arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.3e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::bessely0(
    double x,
    const xparams _params = alglib::xdefault);

`bessely1` function

/*************************************************************************
Bessel function of second kind of order one

Returns Bessel function of the second kind of order one
of the argument.

The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 25 term Chebyshev
expansion is used, and a call to j1() is required.
In the second, the asymptotic trigonometric representation
is employed using two rational functions of degree 5/5.

ACCURACY:

                     Absolute error:
arithmetic   domain      # trials      peak         rms
   IEEE      0, 30       30000       1.0e-15     1.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::bessely1(
    double x,
    const xparams _params = alglib::xdefault);

`besselyn` function

/*************************************************************************
Bessel function of second kind of integer order

Returns Bessel function of order n, where n is a
(possibly negative) integer.

The function is evaluated by forward recurrence on
n, starting with values computed by the routines
y0() and y1().

If n = 0 or 1 the routine for y0 or y1 is called
directly.

ACCURACY:
                     Absolute error, except relative
                     when y > 1:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       3.4e-15     4.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::besselyn(
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`betaf` subpackage

Functions

beta

Examples

`beta` function

/*************************************************************************
Beta function


                  -     -
                 | (a) | (b)
beta( a, b )  =  -----------.
                    -
                   | (a+b)

For large arguments the logarithm of the function is
evaluated using lgam(), then exponentiated.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,30       30000       8.1e-14     1.1e-14

Cephes Math Library Release 2.0:  April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
*************************************************************************/
double alglib::beta(
    double a,
    double b,
    const xparams _params = alglib::xdefault);

`binomialdistr` subpackage

Functions

binomialcdistribution
binomialdistribution
invbinomialdistribution

Examples

`binomialcdistribution` function

/*************************************************************************
Complemented binomial distribution

Returns the sum of the terms k+1 through n of the Binomial
probability density:

  n
  --  ( n )   j      n-j
  >   (   )  p  (1-p)
  --  ( j )
 j=k+1

The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula

y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).

The arguments must be positive, with p ranging from 0 to 1.

ACCURACY:

Tested at random points (a,b,p).

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      6.7e-15     8.2e-16
 For p between 0 and .001:
   IEEE     0,100       100000      1.5e-13     2.7e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::binomialcdistribution(
    ae_int_t k,
    ae_int_t n,
    double p,
    const xparams _params = alglib::xdefault);

`binomialdistribution` function

/*************************************************************************
Binomial distribution

Returns the sum of the terms 0 through k of the Binomial
probability density:

  k
  --  ( n )   j      n-j
  >   (   )  p  (1-p)
  --  ( j )
 j=0

The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula

y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).

The arguments must be positive, with p ranging from 0 to 1.

ACCURACY:

Tested at random points (a,b,p), with p between 0 and 1.

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      4.3e-15     2.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::binomialdistribution(
    ae_int_t k,
    ae_int_t n,
    double p,
    const xparams _params = alglib::xdefault);

`invbinomialdistribution` function

/*************************************************************************
Inverse binomial distribution

Finds the event probability p such that the sum of the
terms 0 through k of the Binomial probability density
is equal to the given cumulative probability y.

This is accomplished using the inverse beta integral
function and the relation

1 - p = incbi( n-k, k+1, y ).

ACCURACY:

Tested at random points (a,b,p).

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      2.3e-14     6.4e-16
   IEEE     0,10000     100000      6.6e-12     1.2e-13
 For p between 10^-6 and 0.001:
   IEEE     0,100       100000      2.0e-12     1.3e-14
   IEEE     0,10000     100000      1.5e-12     3.2e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::invbinomialdistribution(
    ae_int_t k,
    ae_int_t n,
    double y,
    const xparams _params = alglib::xdefault);

`chebyshev` subpackage

Functions

chebyshevcalculate
chebyshevcoefficients
chebyshevsum
fromchebyshev

Examples

`chebyshevcalculate` function

/*************************************************************************
Calculation of the value of the Chebyshev polynomials of the
first and second kinds.

Parameters:
    r   -   polynomial kind, either 1 or 2.
    n   -   degree, n>=0
    x   -   argument, -1 <= x <= 1

Result:
    the value of the Chebyshev polynomial at x
*************************************************************************/
double alglib::chebyshevcalculate(
    ae_int_t r,
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`chebyshevcoefficients` function

/*************************************************************************
Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void alglib::chebyshevcoefficients(
    ae_int_t n,
    real_1d_array& c,
    const xparams _params = alglib::xdefault);

`chebyshevsum` function

/*************************************************************************
Summation of Chebyshev polynomials using Clenshaw's recurrence formula.

This routine calculates
    c[0]*T0(x) + c[1]*T1(x) + ... + c[N]*TN(x)
or
    c[0]*U0(x) + c[1]*U1(x) + ... + c[N]*UN(x)
depending on the R.

Parameters:
    r   -   polynomial kind, either 1 or 2.
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Chebyshev polynomial at x
*************************************************************************/
double alglib::chebyshevsum(
    real_1d_array c,
    ae_int_t r,
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`fromchebyshev` function

/*************************************************************************
Conversion of a series of Chebyshev polynomials to a power series.

Represents A[0]*T0(x) + A[1]*T1(x) + ... + A[N]*Tn(x) as
B[0] + B[1]*X + ... + B[N]*X^N.

Input parameters:
    A   -   Chebyshev series coefficients
    N   -   degree, N>=0

Output parameters
    B   -   power series coefficients
*************************************************************************/
void alglib::fromchebyshev(
    real_1d_array a,
    ae_int_t n,
    real_1d_array& b,
    const xparams _params = alglib::xdefault);

`chisquaredistr` subpackage

Functions

chisquarecdistribution
chisquaredistribution
invchisquaredistribution

Examples

`chisquarecdistribution` function

/*************************************************************************
Complemented Chi-square distribution

Returns the area under the right hand tail (from x to
infinity) of the Chi square probability density function
with v degrees of freedom:

                                 inf.
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  x

where x is the Chi-square variable.

The incomplete gamma integral is used, according to the
formula

y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::chisquarecdistribution(
    double v,
    double x,
    const xparams _params = alglib::xdefault);

`chisquaredistribution` function

/*************************************************************************
Chi-square distribution

Returns the area under the left hand tail (from 0 to x)
of the Chi square probability density function with
v degrees of freedom.


                                  x
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  0

where x is the Chi-square variable.

The incomplete gamma integral is used, according to the
formula

y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function


Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::chisquaredistribution(
    double v,
    double x,
    const xparams _params = alglib::xdefault);

`invchisquaredistribution` function

/*************************************************************************
Inverse of complemented Chi-square distribution

Finds the Chi-square argument x such that the integral
from x to infinity of the Chi-square density is equal
to the given cumulative probability y.

This is accomplished using the inverse gamma integral
function and the relation

   x/2 = igami( df/2, y );

ACCURACY:

See inverse incomplete gamma function


Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::invchisquaredistribution(
    double v,
    double y,
    const xparams _params = alglib::xdefault);

`clustering` subpackage

Classes

ahcreport
clusterizerstate
kmeansreport

Functions

clusterizercreate
clusterizergetdistances
clusterizergetkclusters
clusterizerrunahc
clusterizerrunkmeans
clusterizerseparatedbycorr
clusterizerseparatedbydist
clusterizersetahcalgo
clusterizersetdistances
clusterizersetkmeansinit
clusterizersetkmeanslimits
clusterizersetpoints
clusterizersetseed

Examples

clst_ahc		Simple hierarchical clusterization with Euclidean distance function
clst_distance		Clusterization with different metric types
clst_kclusters		Obtaining K top clusters from clusterization tree
clst_kmeans		Simple k-means clusterization
clst_linkage		Clusterization with different linkage types

`ahcreport` class

/*************************************************************************
This structure  is used to store results of the agglomerative hierarchical
clustering (AHC).

Following information is returned:

* TerminationType - completion code:
  * 1   for successful completion of algorithm
  * -5  inappropriate combination of  clustering  algorithm  and  distance
        function was used. As for now, it  is  possible  only when  Ward's
        method is called for dataset with non-Euclidean distance function.
  In case negative completion code is returned,  other  fields  of  report
  structure are invalid and should not be used.

* NPoints contains number of points in the original dataset

* Z contains information about merges performed  (see below).  Z  contains
  indexes from the original (unsorted) dataset and it can be used when you
  need to know what points were merged. However, it is not convenient when
  you want to build a dendrograd (see below).

* if  you  want  to  build  dendrogram, you  can use Z, but it is not good
  option, because Z contains  indexes from  unsorted  dataset.  Dendrogram
  built from such dataset is likely to have intersections. So, you have to
  reorder you points before building dendrogram.
  Permutation which reorders point is returned in P. Another representation
  of  merges,  which  is  more  convenient for dendorgram construction, is
  returned in PM.

* more information on format of Z, P and PM can be found below and in the
  examples from ALGLIB Reference Manual.

FORMAL DESCRIPTION OF FIELDS:
    NPoints         number of points
    Z               array[NPoints-1,2],  contains   indexes   of  clusters
                    linked in pairs to  form  clustering  tree.  I-th  row
                    corresponds to I-th merge:
                    * Z[I,0] - index of the first cluster to merge
                    * Z[I,1] - index of the second cluster to merge
                    * Z[I,0]<Z[I,1]
                    * clusters are  numbered  from 0 to 2*NPoints-2,  with
                      indexes from 0 to NPoints-1 corresponding to  points
                      of the original dataset, and indexes from NPoints to
                      2*NPoints-2  correspond  to  clusters  generated  by
                      subsequent  merges  (I-th  row  of Z creates cluster
                      with index NPoints+I).

                    IMPORTANT: indexes in Z[] are indexes in the ORIGINAL,
                    unsorted dataset. In addition to  Z algorithm  outputs
                    permutation which rearranges points in such  way  that
                    subsequent merges are  performed  on  adjacent  points
                    (such order is needed if you want to build dendrogram).
                    However,  indexes  in  Z  are  related  to   original,
                    unrearranged sequence of points.

    P               array[NPoints], permutation which reorders points  for
                    dendrogram  construction.  P[i] contains  index of the
                    position  where  we  should  move  I-th  point  of the
                    original dataset in order to apply merges PZ/PM.

    PZ              same as Z, but for permutation of points given  by  P.
                    The  only  thing  which  changed  are  indexes  of the
                    original points; indexes of clusters remained same.

    MergeDist       array[NPoints-1], contains distances between  clusters
                    being merged (MergeDist[i] correspond to merge  stored
                    in Z[i,...]):
                    * CLINK, SLINK and  average  linkage algorithms report
                      "raw", unmodified distance metric.
                    * Ward's   method   reports   weighted   intra-cluster
                      variance, which is equal to ||Ca-Cb||^2 * Sa*Sb/(Sa+Sb).
                      Here  A  and  B  are  clusters being merged, Ca is a
                      center of A, Cb is a center of B, Sa is a size of A,
                      Sb is a size of B.

    PM              array[NPoints-1,6], another representation of  merges,
                    which is suited for dendrogram construction. It  deals
                    with rearranged points (permutation P is applied)  and
                    represents merges in a form which different  from  one
                    used by Z.
                    For each I from 0 to NPoints-2, I-th row of PM represents
                    merge performed on two clusters C0 and C1. Here:
                    * C0 contains points with indexes PM[I,0]...PM[I,1]
                    * C1 contains points with indexes PM[I,2]...PM[I,3]
                    * indexes stored in PM are given for dataset sorted
                      according to permutation P
                    * PM[I,1]=PM[I,2]-1 (only adjacent clusters are merged)
                    * PM[I,0]<=PM[I,1], PM[I,2]<=PM[I,3], i.e. both
                      clusters contain at least one point
                    * heights of "subdendrograms" corresponding  to  C0/C1
                      are stored in PM[I,4]  and  PM[I,5].  Subdendrograms
                      corresponding   to   single-point   clusters    have
                      height=0. Dendrogram of the merge result has  height
                      H=max(H0,H1)+1.

NOTE: there is one-to-one correspondence between merges described by Z and
      PM. I-th row of Z describes same merge of clusters as I-th row of PM,
      with "left" cluster from Z corresponding to the "left" one from PM.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
class ahcreport
{
    ae_int_t             terminationtype;
    ae_int_t             npoints;
    integer_1d_array     p;
    integer_2d_array     z;
    integer_2d_array     pz;
    integer_2d_array     pm;
    real_1d_array        mergedist;
};

`clusterizerstate` class

/*************************************************************************
This structure is a clusterization engine.

You should not try to access its fields directly.
Use ALGLIB functions in order to work with this object.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
class clusterizerstate
{
};

`kmeansreport` class

/*************************************************************************
This  structure   is  used  to  store  results of the  k-means  clustering
algorithm.

Following information is always returned:
* NPoints contains number of points in the original dataset
* TerminationType contains completion code, negative on failure, positive
  on success
* K contains number of clusters

For positive TerminationType we return:
* NFeatures contains number of variables in the original dataset
* C, which contains centers found by algorithm
* CIdx, which maps points of the original dataset to clusters

FORMAL DESCRIPTION OF FIELDS:
    NPoints         number of points, >=0
    NFeatures       number of variables, >=1
    TerminationType completion code:
                    * -5 if  distance  type  is  anything  different  from
                         Euclidean metric
                    * -3 for degenerate dataset: a) less  than  K  distinct
                         points, b) K=0 for non-empty dataset.
                    * +1 for successful completion
    K               number of clusters
    C               array[K,NFeatures], rows of the array store centers
    CIdx            array[NPoints], which contains cluster indexes
    IterationsCount actual number of iterations performed by clusterizer.
                    If algorithm performed more than one random restart,
                    total number of iterations is returned.
    Energy          merit function, "energy", sum  of  squared  deviations
                    from cluster centers

  -- ALGLIB --
     Copyright 27.11.2012 by Bochkanov Sergey
*************************************************************************/
class kmeansreport
{
    ae_int_t             npoints;
    ae_int_t             nfeatures;
    ae_int_t             terminationtype;
    ae_int_t             iterationscount;
    double               energy;
    ae_int_t             k;
    real_2d_array        c;
    integer_1d_array     cidx;
};

`clusterizercreate` function

/*************************************************************************
This function initializes clusterizer object. Newly initialized object  is
empty, i.e. it does not contain dataset. You should use it as follows:
1. creation
2. dataset is added with ClusterizerSetPoints()
3. additional parameters are set
3. clusterization is performed with one of the clustering functions

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizercreate(
    clusterizerstate& s,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`clusterizergetdistances` function

/*************************************************************************
This function returns distance matrix for dataset

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    XY      -   array[NPoints,NFeatures], dataset
    NPoints -   number of points, >=0
    NFeatures-  number of features, >=1
    DistType-   distance function:
                *  0    Chebyshev distance  (L-inf norm)
                *  1    city block distance (L1 norm)
                *  2    Euclidean distance  (L2 norm, non-squared)
                * 10    Pearson correlation:
                        dist(a,b) = 1-corr(a,b)
                * 11    Absolute Pearson correlation:
                        dist(a,b) = 1-|corr(a,b)|
                * 12    Uncentered Pearson correlation (cosine of the angle):
                        dist(a,b) = a'*b/(|a|*|b|)
                * 13    Absolute uncentered Pearson correlation
                        dist(a,b) = |a'*b|/(|a|*|b|)
                * 20    Spearman rank correlation:
                        dist(a,b) = 1-rankcorr(a,b)
                * 21    Absolute Spearman rank correlation
                        dist(a,b) = 1-|rankcorr(a,b)|

OUTPUT PARAMETERS:
    D       -   array[NPoints,NPoints], distance matrix
                (full matrix is returned, with lower and upper triangles)

NOTE:  different distance functions have different performance penalty:
       * Euclidean or Pearson correlation distances are the fastest ones
       * Spearman correlation distance function is a bit slower
       * city block and Chebyshev distances are order of magnitude slower

       The reason behing difference in performance is that correlation-based
       distance functions are computed using optimized linear algebra kernels,
       while Chebyshev and city block distance functions are computed using
       simple nested loops with two branches at each iteration.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizergetdistances(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nfeatures,
    ae_int_t disttype,
    real_2d_array& d,
    const xparams _params = alglib::xdefault);

`clusterizergetkclusters` function

/*************************************************************************
This function takes as input clusterization report Rep,  desired  clusters
count K, and builds top K clusters from hierarchical clusterization  tree.
It returns assignment of points to clusters (array of cluster indexes).

INPUT PARAMETERS:
    Rep     -   report from ClusterizerRunAHC() performed on XY
    K       -   desired number of clusters, 1<=K<=NPoints.
                K can be zero only when NPoints=0.

OUTPUT PARAMETERS:
    CIdx    -   array[NPoints], I-th element contains cluster index  (from
                0 to K-1) for I-th point of the dataset.
    CZ      -   array[K]. This array allows  to  convert  cluster  indexes
                returned by this function to indexes used by  Rep.Z.  J-th
                cluster returned by this function corresponds to  CZ[J]-th
                cluster stored in Rep.Z/PZ/PM.
                It is guaranteed that CZ[I]<CZ[I+1].

NOTE: K clusters built by this subroutine are assumed to have no hierarchy.
      Although  they  were  obtained  by  manipulation with top K nodes of
      dendrogram  (i.e.  hierarchical  decomposition  of  dataset),   this
      function does not return information about hierarchy.  Each  of  the
      clusters stand on its own.

NOTE: Cluster indexes returned by this function  does  not  correspond  to
      indexes returned in Rep.Z/PZ/PM. Either you work  with  hierarchical
      representation of the dataset (dendrogram), or you work with  "flat"
      representation returned by this function.  Each  of  representations
      has its own clusters indexing system (former uses [0, 2*NPoints-2]),
      while latter uses [0..K-1]), although  it  is  possible  to  perform
      conversion from one system to another by means of CZ array, returned
      by this function, which allows you to convert indexes stored in CIdx
      to the numeration system used by Rep.Z.

NOTE: this subroutine is optimized for moderate values of K. Say, for  K=5
      it will perform many times faster than  for  K=100.  Its  worst-case
      performance is O(N*K), although in average case  it  perform  better
      (up to O(N*log(K))).

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizergetkclusters(
    ahcreport rep,
    ae_int_t k,
    integer_1d_array& cidx,
    integer_1d_array& cz,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`clusterizerrunahc` function

/*************************************************************************
This function performs agglomerative hierarchical clustering

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Agglomerative  hierarchical  clustering  algorithm  has two  phases:
      distance matrix calculation and clustering  itself. Only first phase
      (distance matrix  calculation)  is  accelerated  by  Intel  MKL  and
      multithreading. Thus, acceleration is significant only for medium or
      high-dimensional problems.

      Although activating multithreading gives some speedup  over  single-
      threaded execution, you  should  not  expect  nearly-linear  scaling
      with respect to cores count.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()

OUTPUT PARAMETERS:
    Rep     -   clustering results; see description of AHCReport
                structure for more information.

NOTE 1: hierarchical clustering algorithms require large amounts of memory.
        In particular, this implementation needs  sizeof(double)*NPoints^2
        bytes, which are used to store distance matrix. In  case  we  work
        with user-supplied matrix, this amount is multiplied by 2 (we have
        to store original matrix and to work with its copy).

        For example, problem with 10000 points  would require 800M of RAM,
        even when working in a 1-dimensional space.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizerrunahc(
    clusterizerstate s,
    ahcreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`clusterizerrunkmeans` function

/*************************************************************************
This function performs clustering by k-means++ algorithm.

You may change algorithm properties by calling:
* ClusterizerSetKMeansLimits() to change number of restarts or iterations
* ClusterizerSetKMeansInit() to change initialization algorithm

By  default,  one  restart  and  unlimited number of iterations are  used.
Initialization algorithm is chosen automatically.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: k-means clustering  algorithm has two  phases:  selection of initial
      centers and clustering  itself.  ALGLIB  parallelizes  both  phases.
      Parallel version is optimized for the following  scenario: medium or
      high-dimensional problem (8 or more dimensions) with large number of
      points and clusters. However, some speed-up  can  be  obtained  even
      when assumptions above are violated.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    K       -   number of clusters, K>=0.
                K  can  be  zero only when algorithm is called  for  empty
                dataset,  in   this   case   completion  code  is  set  to
                success (+1).
                If  K=0  and  dataset  size  is  non-zero,  we   can   not
                meaningfully assign points to some center  (there  are  no
                centers because K=0) and  return  -3  as  completion  code
                (failure).

OUTPUT PARAMETERS:
    Rep     -   clustering results; see description of KMeansReport
                structure for more information.

NOTE 1: k-means  clustering  can  be  performed  only  for  datasets  with
        Euclidean  distance  function.  Algorithm  will  return   negative
        completion code in Rep.TerminationType in case dataset  was  added
        to clusterizer with DistType other than Euclidean (or dataset  was
        specified by distance matrix instead of explicitly given points).

NOTE 2: by default, k-means uses non-deterministic seed to initialize  RNG
        which is used to select initial centers. As  result,  each  run of
        algorithm may return different values. If you  need  deterministic
        behavior, use ClusterizerSetSeed() function.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizerrunkmeans(
    clusterizerstate s,
    ae_int_t k,
    kmeansreport& rep,
    const xparams _params = alglib::xdefault);

`clusterizerseparatedbycorr` function

/*************************************************************************
This  function  accepts  AHC  report  Rep,  desired  maximum  intercluster
correlation and returns top clusters from hierarchical clusterization tree
which are separated by correlation R or LOWER.

It returns assignment of points to clusters (array of cluster indexes).

There is one more function with similar name - ClusterizerSeparatedByDist,
which returns clusters with intercluster distance equal  to  R  or  HIGHER
(note: higher for distance, lower for correlation).

INPUT PARAMETERS:
    Rep     -   report from ClusterizerRunAHC() performed on XY
    R       -   desired maximum intercluster correlation, -1<=R<=+1

OUTPUT PARAMETERS:
    K       -   number of clusters, 1<=K<=NPoints
    CIdx    -   array[NPoints], I-th element contains cluster index  (from
                0 to K-1) for I-th point of the dataset.
    CZ      -   array[K]. This array allows  to  convert  cluster  indexes
                returned by this function to indexes used by  Rep.Z.  J-th
                cluster returned by this function corresponds to  CZ[J]-th
                cluster stored in Rep.Z/PZ/PM.
                It is guaranteed that CZ[I]<CZ[I+1].

NOTE: K clusters built by this subroutine are assumed to have no hierarchy.
      Although  they  were  obtained  by  manipulation with top K nodes of
      dendrogram  (i.e.  hierarchical  decomposition  of  dataset),   this
      function does not return information about hierarchy.  Each  of  the
      clusters stand on its own.

NOTE: Cluster indexes returned by this function  does  not  correspond  to
      indexes returned in Rep.Z/PZ/PM. Either you work  with  hierarchical
      representation of the dataset (dendrogram), or you work with  "flat"
      representation returned by this function.  Each  of  representations
      has its own clusters indexing system (former uses [0, 2*NPoints-2]),
      while latter uses [0..K-1]), although  it  is  possible  to  perform
      conversion from one system to another by means of CZ array, returned
      by this function, which allows you to convert indexes stored in CIdx
      to the numeration system used by Rep.Z.

NOTE: this subroutine is optimized for moderate values of K. Say, for  K=5
      it will perform many times faster than  for  K=100.  Its  worst-case
      performance is O(N*K), although in average case  it  perform  better
      (up to O(N*log(K))).

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizerseparatedbycorr(
    ahcreport rep,
    double r,
    ae_int_t& k,
    integer_1d_array& cidx,
    integer_1d_array& cz,
    const xparams _params = alglib::xdefault);

`clusterizerseparatedbydist` function

/*************************************************************************
This  function  accepts  AHC  report  Rep,  desired  minimum  intercluster
distance and returns top clusters from  hierarchical  clusterization  tree
which are separated by distance R or HIGHER.

It returns assignment of points to clusters (array of cluster indexes).

There is one more function with similar name - ClusterizerSeparatedByCorr,
which returns clusters with intercluster correlation equal to R  or  LOWER
(note: higher for distance, lower for correlation).

INPUT PARAMETERS:
    Rep     -   report from ClusterizerRunAHC() performed on XY
    R       -   desired minimum intercluster distance, R>=0

OUTPUT PARAMETERS:
    K       -   number of clusters, 1<=K<=NPoints
    CIdx    -   array[NPoints], I-th element contains cluster index  (from
                0 to K-1) for I-th point of the dataset.
    CZ      -   array[K]. This array allows  to  convert  cluster  indexes
                returned by this function to indexes used by  Rep.Z.  J-th
                cluster returned by this function corresponds to  CZ[J]-th
                cluster stored in Rep.Z/PZ/PM.
                It is guaranteed that CZ[I]<CZ[I+1].

NOTE: K clusters built by this subroutine are assumed to have no hierarchy.
      Although  they  were  obtained  by  manipulation with top K nodes of
      dendrogram  (i.e.  hierarchical  decomposition  of  dataset),   this
      function does not return information about hierarchy.  Each  of  the
      clusters stand on its own.

NOTE: Cluster indexes returned by this function  does  not  correspond  to
      indexes returned in Rep.Z/PZ/PM. Either you work  with  hierarchical
      representation of the dataset (dendrogram), or you work with  "flat"
      representation returned by this function.  Each  of  representations
      has its own clusters indexing system (former uses [0, 2*NPoints-2]),
      while latter uses [0..K-1]), although  it  is  possible  to  perform
      conversion from one system to another by means of CZ array, returned
      by this function, which allows you to convert indexes stored in CIdx
      to the numeration system used by Rep.Z.

NOTE: this subroutine is optimized for moderate values of K. Say, for  K=5
      it will perform many times faster than  for  K=100.  Its  worst-case
      performance is O(N*K), although in average case  it  perform  better
      (up to O(N*log(K))).

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizerseparatedbydist(
    ahcreport rep,
    double r,
    ae_int_t& k,
    integer_1d_array& cidx,
    integer_1d_array& cz,
    const xparams _params = alglib::xdefault);

`clusterizersetahcalgo` function

/*************************************************************************
This function sets agglomerative hierarchical clustering algorithm

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    Algo    -   algorithm type:
                * 0     complete linkage (default algorithm)
                * 1     single linkage
                * 2     unweighted average linkage
                * 3     weighted average linkage
                * 4     Ward's method

NOTE: Ward's method works correctly only with Euclidean  distance,  that's
      why algorithm will return negative termination  code  (failure)  for
      any other distance type.

      It is possible, however,  to  use  this  method  with  user-supplied
      distance matrix. It  is  your  responsibility  to pass one which was
      calculated with Euclidean distance function.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizersetahcalgo(
    clusterizerstate s,
    ae_int_t algo,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`clusterizersetdistances` function

/*************************************************************************
This function adds dataset given by distance  matrix  to  the  clusterizer
structure. It is important that dataset is not  given  explicitly  -  only
distance matrix is given.

This function overrides all previous calls  of  ClusterizerSetPoints()  or
ClusterizerSetDistances().

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    D       -   array[NPoints,NPoints], distance matrix given by its upper
                or lower triangle (main diagonal is  ignored  because  its
                entries are expected to be zero).
    NPoints -   number of points
    IsUpper -   whether upper or lower triangle of D is given.

NOTE 1: different clustering algorithms have different limitations:
        * agglomerative hierarchical clustering algorithms may be used with
          any kind of distance metric, including one  which  is  given  by
          distance matrix
        * k-means++ clustering algorithm may be used only  with  Euclidean
          distance function and explicitly given points - it  can  not  be
          used with dataset given by distance matrix
        Thus, if you call this function, you will be unable to use k-means
        clustering algorithm to process your problem.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizersetdistances(
    clusterizerstate s,
    real_2d_array d,
    bool isupper,
    const xparams _params = alglib::xdefault);
void alglib::clusterizersetdistances(
    clusterizerstate s,
    real_2d_array d,
    ae_int_t npoints,
    bool isupper,
    const xparams _params = alglib::xdefault);

Examples: [1]

`clusterizersetkmeansinit` function

/*************************************************************************
This function sets k-means  initialization  algorithm.  Several  different
algorithms can be chosen, including k-means++.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    InitAlgo-   initialization algorithm:
                * 0  automatic selection ( different  versions  of  ALGLIB
                     may select different algorithms)
                * 1  random initialization
                * 2  k-means++ initialization  (best  quality  of  initial
                     centers, but long  non-parallelizable  initialization
                     phase with bad cache locality)
                * 3  "fast-greedy"  algorithm  with  efficient,  easy   to
                     parallelize initialization. Quality of initial centers
                     is  somewhat  worse  than  that  of  k-means++.  This
                     algorithm is a default one in the current version  of
                     ALGLIB.
                *-1  "debug" algorithm which always selects first  K  rows
                     of dataset; this algorithm is used for debug purposes
                     only. Do not use it in the industrial code!

  -- ALGLIB --
     Copyright 21.01.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizersetkmeansinit(
    clusterizerstate s,
    ae_int_t initalgo,
    const xparams _params = alglib::xdefault);

`clusterizersetkmeanslimits` function

/*************************************************************************
This  function  sets k-means properties:  number  of  restarts and maximum
number of iterations per one run.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    Restarts-   restarts count, >=1.
                k-means++ algorithm performs several restarts and  chooses
                best set of centers (one with minimum squared distance).
    MaxIts  -   maximum number of k-means iterations performed during  one
                run. >=0, zero value means that algorithm performs unlimited
                number of iterations.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizersetkmeanslimits(
    clusterizerstate s,
    ae_int_t restarts,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

`clusterizersetpoints` function

/*************************************************************************
This function adds dataset to the clusterizer structure.

This function overrides all previous calls  of  ClusterizerSetPoints()  or
ClusterizerSetDistances().

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    XY      -   array[NPoints,NFeatures], dataset
    NPoints -   number of points, >=0
    NFeatures-  number of features, >=1
    DistType-   distance function:
                *  0    Chebyshev distance  (L-inf norm)
                *  1    city block distance (L1 norm)
                *  2    Euclidean distance  (L2 norm), non-squared
                * 10    Pearson correlation:
                        dist(a,b) = 1-corr(a,b)
                * 11    Absolute Pearson correlation:
                        dist(a,b) = 1-|corr(a,b)|
                * 12    Uncentered Pearson correlation (cosine of the angle):
                        dist(a,b) = a'*b/(|a|*|b|)
                * 13    Absolute uncentered Pearson correlation
                        dist(a,b) = |a'*b|/(|a|*|b|)
                * 20    Spearman rank correlation:
                        dist(a,b) = 1-rankcorr(a,b)
                * 21    Absolute Spearman rank correlation
                        dist(a,b) = 1-|rankcorr(a,b)|

NOTE 1: different distance functions have different performance penalty:
        * Euclidean or Pearson correlation distances are the fastest ones
        * Spearman correlation distance function is a bit slower
        * city block and Chebyshev distances are order of magnitude slower

        The reason behing difference in performance is that correlation-based
        distance functions are computed using optimized linear algebra kernels,
        while Chebyshev and city block distance functions are computed using
        simple nested loops with two branches at each iteration.

NOTE 2: different clustering algorithms have different limitations:
        * agglomerative hierarchical clustering algorithms may be used with
          any kind of distance metric
        * k-means++ clustering algorithm may be used only  with  Euclidean
          distance function
        Thus, list of specific clustering algorithms you may  use  depends
        on distance function you specify when you set your dataset.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizersetpoints(
    clusterizerstate s,
    real_2d_array xy,
    ae_int_t disttype,
    const xparams _params = alglib::xdefault);
void alglib::clusterizersetpoints(
    clusterizerstate s,
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nfeatures,
    ae_int_t disttype,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`clusterizersetseed` function

/*************************************************************************
This  function  sets  seed  which  is  used to initialize internal RNG. By
default, deterministic seed is used - same for each run of clusterizer. If
you specify non-deterministic  seed  value,  then  some  algorithms  which
depend on random initialization (in current version: k-means)  may  return
slightly different results after each run.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    Seed    -   seed:
                * positive values = use deterministic seed for each run of
                  algorithms which depend on random initialization
                * zero or negative values = use non-deterministic seed

  -- ALGLIB --
     Copyright 08.06.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::clusterizersetseed(
    clusterizerstate s,
    ae_int_t seed,
    const xparams _params = alglib::xdefault);

clst_ahc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // The very simple clusterization example
    //
    // We have a set of points in 2D space:
    //     (P0,P1,P2,P3,P4) = ((1,1),(1,2),(4,1),(2,3),(4,1.5))
    //
    //  |
    //  |     P3
    //  |
    //  | P1          
    //  |             P4
    //  | P0          P2
    //  |-------------------------
    //
    // We want to perform Agglomerative Hierarchic Clusterization (AHC),
    // using complete linkage (default algorithm) and Euclidean distance
    // (default metric).
    //
    // In order to do that, we:
    // * create clusterizer with clusterizercreate()
    // * set points XY and metric (2=Euclidean) with clusterizersetpoints()
    // * run AHC algorithm with clusterizerrunahc
    //
    // You may see that clusterization itself is a minor part of the example,
    // most of which is dominated by comments :)
    //
    clusterizerstate s;
    ahcreport rep;
    real_2d_array xy = "[[1,1],[1,2],[4,1],[2,3],[4,1.5]]";

    clusterizercreate(s);
    clusterizersetpoints(s, xy, 2);
    clusterizerrunahc(s, rep);

    //
    // Now we've built our clusterization tree. Rep.z contains information which
    // is required to build dendrogram. I-th row of rep.z represents one merge
    // operation, with first cluster to merge having index rep.z[I,0] and second
    // one having index rep.z[I,1]. Merge result has index NPoints+I.
    //
    // Clusters with indexes less than NPoints are single-point initial clusters,
    // while ones with indexes from NPoints to 2*NPoints-2 are multi-point
    // clusters created during merges.
    //
    // In our example, Z=[[2,4], [0,1], [3,6], [5,7]]
    //
    // It means that:
    // * first, we merge C2=(P2) and C4=(P4),    and create C5=(P2,P4)
    // * then, we merge  C2=(P0) and C1=(P1),    and create C6=(P0,P1)
    // * then, we merge  C3=(P3) and C6=(P0,P1), and create C7=(P0,P1,P3)
    // * finally, we merge C5 and C7 and create C8=(P0,P1,P2,P3,P4)
    //
    // Thus, we have following dendrogram:
    //  
    //      ------8-----
    //      |          |
    //      |      ----7----
    //      |      |       |
    //   ---5---   |    ---6---
    //   |     |   |    |     |
    //   P2   P4   P3   P0   P1
    //
    printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[2,4],[0,1],[3,6],[5,7]]

    //
    // We've built dendrogram above by reordering our dataset.
    //
    // Without such reordering it would be impossible to build dendrogram without
    // intersections. Luckily, ahcreport structure contains two additional fields
    // which help to build dendrogram from your data:
    // * rep.p, which contains permutation applied to dataset
    // * rep.pm, which contains another representation of merges 
    //
    // In our example we have:
    // * P=[3,4,0,2,1]
    // * PZ=[[0,0,1,1,0,0],[3,3,4,4,0,0],[2,2,3,4,0,1],[0,1,2,4,1,2]]
    //
    // Permutation array P tells us that P0 should be moved to position 3,
    // P1 moved to position 4, P2 moved to position 0 and so on:
    //
    //   (P0 P1 P2 P3 P4) => (P2 P4 P3 P0 P1)
    //
    // Merges array PZ tells us how to perform merges on the sorted dataset.
    // One row of PZ corresponds to one merge operations, with first pair of
    // elements denoting first of the clusters to merge (start index, end
    // index) and next pair of elements denoting second of the clusters to
    // merge. Clusters being merged are always adjacent, with first one on
    // the left and second one on the right.
    //
    // For example, first row of PZ tells us that clusters [0,0] and [1,1] are
    // merged (single-point clusters, with first one containing P2 and second
    // one containing P4). Third row of PZ tells us that we merge one single-
    // point cluster [2,2] with one two-point cluster [3,4].
    //
    // There are two more elements in each row of PZ. These are the helper
    // elements, which denote HEIGHT (not size) of left and right subdendrograms.
    // For example, according to PZ, first two merges are performed on clusterization
    // trees of height 0, while next two merges are performed on 0-1 and 1-2
    // pairs of trees correspondingly.
    //
    printf("%s\n", rep.p.tostring().c_str()); // EXPECTED: [3,4,0,2,1]
    printf("%s\n", rep.pm.tostring().c_str()); // EXPECTED: [[0,0,1,1,0,0],[3,3,4,4,0,0],[2,2,3,4,0,1],[0,1,2,4,1,2]]
    return 0;
}

clst_distance example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We have three points in 4D space:
    //     (P0,P1,P2) = ((1, 2, 1, 2), (6, 7, 6, 7), (7, 6, 7, 6))
    //
    // We want to try clustering them with different distance functions.
    // Distance function is chosen when we add dataset to the clusterizer.
    // We can choose several distance types - Euclidean, city block, Chebyshev,
    // several correlation measures or user-supplied distance matrix.
    //
    // Here we'll try three distances: Euclidean, Pearson correlation,
    // user-supplied distance matrix. Different distance functions lead
    // to different choices being made by algorithm during clustering.
    //
    clusterizerstate s;
    ahcreport rep;
    ae_int_t disttype;
    real_2d_array xy = "[[1, 2, 1, 2], [6, 7, 6, 7], [7, 6, 7, 6]]";
    clusterizercreate(s);

    // With Euclidean distance function (disttype=2) two closest points
    // are P1 and P2, thus:
    // * first, we merge P1 and P2 to form C3=[P1,P2]
    // * second, we merge P0 and C3 to form C4=[P0,P1,P2]
    disttype = 2;
    clusterizersetpoints(s, xy, disttype);
    clusterizerrunahc(s, rep);
    printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[1,2],[0,3]]

    // With Pearson correlation distance function (disttype=10) situation
    // is different - distance between P0 and P1 is zero, thus:
    // * first, we merge P0 and P1 to form C3=[P0,P1]
    // * second, we merge P2 and C3 to form C4=[P0,P1,P2]
    disttype = 10;
    clusterizersetpoints(s, xy, disttype);
    clusterizerrunahc(s, rep);
    printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[0,1],[2,3]]

    // Finally, we try clustering with user-supplied distance matrix:
    //     [ 0 3 1 ]
    // P = [ 3 0 3 ], where P[i,j] = dist(Pi,Pj)
    //     [ 1 3 0 ]
    //
    // * first, we merge P0 and P2 to form C3=[P0,P2]
    // * second, we merge P1 and C3 to form C4=[P0,P1,P2]
    real_2d_array d = "[[0,3,1],[3,0,3],[1,3,0]]";
    clusterizersetdistances(s, d, true);
    clusterizerrunahc(s, rep);
    printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[0,2],[1,3]]
    return 0;
}

clst_kclusters example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We have a set of points in 2D space:
    //     (P0,P1,P2,P3,P4) = ((1,1),(1,2),(4,1),(2,3),(4,1.5))
    //
    //  |
    //  |     P3
    //  |
    //  | P1          
    //  |             P4
    //  | P0          P2
    //  |-------------------------
    //
    // We perform Agglomerative Hierarchic Clusterization (AHC) and we want
    // to get top K clusters from clusterization tree for different K.
    //
    clusterizerstate s;
    ahcreport rep;
    real_2d_array xy = "[[1,1],[1,2],[4,1],[2,3],[4,1.5]]";
    integer_1d_array cidx;
    integer_1d_array cz;

    clusterizercreate(s);
    clusterizersetpoints(s, xy, 2);
    clusterizerrunahc(s, rep);

    // with K=5, every points is assigned to its own cluster:
    // C0=P0, C1=P1 and so on...
    clusterizergetkclusters(rep, 5, cidx, cz);
    printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [0,1,2,3,4]

    // with K=1 we have one large cluster C0=[P0,P1,P2,P3,P4,P5]
    clusterizergetkclusters(rep, 1, cidx, cz);
    printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [0,0,0,0,0]

    // with K=3 we have three clusters C0=[P3], C1=[P2,P4], C2=[P0,P1]
    clusterizergetkclusters(rep, 3, cidx, cz);
    printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [2,2,1,0,1]
    return 0;
}

clst_kmeans example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // The very simple clusterization example
    //
    // We have a set of points in 2D space:
    //     (P0,P1,P2,P3,P4) = ((1,1),(1,2),(4,1),(2,3),(4,1.5))
    //
    //  |
    //  |     P3
    //  |
    //  | P1          
    //  |             P4
    //  | P0          P2
    //  |-------------------------
    //
    // We want to perform k-means++ clustering with K=2.
    //
    // In order to do that, we:
    // * create clusterizer with clusterizercreate()
    // * set points XY and metric (must be Euclidean, distype=2) with clusterizersetpoints()
    // * (optional) set number of restarts from random positions to 5
    // * run k-means algorithm with clusterizerrunkmeans()
    //
    // You may see that clusterization itself is a minor part of the example,
    // most of which is dominated by comments :)
    //
    clusterizerstate s;
    kmeansreport rep;
    real_2d_array xy = "[[1,1],[1,2],[4,1],[2,3],[4,1.5]]";

    clusterizercreate(s);
    clusterizersetpoints(s, xy, 2);
    clusterizersetkmeanslimits(s, 5, 0);
    clusterizerrunkmeans(s, 2, rep);

    //
    // We've performed clusterization, and it succeeded (completion code is +1).
    //
    // Now first center is stored in the first row of rep.c, second one is stored
    // in the second row. rep.cidx can be used to determine which center is
    // closest to some specific point of the dataset.
    //
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1

    // We called clusterizersetpoints() with disttype=2 because k-means++
    // algorithm does NOT support metrics other than Euclidean. But what if we
    // try to use some other metric?
    //
    // We change metric type by calling clusterizersetpoints() one more time,
    // and try to run k-means algo again. It fails.
    //
    clusterizersetpoints(s, xy, 0);
    clusterizerrunkmeans(s, 2, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: -5
    return 0;
}

clst_linkage example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We have a set of points in 1D space:
    //     (P0,P1,P2,P3,P4) = (1, 3, 10, 16, 20)
    //
    // We want to perform Agglomerative Hierarchic Clusterization (AHC),
    // using either complete or single linkage and Euclidean distance
    // (default metric).
    //
    // First two steps merge P0/P1 and P3/P4 independently of the linkage type.
    // However, third step depends on linkage type being used:
    // * in case of complete linkage P2=10 is merged with [P0,P1]
    // * in case of single linkage P2=10 is merged with [P3,P4]
    //
    clusterizerstate s;
    ahcreport rep;
    real_2d_array xy = "[[1],[3],[10],[16],[20]]";
    integer_1d_array cidx;
    integer_1d_array cz;

    clusterizercreate(s);
    clusterizersetpoints(s, xy, 2);

    // use complete linkage, reduce set down to 2 clusters.
    // print clusterization with clusterizergetkclusters(2).
    // P2 must belong to [P0,P1]
    clusterizersetahcalgo(s, 0);
    clusterizerrunahc(s, rep);
    clusterizergetkclusters(rep, 2, cidx, cz);
    printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [1,1,1,0,0]

    // use single linkage, reduce set down to 2 clusters.
    // print clusterization with clusterizergetkclusters(2).
    // P2 must belong to [P2,P3]
    clusterizersetahcalgo(s, 1);
    clusterizerrunahc(s, rep);
    clusterizergetkclusters(rep, 2, cidx, cz);
    printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [0,0,1,1,1]
    return 0;
}

`conv` subpackage

Functions

convc1d
convc1dcircular
convc1dcircularinv
convc1dinv
convr1d
convr1dcircular
convr1dcircularinv
convr1dinv

Examples

`convc1d` function

/*************************************************************************
1-dimensional complex convolution.

For given A/B returns conv(A,B) (non-circular). Subroutine can automatically
choose between three implementations: straightforward O(M*N)  formula  for
very small N (or M), overlap-add algorithm for  cases  where  max(M,N)  is
significantly larger than min(M,N), but O(M*N) algorithm is too slow,  and
general FFT-based formula for cases where two previois algorithms are  too
slow.

Algorithm has max(M,N)*log(max(M,N)) complexity for any M/N.

INPUT PARAMETERS
    A   -   array[0..M-1] - complex function to be transformed
    M   -   problem size
    B   -   array[0..N-1] - complex function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..N+M-2].

NOTE:
    It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
functions have non-zero values at negative T's, you  can  still  use  this
subroutine - just shift its result correspondingly.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::convc1d(
    complex_1d_array a,
    ae_int_t m,
    complex_1d_array b,
    ae_int_t n,
    complex_1d_array& r,
    const xparams _params = alglib::xdefault);

`convc1dcircular` function

/*************************************************************************
1-dimensional circular complex convolution.

For given S/R returns conv(S,R) (circular). Algorithm has linearithmic
complexity for any M/N.

IMPORTANT:  normal convolution is commutative,  i.e.   it  is symmetric  -
conv(A,B)=conv(B,A).  Cyclic convolution IS NOT.  One function - S - is  a
signal,  periodic function, and another - R - is a response,  non-periodic
function with limited length.

INPUT PARAMETERS
    S   -   array[0..M-1] - complex periodic signal
    M   -   problem size
    B   -   array[0..N-1] - complex non-periodic response
    N   -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..M-1].

NOTE:
    It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
negative T's, you can still use this subroutine - just  shift  its  result
correspondingly.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::convc1dcircular(
    complex_1d_array s,
    ae_int_t m,
    complex_1d_array r,
    ae_int_t n,
    complex_1d_array& c,
    const xparams _params = alglib::xdefault);

`convc1dcircularinv` function

/*************************************************************************
1-dimensional circular complex deconvolution (inverse of ConvC1DCircular()).

Algorithm has M*log(M)) complexity for any M (composite or prime).

INPUT PARAMETERS
    A   -   array[0..M-1] - convolved periodic signal, A = conv(R, B)
    M   -   convolved signal length
    B   -   array[0..N-1] - non-periodic response
    N   -   response length

OUTPUT PARAMETERS
    R   -   deconvolved signal. array[0..M-1].

NOTE:
    deconvolution is unstable process and may result in division  by  zero
(if your response function is degenerate, i.e. has zero Fourier coefficient).

NOTE:
    It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
negative T's, you can still use this subroutine - just  shift  its  result
correspondingly.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::convc1dcircularinv(
    complex_1d_array a,
    ae_int_t m,
    complex_1d_array b,
    ae_int_t n,
    complex_1d_array& r,
    const xparams _params = alglib::xdefault);

`convc1dinv` function

/*************************************************************************
1-dimensional complex non-circular deconvolution (inverse of ConvC1D()).

Algorithm has M*log(M)) complexity for any M (composite or prime).

INPUT PARAMETERS
    A   -   array[0..M-1] - convolved signal, A = conv(R, B)
    M   -   convolved signal length
    B   -   array[0..N-1] - response
    N   -   response length, N<=M

OUTPUT PARAMETERS
    R   -   deconvolved signal. array[0..M-N].

NOTE:
    deconvolution is unstable process and may result in division  by  zero
(if your response function is degenerate, i.e. has zero Fourier coefficient).

NOTE:
    It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
functions have non-zero values at negative T's, you  can  still  use  this
subroutine - just shift its result correspondingly.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::convc1dinv(
    complex_1d_array a,
    ae_int_t m,
    complex_1d_array b,
    ae_int_t n,
    complex_1d_array& r,
    const xparams _params = alglib::xdefault);

`convr1d` function

/*************************************************************************
1-dimensional real convolution.

Analogous to ConvC1D(), see ConvC1D() comments for more details.

INPUT PARAMETERS
    A   -   array[0..M-1] - real function to be transformed
    M   -   problem size
    B   -   array[0..N-1] - real function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..N+M-2].

NOTE:
    It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
functions have non-zero values at negative T's, you  can  still  use  this
subroutine - just shift its result correspondingly.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::convr1d(
    real_1d_array a,
    ae_int_t m,
    real_1d_array b,
    ae_int_t n,
    real_1d_array& r,
    const xparams _params = alglib::xdefault);

`convr1dcircular` function

/*************************************************************************
1-dimensional circular real convolution.

Analogous to ConvC1DCircular(), see ConvC1DCircular() comments for more details.

INPUT PARAMETERS
    S   -   array[0..M-1] - real signal
    M   -   problem size
    B   -   array[0..N-1] - real response
    N   -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..M-1].

NOTE:
    It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
negative T's, you can still use this subroutine - just  shift  its  result
correspondingly.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::convr1dcircular(
    real_1d_array s,
    ae_int_t m,
    real_1d_array r,
    ae_int_t n,
    real_1d_array& c,
    const xparams _params = alglib::xdefault);

`convr1dcircularinv` function

/*************************************************************************
1-dimensional complex deconvolution (inverse of ConvC1D()).

Algorithm has M*log(M)) complexity for any M (composite or prime).

INPUT PARAMETERS
    A   -   array[0..M-1] - convolved signal, A = conv(R, B)
    M   -   convolved signal length
    B   -   array[0..N-1] - response
    N   -   response length

OUTPUT PARAMETERS
    R   -   deconvolved signal. array[0..M-N].

NOTE:
    deconvolution is unstable process and may result in division  by  zero
(if your response function is degenerate, i.e. has zero Fourier coefficient).

NOTE:
    It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
negative T's, you can still use this subroutine - just  shift  its  result
correspondingly.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::convr1dcircularinv(
    real_1d_array a,
    ae_int_t m,
    real_1d_array b,
    ae_int_t n,
    real_1d_array& r,
    const xparams _params = alglib::xdefault);

`convr1dinv` function

/*************************************************************************
1-dimensional real deconvolution (inverse of ConvC1D()).

Algorithm has M*log(M)) complexity for any M (composite or prime).

INPUT PARAMETERS
    A   -   array[0..M-1] - convolved signal, A = conv(R, B)
    M   -   convolved signal length
    B   -   array[0..N-1] - response
    N   -   response length, N<=M

OUTPUT PARAMETERS
    R   -   deconvolved signal. array[0..M-N].

NOTE:
    deconvolution is unstable process and may result in division  by  zero
(if your response function is degenerate, i.e. has zero Fourier coefficient).

NOTE:
    It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
functions have non-zero values at negative T's, you  can  still  use  this
subroutine - just shift its result correspondingly.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::convr1dinv(
    real_1d_array a,
    ae_int_t m,
    real_1d_array b,
    ae_int_t n,
    real_1d_array& r,
    const xparams _params = alglib::xdefault);

`corr` subpackage

Functions

corrc1d
corrc1dcircular
corrr1d
corrr1dcircular

Examples

`corrc1d` function

/*************************************************************************
1-dimensional complex cross-correlation.

For given Pattern/Signal returns corr(Pattern,Signal) (non-circular).

Correlation is calculated using reduction to  convolution.  Algorithm with
max(N,N)*log(max(N,N)) complexity is used (see  ConvC1D()  for  more  info
about performance).

IMPORTANT:
    for  historical reasons subroutine accepts its parameters in  reversed
    order: CorrC1D(Signal, Pattern) = Pattern x Signal (using  traditional
    definition of cross-correlation, denoting cross-correlation as "x").

INPUT PARAMETERS
    Signal  -   array[0..N-1] - complex function to be transformed,
                signal containing pattern
    N       -   problem size
    Pattern -   array[0..M-1] - complex function to be transformed,
                pattern to search withing signal
    M       -   problem size

OUTPUT PARAMETERS
    R       -   cross-correlation, array[0..N+M-2]:
                * positive lags are stored in R[0..N-1],
                  R[i] = sum(conj(pattern[j])*signal[i+j]
                * negative lags are stored in R[N..N+M-2],
                  R[N+M-1-i] = sum(conj(pattern[j])*signal[-i+j]

NOTE:
    It is assumed that pattern domain is [0..M-1].  If Pattern is non-zero
on [-K..M-1],  you can still use this subroutine, just shift result by K.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::corrc1d(
    complex_1d_array signal,
    ae_int_t n,
    complex_1d_array pattern,
    ae_int_t m,
    complex_1d_array& r,
    const xparams _params = alglib::xdefault);

`corrc1dcircular` function

/*************************************************************************
1-dimensional circular complex cross-correlation.

For given Pattern/Signal returns corr(Pattern,Signal) (circular).
Algorithm has linearithmic complexity for any M/N.

IMPORTANT:
    for  historical reasons subroutine accepts its parameters in  reversed
    order:   CorrC1DCircular(Signal, Pattern) = Pattern x Signal    (using
    traditional definition of cross-correlation, denoting cross-correlation
    as "x").

INPUT PARAMETERS
    Signal  -   array[0..N-1] - complex function to be transformed,
                periodic signal containing pattern
    N       -   problem size
    Pattern -   array[0..M-1] - complex function to be transformed,
                non-periodic pattern to search withing signal
    M       -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..M-1].


  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::corrc1dcircular(
    complex_1d_array signal,
    ae_int_t m,
    complex_1d_array pattern,
    ae_int_t n,
    complex_1d_array& c,
    const xparams _params = alglib::xdefault);

`corrr1d` function

/*************************************************************************
1-dimensional real cross-correlation.

For given Pattern/Signal returns corr(Pattern,Signal) (non-circular).

Correlation is calculated using reduction to  convolution.  Algorithm with
max(N,N)*log(max(N,N)) complexity is used (see  ConvC1D()  for  more  info
about performance).

IMPORTANT:
    for  historical reasons subroutine accepts its parameters in  reversed
    order: CorrR1D(Signal, Pattern) = Pattern x Signal (using  traditional
    definition of cross-correlation, denoting cross-correlation as "x").

INPUT PARAMETERS
    Signal  -   array[0..N-1] - real function to be transformed,
                signal containing pattern
    N       -   problem size
    Pattern -   array[0..M-1] - real function to be transformed,
                pattern to search withing signal
    M       -   problem size

OUTPUT PARAMETERS
    R       -   cross-correlation, array[0..N+M-2]:
                * positive lags are stored in R[0..N-1],
                  R[i] = sum(pattern[j]*signal[i+j]
                * negative lags are stored in R[N..N+M-2],
                  R[N+M-1-i] = sum(pattern[j]*signal[-i+j]

NOTE:
    It is assumed that pattern domain is [0..M-1].  If Pattern is non-zero
on [-K..M-1],  you can still use this subroutine, just shift result by K.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::corrr1d(
    real_1d_array signal,
    ae_int_t n,
    real_1d_array pattern,
    ae_int_t m,
    real_1d_array& r,
    const xparams _params = alglib::xdefault);

`corrr1dcircular` function

/*************************************************************************
1-dimensional circular real cross-correlation.

For given Pattern/Signal returns corr(Pattern,Signal) (circular).
Algorithm has linearithmic complexity for any M/N.

IMPORTANT:
    for  historical reasons subroutine accepts its parameters in  reversed
    order:   CorrR1DCircular(Signal, Pattern) = Pattern x Signal    (using
    traditional definition of cross-correlation, denoting cross-correlation
    as "x").

INPUT PARAMETERS
    Signal  -   array[0..N-1] - real function to be transformed,
                periodic signal containing pattern
    N       -   problem size
    Pattern -   array[0..M-1] - real function to be transformed,
                non-periodic pattern to search withing signal
    M       -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..M-1].


  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::corrr1dcircular(
    real_1d_array signal,
    ae_int_t m,
    real_1d_array pattern,
    ae_int_t n,
    real_1d_array& c,
    const xparams _params = alglib::xdefault);

`correlationtests` subpackage

Functions

pearsoncorrelationsignificance
spearmanrankcorrelationsignificance

Examples

`pearsoncorrelationsignificance` function

/*************************************************************************
Pearson's correlation coefficient significance test

This test checks hypotheses about whether X  and  Y  are  samples  of  two
continuous  distributions  having  zero  correlation  or   whether   their
correlation is non-zero.

The following tests are performed:
    * two-tailed test (null hypothesis - X and Y have zero correlation)
    * left-tailed test (null hypothesis - the correlation  coefficient  is
      greater than or equal to 0)
    * right-tailed test (null hypothesis - the correlation coefficient  is
      less than or equal to 0).

Requirements:
    * the number of elements in each sample is not less than 5
    * normality of distributions of X and Y.

Input parameters:
    R   -   Pearson's correlation coefficient for X and Y
    N   -   number of elements in samples, N>=5.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::pearsoncorrelationsignificance(
    double r,
    ae_int_t n,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`spearmanrankcorrelationsignificance` function

/*************************************************************************
Spearman's rank correlation coefficient significance test

This test checks hypotheses about whether X  and  Y  are  samples  of  two
continuous  distributions  having  zero  correlation  or   whether   their
correlation is non-zero.

The following tests are performed:
    * two-tailed test (null hypothesis - X and Y have zero correlation)
    * left-tailed test (null hypothesis - the correlation  coefficient  is
      greater than or equal to 0)
    * right-tailed test (null hypothesis - the correlation coefficient  is
      less than or equal to 0).

Requirements:
    * the number of elements in each sample is not less than 5.

The test is non-parametric and doesn't require distributions X and Y to be
normal.

Input parameters:
    R   -   Spearman's rank correlation coefficient for X and Y
    N   -   number of elements in samples, N>=5.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spearmanrankcorrelationsignificance(
    double r,
    ae_int_t n,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`datacomp` subpackage

Functions

kmeansgenerate

Examples

`kmeansgenerate` function

/*************************************************************************
k-means++ clusterization.
Backward compatibility function, we recommend to use CLUSTERING subpackage
as better replacement.

  -- ALGLIB --
     Copyright 21.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::kmeansgenerate(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t k,
    ae_int_t restarts,
    ae_int_t& info,
    real_2d_array& c,
    integer_1d_array& xyc,
    const xparams _params = alglib::xdefault);

`dawson` subpackage

Functions

dawsonintegral

Examples

`dawsonintegral` function

/*************************************************************************
Dawson's Integral

Approximates the integral

                            x
                            -
                     2     | |        2
 dawsn(x)  =  exp( -x  )   |    exp( t  ) dt
                         | |
                          -
                          0

Three different rational approximations are employed, for
the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,10        10000       6.9e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::dawsonintegral(
    double x,
    const xparams _params = alglib::xdefault);

`dforest` subpackage

Classes

decisionforest
decisionforestbuilder
dfreport

Functions

dfavgce
dfavgerror
dfavgrelerror
dfbuilderbuildrandomforest
dfbuildercreate
dfbuildergetprogress
dfbuildersetdataset
dfbuildersetrdfalgo
dfbuildersetrdfsplitstrength
dfbuildersetrndvars
dfbuildersetrndvarsauto
dfbuildersetrndvarsratio
dfbuildersetseed
dfbuildersetsubsampleratio
dfbuildrandomdecisionforest
dfbuildrandomdecisionforestx1
dfprocess
dfprocessi
dfrelclserror
dfrmserror
dfserialize
dfunserialize

Examples

randomforest_cls

Simple classification with random forests

`decisionforest` class

/*************************************************************************
Decision forest (random forest) model.
*************************************************************************/
class decisionforest
{
};

`decisionforestbuilder` class

/*************************************************************************
A random forest (decision forest) builder object.

Used to store dataset and specify random forest training algorithm settings.
*************************************************************************/
class decisionforestbuilder
{
};

`dfreport` class

/*************************************************************************
Decision forest training report.

Following fields store training set errors:
* relclserror       -   fraction of misclassified cases, [0,1]
* avgce             -   average cross-entropy in bits per symbol
* rmserror          -   root-mean-square error
* avgerror          -   average error
* avgrelerror       -   average relative error

Out-of-bag estimates are stored in fields with same names, but "oob" prefix.

For classification problems:
* RMS, AVG and AVGREL errors are calculated for posterior probabilities

For regression problems:
* RELCLS and AVGCE errors are zero
*************************************************************************/
class dfreport
{
    double               relclserror;
    double               avgce;
    double               rmserror;
    double               avgerror;
    double               avgrelerror;
    double               oobrelclserror;
    double               oobavgce;
    double               oobrmserror;
    double               oobavgerror;
    double               oobavgrelerror;
};

`dfavgce` function

/*************************************************************************
Average cross-entropy (in bits per element) on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    CrossEntropy/(NPoints*LN(2)).
    Zero if model solves regression task.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::dfavgce(
    decisionforest df,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`dfavgerror` function

/*************************************************************************
Average error on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    Its meaning for regression task is obvious. As for
    classification task, it means average error when estimating posterior
    probabilities.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::dfavgerror(
    decisionforest df,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`dfavgrelerror` function

/*************************************************************************
Average relative error on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    Its meaning for regression task is obvious. As for
    classification task, it means average relative error when estimating
    posterior probability of belonging to the correct class.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::dfavgrelerror(
    decisionforest df,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`dfbuilderbuildrandomforest` function

/*************************************************************************
This subroutine builds random forest according to current settings,  using
dataset internally stored in the builder object. Dense algorithm is used.

NOTE: this   function   uses   dense  algorithm  for  forest  construction
      independently from the dataset format (dense or sparse).

Default settings are used by the algorithm; you can tweak  them  with  the
help of the following functions:
* dfbuildersetrfactor() - to control a fraction of the  dataset  used  for
  subsampling
* dfbuildersetrandomvars() - to control number of variables randomly chosen
  for decision rule creation

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S           -   decision forest builder object
    NTrees      -   NTrees>=1, number of trees to train

OUTPUT PARAMETERS:
    DF          -   decision forest
    Rep         -   report

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuilderbuildrandomforest(
    decisionforestbuilder s,
    ae_int_t ntrees,
    decisionforest& df,
    dfreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildercreate` function

/*************************************************************************
This subroutine creates DecisionForestBuilder  object  which  is  used  to
train random forests.

By default, new builder stores empty dataset and some  reasonable  default
settings. At the very least, you should specify dataset prior to  building
decision forest. You can also tweak settings of  the  forest  construction
algorithm (recommended, although default setting should work well).

Following actions are mandatory:
* calling dfbuildersetdataset() to specify dataset
* calling dfbuilderbuildrandomforest() to build random forest using current
  dataset and default settings

Additionally, you may call:
* dfbuildersetrndvars() or dfbuildersetrndvarsratio() to specify number of
  variables randomly chosen for each split
* dfbuildersetsubsampleratio() to specify fraction of the dataset randomly
  subsampled to build each tree
* dfbuildersetseed() to control random seed chosen for tree construction

INPUT PARAMETERS:
    none

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildercreate(
    decisionforestbuilder& s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildergetprogress` function

/*************************************************************************
This function is used to peek into random forest construction process from
other thread and get current progress indicator. It returns value in [0,1].

INPUT PARAMETERS:
    S           -   decision forest builder object used  to  build  random
                    forest in some other thread

RESULT:
    progress value, in [0,1]

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
double alglib::dfbuildergetprogress(
    decisionforestbuilder s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildersetdataset` function

/*************************************************************************
This subroutine adds dense dataset to the internal storage of the  builder
object. Specifying your dataset in the dense format means that  the  dense
version of the forest construction algorithm will be invoked.

INPUT PARAMETERS:
    S           -   decision forest builder object
    XY          -   array[NPoints,NVars+1] (minimum size; actual size  can
                    be larger, only leading part is used anyway), dataset:
                    * first NVars elements of each row store values of the
                      independent variables
                    * last  column  store class number (in 0...NClasses-1)
                      or real value of the dependent variable
    NPoints     -   number of rows in the dataset, NPoints>=1
    NVars       -   number of independent variables, NVars>=1
    NClasses    -   indicates type of the problem being solved:
                    * NClasses>=2 means  that  classification  problem  is
                      solved  (last  column  of  the  dataset stores class
                      number)
                    * NClasses=1 means that regression problem  is  solved
                      (last column of the dataset stores variable value)

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildersetdataset(
    decisionforestbuilder s,
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t nclasses,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildersetrdfalgo` function

/*************************************************************************
This function sets random decision forest construction algorithm.

As for now, only one random forest construction algorithm is  supported  -
a dense "baseline" RDF algorithm.

INPUT PARAMETERS:
    S           -   decision forest builder object
    AlgoType    -   algorithm type:
                    * 0 = baseline dense RDF

OUTPUT PARAMETERS:
    S           -   decision forest builder, see

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildersetrdfalgo(
    decisionforestbuilder s,
    ae_int_t algotype,
    const xparams _params = alglib::xdefault);

`dfbuildersetrdfsplitstrength` function

/*************************************************************************
This  function  sets  split  selection  algorithm  used  by random forests
classifier. You may choose several algorithms, with  different  speed  and
quality of the results.

INPUT PARAMETERS:
    S           -   decision forest builder object
    SplitStrength-  split type:
                    * 0 = split at the random position, fastest one
                    * 1 = split at the middle of the range
                    * 2 = strong split at the best point of the range (default)

OUTPUT PARAMETERS:
    S           -   decision forest builder, see

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildersetrdfsplitstrength(
    decisionforestbuilder s,
    ae_int_t splitstrength,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildersetrndvars` function

/*************************************************************************
This function sets number of variables (in [1,NVars] range) used by random
forest construction algorithm.

The default option is to use roughly sqrt(NVars) variables.

INPUT PARAMETERS:
    S           -   decision forest builder object
    RndVars     -   number of randomly selected variables; values  outside
                    of [1,NVars] range are silently clipped.

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildersetrndvars(
    decisionforestbuilder s,
    ae_int_t rndvars,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildersetrndvarsauto` function

/*************************************************************************
This function tells random forest builder to automatically  choose  number
of  variables  used  by  random  forest  construction  algorithm.  Roughly
sqrt(NVars) variables will be used.

INPUT PARAMETERS:
    S           -   decision forest builder object

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildersetrndvarsauto(
    decisionforestbuilder s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildersetrndvarsratio` function

/*************************************************************************
This function sets number of variables used by random forest  construction
algorithm as a fraction of total variable count (0,1) range.

The default option is to use roughly sqrt(NVars) variables.

INPUT PARAMETERS:
    S           -   decision forest builder object
    F           -   round(NVars*F) variables are selected

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildersetrndvarsratio(
    decisionforestbuilder s,
    double f,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildersetseed` function

/*************************************************************************
This function sets seed used by internal RNG for  random  subsampling  and
random selection of variable subsets.

By default, random seed is used, i.e. every time you build random  forest,
we seed generator with new value  obtained  from  system-wide  RNG.  Thus,
decision forest builder returns non-deterministic results. You can  change
such behavior by specyfing fixed positive seed value.

INPUT PARAMETERS:
    S           -   decision forest builder object
    SeedVal     -   seed value:
                    * positive values are used for seeding RNG with fixed
                      seed, i.e. subsequent runs on same data will return
                      same random forests
                    * non-positive seed means that random seed is used
                      for every run of builder, i.e. subsequent  runs  on
                      same datasets will return slightly different random
                      forests

OUTPUT PARAMETERS:
    S           -   decision forest builder, see

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildersetseed(
    decisionforestbuilder s,
    ae_int_t seedval,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildersetsubsampleratio` function

/*************************************************************************
This function sets size of dataset subsample generated the  random  forest
construction algorithm. Size is specified as a fraction of  total  dataset
size.

The default option is to use 50% of the dataset for training, 50% for  the
OOB estimates. You can decrease fraction F down to 10%, 1% or  even  below
in order to reduce overfitting.

INPUT PARAMETERS:
    S           -   decision forest builder object
    F           -   fraction of the dataset to use, in (0,1] range. Values
                    outside of this range will  be  silently  clipped.  At
                    least one element is always selected for the  training
                    set.

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildersetsubsampleratio(
    decisionforestbuilder s,
    double f,
    const xparams _params = alglib::xdefault);

Examples: [1]

`dfbuildrandomdecisionforest` function

/*************************************************************************
This subroutine builds random decision forest.

--------- DEPRECATED VERSION! USE DECISION FOREST BUILDER OBJECT ---------

  -- ALGLIB --
     Copyright 19.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildrandomdecisionforest(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t nclasses,
    ae_int_t ntrees,
    double r,
    ae_int_t& info,
    decisionforest& df,
    dfreport& rep,
    const xparams _params = alglib::xdefault);

`dfbuildrandomdecisionforestx1` function

/*************************************************************************
This subroutine builds random decision forest.

--------- DEPRECATED VERSION! USE DECISION FOREST BUILDER OBJECT ---------

  -- ALGLIB --
     Copyright 19.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::dfbuildrandomdecisionforestx1(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t nclasses,
    ae_int_t ntrees,
    ae_int_t nrndvars,
    double r,
    ae_int_t& info,
    decisionforest& df,
    dfreport& rep,
    const xparams _params = alglib::xdefault);

`dfprocess` function

/*************************************************************************
Procesing

INPUT PARAMETERS:
    DF      -   decision forest model
    X       -   input vector,  array[0..NVars-1].

OUTPUT PARAMETERS:
    Y       -   result. Regression estimate when solving regression  task,
                vector of posterior probabilities for classification task.

See also DFProcessI.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::dfprocess(
    decisionforest df,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`dfprocessi` function

/*************************************************************************
'interactive' variant of DFProcess for languages like Python which support
constructs like "Y = DFProcessI(DF,X)" and interactive mode of interpreter

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::dfprocessi(
    decisionforest df,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`dfrelclserror` function

/*************************************************************************
Relative classification error on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    percent of incorrectly classified cases.
    Zero if model solves regression task.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::dfrelclserror(
    decisionforest df,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`dfrmserror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    root mean square error.
    Its meaning for regression task is obvious. As for
    classification task, RMS error means error when estimating posterior
    probabilities.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::dfrmserror(
    decisionforest df,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`dfserialize` function

/*************************************************************************
This function serializes data structure to string.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string  in  case you want to include it into text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor you should change case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize  it  in  C++ version of ALGLIB and unserialize in C# one, 
  and vice versa.
*************************************************************************/
void dfserialize(decisionforest &obj, std::string &s_out);
void dfserialize(decisionforest &obj, std::ostream &s_out);

`dfunserialize` function

/*************************************************************************
This function unserializes data structure from string.
*************************************************************************/
void dfunserialize(const std::string &s_in, decisionforest &obj);
void dfunserialize(const std::istream &s_in, decisionforest &obj);

randomforest_cls example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // The very simple classification example: classify points (x,y) in 2D space
    // as ones with x>=0 and ones with x<0 (y is ignored, but our classifier
    // has to find out it).
    //
    // First, we have to create decision forest builder object, load dataset and
    // specify training settings. Our dataset is specified as matrix, which has
    // following format:
    //
    //     x0 y0 class0
    //     x1 y1 class1
    //     x2 y2 class2
    //     ....
    //
    // Here xi and yi can be any values (and in fact you can have any number of
    // independent variables), and classi MUST be integer number in [0,NClasses)
    // range. In our example we denote points with x>=0 as class #0, and
    // ones with negative xi as class #1.
    //
    // NOTE: if you want to solve regression problem, specify NClasses=1. In
    //       this case last column of xy can be any numeric value.
    //
    // For the sake of simplicity, our example includes only 4-point dataset.
    // However, random forests are able to cope with extremely large datasets
    // having millions of examples.
    //
    decisionforestbuilder builder;
    ae_int_t nvars = 2;
    ae_int_t nclasses = 2;
    ae_int_t npoints = 4;
    real_2d_array xy = "[[1,1,0],[1,-1,0],[-1,1,1],[-1,-1,1]]";

    dfbuildercreate(builder);
    dfbuildersetdataset(builder, xy, npoints, nvars, nclasses);

    // in our example we train decision forest using full sample - it allows us
    // to get zero classification error. However, in practical applications smaller
    // values are used: 50%, 25%, 5% or even less.
    dfbuildersetsubsampleratio(builder, 1.0);

    // we train random forest with just one tree; again, in real life situations
    // you typically need from 50 to 500 trees.
    ae_int_t ntrees = 1;
    decisionforest forest;
    dfreport rep;
    dfbuilderbuildrandomforest(builder, ntrees, forest, rep);

    // with such settings (100% of the training set is used) you can expect
    // zero classification error. Beautiful results, but remember - in real life
    // you do not need zero TRAINING SET error, you need good generalization.

    printf("%.4f\n", double(rep.relclserror)); // EXPECTED: 0.0000
    return 0;
}

`directdensesolvers` subpackage

Classes

densesolverlsreport
densesolverreport

Functions

Examples

`densesolverlsreport` class

/*************************************************************************

*************************************************************************/
class densesolverlsreport
{
    double               r2;
    real_2d_array        cx;
    ae_int_t             n;
    ae_int_t             k;
};

`densesolverreport` class

/*************************************************************************

*************************************************************************/
class densesolverreport
{
    double               r1;
    double               rinf;
};

`cmatrixlusolve` function

/*************************************************************************
Complex dense linear solver for A*x=b with complex N*N A  given  by its LU
decomposition and N*1 vectors x and b. This is  "slow-but-robust"  version
of  the  complex  linear  solver  with  additional  features   which   add
significant performance overhead. Faster version  is  CMatrixLUSolveFast()
function.

Algorithm features:
* automatic detection of degenerate cases
* O(N^2) complexity
* condition number estimation

No iterative refinement is provided because exact form of original matrix
is not known to subroutine. Use CMatrixSolve or CMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which results in 10-15x  performance  penalty  when  compared
           ! with "fast" version which just calls triangular solver.
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems.
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! CMatrixLUSolveFast() function.

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
    P       -   array[0..N-1], pivots array, CMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixlusolve(
    complex_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    complex_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    complex_1d_array& x,
    const xparams _params = alglib::xdefault);

`cmatrixlusolvefast` function

/*************************************************************************
Complex dense linear solver for A*x=b with N*N complex A given by  its  LU
decomposition and N*1 vectors x and b. This is  fast  lightweight  version
of solver, which is significantly faster than CMatrixLUSolve(),  but  does
not provide additional information (like condition numbers).

Algorithm features:
* O(N^2) complexity
* no additional time-consuming features, just triangular solver

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
    P       -   array[0..N-1], pivots array, CMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is exactly singular (ill conditioned matrices
                        are not recognized).
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[N]:
                * info>0    =>  overwritten by solution
                * info=-3   =>  filled by zeros

NOTE: unlike  CMatrixLUSolve(),  this   function   does   NOT   check  for
      near-degeneracy of input matrix. It  checks  for  EXACT  degeneracy,
      because this check is easy to do. However,  very  badly  conditioned
      matrices may went unnoticed.


  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixlusolvefast(
    complex_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    complex_1d_array& b,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`cmatrixlusolvem` function

/*************************************************************************
Dense solver for A*X=B with N*N complex A given by its  LU  decomposition,
and N*M matrices X and B (multiple right sides).   "Slow-but-feature-rich"
version of the solver.

Algorithm features:
* automatic detection of degenerate cases
* O(M*N^2) complexity
* condition number estimation

No iterative refinement  is provided because exact form of original matrix
is not known to subroutine. Use CMatrixSolve or CMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver.
           !
           ! This performance penalty is especially apparent when you  use
           ! ALGLIB parallel capabilities (condition number estimation  is
           ! inherently  sequential).  It   also   becomes significant for
           ! small-scale problems.
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! CMatrixLUSolveMFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixlusolvem(
    complex_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    complex_2d_array b,
    ae_int_t m,
    ae_int_t& info,
    densesolverreport& rep,
    complex_2d_array& x,
    const xparams _params = alglib::xdefault);

`cmatrixlusolvemfast` function

/*************************************************************************
Dense solver for A*X=B with N*N complex A given by its  LU  decomposition,
and N*M matrices X and B (multiple  right  sides).  "Fast-but-lightweight"
version of the solver.

Algorithm features:
* O(M*N^2) complexity
* no additional time-consuming features

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is exactly singular (ill conditioned matrices
                        are not recognized).
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[N,M]:
                * info>0    =>  overwritten by solution
                * info=-3   =>  filled by zeros


  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixlusolvemfast(
    complex_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    complex_2d_array& b,
    ae_int_t m,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`cmatrixmixedsolve` function

/*************************************************************************
Dense solver. Same as RMatrixMixedSolve(), but for complex matrices.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^2) complexity

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
    P       -   array[0..N-1], pivots array, CMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixmixedsolve(
    complex_2d_array a,
    complex_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    complex_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    complex_1d_array& x,
    const xparams _params = alglib::xdefault);

`cmatrixmixedsolvem` function

/*************************************************************************
Dense solver. Same as RMatrixMixedSolveM(), but for complex matrices.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(M*N^2) complexity

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
    P       -   array[0..N-1], pivots array, CMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixmixedsolvem(
    complex_2d_array a,
    complex_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    complex_2d_array b,
    ae_int_t m,
    ae_int_t& info,
    densesolverreport& rep,
    complex_2d_array& x,
    const xparams _params = alglib::xdefault);

`cmatrixsolve` function

/*************************************************************************
Complex dense solver for A*x=B with N*N complex matrix A and  N*1  complex
vectors x and b. "Slow-but-feature-rich" version of the solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^3) complexity

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear  system
           ! and  performs  iterative   refinement,   which   results   in
           ! significant performance penalty  when  compared  with  "fast"
           ! version  which  just  performs  LU  decomposition  and  calls
           ! triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, CMatrixSolveFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixsolve(
    complex_2d_array a,
    ae_int_t n,
    complex_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    complex_1d_array& x,
    const xparams _params = alglib::xdefault);

`cmatrixsolvefast` function

/*************************************************************************
Complex dense solver for A*x=B with N*N complex matrix A and  N*1  complex
vectors x and b. "Fast-but-lightweight" version of the solver.

Algorithm features:
* O(N^3) complexity
* no additional time consuming features, just triangular solver

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS:
    Info    -   return code:
                * -3    matrix is exactly singular (ill conditioned matrices
                        are not recognized).
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[N]:
                * info>0    =>  overwritten by solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixsolvefast(
    complex_2d_array a,
    ae_int_t n,
    complex_1d_array& b,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`cmatrixsolvem` function

/*************************************************************************
Complex dense solver for A*X=B with N*N  complex  matrix  A,  N*M  complex
matrices  X  and  B.  "Slow-but-feature-rich"   version   which   provides
additional functions, at the cost of slower  performance.  Faster  version
may be invoked with CMatrixSolveMFast() function.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^3+M*N^2) complexity

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear  system
           ! and  performs  iterative   refinement,   which   results   in
           ! significant performance penalty  when  compared  with  "fast"
           ! version  which  just  performs  LU  decomposition  and  calls
           ! triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, CMatrixSolveMFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size
    RFS     -   iterative refinement switch:
                * True - refinement is used.
                  Less performance, more precision.
                * False - refinement is not used.
                  More performance, less precision.

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixsolvem(
    complex_2d_array a,
    ae_int_t n,
    complex_2d_array b,
    ae_int_t m,
    bool rfs,
    ae_int_t& info,
    densesolverreport& rep,
    complex_2d_array& x,
    const xparams _params = alglib::xdefault);

`cmatrixsolvemfast` function

/*************************************************************************
Complex dense solver for A*X=B with N*N  complex  matrix  A,  N*M  complex
matrices  X  and  B.  "Fast-but-lightweight" version which  provides  just
triangular solver - and no additional functions like iterative  refinement
or condition number estimation.

Algorithm features:
* O(N^3+M*N^2) complexity
* no additional time consuming functions

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS:
    Info    -   return code:
                * -3    matrix is exactly singular (ill conditioned matrices
                        are not recognized).
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[N,M]:
                * info>0    =>  overwritten by solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 16.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixsolvemfast(
    complex_2d_array a,
    ae_int_t n,
    complex_2d_array& b,
    ae_int_t m,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`hpdmatrixcholeskysolve` function

/*************************************************************************
Dense solver for A*x=b with N*N Hermitian positive definite matrix A given
by its Cholesky decomposition, and N*1 complex vectors x and  b.  This  is
"slow-but-feature-rich" version of the solver  which  estimates  condition
number of the system.

Algorithm features:
* automatic detection of degenerate cases
* O(N^2) complexity
* condition number estimation
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which results in 10-15x  performance  penalty  when  compared
           ! with "fast" version which just calls triangular solver.
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems (N<50).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! HPDMatrixCholeskySolveFast() function.

INPUT PARAMETERS
    CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of A
    IsUpper -   what half of CHA is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly singular or ill conditioned
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N]:
                * for info>0  - solution
                * for info=-3 - filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixcholeskysolve(
    complex_2d_array cha,
    ae_int_t n,
    bool isupper,
    complex_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    complex_1d_array& x,
    const xparams _params = alglib::xdefault);

`hpdmatrixcholeskysolvefast` function

/*************************************************************************
Dense solver for A*x=b with N*N Hermitian positive definite matrix A given
by its Cholesky decomposition, and N*1 complex vectors x and  b.  This  is
"fast-but-lightweight" version of the solver.

Algorithm features:
* O(N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional time-consuming features

INPUT PARAMETERS
    CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of A
    IsUpper -   what half of CHA is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly singular or ill conditioned
                        B is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[N]:
                * for info>0  - overwritten by solution
                * for info=-3 - filled by zeros

  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixcholeskysolvefast(
    complex_2d_array cha,
    ae_int_t n,
    bool isupper,
    complex_1d_array& b,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`hpdmatrixcholeskysolvem` function

/*************************************************************************
Dense solver for A*X=B with N*N Hermitian positive definite matrix A given
by its Cholesky decomposition and N*M complex matrices X  and  B.  This is
"slow-but-feature-rich" version of the solver which, in  addition  to  the
solution, estimates condition number of the system.

Algorithm features:
* automatic detection of degenerate cases
* O(M*N^2) complexity
* condition number estimation
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver. Amount of  overhead  introduced  depends  on  M  (the
           ! larger - the more efficient).
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large Cholesky decomposition.  However,  if  you call
           ! this  function  many  times  for  the same  left  side,  this
           ! overhead BECOMES significant. It  also   becomes  significant
           ! for small-scale problems (N<50).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! HPDMatrixCholeskySolveMFast() function.


INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                HPDMatrixCholesky result
    N       -   size of CHA
    IsUpper -   what half of CHA is provided
    B       -   array[N,M], right part
    M       -   right part size

OUTPUT PARAMETERS:
    Info    -   return code:
                * -3    A is singular, or VERY close to singular.
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task was solved
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N]:
                * for info>0 contains solution
                * for info=-3 filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixcholeskysolvem(
    complex_2d_array cha,
    ae_int_t n,
    bool isupper,
    complex_2d_array b,
    ae_int_t m,
    ae_int_t& info,
    densesolverreport& rep,
    complex_2d_array& x,
    const xparams _params = alglib::xdefault);

`hpdmatrixcholeskysolvemfast` function

/*************************************************************************
Dense solver for A*X=B with N*N Hermitian positive definite matrix A given
by its Cholesky decomposition and N*M complex matrices X  and  B.  This is
"fast-but-lightweight" version of the solver.

Algorithm features:
* O(M*N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional time-consuming features

INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                HPDMatrixCholesky result
    N       -   size of CHA
    IsUpper -   what half of CHA is provided
    B       -   array[N,M], right part
    M       -   right part size

OUTPUT PARAMETERS:
    Info    -   return code:
                * -3    A is singular, or VERY close to singular.
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task was solved
    B       -   array[N]:
                * for info>0 overwritten by solution
                * for info=-3 filled by zeros

  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixcholeskysolvemfast(
    complex_2d_array cha,
    ae_int_t n,
    bool isupper,
    complex_2d_array& b,
    ae_int_t m,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`hpdmatrixsolve` function

/*************************************************************************
Dense solver for A*x=b, with N*N Hermitian positive definite matrix A, and
N*1 complex vectors  x  and  b.  "Slow-but-feature-rich"  version  of  the
solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* O(N^3) complexity
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant   performance   penalty  when
           ! compared with "fast" version  which  just  performs  Cholesky
           ! decomposition and calls triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, HPDMatrixSolveFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   same as in RMatrixSolve
                Returns -3 for non-HPD matrices.
    Rep     -   same as in RMatrixSolve
    X       -   same as in RMatrixSolve

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixsolve(
    complex_2d_array a,
    ae_int_t n,
    bool isupper,
    complex_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    complex_1d_array& x,
    const xparams _params = alglib::xdefault);

`hpdmatrixsolvefast` function

/*************************************************************************
Dense solver for A*x=b, with N*N Hermitian positive definite matrix A, and
N*1 complex vectors  x  and  b.  "Fast-but-lightweight"  version  of   the
solver without additional functions.

Algorithm features:
* O(N^3) complexity
* matrix is represented by its upper or lower triangle
* no additional time consuming functions

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly singular or not positive definite
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task was solved
    B       -   array[0..N-1]:
                * overwritten by solution
                * zeros, if A is exactly singular (diagonal of its LU
                  decomposition has exact zeros).

  -- ALGLIB --
     Copyright 17.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixsolvefast(
    complex_2d_array a,
    ae_int_t n,
    bool isupper,
    complex_1d_array& b,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`hpdmatrixsolvem` function

/*************************************************************************
Dense solver for A*X=B, with N*N Hermitian positive definite matrix A  and
N*M  complex  matrices  X  and  B.  "Slow-but-feature-rich" version of the
solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* O(N^3+M*N^2) complexity
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver.
           !
           ! This performance penalty is especially apparent when you  use
           ! ALGLIB parallel capabilities (condition number estimation  is
           ! inherently  sequential).  It   also   becomes significant for
           ! small-scale problems (N<100).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! HPDMatrixSolveMFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   same as in RMatrixSolve.
                Returns -3 for non-HPD matrices.
    Rep     -   same as in RMatrixSolve
    X       -   same as in RMatrixSolve

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixsolvem(
    complex_2d_array a,
    ae_int_t n,
    bool isupper,
    complex_2d_array b,
    ae_int_t m,
    ae_int_t& info,
    densesolverreport& rep,
    complex_2d_array& x,
    const xparams _params = alglib::xdefault);

`hpdmatrixsolvemfast` function

/*************************************************************************
Dense solver for A*X=B, with N*N Hermitian positive definite matrix A  and
N*M complex matrices X and B. "Fast-but-lightweight" version of the solver.

Algorithm features:
* O(N^3+M*N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional time consuming features like condition number estimation

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly  singular or is not positive definite.
                        B is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[0..N-1]:
                * overwritten by solution
                * zeros, if problem was not solved

  -- ALGLIB --
     Copyright 17.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixsolvemfast(
    complex_2d_array a,
    ae_int_t n,
    bool isupper,
    complex_2d_array& b,
    ae_int_t m,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`rmatrixlusolve` function

/*************************************************************************
Dense solver.

This  subroutine  solves  a  system  A*x=b,  where A is NxN non-denegerate
real matrix given by its LU decomposition, x and b are real vectors.  This
is "slow-but-robust" version of the linear LU-based solver. Faster version
is RMatrixLUSolveFast() function.

Algorithm features:
* automatic detection of degenerate cases
* O(N^2) complexity
* condition number estimation

No iterative refinement  is provided because exact form of original matrix
is not known to subroutine. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which results in 10-15x  performance  penalty  when  compared
           ! with "fast" version which just calls triangular solver.
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems.
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! RMatrixLUSolveFast() function.

INPUT PARAMETERS
    LUA     -   array[N,N], LU decomposition, RMatrixLU result
    P       -   array[N], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[N], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros


  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixlusolve(
    real_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    real_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    real_1d_array& x,
    const xparams _params = alglib::xdefault);

`rmatrixlusolvefast` function

/*************************************************************************
Dense solver.

This  subroutine  solves  a  system  A*x=b,  where A is NxN non-denegerate
real matrix given by its LU decomposition, x and b are real vectors.  This
is "fast-without-any-checks" version of the linear LU-based solver. Slower
but more robust version is RMatrixLUSolve() function.

Algorithm features:
* O(N^2) complexity
* fast algorithm without ANY additional checks, just triangular solver

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is exactly singular (ill conditioned matrices
                        are not recognized).
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[N]:
                * info>0    =>  overwritten by solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixlusolvefast(
    real_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    real_1d_array& b,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`rmatrixlusolvem` function

/*************************************************************************
Dense solver.

Similar to RMatrixLUSolve() but solves  task  with  multiple  right  parts
(where b and x are NxM matrices). This  is  "robust-but-slow"  version  of
LU-based solver which performs additional  checks  for  non-degeneracy  of
inputs (condition number estimation). If you need  best  performance,  use
"fast-without-any-checks" version, RMatrixLUSolveMFast().

Algorithm features:
* automatic detection of degenerate cases
* O(M*N^2) complexity
* condition number estimation

No iterative refinement  is provided because exact form of original matrix
is not known to subroutine. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver.
           !
           ! This performance penalty is especially apparent when you  use
           ! ALGLIB parallel capabilities (condition number estimation  is
           ! inherently  sequential).  It   also   becomes significant for
           ! small-scale problems.
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! RMatrixLUSolveMFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    LUA     -   array[N,N], LU decomposition, RMatrixLU result
    P       -   array[N], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros


  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixlusolvem(
    real_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    real_2d_array b,
    ae_int_t m,
    ae_int_t& info,
    densesolverreport& rep,
    real_2d_array& x,
    const xparams _params = alglib::xdefault);

`rmatrixlusolvemfast` function

/*************************************************************************
Dense solver.

Similar to RMatrixLUSolve() but solves  task  with  multiple  right parts,
where b and x are NxM matrices.  This is "fast-without-any-checks" version
of LU-based solver. It does not estimate  condition number  of  a  system,
so it is extremely fast. If you need better detection  of  near-degenerate
cases, use RMatrixLUSolveM() function.

Algorithm features:
* O(M*N^2) complexity
* fast algorithm without ANY additional checks, just triangular solver

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS:
    Info    -   return code:
                * -3    matrix is exactly singular (ill conditioned matrices
                        are not recognized).
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[N,M]:
                * info>0    =>  overwritten by solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixlusolvemfast(
    real_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    real_2d_array& b,
    ae_int_t m,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`rmatrixmixedsolve` function

/*************************************************************************
Dense solver.

This  subroutine  solves  a  system  A*x=b,  where BOTH ORIGINAL A AND ITS
LU DECOMPOSITION ARE KNOWN. You can use it if for some  reasons  you  have
both A and its LU decomposition.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^2) complexity

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixmixedsolve(
    real_2d_array a,
    real_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    real_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    real_1d_array& x,
    const xparams _params = alglib::xdefault);

`rmatrixmixedsolvem` function

/*************************************************************************
Dense solver.

Similar to RMatrixMixedSolve() but  solves task with multiple right  parts
(where b and x are NxM matrices).

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(M*N^2) complexity

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixmixedsolvem(
    real_2d_array a,
    real_2d_array lua,
    integer_1d_array p,
    ae_int_t n,
    real_2d_array b,
    ae_int_t m,
    ae_int_t& info,
    densesolverreport& rep,
    real_2d_array& x,
    const xparams _params = alglib::xdefault);

`rmatrixsolve` function

/*************************************************************************
Dense solver for A*x=b with N*N real matrix A and N*1 real vectorx  x  and
b. This is "slow-but-feature rich" version of the  linear  solver.  Faster
version is RMatrixSolveFast() function.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^3) complexity

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear  system
           ! and  performs  iterative   refinement,   which   results   in
           ! significant performance penalty  when  compared  with  "fast"
           ! version  which  just  performs  LU  decomposition  and  calls
           ! triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations. It is also very significant on small matrices.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, RMatrixSolveFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or exactly singular.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixsolve(
    real_2d_array a,
    ae_int_t n,
    real_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    real_1d_array& x,
    const xparams _params = alglib::xdefault);

`rmatrixsolvefast` function

/*************************************************************************
Dense solver.

This  subroutine  solves  a  system  A*x=b,  where A is NxN non-denegerate
real matrix, x  and  b  are  vectors.  This is a "fast" version of  linear
solver which does NOT provide  any  additional  functions  like  condition
number estimation or iterative refinement.

Algorithm features:
* efficient algorithm O(N^3) complexity
* no performance overhead from additional functionality

If you need condition number estimation or iterative refinement, use  more
feature-rich version - RMatrixSolve().

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is exactly singular (ill conditioned matrices
                        are not recognized).
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[N]:
                * info>0    =>  overwritten by solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 16.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixsolvefast(
    real_2d_array a,
    ae_int_t n,
    real_1d_array& b,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`rmatrixsolvels` function

/*************************************************************************
Dense solver.

This subroutine finds solution of the linear system A*X=B with non-square,
possibly degenerate A.  System  is  solved in the least squares sense, and
general least squares solution  X = X0 + CX*y  which  minimizes |A*X-B| is
returned. If A is non-degenerate, solution in the usual sense is returned.

Algorithm features:
* automatic detection (and correct handling!) of degenerate cases
* iterative refinement
* O(N^3) complexity

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..NRows-1,0..NCols-1], system matrix
    NRows   -   vertical size of A
    NCols   -   horizontal size of A
    B       -   array[0..NCols-1], right part
    Threshold-  a number in [0,1]. Singular values  beyond  Threshold  are
                considered  zero.  Set  it to 0.0, if you don't understand
                what it means, so the solver will choose good value on its
                own.

OUTPUT PARAMETERS
    Info    -   return code:
                * -4    SVD subroutine failed
                * -1    if NRows<=0 or NCols<=0 or Threshold<0 was passed
                *  1    if task is solved
    Rep     -   solver report, see below for more info
    X       -   array[0..N-1,0..M-1], it contains:
                * solution of A*X=B (even for singular A)
                * zeros, if SVD subroutine failed

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* R2        reciprocal of condition number: 1/cond(A), 2-norm.
* N         = NCols
* K         dim(Null(A))
* CX        array[0..N-1,0..K-1], kernel of A.
            Columns of CX store such vectors that A*CX[i]=0.

  -- ALGLIB --
     Copyright 24.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixsolvels(
    real_2d_array a,
    ae_int_t nrows,
    ae_int_t ncols,
    real_1d_array b,
    double threshold,
    ae_int_t& info,
    densesolverlsreport& rep,
    real_1d_array& x,
    const xparams _params = alglib::xdefault);

`rmatrixsolvem` function

/*************************************************************************
Dense solver.

Similar to RMatrixSolve() but solves task with multiple right parts (where
b and x are NxM matrices). This is  "slow-but-robust"  version  of  linear
solver with additional functionality  like  condition  number  estimation.
There also exists faster version - RMatrixSolveMFast().

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* optional iterative refinement
* O(N^3+M*N^2) complexity

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear  system
           ! and  performs  iterative   refinement,   which   results   in
           ! significant performance penalty  when  compared  with  "fast"
           ! version  which  just  performs  LU  decomposition  and  calls
           ! triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations. It also very significant on small matrices.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, RMatrixSolveMFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size
    RFS     -   iterative refinement switch:
                * True - refinement is used.
                  Less performance, more precision.
                * False - refinement is not used.
                  More performance, less precision.

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is ill conditioned or singular.
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros


  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixsolvem(
    real_2d_array a,
    ae_int_t n,
    real_2d_array b,
    ae_int_t m,
    bool rfs,
    ae_int_t& info,
    densesolverreport& rep,
    real_2d_array& x,
    const xparams _params = alglib::xdefault);

`rmatrixsolvemfast` function

/*************************************************************************
Dense solver.

Similar to RMatrixSolve() but solves task with multiple right parts (where
b and x are NxM matrices). This is "fast" version of linear  solver  which
does NOT offer additional functions like condition  number  estimation  or
iterative refinement.

Algorithm features:
* O(N^3+M*N^2) complexity
* no additional functionality, highest performance

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size
    RFS     -   iterative refinement switch:
                * True - refinement is used.
                  Less performance, more precision.
                * False - refinement is not used.
                  More performance, less precision.

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is exactly singular (ill conditioned matrices
                        are not recognized).
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    B       -   array[N]:
                * info>0    =>  overwritten by solution
                * info=-3   =>  filled by zeros


  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixsolvemfast(
    real_2d_array a,
    ae_int_t n,
    real_2d_array& b,
    ae_int_t m,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskysolve` function

/*************************************************************************
Dense solver for A*x=b with N*N symmetric positive definite matrix A given
by its Cholesky decomposition, and N*1 real vectors x and b. This is "slow-
but-feature-rich"  version  of  the  solver  which,  in  addition  to  the
solution, performs condition number estimation.

Algorithm features:
* automatic detection of degenerate cases
* O(N^2) complexity
* condition number estimation
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which results in 10-15x  performance  penalty  when  compared
           ! with "fast" version which just calls triangular solver.
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems (N<50).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! SPDMatrixCholeskySolveFast() function.

INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of A
    IsUpper -   what half of CHA is provided
    B       -   array[N], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly singular or ill conditioned
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N]:
                * for info>0  - solution
                * for info=-3 - filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixcholeskysolve(
    real_2d_array cha,
    ae_int_t n,
    bool isupper,
    real_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    real_1d_array& x,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskysolvefast` function

/*************************************************************************
Dense solver for A*x=b with N*N symmetric positive definite matrix A given
by its Cholesky decomposition, and N*1 real vectors x and b. This is "fast-
but-lightweight" version of the solver.

Algorithm features:
* O(N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional features

INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of A
    IsUpper -   what half of CHA is provided
    B       -   array[N], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly singular or ill conditioned
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task is solved
    B       -   array[N]:
                * for info>0  - overwritten by solution
                * for info=-3 - filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixcholeskysolvefast(
    real_2d_array cha,
    ae_int_t n,
    bool isupper,
    real_1d_array& b,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskysolvem` function

/*************************************************************************
Dense solver for A*X=B with N*N symmetric positive definite matrix A given
by its Cholesky decomposition, and N*M vectors X and B. It  is  "slow-but-
feature-rich" version of the solver which estimates  condition  number  of
the system.

Algorithm features:
* automatic detection of degenerate cases
* O(M*N^2) complexity
* condition number estimation
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver. Amount of  overhead  introduced  depends  on  M  (the
           ! larger - the more efficient).
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems (N<50).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! SPDMatrixCholeskySolveMFast() function.

INPUT PARAMETERS
    CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of CHA
    IsUpper -   what half of CHA is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly singular or badly conditioned
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task was solved
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N]:
                * for info>0 contains solution
                * for info=-3 filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixcholeskysolvem(
    real_2d_array cha,
    ae_int_t n,
    bool isupper,
    real_2d_array b,
    ae_int_t m,
    ae_int_t& info,
    densesolverreport& rep,
    real_2d_array& x,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskysolvemfast` function

/*************************************************************************
Dense solver for A*X=B with N*N symmetric positive definite matrix A given
by its Cholesky decomposition, and N*M vectors X and B. It  is  "fast-but-
lightweight" version of  the  solver  which  just  solves  linear  system,
without any additional functions.

Algorithm features:
* O(M*N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional functionality

INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of CHA
    IsUpper -   what half of CHA is provided
    B       -   array[N,M], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly singular or badly conditioned
                        X is filled by zeros in such cases.
                * -1    N<=0 was passed
                *  1    task was solved
    B       -   array[N]:
                * for info>0 overwritten by solution
                * for info=-3 filled by zeros

  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixcholeskysolvemfast(
    real_2d_array cha,
    ae_int_t n,
    bool isupper,
    real_2d_array& b,
    ae_int_t m,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`spdmatrixsolve` function

/*************************************************************************
Dense linear solver for A*x=b with N*N real  symmetric  positive  definite
matrix A,  N*1 vectors x and b.  "Slow-but-feature-rich"  version  of  the
solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* O(N^3) complexity
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant   performance   penalty  when
           ! compared with "fast" version  which  just  performs  Cholesky
           ! decomposition and calls triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, SPDMatrixSolveFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or non-SPD.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixsolve(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    real_1d_array b,
    ae_int_t& info,
    densesolverreport& rep,
    real_1d_array& x,
    const xparams _params = alglib::xdefault);

`spdmatrixsolvefast` function

/*************************************************************************
Dense linear solver for A*x=b with N*N real  symmetric  positive  definite
matrix A,  N*1 vectors x and  b.  "Fast-but-lightweight"  version  of  the
solver.

Algorithm features:
* O(N^3) complexity
* matrix is represented by its upper or lower triangle
* no additional time consuming features like condition number estimation

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly singular or non-SPD
                * -1    N<=0 was passed
                *  1    task was solved
    B       -   array[N], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 17.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixsolvefast(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    real_1d_array& b,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`spdmatrixsolvem` function

/*************************************************************************
Dense solver for A*X=B with N*N symmetric positive definite matrix A,  and
N*M vectors X and B. It is "slow-but-feature-rich" version of the solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* O(N^3+M*N^2) complexity
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant   performance   penalty  when
           ! compared with "fast" version  which  just  performs  Cholesky
           ! decomposition and calls triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, SPDMatrixSolveMFast() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    matrix is very badly conditioned or non-SPD.
                * -1    N<=0 was passed
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixsolvem(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    real_2d_array b,
    ae_int_t m,
    ae_int_t& info,
    densesolverreport& rep,
    real_2d_array& x,
    const xparams _params = alglib::xdefault);

`spdmatrixsolvemfast` function

/*************************************************************************
Dense solver for A*X=B with N*N symmetric positive definite matrix A,  and
N*M vectors X and B. It is "fast-but-lightweight" version of the solver.

Algorithm features:
* O(N^3+M*N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional time consuming features

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Info    -   return code:
                * -3    A is is exactly singular
                * -1    N<=0 was passed
                *  1    task was solved
    B       -   array[N,M], it contains:
                * info>0    =>  solution
                * info=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 17.03.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixsolvemfast(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    real_2d_array& b,
    ae_int_t m,
    ae_int_t& info,
    const xparams _params = alglib::xdefault);

`directsparsesolvers` subpackage

Classes

sparsesolverreport

Functions

sparsecholeskysolvesks
sparsesolvesks

Examples

solvesks_d_1

Solving positive definite sparse system using Skyline (SKS) solver

`sparsesolverreport` class

/*************************************************************************
This structure is a sparse solver report.

Following fields can be accessed by users:
*************************************************************************/
class sparsesolverreport
{
    ae_int_t             terminationtype;
};

`sparsecholeskysolvesks` function

/*************************************************************************
Sparse linear solver for A*x=b with N*N real  symmetric  positive definite
matrix A given by its Cholesky decomposition, and N*1 vectors x and b.

IMPORTANT: this solver requires input matrix to be in  the  SKS  (Skyline)
           sparse storage format. An exception will be  generated  if  you
           pass matrix in some other format (HASH or CRS).

INPUT PARAMETERS
    A       -   sparse NxN matrix stored in SKS format, must be NxN exactly
    N       -   size of A, N>0
    IsUpper -   which half of A is provided (another half is ignored)
    B       -   array[N], right part

OUTPUT PARAMETERS
    Rep     -   solver report, following fields are set:
                * rep.terminationtype - solver status; >0 for success,
                  set to -3 on failure (degenerate or non-SPD system).
    X       -   array[N], it contains:
                * rep.terminationtype>0    =>  solution
                * rep.terminationtype=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 26.12.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecholeskysolvesks(
    sparsematrix a,
    ae_int_t n,
    bool isupper,
    real_1d_array b,
    sparsesolverreport& rep,
    real_1d_array& x,
    const xparams _params = alglib::xdefault);

Examples: [1]

`sparsesolvesks` function

/*************************************************************************
Sparse linear solver for A*x=b with N*N  sparse  real  symmetric  positive
definite matrix A, N*1 vectors x and b.

This solver  converts  input  matrix  to  SKS  format,  performs  Cholesky
factorization using  SKS  Cholesky  subroutine  (works  well  for  limited
bandwidth matrices) and uses sparse triangular solvers to get solution  of
the original system.

INPUT PARAMETERS
    A       -   sparse matrix, must be NxN exactly
    N       -   size of A, N>0
    IsUpper -   which half of A is provided (another half is ignored)
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Rep     -   solver report, following fields are set:
                * rep.terminationtype - solver status; >0 for success,
                  set to -3 on failure (degenerate or non-SPD system).
    X       -   array[N], it contains:
                * rep.terminationtype>0    =>  solution
                * rep.terminationtype=-3   =>  filled by zeros

  -- ALGLIB --
     Copyright 26.12.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsesolvesks(
    sparsematrix a,
    ae_int_t n,
    bool isupper,
    real_1d_array b,
    sparsesolverreport& rep,
    real_1d_array& x,
    const xparams _params = alglib::xdefault);

Examples: [1]

solvesks_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates creation/initialization of the sparse matrix
    // in the SKS (Skyline) storage format and solution using SKS-based direct
    // solver.
    //
    // First, we have to create matrix and initialize it. Matrix is created
    // in the SKS format, using fixed bandwidth initialization function.
    // Several points should be noted:
    //
    // 1. SKS sparse storage format also allows variable bandwidth matrices;
    //    we just do not want to overcomplicate this example.
    //
    // 2. SKS format requires you to specify matrix geometry prior to
    //    initialization of its elements with sparseset(). If you specified
    //    bandwidth=1, you can not change your mind afterwards and call
    //    sparseset() for non-existent elements.
    // 
    // 3. Because SKS solver need just one triangle of SPD matrix, we can
    //    omit initialization of the lower triangle of our matrix.
    //
    ae_int_t n = 4;
    ae_int_t bandwidth = 1;
    sparsematrix s;
    sparsecreatesksband(n, n, bandwidth, s);
    sparseset(s, 0, 0, 2.0);
    sparseset(s, 0, 1, 1.0);
    sparseset(s, 1, 1, 3.0);
    sparseset(s, 1, 2, 1.0);
    sparseset(s, 2, 2, 3.0);
    sparseset(s, 2, 3, 1.0);
    sparseset(s, 3, 3, 2.0);

    //
    // Now we have symmetric positive definite 4x4 system width bandwidth=1:
    //
    //     [ 2 1     ]   [ x0]]   [  4 ]
    //     [ 1 3 1   ]   [ x1 ]   [ 10 ]
    //     [   1 3 1 ] * [ x2 ] = [ 15 ]
    //     [     1 2 ]   [ x3 ]   [ 11 ]
    //
    // After successful creation we can call SKS solver.
    //
    real_1d_array b = "[4,10,15,11]";
    sparsesolverreport rep;
    real_1d_array x;
    bool isuppertriangle = true;
    sparsesolvesks(s, n, isuppertriangle, b, rep, x);
    printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [1.0000, 2.0000, 3.0000, 4.0000]
    return 0;
}

`elliptic` subpackage

Functions

ellipticintegrale
ellipticintegralk
ellipticintegralkhighprecision
incompleteellipticintegrale
incompleteellipticintegralk

Examples

`ellipticintegrale` function

/*************************************************************************
Complete elliptic integral of the second kind

Approximates the integral


           pi/2
            -
           | |                 2
E(m)  =    |    sqrt( 1 - m sin t ) dt
         | |
          -
           0

using the approximation

     P(x)  -  x log x Q(x).

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0, 1       10000       2.1e-16     7.3e-17

Cephes Math Library, Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::ellipticintegrale(
    double m,
    const xparams _params = alglib::xdefault);

`ellipticintegralk` function

/*************************************************************************
Complete elliptic integral of the first kind

Approximates the integral



           pi/2
            -
           | |
           |           dt
K(m)  =    |    ------------------
           |                   2
         | |    sqrt( 1 - m sin t )
          -
           0

using the approximation

    P(x)  -  log x Q(x).

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,1        30000       2.5e-16     6.8e-17

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::ellipticintegralk(
    double m,
    const xparams _params = alglib::xdefault);

`ellipticintegralkhighprecision` function

/*************************************************************************
Complete elliptic integral of the first kind

Approximates the integral



           pi/2
            -
           | |
           |           dt
K(m)  =    |    ------------------
           |                   2
         | |    sqrt( 1 - m sin t )
          -
           0

where m = 1 - m1, using the approximation

    P(x)  -  log x Q(x).

The argument m1 is used rather than m so that the logarithmic
singularity at m = 1 will be shifted to the origin; this
preserves maximum accuracy.

K(0) = pi/2.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,1        30000       2.5e-16     6.8e-17

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::ellipticintegralkhighprecision(
    double m1,
    const xparams _params = alglib::xdefault);

`incompleteellipticintegrale` function

/*************************************************************************
Incomplete elliptic integral of the second kind

Approximates the integral


               phi
                -
               | |
               |                   2
E(phi_\m)  =    |    sqrt( 1 - m sin t ) dt
               |
             | |
              -
               0

of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.

ACCURACY:

Tested at random arguments with phi in [-10, 10] and m in
[0, 1].
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -10,10      150000       3.3e-15     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::incompleteellipticintegrale(
    double phi,
    double m,
    const xparams _params = alglib::xdefault);

`incompleteellipticintegralk` function

/*************************************************************************
Incomplete elliptic integral of the first kind F(phi|m)

Approximates the integral



               phi
                -
               | |
               |           dt
F(phi_\m)  =    |    ------------------
               |                   2
             | |    sqrt( 1 - m sin t )
              -
               0

of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.




ACCURACY:

Tested at random points with m in [0, 1] and phi as indicated.

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -10,10       200000      7.4e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::incompleteellipticintegralk(
    double phi,
    double m,
    const xparams _params = alglib::xdefault);

`evd` subpackage

Classes

eigsubspacereport
eigsubspacestate

Functions

eigsubspacecreate
eigsubspacecreatebuf
eigsubspaceooccontinue
eigsubspaceoocgetrequestdata
eigsubspaceoocgetrequestinfo
eigsubspaceoocsendresult
eigsubspaceoocstart
eigsubspaceoocstop
eigsubspacesetcond
eigsubspacesetwarmstart
eigsubspacesolvedenses
eigsubspacesolvesparses
hmatrixevd
hmatrixevdi
hmatrixevdr
rmatrixevd
smatrixevd
smatrixevdi
smatrixevdr
smatrixtdevd
smatrixtdevdi
smatrixtdevdr

Examples

`eigsubspacereport` class

/*************************************************************************
This object stores state of the subspace iteration algorithm.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class eigsubspacereport
{
    ae_int_t             iterationscount;
};

`eigsubspacestate` class

/*************************************************************************
This object stores state of the subspace iteration algorithm.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class eigsubspacestate
{
};

`eigsubspacecreate` function

/*************************************************************************
This function initializes subspace iteration solver. This solver  is  used
to solve symmetric real eigenproblems where just a few (top K) eigenvalues
and corresponding eigenvectors is required.

This solver can be significantly faster than  complete  EVD  decomposition
in the following case:
* when only just a small fraction  of  top  eigenpairs  of dense matrix is
  required. When K approaches N, this solver is slower than complete dense
  EVD
* when problem matrix is sparse (and/or is not known explicitly, i.e. only
  matrix-matrix product can be performed)

USAGE (explicit dense/sparse matrix):
1. User initializes algorithm state with eigsubspacecreate() call
2. [optional] User tunes solver parameters by calling eigsubspacesetcond()
   or other functions
3. User  calls  eigsubspacesolvedense() or eigsubspacesolvesparse() methods,
   which take algorithm state and 2D array or alglib.sparsematrix object.

USAGE (out-of-core mode):
1. User initializes algorithm state with eigsubspacecreate() call
2. [optional] User tunes solver parameters by calling eigsubspacesetcond()
   or other functions
3. User activates out-of-core mode of  the  solver  and  repeatedly  calls
   communication functions in a loop like below:
   > alglib.eigsubspaceoocstart(state)
   > while alglib.eigsubspaceooccontinue(state) do
   >     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
   >     alglib.eigsubspaceoocgetrequestdata(state, out X)
   >     [calculate  Y=A*X, with X=R^NxM]
   >     alglib.eigsubspaceoocsendresult(state, in Y)
   > alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    N       -   problem dimensionality, N>0
    K       -   number of top eigenvector to calculate, 0<K<=N.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE: if you solve many similar EVD problems you may  find  it  useful  to
      reuse previous subspace as warm-start point for new EVD problem.  It
      can be done with eigsubspacesetwarmstart() function.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspacecreate(
    ae_int_t n,
    ae_int_t k,
    eigsubspacestate& state,
    const xparams _params = alglib::xdefault);

`eigsubspacecreatebuf` function

/*************************************************************************
Buffered version of constructor which aims to reuse  previously  allocated
memory as much as possible.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspacecreatebuf(
    ae_int_t n,
    ae_int_t k,
    eigsubspacestate state,
    const xparams _params = alglib::xdefault);

`eigsubspaceooccontinue` function

/*************************************************************************
This function performs subspace iteration  in  the  out-of-core  mode.  It
should be used in conjunction with other out-of-core-related functions  of
this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
bool alglib::eigsubspaceooccontinue(
    eigsubspacestate state,
    const xparams _params = alglib::xdefault);

`eigsubspaceoocgetrequestdata` function

/*************************************************************************
This function is used to retrieve information  about  out-of-core  request
sent by solver to user code: matrix X (array[N,RequestSize) which have  to
be multiplied by out-of-core matrix A in a product A*X.

This function returns just request data; in order to get size of  the data
prior to processing requestm, use eigsubspaceoocgetrequestinfo().

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode
    X               -   possibly  preallocated   storage;  reallocated  if
                        needed, left unchanged, if large enough  to  store
                        request data.

OUTPUT PARAMETERS:
    X               -   array[N,RequestSize] or larger, leading  rectangle
                        is filled with dense matrix X.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspaceoocgetrequestdata(
    eigsubspacestate state,
    real_2d_array& x,
    const xparams _params = alglib::xdefault);

`eigsubspaceoocgetrequestinfo` function

/*************************************************************************
This function is used to retrieve information  about  out-of-core  request
sent by solver to user code: request type (current version  of  the solver
sends only requests for matrix-matrix products) and request size (size  of
the matrices being multiplied).

This function returns just request metrics; in order  to  get contents  of
the matrices being multiplied, use eigsubspaceoocgetrequestdata().

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode

OUTPUT PARAMETERS:
    RequestType     -   type of the request to process:
                        * 0 - for matrix-matrix product A*X, with A  being
                          NxN matrix whose eigenvalues/vectors are needed,
                          and X being NxREQUESTSIZE one which is  returned
                          by the eigsubspaceoocgetrequestdata().
    RequestSize     -   size of the X matrix (number of columns),  usually
                        it is several times larger than number of  vectors
                        K requested by user.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspaceoocgetrequestinfo(
    eigsubspacestate state,
    ae_int_t& requesttype,
    ae_int_t& requestsize,
    const xparams _params = alglib::xdefault);

`eigsubspaceoocsendresult` function

/*************************************************************************
This function is used to send user reply to out-of-core  request  sent  by
solver. Usually it is product A*X for returned by solver matrix X.

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode
    AX              -   array[N,RequestSize] or larger, leading  rectangle
                        is filled with product A*X.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspaceoocsendresult(
    eigsubspacestate state,
    real_2d_array ax,
    const xparams _params = alglib::xdefault);

`eigsubspaceoocstart` function

/*************************************************************************
This  function  initiates  out-of-core  mode  of  subspace eigensolver. It
should be used in conjunction with other out-of-core-related functions  of
this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State       -   solver object
    MType       -   matrix type:
                    * 0 for real  symmetric  matrix  (solver  assumes that
                      matrix  being   processed  is  symmetric;  symmetric
                      direct eigensolver is used for  smaller  subproblems
                      arising during solution of larger "full" task)
                    Future versions of ALGLIB may  introduce  support  for
                    other  matrix   types;   for   now,   only   symmetric
                    eigenproblems are supported.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspaceoocstart(
    eigsubspacestate state,
    ae_int_t mtype,
    const xparams _params = alglib::xdefault);

`eigsubspaceoocstop` function

/*************************************************************************
This  function  finalizes out-of-core  mode  of  subspace eigensolver.  It
should be used in conjunction with other out-of-core-related functions  of
this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State       -   solver state

OUTPUT PARAMETERS:
    W           -   array[K], depending on solver settings:
                    * top  K  eigenvalues ordered  by  descending   -   if
                      eigenvectors are returned in Z
                    * zeros - if invariant subspace is returned in Z
    Z           -   array[N,K], depending on solver settings either:
                    * matrix of eigenvectors found
                    * orthogonal basis of K-dimensional invariant subspace
    Rep         -   report with additional parameters

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspaceoocstop(
    eigsubspacestate state,
    real_1d_array& w,
    real_2d_array& z,
    eigsubspacereport& rep,
    const xparams _params = alglib::xdefault);

`eigsubspacesetcond` function

/*************************************************************************
This function sets stopping critera for the solver:
* error in eigenvector/value allowed by solver
* maximum number of iterations to perform

INPUT PARAMETERS:
    State       -   solver structure
    Eps         -   eps>=0,  with non-zero value used to tell solver  that
                    it can  stop  after  all  eigenvalues  converged  with
                    error  roughly  proportional  to  eps*MAX(LAMBDA_MAX),
                    where LAMBDA_MAX is a maximum eigenvalue.
                    Zero  value  means  that  no  check  for  precision is
                    performed.
    MaxIts      -   maxits>=0,  with non-zero value used  to  tell  solver
                    that it can stop after maxits  steps  (no  matter  how
                    precise current estimate is)

NOTE: passing  eps=0  and  maxits=0  results  in  automatic  selection  of
      moderate eps as stopping criteria (1.0E-6 in current implementation,
      but it may change without notice).

NOTE: very small values of eps are possible (say, 1.0E-12),  although  the
      larger problem you solve (N and/or K), the  harder  it  is  to  find
      precise eigenvectors because rounding errors tend to accumulate.

NOTE: passing non-zero eps results in  some performance  penalty,  roughly
      equal to 2N*(2K)^2 FLOPs per iteration. These additional computations
      are required in order to estimate current error in  eigenvalues  via
      Rayleigh-Ritz process.
      Most of this additional time is  spent  in  construction  of  ~2Kx2K
      symmetric  subproblem  whose  eigenvalues  are  checked  with  exact
      eigensolver.
      This additional time is negligible if you search for eigenvalues  of
      the large dense matrix, but may become noticeable on  highly  sparse
      EVD problems, where cost of matrix-matrix product is low.
      If you set eps to exactly zero,  Rayleigh-Ritz  phase  is completely
      turned off.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspacesetcond(
    eigsubspacestate state,
    double eps,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

`eigsubspacesetwarmstart` function

/*************************************************************************
This function sets warm-start mode of the solver: next call to the  solver
will reuse previous subspace as warm-start  point.  It  can  significantly
speed-up convergence when you solve many similar eigenproblems.

INPUT PARAMETERS:
    State       -   solver structure
    UseWarmStart-   either True or False

  -- ALGLIB --
     Copyright 12.11.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspacesetwarmstart(
    eigsubspacestate state,
    bool usewarmstart,
    const xparams _params = alglib::xdefault);

`eigsubspacesolvedenses` function

/*************************************************************************
This  function runs eigensolver for dense NxN symmetric matrix A, given by
upper or lower triangle.

This function can not process nonsymmetric matrices.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    State       -   solver state
    A           -   array[N,N], symmetric NxN matrix given by one  of  its
                    triangles
    IsUpper     -   whether upper or lower triangle of  A  is  given  (the
                    other one is not referenced at all).

OUTPUT PARAMETERS:
    W           -   array[K], top  K  eigenvalues ordered  by   descending
                    of their absolute values
    Z           -   array[N,K], matrix of eigenvectors found
    Rep         -   report with additional parameters

NOTE: internally this function allocates a copy of NxN dense A. You should
      take it into account when working with very large matrices occupying
      almost all RAM.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspacesolvedenses(
    eigsubspacestate state,
    real_2d_array a,
    bool isupper,
    real_1d_array& w,
    real_2d_array& z,
    eigsubspacereport& rep,
    const xparams _params = alglib::xdefault);

`eigsubspacesolvesparses` function

/*************************************************************************
This  function runs eigensolver for dense NxN symmetric matrix A, given by
upper or lower triangle.

This function can not process nonsymmetric matrices.

INPUT PARAMETERS:
    State       -   solver state
    A           -   NxN symmetric matrix given by one of its triangles
    IsUpper     -   whether upper or lower triangle of  A  is  given  (the
                    other one is not referenced at all).

OUTPUT PARAMETERS:
    W           -   array[K], top  K  eigenvalues ordered  by   descending
                    of their absolute values
    Z           -   array[N,K], matrix of eigenvectors found
    Rep         -   report with additional parameters

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::eigsubspacesolvesparses(
    eigsubspacestate state,
    sparsematrix a,
    bool isupper,
    real_1d_array& w,
    real_2d_array& z,
    eigsubspacereport& rep,
    const xparams _params = alglib::xdefault);

`hmatrixevd` function

/*************************************************************************
Finding the eigenvalues and eigenvectors of a Hermitian matrix

The algorithm finds eigen pairs of a Hermitian matrix by  reducing  it  to
real tridiagonal form and using the QL/QR algorithm.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   Hermitian matrix which is given  by  its  upper  or  lower
                triangular part.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format.
    ZNeeded -   flag controlling whether the eigenvectors  are  needed  or
                not. If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.

Output parameters:
    D       -   eigenvalues in ascending order.
                Array whose index ranges within [0..N-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains the eigenvectors.
                Array whose indexes range within [0..N-1, 0..N-1].
                The eigenvectors are stored in the matrix columns.

Result:
    True, if the algorithm has converged.
    False, if the algorithm hasn't converged (rare case).

Note:
    eigenvectors of Hermitian matrix are defined up to  multiplication  by
    a complex number L, such that |L|=1.

  -- ALGLIB --
     Copyright 2005, 23 March 2007 by Bochkanov Sergey
*************************************************************************/
bool alglib::hmatrixevd(
    complex_2d_array a,
    ae_int_t n,
    ae_int_t zneeded,
    bool isupper,
    real_1d_array& d,
    complex_2d_array& z,
    const xparams _params = alglib::xdefault);

`hmatrixevdi` function

/*************************************************************************
Subroutine for finding the eigenvalues and  eigenvectors  of  a  Hermitian
matrix with given indexes by using bisection and inverse iteration methods

Input parameters:
    A       -   Hermitian matrix which is given  by  its  upper  or  lower
                triangular part.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors  are  needed  or
                not. If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpperA -  storage format of matrix A.
    I1, I2 -    index interval for searching (from I1 to I2).
                0 <= I1 <= I2 <= N-1.

Output parameters:
    W       -   array of the eigenvalues found.
                Array whose index ranges within [0..I2-I1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains eigenvectors.
                Array whose indexes range within [0..N-1, 0..I2-I1].
                In  that  case,  the eigenvectors are stored in the matrix
                columns.

Result:
    True, if successful. W contains the eigenvalues, Z contains the
    eigenvectors (if needed).

    False, if the bisection method subroutine  wasn't  able  to  find  the
    eigenvalues  in  the  given  interval  or  if  the  inverse  iteration
    subroutine wasn't able to find  all  the  corresponding  eigenvectors.
    In that case, the eigenvalues and eigenvectors are not returned.

Note:
    eigen vectors of Hermitian matrix are defined up to multiplication  by
    a complex number L, such as |L|=1.

  -- ALGLIB --
     Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey.
*************************************************************************/
bool alglib::hmatrixevdi(
    complex_2d_array a,
    ae_int_t n,
    ae_int_t zneeded,
    bool isupper,
    ae_int_t i1,
    ae_int_t i2,
    real_1d_array& w,
    complex_2d_array& z,
    const xparams _params = alglib::xdefault);

`hmatrixevdr` function

/*************************************************************************
Subroutine for finding the eigenvalues (and eigenvectors) of  a  Hermitian
matrix  in  a  given half-interval (A, B] by using a bisection and inverse
iteration

Input parameters:
    A       -   Hermitian matrix which is given  by  its  upper  or  lower
                triangular  part.  Array  whose   indexes   range   within
                [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors  are  needed  or
                not. If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpperA -  storage format of matrix A.
    B1, B2 -    half-interval (B1, B2] to search eigenvalues in.

Output parameters:
    M       -   number of eigenvalues found in a given half-interval, M>=0
    W       -   array of the eigenvalues found.
                Array whose index ranges within [0..M-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains eigenvectors.
                Array whose indexes range within [0..N-1, 0..M-1].
                The eigenvectors are stored in the matrix columns.

Result:
    True, if successful. M contains the number of eigenvalues in the given
    half-interval (could be equal to 0), W contains the eigenvalues,
    Z contains the eigenvectors (if needed).

    False, if the bisection method subroutine  wasn't  able  to  find  the
    eigenvalues  in  the  given  interval  or  if  the  inverse  iteration
    subroutine  wasn't  able  to  find all the corresponding eigenvectors.
    In that case, the eigenvalues and eigenvectors are not returned, M  is
    equal to 0.

Note:
    eigen vectors of Hermitian matrix are defined up to multiplication  by
    a complex number L, such as |L|=1.

  -- ALGLIB --
     Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey.
*************************************************************************/
bool alglib::hmatrixevdr(
    complex_2d_array a,
    ae_int_t n,
    ae_int_t zneeded,
    bool isupper,
    double b1,
    double b2,
    ae_int_t& m,
    real_1d_array& w,
    complex_2d_array& z,
    const xparams _params = alglib::xdefault);

`rmatrixevd` function

/*************************************************************************
Finding eigenvalues and eigenvectors of a general (unsymmetric) matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

The algorithm finds eigenvalues and eigenvectors of a general matrix by
using the QR algorithm with multiple shifts. The algorithm can find
eigenvalues and both left and right eigenvectors.

The right eigenvector is a vector x such that A*x = w*x, and the left
eigenvector is a vector y such that y'*A = w*y' (here y' implies a complex
conjugate transposition of vector y).

Input parameters:
    A       -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    VNeeded -   flag controlling whether eigenvectors are needed or not.
                If VNeeded is equal to:
                 * 0, eigenvectors are not returned;
                 * 1, right eigenvectors are returned;
                 * 2, left eigenvectors are returned;
                 * 3, both left and right eigenvectors are returned.

Output parameters:
    WR      -   real parts of eigenvalues.
                Array whose index ranges within [0..N-1].
    WR      -   imaginary parts of eigenvalues.
                Array whose index ranges within [0..N-1].
    VL, VR  -   arrays of left and right eigenvectors (if they are needed).
                If WI[i]=0, the respective eigenvalue is a real number,
                and it corresponds to the column number I of matrices VL/VR.
                If WI[i]>0, we have a pair of complex conjugate numbers with
                positive and negative imaginary parts:
                    the first eigenvalue WR[i] + sqrt(-1)*WI[i];
                    the second eigenvalue WR[i+1] + sqrt(-1)*WI[i+1];
                    WI[i]>0
                    WI[i+1] = -WI[i] < 0
                In that case, the eigenvector  corresponding to the first
                eigenvalue is located in i and i+1 columns of matrices
                VL/VR (the column number i contains the real part, and the
                column number i+1 contains the imaginary part), and the vector
                corresponding to the second eigenvalue is a complex conjugate to
                the first vector.
                Arrays whose indexes range within [0..N-1, 0..N-1].

Result:
    True, if the algorithm has converged.
    False, if the algorithm has not converged.

Note 1:
    Some users may ask the following question: what if WI[N-1]>0?
    WI[N] must contain an eigenvalue which is complex conjugate to the
    N-th eigenvalue, but the array has only size N?
    The answer is as follows: such a situation cannot occur because the
    algorithm finds a pairs of eigenvalues, therefore, if WI[i]>0, I is
    strictly less than N-1.

Note 2:
    The algorithm performance depends on the value of the internal parameter
    NS of the InternalSchurDecomposition subroutine which defines the number
    of shifts in the QR algorithm (similarly to the block width in block-matrix
    algorithms of linear algebra). If you require maximum performance
    on your machine, it is recommended to adjust this parameter manually.


See also the InternalTREVC subroutine.

The algorithm is based on the LAPACK 3.0 library.
*************************************************************************/
bool alglib::rmatrixevd(
    real_2d_array a,
    ae_int_t n,
    ae_int_t vneeded,
    real_1d_array& wr,
    real_1d_array& wi,
    real_2d_array& vl,
    real_2d_array& vr,
    const xparams _params = alglib::xdefault);

`smatrixevd` function

/*************************************************************************
Finding the eigenvalues and eigenvectors of a symmetric matrix

The algorithm finds eigen pairs of a symmetric matrix by reducing it to
tridiagonal form and using the QL/QR algorithm.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   symmetric matrix which is given by its upper or lower
                triangular part.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpper -   storage format.

Output parameters:
    D       -   eigenvalues in ascending order.
                Array whose index ranges within [0..N-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains the eigenvectors.
                Array whose indexes range within [0..N-1, 0..N-1].
                The eigenvectors are stored in the matrix columns.

Result:
    True, if the algorithm has converged.
    False, if the algorithm hasn't converged (rare case).

  -- ALGLIB --
     Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
bool alglib::smatrixevd(
    real_2d_array a,
    ae_int_t n,
    ae_int_t zneeded,
    bool isupper,
    real_1d_array& d,
    real_2d_array& z,
    const xparams _params = alglib::xdefault);

`smatrixevdi` function

/*************************************************************************
Subroutine for finding the eigenvalues and  eigenvectors  of  a  symmetric
matrix with given indexes by using bisection and inverse iteration methods.

Input parameters:
    A       -   symmetric matrix which is given by its upper or lower
                triangular part. Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpperA -  storage format of matrix A.
    I1, I2 -    index interval for searching (from I1 to I2).
                0 <= I1 <= I2 <= N-1.

Output parameters:
    W       -   array of the eigenvalues found.
                Array whose index ranges within [0..I2-I1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains eigenvectors.
                Array whose indexes range within [0..N-1, 0..I2-I1].
                In that case, the eigenvectors are stored in the matrix columns.

Result:
    True, if successful. W contains the eigenvalues, Z contains the
    eigenvectors (if needed).

    False, if the bisection method subroutine wasn't able to find the
    eigenvalues in the given interval or if the inverse iteration subroutine
    wasn't able to find all the corresponding eigenvectors.
    In that case, the eigenvalues and eigenvectors are not returned.

  -- ALGLIB --
     Copyright 07.01.2006 by Bochkanov Sergey
*************************************************************************/
bool alglib::smatrixevdi(
    real_2d_array a,
    ae_int_t n,
    ae_int_t zneeded,
    bool isupper,
    ae_int_t i1,
    ae_int_t i2,
    real_1d_array& w,
    real_2d_array& z,
    const xparams _params = alglib::xdefault);

`smatrixevdr` function

/*************************************************************************
Subroutine for finding the eigenvalues (and eigenvectors) of  a  symmetric
matrix  in  a  given half open interval (A, B] by using  a  bisection  and
inverse iteration

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   symmetric matrix which is given by its upper or lower
                triangular part. Array [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpperA -  storage format of matrix A.
    B1, B2 -    half open interval (B1, B2] to search eigenvalues in.

Output parameters:
    M       -   number of eigenvalues found in a given half-interval (M>=0).
    W       -   array of the eigenvalues found.
                Array whose index ranges within [0..M-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains eigenvectors.
                Array whose indexes range within [0..N-1, 0..M-1].
                The eigenvectors are stored in the matrix columns.

Result:
    True, if successful. M contains the number of eigenvalues in the given
    half-interval (could be equal to 0), W contains the eigenvalues,
    Z contains the eigenvectors (if needed).

    False, if the bisection method subroutine wasn't able to find the
    eigenvalues in the given interval or if the inverse iteration subroutine
    wasn't able to find all the corresponding eigenvectors.
    In that case, the eigenvalues and eigenvectors are not returned,
    M is equal to 0.

  -- ALGLIB --
     Copyright 07.01.2006 by Bochkanov Sergey
*************************************************************************/
bool alglib::smatrixevdr(
    real_2d_array a,
    ae_int_t n,
    ae_int_t zneeded,
    bool isupper,
    double b1,
    double b2,
    ae_int_t& m,
    real_1d_array& w,
    real_2d_array& z,
    const xparams _params = alglib::xdefault);

`smatrixtdevd` function

/*************************************************************************
Finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix

The algorithm finds the eigen pairs of a tridiagonal symmetric matrix by
using an QL/QR algorithm with implicit shifts.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    D       -   the main diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-1].
    E       -   the secondary diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-2].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not needed;
                 * 1, the eigenvectors of a tridiagonal matrix
                   are multiplied by the square matrix Z. It is used if the
                   tridiagonal matrix is obtained by the similarity
                   transformation of a symmetric matrix;
                 * 2, the eigenvectors of a tridiagonal matrix replace the
                   square matrix Z;
                 * 3, matrix Z contains the first row of the eigenvectors
                   matrix.
    Z       -   if ZNeeded=1, Z contains the square matrix by which the
                eigenvectors are multiplied.
                Array whose indexes range within [0..N-1, 0..N-1].

Output parameters:
    D       -   eigenvalues in ascending order.
                Array whose index ranges within [0..N-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains the product of a given matrix (from the left)
                   and the eigenvectors matrix (from the right);
                 * 2, Z contains the eigenvectors.
                 * 3, Z contains the first row of the eigenvectors matrix.
                If ZNeeded<3, Z is the array whose indexes range within [0..N-1, 0..N-1].
                In that case, the eigenvectors are stored in the matrix columns.
                If ZNeeded=3, Z is the array whose indexes range within [0..0, 0..N-1].

Result:
    True, if the algorithm has converged.
    False, if the algorithm hasn't converged.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994
*************************************************************************/
bool alglib::smatrixtdevd(
    real_1d_array& d,
    real_1d_array e,
    ae_int_t n,
    ae_int_t zneeded,
    real_2d_array& z,
    const xparams _params = alglib::xdefault);

`smatrixtdevdi` function

/*************************************************************************
Subroutine for finding tridiagonal matrix eigenvalues/vectors with given
indexes (in ascending order) by using the bisection and inverse iteraion.

Input parameters:
    D       -   the main diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-1].
    E       -   the secondary diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-2].
    N       -   size of matrix. N>=0.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not needed;
                 * 1, the eigenvectors of a tridiagonal matrix are multiplied
                   by the square matrix Z. It is used if the
                   tridiagonal matrix is obtained by the similarity transformation
                   of a symmetric matrix.
                 * 2, the eigenvectors of a tridiagonal matrix replace
                   matrix Z.
    I1, I2  -   index interval for searching (from I1 to I2).
                0 <= I1 <= I2 <= N-1.
    Z       -   if ZNeeded is equal to:
                 * 0, Z isn't used and remains unchanged;
                 * 1, Z contains the square matrix (array whose indexes range within [0..N-1, 0..N-1])
                   which reduces the given symmetric matrix to  tridiagonal form;
                 * 2, Z isn't used (but changed on the exit).

Output parameters:
    D       -   array of the eigenvalues found.
                Array whose index ranges within [0..I2-I1].
    Z       -   if ZNeeded is equal to:
                 * 0, doesn't contain any information;
                 * 1, contains the product of a given NxN matrix Z (from the left) and
                   Nx(I2-I1) matrix of the eigenvectors found (from the right).
                   Array whose indexes range within [0..N-1, 0..I2-I1].
                 * 2, contains the matrix of the eigenvalues found.
                   Array whose indexes range within [0..N-1, 0..I2-I1].


Result:

    True, if successful. In that case, D contains the eigenvalues,
    Z contains the eigenvectors (if needed).
    It should be noted that the subroutine changes the size of arrays D and Z.

    False, if the bisection method subroutine wasn't able to find the eigenvalues
    in the given interval or if the inverse iteration subroutine wasn't able
    to find all the corresponding eigenvectors. In that case, the eigenvalues
    and eigenvectors are not returned.

  -- ALGLIB --
     Copyright 25.12.2005 by Bochkanov Sergey
*************************************************************************/
bool alglib::smatrixtdevdi(
    real_1d_array& d,
    real_1d_array e,
    ae_int_t n,
    ae_int_t zneeded,
    ae_int_t i1,
    ae_int_t i2,
    real_2d_array& z,
    const xparams _params = alglib::xdefault);

`smatrixtdevdr` function

/*************************************************************************
Subroutine for finding the tridiagonal matrix eigenvalues/vectors in a
given half-interval (A, B] by using bisection and inverse iteration.

Input parameters:
    D       -   the main diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-1].
    E       -   the secondary diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-2].
    N       -   size of matrix, N>=0.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not needed;
                 * 1, the eigenvectors of a tridiagonal matrix are multiplied
                   by the square matrix Z. It is used if the tridiagonal
                   matrix is obtained by the similarity transformation
                   of a symmetric matrix.
                 * 2, the eigenvectors of a tridiagonal matrix replace matrix Z.
    A, B    -   half-interval (A, B] to search eigenvalues in.
    Z       -   if ZNeeded is equal to:
                 * 0, Z isn't used and remains unchanged;
                 * 1, Z contains the square matrix (array whose indexes range
                   within [0..N-1, 0..N-1]) which reduces the given symmetric
                   matrix to tridiagonal form;
                 * 2, Z isn't used (but changed on the exit).

Output parameters:
    D       -   array of the eigenvalues found.
                Array whose index ranges within [0..M-1].
    M       -   number of eigenvalues found in the given half-interval (M>=0).
    Z       -   if ZNeeded is equal to:
                 * 0, doesn't contain any information;
                 * 1, contains the product of a given NxN matrix Z (from the
                   left) and NxM matrix of the eigenvectors found (from the
                   right). Array whose indexes range within [0..N-1, 0..M-1].
                 * 2, contains the matrix of the eigenvectors found.
                   Array whose indexes range within [0..N-1, 0..M-1].

Result:

    True, if successful. In that case, M contains the number of eigenvalues
    in the given half-interval (could be equal to 0), D contains the eigenvalues,
    Z contains the eigenvectors (if needed).
    It should be noted that the subroutine changes the size of arrays D and Z.

    False, if the bisection method subroutine wasn't able to find the
    eigenvalues in the given interval or if the inverse iteration subroutine
    wasn't able to find all the corresponding eigenvectors. In that case,
    the eigenvalues and eigenvectors are not returned, M is equal to 0.

  -- ALGLIB --
     Copyright 31.03.2008 by Bochkanov Sergey
*************************************************************************/
bool alglib::smatrixtdevdr(
    real_1d_array& d,
    real_1d_array e,
    ae_int_t n,
    ae_int_t zneeded,
    double a,
    double b,
    ae_int_t& m,
    real_2d_array& z,
    const xparams _params = alglib::xdefault);

`expintegrals` subpackage

Functions

exponentialintegralei
exponentialintegralen

Examples

`exponentialintegralei` function

/*************************************************************************
Exponential integral Ei(x)

              x
               -     t
              | |   e
   Ei(x) =   -|-   ---  dt .
            | |     t
             -
            -inf

Not defined for x <= 0.
See also expn.c.



ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,100       50000      8.6e-16     1.3e-16

Cephes Math Library Release 2.8:  May, 1999
Copyright 1999 by Stephen L. Moshier
*************************************************************************/
double alglib::exponentialintegralei(
    double x,
    const xparams _params = alglib::xdefault);

`exponentialintegralen` function

/*************************************************************************
Exponential integral En(x)

Evaluates the exponential integral

                inf.
                  -
                 | |   -xt
                 |    e
     E (x)  =    |    ----  dt.
      n          |      n
               | |     t
                -
                 1


Both n and x must be nonnegative.

The routine employs either a power series, a continued
fraction, or an asymptotic formula depending on the
relative values of n and x.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       10000       1.7e-15     3.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::exponentialintegralen(
    double x,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`fdistr` subpackage

Functions

fcdistribution
fdistribution
invfdistribution

Examples

`fcdistribution` function

/*************************************************************************
Complemented F distribution

Returns the area from x to infinity under the F density
function (also known as Snedcor's density or the
variance ratio density).


                     inf.
                      -
             1       | |  a-1      b-1
1-P(x)  =  ------    |   t    (1-t)    dt
           B(a,b)  | |
                    -
                     x


The incomplete beta integral is used, according to the
formula

P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).


ACCURACY:

Tested at random points (a,b,x) in the indicated intervals.
               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    1,100       100000      3.7e-14     5.9e-16
   IEEE      1,5    1,100       100000      8.0e-15     1.6e-15
   IEEE      0,1    1,10000     100000      1.8e-11     3.5e-13
   IEEE      1,5    1,10000     100000      2.0e-11     3.0e-12

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::fcdistribution(
    ae_int_t a,
    ae_int_t b,
    double x,
    const xparams _params = alglib::xdefault);

`fdistribution` function

/*************************************************************************
F distribution

Returns the area from zero to x under the F density
function (also known as Snedcor's density or the
variance ratio density).  This is the density
of x = (u1/df1)/(u2/df2), where u1 and u2 are random
variables having Chi square distributions with df1
and df2 degrees of freedom, respectively.
The incomplete beta integral is used, according to the
formula

P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).


The arguments a and b are greater than zero, and x is
nonnegative.

ACCURACY:

Tested at random points (a,b,x).

               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    0,100       100000      9.8e-15     1.7e-15
   IEEE      1,5    0,100       100000      6.5e-15     3.5e-16
   IEEE      0,1    1,10000     100000      2.2e-11     3.3e-12
   IEEE      1,5    1,10000     100000      1.1e-11     1.7e-13

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::fdistribution(
    ae_int_t a,
    ae_int_t b,
    double x,
    const xparams _params = alglib::xdefault);

`invfdistribution` function

/*************************************************************************
Inverse of complemented F distribution

Finds the F density argument x such that the integral
from x to infinity of the F density is equal to the
given probability p.

This is accomplished using the inverse beta integral
function and the relations

     z = incbi( df2/2, df1/2, p )
     x = df2 (1-z) / (df1 z).

Note: the following relations hold for the inverse of
the uncomplemented F distribution:

     z = incbi( df1/2, df2/2, p )
     x = df2 z / (df1 (1-z)).

ACCURACY:

Tested at random points (a,b,p).

             a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between .001 and 1:
   IEEE     1,100       100000      8.3e-15     4.7e-16
   IEEE     1,10000     100000      2.1e-11     1.4e-13
 For p between 10^-6 and 10^-3:
   IEEE     1,100        50000      1.3e-12     8.4e-15
   IEEE     1,10000      50000      3.0e-12     4.8e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::invfdistribution(
    ae_int_t a,
    ae_int_t b,
    double y,
    const xparams _params = alglib::xdefault);

`fft` subpackage

Functions

fftc1d
fftc1dinv
fftr1d
fftr1dinv

Examples

fft_complex_d1		Complex FFT: simple example
fft_complex_d2		Complex FFT: advanced example
fft_real_d1		Real FFT: simple example
fft_real_d2		Real FFT: advanced example

`fftc1d` function

/*************************************************************************
1-dimensional complex FFT.

Array size N may be arbitrary number (composite or prime).  Composite  N's
are handled with cache-oblivious variation of  a  Cooley-Tukey  algorithm.
Small prime-factors are transformed using hard coded  codelets (similar to
FFTW codelets, but without low-level  optimization),  large  prime-factors
are handled with Bluestein's algorithm.

Fastests transforms are for smooth N's (prime factors are 2, 3,  5  only),
most fast for powers of 2. When N have prime factors  larger  than  these,
but orders of magnitude smaller than N, computations will be about 4 times
slower than for nearby highly composite N's. When N itself is prime, speed
will be 6 times lower.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    A   -   array[0..N-1] - complex function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    A   -   DFT of a input array, array[0..N-1]
            A_out[j] = SUM(A_in[k]*exp(-2*pi*sqrt(-1)*j*k/N), k = 0..N-1)


  -- ALGLIB --
     Copyright 29.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::fftc1d(
    complex_1d_array& a,
    const xparams _params = alglib::xdefault);
void alglib::fftc1d(
    complex_1d_array& a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`fftc1dinv` function

/*************************************************************************
1-dimensional complex inverse FFT.

Array size N may be arbitrary number (composite or prime).  Algorithm  has
O(N*logN) complexity for any N (composite or prime).

See FFTC1D() description for more information about algorithm performance.

INPUT PARAMETERS
    A   -   array[0..N-1] - complex array to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    A   -   inverse DFT of a input array, array[0..N-1]
            A_out[j] = SUM(A_in[k]/N*exp(+2*pi*sqrt(-1)*j*k/N), k = 0..N-1)


  -- ALGLIB --
     Copyright 29.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::fftc1dinv(
    complex_1d_array& a,
    const xparams _params = alglib::xdefault);
void alglib::fftc1dinv(
    complex_1d_array& a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`fftr1d` function

/*************************************************************************
1-dimensional real FFT.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    A   -   array[0..N-1] - real function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    F   -   DFT of a input array, array[0..N-1]
            F[j] = SUM(A[k]*exp(-2*pi*sqrt(-1)*j*k/N), k = 0..N-1)

NOTE:
    F[] satisfies symmetry property F[k] = conj(F[N-k]),  so just one half
of  array  is  usually needed. But for convinience subroutine returns full
complex array (with frequencies above N/2), so its result may be  used  by
other FFT-related subroutines.


  -- ALGLIB --
     Copyright 01.06.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::fftr1d(
    real_1d_array a,
    complex_1d_array& f,
    const xparams _params = alglib::xdefault);
void alglib::fftr1d(
    real_1d_array a,
    ae_int_t n,
    complex_1d_array& f,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`fftr1dinv` function

/*************************************************************************
1-dimensional real inverse FFT.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    F   -   array[0..floor(N/2)] - frequencies from forward real FFT
    N   -   problem size

OUTPUT PARAMETERS
    A   -   inverse DFT of a input array, array[0..N-1]

NOTE:
    F[] should satisfy symmetry property F[k] = conj(F[N-k]), so just  one
half of frequencies array is needed - elements from 0 to floor(N/2).  F[0]
is ALWAYS real. If N is even F[floor(N/2)] is real too. If N is odd,  then
F[floor(N/2)] has no special properties.

Relying on properties noted above, FFTR1DInv subroutine uses only elements
from 0th to floor(N/2)-th. It ignores imaginary part of F[0],  and in case
N is even it ignores imaginary part of F[floor(N/2)] too.

When you call this function using full arguments list - "FFTR1DInv(F,N,A)"
- you can pass either either frequencies array with N elements or  reduced
array with roughly N/2 elements - subroutine will  successfully  transform
both.

If you call this function using reduced arguments list -  "FFTR1DInv(F,A)"
- you must pass FULL array with N elements (although higher  N/2 are still
not used) because array size is used to automatically determine FFT length


  -- ALGLIB --
     Copyright 01.06.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::fftr1dinv(
    complex_1d_array f,
    real_1d_array& a,
    const xparams _params = alglib::xdefault);
void alglib::fftr1dinv(
    complex_1d_array f,
    ae_int_t n,
    real_1d_array& a,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

fft_complex_d1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // first we demonstrate forward FFT:
    // [1i,1i,1i,1i] is converted to [4i, 0, 0, 0]
    //
    complex_1d_array z = "[1i,1i,1i,1i]";
    fftc1d(z);
    printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [4i,0,0,0]

    //
    // now we convert [4i, 0, 0, 0] back to [1i,1i,1i,1i]
    // with backward FFT
    //
    fftc1dinv(z);
    printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [1i,1i,1i,1i]
    return 0;
}

fft_complex_d2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // first we demonstrate forward FFT:
    // [0,1,0,1i] is converted to [1+1i, -1-1i, -1-1i, 1+1i]
    //
    complex_1d_array z = "[0,1,0,1i]";
    fftc1d(z);
    printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [1+1i, -1-1i, -1-1i, 1+1i]

    //
    // now we convert result back with backward FFT
    //
    fftc1dinv(z);
    printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [0,1,0,1i]
    return 0;
}

fft_real_d1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // first we demonstrate forward FFT:
    // [1,1,1,1] is converted to [4, 0, 0, 0]
    //
    real_1d_array x = "[1,1,1,1]";
    complex_1d_array f;
    real_1d_array x2;
    fftr1d(x, f);
    printf("%s\n", f.tostring(3).c_str()); // EXPECTED: [4,0,0,0]

    //
    // now we convert [4, 0, 0, 0] back to [1,1,1,1]
    // with backward FFT
    //
    fftr1dinv(f, x2);
    printf("%s\n", x2.tostring(3).c_str()); // EXPECTED: [1,1,1,1]
    return 0;
}

fft_real_d2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // first we demonstrate forward FFT:
    // [1,2,3,4] is converted to [10, -2+2i, -2, -2-2i]
    //
    // note that output array is self-adjoint:
    // * f[0] = conj(f[0])
    // * f[1] = conj(f[3])
    // * f[2] = conj(f[2])
    //
    real_1d_array x = "[1,2,3,4]";
    complex_1d_array f;
    real_1d_array x2;
    fftr1d(x, f);
    printf("%s\n", f.tostring(3).c_str()); // EXPECTED: [10, -2+2i, -2, -2-2i]

    //
    // now we convert [10, -2+2i, -2, -2-2i] back to [1,2,3,4]
    //
    fftr1dinv(f, x2);
    printf("%s\n", x2.tostring(3).c_str()); // EXPECTED: [1,2,3,4]

    //
    // remember that F is self-adjoint? It means that we can pass just half
    // (slightly larger than half) of F to inverse real FFT and still get our result.
    //
    // I.e. instead [10, -2+2i, -2, -2-2i] we pass just [10, -2+2i, -2] and everything works!
    //
    // NOTE: in this case we should explicitly pass array length (which is 4) to ALGLIB;
    // if not, it will automatically use array length to determine FFT size and
    // will erroneously make half-length FFT.
    //
    f = "[10, -2+2i, -2]";
    fftr1dinv(f, 4, x2);
    printf("%s\n", x2.tostring(3).c_str()); // EXPECTED: [1,2,3,4]
    return 0;
}

`fht` subpackage

Functions

fhtr1d
fhtr1dinv

Examples

`fhtr1d` function

/*************************************************************************
1-dimensional Fast Hartley Transform.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    A   -   array[0..N-1] - real function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    A   -   FHT of a input array, array[0..N-1],
            A_out[k] = sum(A_in[j]*(cos(2*pi*j*k/N)+sin(2*pi*j*k/N)), j=0..N-1)


  -- ALGLIB --
     Copyright 04.06.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::fhtr1d(
    real_1d_array& a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`fhtr1dinv` function

/*************************************************************************
1-dimensional inverse FHT.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    A   -   array[0..N-1] - complex array to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    A   -   inverse FHT of a input array, array[0..N-1]


  -- ALGLIB --
     Copyright 29.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::fhtr1dinv(
    real_1d_array& a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`filters` subpackage

Functions

filterema
filterlrma
filtersma

Examples

filters_d_ema		EMA(alpha) filter
filters_d_lrma		LRMA(k) filter
filters_d_sma		SMA(k) filter

`filterema` function

/*************************************************************************
Filters: exponential moving averages.

This filter replaces array by results of EMA(alpha) filter. EMA(alpha) is
defined as filter which replaces X[] by S[]:
    S[0] = X[0]
    S[t] = alpha*X[t] + (1-alpha)*S[t-1]

INPUT PARAMETERS:
    X           -   array[N], array to process. It can be larger than N,
                    in this case only first N points are processed.
    N           -   points count, N>=0
    alpha       -   0<alpha<=1, smoothing parameter.

OUTPUT PARAMETERS:
    X           -   array, whose first N elements were processed
                    with EMA(alpha)

NOTE 1: this function uses efficient in-place  algorithm  which  does not
        allocate temporary arrays.

NOTE 2: this algorithm uses BOTH previous points and  current  one,  i.e.
        new value of X[i] depends on BOTH previous point and X[i] itself.

NOTE 3: technical analytis users quite often work  with  EMA  coefficient
        expressed in DAYS instead of fractions. If you want to  calculate
        EMA(N), where N is a number of days, you can use alpha=2/(N+1).

  -- ALGLIB --
     Copyright 25.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::filterema(
    real_1d_array& x,
    double alpha,
    const xparams _params = alglib::xdefault);
void alglib::filterema(
    real_1d_array& x,
    ae_int_t n,
    double alpha,
    const xparams _params = alglib::xdefault);

Examples: [1]

`filterlrma` function

/*************************************************************************
Filters: linear regression moving averages.

This filter replaces array by results of LRMA(K) filter.

LRMA(K) is defined as filter which, for each data  point,  builds  linear
regression  model  using  K  prevous  points (point itself is included in
these K points) and calculates value of this linear model at the point in
question.

INPUT PARAMETERS:
    X           -   array[N], array to process. It can be larger than N,
                    in this case only first N points are processed.
    N           -   points count, N>=0
    K           -   K>=1 (K can be larger than N ,  such  cases  will  be
                    correctly handled). Window width. K=1 corresponds  to
                    identity transformation (nothing changes).

OUTPUT PARAMETERS:
    X           -   array, whose first N elements were processed with SMA(K)

NOTE 1: this function uses efficient in-place  algorithm  which  does not
        allocate temporary arrays.

NOTE 2: this algorithm makes only one pass through array and uses running
        sum  to speed-up calculation of the averages. Additional measures
        are taken to ensure that running sum on a long sequence  of  zero
        elements will be correctly reset to zero even in the presence  of
        round-off error.

NOTE 3: this  is  unsymmetric version of the algorithm,  which  does  NOT
        averages points after the current one. Only X[i], X[i-1], ... are
        used when calculating new value of X[i]. We should also note that
        this algorithm uses BOTH previous points and  current  one,  i.e.
        new value of X[i] depends on BOTH previous point and X[i] itself.

  -- ALGLIB --
     Copyright 25.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::filterlrma(
    real_1d_array& x,
    ae_int_t k,
    const xparams _params = alglib::xdefault);
void alglib::filterlrma(
    real_1d_array& x,
    ae_int_t n,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

Examples: [1]

`filtersma` function

/*************************************************************************
Filters: simple moving averages (unsymmetric).

This filter replaces array by results of SMA(K) filter. SMA(K) is defined
as filter which averages at most K previous points (previous - not points
AROUND central point) - or less, in case of the first K-1 points.

INPUT PARAMETERS:
    X           -   array[N], array to process. It can be larger than N,
                    in this case only first N points are processed.
    N           -   points count, N>=0
    K           -   K>=1 (K can be larger than N ,  such  cases  will  be
                    correctly handled). Window width. K=1 corresponds  to
                    identity transformation (nothing changes).

OUTPUT PARAMETERS:
    X           -   array, whose first N elements were processed with SMA(K)

NOTE 1: this function uses efficient in-place  algorithm  which  does not
        allocate temporary arrays.

NOTE 2: this algorithm makes only one pass through array and uses running
        sum  to speed-up calculation of the averages. Additional measures
        are taken to ensure that running sum on a long sequence  of  zero
        elements will be correctly reset to zero even in the presence  of
        round-off error.

NOTE 3: this  is  unsymmetric version of the algorithm,  which  does  NOT
        averages points after the current one. Only X[i], X[i-1], ... are
        used when calculating new value of X[i]. We should also note that
        this algorithm uses BOTH previous points and  current  one,  i.e.
        new value of X[i] depends on BOTH previous point and X[i] itself.

  -- ALGLIB --
     Copyright 25.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::filtersma(
    real_1d_array& x,
    ae_int_t k,
    const xparams _params = alglib::xdefault);
void alglib::filtersma(
    real_1d_array& x,
    ae_int_t n,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

Examples: [1]

filters_d_ema example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Here we demonstrate EMA(0.5) filtering for time series.
    //
    real_1d_array x = "[5,6,7,8]";

    //
    // Apply filter.
    // We should get [5, 5.5, 6.25, 7.125] as result
    //
    filterema(x, 0.5);
    printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [5,5.5,6.25,7.125]
    return 0;
}

filters_d_lrma example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Here we demonstrate LRMA(3) filtering for time series.
    //
    real_1d_array x = "[7,8,8,9,12,12]";

    //
    // Apply filter.
    // We should get [7.0000, 8.0000, 8.1667, 8.8333, 11.6667, 12.5000] as result
    //    
    filterlrma(x, 3);
    printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [7.0000,8.0000,8.1667,8.8333,11.6667,12.5000]
    return 0;
}

filters_d_sma example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Here we demonstrate SMA(k) filtering for time series.
    //
    real_1d_array x = "[5,6,7,8]";

    //
    // Apply filter.
    // We should get [5, 5.5, 6.5, 7.5] as result
    //
    filtersma(x, 2);
    printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [5,5.5,6.5,7.5]
    return 0;
}

`fitsphere` subpackage

Functions

fitspherels
fitspheremc
fitspheremi
fitspheremz
fitspherex

Examples

`fitspherels` function

/*************************************************************************
Fits least squares (LS) circle (or NX-dimensional sphere) to data  (a  set
of points in NX-dimensional space).

Least squares circle minimizes sum of squared deviations between distances
from points to the center and  some  "candidate"  radius,  which  is  also
fitted to the data.

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    R       -   radius

  -- ALGLIB --
     Copyright 07.05.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::fitspherels(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nx,
    real_1d_array& cx,
    double& r,
    const xparams _params = alglib::xdefault);

`fitspheremc` function

/*************************************************************************
Fits minimum circumscribed (MC) circle (or NX-dimensional sphere) to  data
(a set of points in NX-dimensional space).

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    RHi     -   radius

NOTE: this function is an easy-to-use wrapper around more powerful "expert"
      function fitspherex().

      This  wrapper  is optimized  for  ease of use and stability - at the
      cost of somewhat lower  performance  (we  have  to  use  very  tight
      stopping criteria for inner optimizer because we want to  make  sure
      that it will converge on any dataset).

      If you are ready to experiment with settings of  "expert"  function,
      you can achieve ~2-4x speedup over standard "bulletproof" settings.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::fitspheremc(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nx,
    real_1d_array& cx,
    double& rhi,
    const xparams _params = alglib::xdefault);

`fitspheremi` function

/*************************************************************************
Fits maximum inscribed circle (or NX-dimensional sphere) to data (a set of
points in NX-dimensional space).

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    RLo     -   radius

NOTE: this function is an easy-to-use wrapper around more powerful "expert"
      function fitspherex().

      This  wrapper  is optimized  for  ease of use and stability - at the
      cost of somewhat lower  performance  (we  have  to  use  very  tight
      stopping criteria for inner optimizer because we want to  make  sure
      that it will converge on any dataset).

      If you are ready to experiment with settings of  "expert"  function,
      you can achieve ~2-4x speedup over standard "bulletproof" settings.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::fitspheremi(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nx,
    real_1d_array& cx,
    double& rlo,
    const xparams _params = alglib::xdefault);

`fitspheremz` function

/*************************************************************************
Fits minimum zone circle (or NX-dimensional sphere)  to  data  (a  set  of
points in NX-dimensional space).

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    RLo     -   radius of inscribed circle
    RHo     -   radius of circumscribed circle

NOTE: this function is an easy-to-use wrapper around more powerful "expert"
      function fitspherex().

      This  wrapper  is optimized  for  ease of use and stability - at the
      cost of somewhat lower  performance  (we  have  to  use  very  tight
      stopping criteria for inner optimizer because we want to  make  sure
      that it will converge on any dataset).

      If you are ready to experiment with settings of  "expert"  function,
      you can achieve ~2-4x speedup over standard "bulletproof" settings.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::fitspheremz(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nx,
    real_1d_array& cx,
    double& rlo,
    double& rhi,
    const xparams _params = alglib::xdefault);

`fitspherex` function

/*************************************************************************
Fitting minimum circumscribed, maximum inscribed or minimum  zone  circles
(or NX-dimensional spheres)  to  data  (a  set of points in NX-dimensional
space).

This  is  expert  function  which  allows  to  tweak  many  parameters  of
underlying nonlinear solver:
* stopping criteria for inner iterations
* number of outer iterations
* penalty coefficient used to handle  nonlinear  constraints  (we  convert
  unconstrained nonsmooth optimization problem ivolving max() and/or min()
  operations to quadratically constrained smooth one).

You may tweak all these parameters or only some  of  them,  leaving  other
ones at their default state - just specify zero  value,  and  solver  will
fill it with appropriate default one.

These comments also include some discussion of  approach  used  to  handle
such unusual fitting problem,  its  stability,  drawbacks  of  alternative
methods, and convergence properties.

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)
    ProblemType-used to encode problem type:
                * 0 for least squares circle
                * 1 for minimum circumscribed circle/sphere fitting (MC)
                * 2 for  maximum inscribed circle/sphere fitting (MI)
                * 3 for minimum zone circle fitting (difference between
                    Rhi and Rlo is minimized), denoted as MZ
    EpsX    -   stopping condition for NLC optimizer:
                * must be non-negative
                * use 0 to choose default value (1.0E-12 is used by default)
                * you may specify larger values, up to 1.0E-6, if you want
                  to   speed-up   solver;   NLC   solver  performs several
                  preconditioned  outer  iterations,   so   final   result
                  typically has precision much better than EpsX.
    AULIts  -   number of outer iterations performed by NLC optimizer:
                * must be non-negative
                * use 0 to choose default value (20 is used by default)
                * you may specify values smaller than 20 if you want to
                  speed up solver; 10 often results in good combination of
                  precision and speed; sometimes you may get good results
                  with just 6 outer iterations.
                Ignored for ProblemType=0.
    Penalty -   penalty coefficient for NLC optimizer:
                * must be non-negative
                * use 0 to choose default value (1.0E6 in current version)
                * it should be really large, 1.0E6...1.0E7 is a good value
                  to start from;
                * generally, default value is good enough
                Ignored for ProblemType=0.

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    RLo     -   radius:
                * for ProblemType=2,3, radius of the inscribed sphere
                * for ProblemType=0 - radius of the least squares sphere
                * for ProblemType=1 - zero
    RHo     -   radius:
                * for ProblemType=1,3, radius of the circumscribed sphere
                * for ProblemType=0 - radius of the least squares sphere
                * for ProblemType=2 - zero

NOTE: ON THE UNIQUENESS OF SOLUTIONS

ALGLIB provides solution to several related circle fitting  problems:   MC
(minimum circumscribed), MI (maximum inscribed)   and   MZ  (minimum zone)
fitting, LS (least squares) fitting.

It  is  important  to  note  that  among these problems only MC and LS are
convex and have unique solution independently from starting point.

As  for MI,  it  may (or  may  not, depending on dataset properties)  have
multiple solutions, and it always  has  one degenerate solution C=infinity
which corresponds to infinitely large radius. Thus, there are no guarantees
that solution to  MI returned by this solver will be the best one (and  no
one can provide you with such guarantee because problem is  NP-hard).  The
only guarantee you have is that this solution is locally optimal, i.e.  it
can not be improved by infinitesimally small tweaks in the parameters.

It  is  also  possible  to "run away" to infinity when  started  from  bad
initial point located outside of point cloud (or when point cloud does not
span entire circumference/surface of the sphere).

Finally,  MZ (minimum zone circle) stands somewhere between MC  and  MI in
stability. It is somewhat regularized by "circumscribed" term of the merit
function; however, solutions to  MZ may be non-unique, and in some unlucky
cases it is also possible to "run away to infinity".


NOTE: ON THE NONLINEARLY CONSTRAINED PROGRAMMING APPROACH

The problem formulation for MC  (minimum circumscribed   circle;  for  the
sake of simplicity we omit MZ and MI here) is:

        [     [         ]2 ]
    min [ max [ XY[i]-C ]  ]
     C  [  i  [         ]  ]

i.e. it is unconstrained nonsmooth optimization problem of finding  "best"
central point, with radius R being unambiguously  determined  from  C.  In
order to move away from non-smoothness we use following reformulation:

        [   ]                  [         ]2
    min [ R ] subject to R>=0, [ XY[i]-C ]  <= R^2
    C,R [   ]                  [         ]

i.e. it becomes smooth quadratically constrained optimization problem with
linear target function. Such problem statement is 100% equivalent  to  the
original nonsmooth one, but much easier  to  approach.  We solve  it  with
MinNLC solver provided by ALGLIB.


NOTE: ON INSTABILITY OF SEQUENTIAL LINEAR PROGRAMMING APPROACJ

ALGLIB  has  nonlinearly  constrained  solver which proved to be stable on
such problems. However, some authors proposed to linearize constraints  in
the vicinity of current approximation (Ci,Ri) and to get next  approximate
solution (Ci+1,Ri+1) as solution to linear programming problem. Obviously,
LP problems are easier than nonlinearly constrained ones.

Indeed,  SLP  approach  to   MC/MI/MZ   resulted   in  ~10-20x increase in
performance (when compared with NLC solver). However, it turned  out  that
in some cases linearized model fails to predict correct direction for next
step and tells us that we converged to solution even when we are still 2-4
digits of precision away from it.

It is important that it is not failure of LP solver - it is failure of the
linear model;  even  when  solved  exactly,  it  fails  to  handle  subtle
nonlinearities which arise near the solution. We validated it by comparing
results returned by ALGLIB linear solver with that of MATLAB.

In our experiments with SLP solver:
* MC failed most often, at both realistic and synthetic datasets
* MI sometimes failed, but sometimes succeeded
* MZ often  succeeded; our guess is that presence of two independent  sets
  of constraints (one set for Rlo and another one for Rhi) and  two  terms
  in the target function (Rlo and Rhi) regularizes task,  so  when  linear
  model fails to handle nonlinearities from Rlo, it uses  Rhi  as  a  hint
  (and vice versa).

Because SLP approach failed to achieve stable results, we do  not  include
it in ALGLIB.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::fitspherex(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nx,
    ae_int_t problemtype,
    double epsx,
    ae_int_t aulits,
    double penalty,
    real_1d_array& cx,
    double& rlo,
    double& rhi,
    const xparams _params = alglib::xdefault);

`fresnel` subpackage

Functions

fresnelintegral

Examples

`fresnelintegral` function

/*************************************************************************
Fresnel integral

Evaluates the Fresnel integrals

          x
          -
         | |
C(x) =   |   cos(pi/2 t**2) dt,
       | |
        -
         0

          x
          -
         | |
S(x) =   |   sin(pi/2 t**2) dt.
       | |
        -
         0


The integrals are evaluated by a power series for x < 1.
For x >= 1 auxiliary functions f(x) and g(x) are employed
such that

C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )



ACCURACY:

 Relative error.

Arithmetic  function   domain     # trials      peak         rms
  IEEE       S(x)      0, 10       10000       2.0e-15     3.2e-16
  IEEE       C(x)      0, 10       10000       1.8e-15     3.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void alglib::fresnelintegral(
    double x,
    double& c,
    double& s,
    const xparams _params = alglib::xdefault);

`gammafunc` subpackage

Functions

gammafunction
lngamma

Examples

`gammafunction` function

/*************************************************************************
Gamma function

Input parameters:
    X   -   argument

Domain:
    0 < X < 171.6
    -170 < X < 0, X is not an integer.

Relative error:
 arithmetic   domain     # trials      peak         rms
    IEEE    -170,-33      20000       2.3e-15     3.3e-16
    IEEE     -33,  33     20000       9.4e-16     2.2e-16
    IEEE      33, 171.6   20000       2.3e-15     3.2e-16

Cephes Math Library Release 2.8:  June, 2000
Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double alglib::gammafunction(
    double x,
    const xparams _params = alglib::xdefault);

`lngamma` function

/*************************************************************************
Natural logarithm of gamma function

Input parameters:
    X       -   argument

Result:
    logarithm of the absolute value of the Gamma(X).

Output parameters:
    SgnGam  -   sign(Gamma(X))

Domain:
    0 < X < 2.55e305
    -2.55e305 < X < 0, X is not an integer.

ACCURACY:
arithmetic      domain        # trials     peak         rms
   IEEE    0, 3                 28000     5.4e-16     1.1e-16
   IEEE    2.718, 2.556e305     40000     3.5e-16     8.3e-17
The error criterion was relative when the function magnitude
was greater than one but absolute when it was less than one.

The following test used the relative error criterion, though
at certain points the relative error could be much higher than
indicated.
   IEEE    -200, -4             10000     4.8e-16     1.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double alglib::lngamma(
    double x,
    double& sgngam,
    const xparams _params = alglib::xdefault);

`gkq` subpackage

Functions

gkqgenerategaussjacobi
gkqgenerategausslegendre
gkqgeneraterec
gkqlegendrecalc
gkqlegendretbl

Examples

`gkqgenerategaussjacobi` function

/*************************************************************************
Returns   Gauss   and   Gauss-Kronrod   nodes/weights   for   Gauss-Jacobi
quadrature on [-1,1] with weight function

    W(x)=Power(1-x,Alpha)*Power(1+x,Beta).

INPUT PARAMETERS:
    N           -   number of Kronrod nodes, must be odd number, >=3.
    Alpha       -   power-law coefficient, Alpha>-1
    Beta        -   power-law coefficient, Beta>-1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -5    no real and positive Gauss-Kronrod formula can
                            be created for such a weight function  with  a
                            given number of nodes.
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes. Alpha or  Beta  are  too  close
                            to -1 to obtain weights/nodes with high enough
                            accuracy, or, may be, N is too large.  Try  to
                            use multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N was passed
                    * +1    OK
                    * +2    OK, but quadrature rule have exterior  nodes,
                            x[0]<-1 or x[n-1]>+1
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gkqgenerategaussjacobi(
    ae_int_t n,
    double alpha,
    double beta,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& wkronrod,
    real_1d_array& wgauss,
    const xparams _params = alglib::xdefault);

`gkqgenerategausslegendre` function

/*************************************************************************
Returns   Gauss   and   Gauss-Kronrod   nodes/weights  for  Gauss-Legendre
quadrature with N points.

GKQLegendreCalc (calculation) or  GKQLegendreTbl  (precomputed  table)  is
used depending on machine precision and number of nodes.

INPUT PARAMETERS:
    N           -   number of Kronrod nodes, must be odd number, >=3.

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error   was   detected   when  calculating
                            weights/nodes.  N  is  too  large   to  obtain
                            weights/nodes  with  high   enough   accuracy.
                            Try  to   use   multiple   precision  version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gkqgenerategausslegendre(
    ae_int_t n,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& wkronrod,
    real_1d_array& wgauss,
    const xparams _params = alglib::xdefault);

`gkqgeneraterec` function

/*************************************************************************
Computation of nodes and weights of a Gauss-Kronrod quadrature formula

The algorithm generates the N-point Gauss-Kronrod quadrature formula  with
weight  function  given  by  coefficients  alpha  and beta of a recurrence
relation which generates a system of orthogonal polynomials:

    P-1(x)   =  0
    P0(x)    =  1
    Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zero moment Mu0

    Mu0 = integral(W(x)dx,a,b)


INPUT PARAMETERS:
    Alpha       -   alpha coefficients, array[0..floor(3*K/2)].
    Beta        -   beta coefficients,  array[0..ceil(3*K/2)].
                    Beta[0] is not used and may be arbitrary.
                    Beta[I]>0.
    Mu0         -   zeroth moment of the weight function.
    N           -   number of nodes of the Gauss-Kronrod quadrature formula,
                    N >= 3,
                    N =  2*K+1.

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -5    no real and positive Gauss-Kronrod formula can
                            be created for such a weight function  with  a
                            given number of nodes.
                    * -4    N is too large, task may be ill  conditioned -
                            x[i]=x[i+1] found.
                    * -3    internal eigenproblem solver hasn't converged
                    * -2    Beta[i]<=0
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).

  -- ALGLIB --
     Copyright 08.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gkqgeneraterec(
    real_1d_array alpha,
    real_1d_array beta,
    double mu0,
    ae_int_t n,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& wkronrod,
    real_1d_array& wgauss,
    const xparams _params = alglib::xdefault);

`gkqlegendrecalc` function

/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes for quadrature with N points.

Reduction to tridiagonal eigenproblem is used.

INPUT PARAMETERS:
    N           -   number of Kronrod nodes, must be odd number, >=3.

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error   was   detected   when  calculating
                            weights/nodes.  N  is  too  large   to  obtain
                            weights/nodes  with  high   enough   accuracy.
                            Try  to   use   multiple   precision  version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).

  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gkqlegendrecalc(
    ae_int_t n,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& wkronrod,
    real_1d_array& wgauss,
    const xparams _params = alglib::xdefault);

`gkqlegendretbl` function

/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes for quadrature with N  points  using
pre-calculated table. Nodes/weights were  computed  with  accuracy  up  to
1.0E-32 (if MPFR version of ALGLIB is used). In standard double  precision
accuracy reduces to something about 2.0E-16 (depending  on your compiler's
handling of long floating point constants).

INPUT PARAMETERS:
    N           -   number of Kronrod nodes.
                    N can be 15, 21, 31, 41, 51, 61.

OUTPUT PARAMETERS:
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gkqlegendretbl(
    ae_int_t n,
    real_1d_array& x,
    real_1d_array& wkronrod,
    real_1d_array& wgauss,
    double& eps,
    const xparams _params = alglib::xdefault);

`gq` subpackage

Functions

gqgenerategausshermite
gqgenerategaussjacobi
gqgenerategausslaguerre
gqgenerategausslegendre
gqgenerategausslobattorec
gqgenerategaussradaurec
gqgeneraterec

Examples

`gqgenerategausshermite` function

/*************************************************************************
Returns  nodes/weights  for  Gauss-Hermite  quadrature on (-inf,+inf) with
weight function W(x)=Exp(-x*x)

INPUT PARAMETERS:
    N           -   number of nodes, >=1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes.  May be, N is too large. Try to
                            use multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N/Alpha was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gqgenerategausshermite(
    ae_int_t n,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& w,
    const xparams _params = alglib::xdefault);

`gqgenerategaussjacobi` function

/*************************************************************************
Returns  nodes/weights  for  Gauss-Jacobi quadrature on [-1,1] with weight
function W(x)=Power(1-x,Alpha)*Power(1+x,Beta).

INPUT PARAMETERS:
    N           -   number of nodes, >=1
    Alpha       -   power-law coefficient, Alpha>-1
    Beta        -   power-law coefficient, Beta>-1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes. Alpha or  Beta  are  too  close
                            to -1 to obtain weights/nodes with high enough
                            accuracy, or, may be, N is too large.  Try  to
                            use multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N/Alpha/Beta was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gqgenerategaussjacobi(
    ae_int_t n,
    double alpha,
    double beta,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& w,
    const xparams _params = alglib::xdefault);

`gqgenerategausslaguerre` function

/*************************************************************************
Returns  nodes/weights  for  Gauss-Laguerre  quadrature  on  [0,+inf) with
weight function W(x)=Power(x,Alpha)*Exp(-x)

INPUT PARAMETERS:
    N           -   number of nodes, >=1
    Alpha       -   power-law coefficient, Alpha>-1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes. Alpha is too  close  to  -1  to
                            obtain weights/nodes with high enough accuracy
                            or, may  be,  N  is  too  large.  Try  to  use
                            multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N/Alpha was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gqgenerategausslaguerre(
    ae_int_t n,
    double alpha,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& w,
    const xparams _params = alglib::xdefault);

`gqgenerategausslegendre` function

/*************************************************************************
Returns nodes/weights for Gauss-Legendre quadrature on [-1,1] with N
nodes.

INPUT PARAMETERS:
    N           -   number of nodes, >=1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error   was   detected   when  calculating
                            weights/nodes.  N  is  too  large   to  obtain
                            weights/nodes  with  high   enough   accuracy.
                            Try  to   use   multiple   precision  version.
                    * -3    internal eigenproblem solver hasn't  converged
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gqgenerategausslegendre(
    ae_int_t n,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& w,
    const xparams _params = alglib::xdefault);

`gqgenerategausslobattorec` function

/*************************************************************************
Computation of nodes and weights for a Gauss-Lobatto quadrature formula

The algorithm generates the N-point Gauss-Lobatto quadrature formula  with
weight function given by coefficients alpha and beta of a recurrence which
generates a system of orthogonal polynomials.

P-1(x)   =  0
P0(x)    =  1
Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zeroth moment Mu0

Mu0 = integral(W(x)dx,a,b)

INPUT PARAMETERS:
    Alpha   -   array[0..N-2], alpha coefficients
    Beta    -   array[0..N-2], beta coefficients.
                Zero-indexed element is not used, may be arbitrary.
                Beta[I]>0
    Mu0     -   zeroth moment of the weighting function.
    A       -   left boundary of the integration interval.
    B       -   right boundary of the integration interval.
    N       -   number of nodes of the quadrature formula, N>=3
                (including the left and right boundary nodes).

OUTPUT PARAMETERS:
    Info    -   error code:
                * -3    internal eigenproblem solver hasn't converged
                * -2    Beta[i]<=0
                * -1    incorrect N was passed
                *  1    OK
    X       -   array[0..N-1] - array of quadrature nodes,
                in ascending order.
    W       -   array[0..N-1] - array of quadrature weights.

  -- ALGLIB --
     Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gqgenerategausslobattorec(
    real_1d_array alpha,
    real_1d_array beta,
    double mu0,
    double a,
    double b,
    ae_int_t n,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& w,
    const xparams _params = alglib::xdefault);

`gqgenerategaussradaurec` function

/*************************************************************************
Computation of nodes and weights for a Gauss-Radau quadrature formula

The algorithm generates the N-point Gauss-Radau  quadrature  formula  with
weight function given by the coefficients alpha and  beta  of a recurrence
which generates a system of orthogonal polynomials.

P-1(x)   =  0
P0(x)    =  1
Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zeroth moment Mu0

Mu0 = integral(W(x)dx,a,b)

INPUT PARAMETERS:
    Alpha   -   array[0..N-2], alpha coefficients.
    Beta    -   array[0..N-1], beta coefficients
                Zero-indexed element is not used.
                Beta[I]>0
    Mu0     -   zeroth moment of the weighting function.
    A       -   left boundary of the integration interval.
    N       -   number of nodes of the quadrature formula, N>=2
                (including the left boundary node).

OUTPUT PARAMETERS:
    Info    -   error code:
                * -3    internal eigenproblem solver hasn't converged
                * -2    Beta[i]<=0
                * -1    incorrect N was passed
                *  1    OK
    X       -   array[0..N-1] - array of quadrature nodes,
                in ascending order.
    W       -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gqgenerategaussradaurec(
    real_1d_array alpha,
    real_1d_array beta,
    double mu0,
    double a,
    ae_int_t n,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& w,
    const xparams _params = alglib::xdefault);

`gqgeneraterec` function

/*************************************************************************
Computation of nodes and weights for a Gauss quadrature formula

The algorithm generates the N-point Gauss quadrature formula  with  weight
function given by coefficients alpha and beta  of  a  recurrence  relation
which generates a system of orthogonal polynomials:

P-1(x)   =  0
P0(x)    =  1
Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zeroth moment Mu0

Mu0 = integral(W(x)dx,a,b)

INPUT PARAMETERS:
    Alpha   -   array[0..N-1], alpha coefficients
    Beta    -   array[0..N-1], beta coefficients
                Zero-indexed element is not used and may be arbitrary.
                Beta[I]>0.
    Mu0     -   zeroth moment of the weight function.
    N       -   number of nodes of the quadrature formula, N>=1

OUTPUT PARAMETERS:
    Info    -   error code:
                * -3    internal eigenproblem solver hasn't converged
                * -2    Beta[i]<=0
                * -1    incorrect N was passed
                *  1    OK
    X       -   array[0..N-1] - array of quadrature nodes,
                in ascending order.
    W       -   array[0..N-1] - array of quadrature weights.

  -- ALGLIB --
     Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
void alglib::gqgeneraterec(
    real_1d_array alpha,
    real_1d_array beta,
    double mu0,
    ae_int_t n,
    ae_int_t& info,
    real_1d_array& x,
    real_1d_array& w,
    const xparams _params = alglib::xdefault);

`hermite` subpackage

Functions

hermitecalculate
hermitecoefficients
hermitesum

Examples

`hermitecalculate` function

/*************************************************************************
Calculation of the value of the Hermite polynomial.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Hermite polynomial Hn at x
*************************************************************************/
double alglib::hermitecalculate(
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`hermitecoefficients` function

/*************************************************************************
Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void alglib::hermitecoefficients(
    ae_int_t n,
    real_1d_array& c,
    const xparams _params = alglib::xdefault);

`hermitesum` function

/*************************************************************************
Summation of Hermite polynomials using Clenshaw's recurrence formula.

This routine calculates
    c[0]*H0(x) + c[1]*H1(x) + ... + c[N]*HN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Hermite polynomial at x
*************************************************************************/
double alglib::hermitesum(
    real_1d_array c,
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`hqrnd` subpackage

Classes

hqrndstate

Functions

hqrndcontinuous
hqrnddiscrete
hqrndexponential
hqrndnormal
hqrndnormal2
hqrndrandomize
hqrndseed
hqrnduniformi
hqrnduniformr
hqrndunit2

Examples

`hqrndstate` class

/*************************************************************************
Portable high quality random number generator state.
Initialized with HQRNDRandomize() or HQRNDSeed().

Fields:
    S1, S2      -   seed values
    V           -   precomputed value
    MagicV      -   'magic' value used to determine whether State structure
                    was correctly initialized.
*************************************************************************/
class hqrndstate
{
};

`hqrndcontinuous` function

/*************************************************************************
This function generates random number from continuous  distribution  given
by finite sample X.

INPUT PARAMETERS
    State   -   high quality random number generator, must be
                initialized with HQRNDRandomize() or HQRNDSeed().
        X   -   finite sample, array[N] (can be larger, in this  case only
                leading N elements are used). THIS ARRAY MUST BE SORTED BY
                ASCENDING.
        N   -   number of elements to use, N>=1

RESULT
    this function returns random number from continuous distribution which
    tries to approximate X as mush as possible. min(X)<=Result<=max(X).

  -- ALGLIB --
     Copyright 08.11.2011 by Bochkanov Sergey
*************************************************************************/
double alglib::hqrndcontinuous(
    hqrndstate state,
    real_1d_array x,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`hqrnddiscrete` function

/*************************************************************************
This function generates  random number from discrete distribution given by
finite sample X.

INPUT PARAMETERS
    State   -   high quality random number generator, must be
                initialized with HQRNDRandomize() or HQRNDSeed().
        X   -   finite sample
        N   -   number of elements to use, N>=1

RESULT
    this function returns one of the X[i] for random i=0..N-1

  -- ALGLIB --
     Copyright 08.11.2011 by Bochkanov Sergey
*************************************************************************/
double alglib::hqrnddiscrete(
    hqrndstate state,
    real_1d_array x,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`hqrndexponential` function

/*************************************************************************
Random number generator: exponential distribution

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 11.08.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::hqrndexponential(
    hqrndstate state,
    double lambdav,
    const xparams _params = alglib::xdefault);

`hqrndnormal` function

/*************************************************************************
Random number generator: normal numbers

This function generates one random number from normal distribution.
Its performance is equal to that of HQRNDNormal2()

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::hqrndnormal(
    hqrndstate state,
    const xparams _params = alglib::xdefault);

`hqrndnormal2` function

/*************************************************************************
Random number generator: normal numbers

This function generates two independent random numbers from normal
distribution. Its performance is equal to that of HQRNDNormal()

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::hqrndnormal2(
    hqrndstate state,
    double& x1,
    double& x2,
    const xparams _params = alglib::xdefault);

`hqrndrandomize` function

/*************************************************************************
HQRNDState  initialization  with  random  values  which come from standard
RNG.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::hqrndrandomize(
    hqrndstate& state,
    const xparams _params = alglib::xdefault);

`hqrndseed` function

/*************************************************************************
HQRNDState initialization with seed values

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::hqrndseed(
    ae_int_t s1,
    ae_int_t s2,
    hqrndstate& state,
    const xparams _params = alglib::xdefault);

`hqrnduniformi` function

/*************************************************************************
This function generates random integer number in [0, N)

1. State structure must be initialized with HQRNDRandomize() or HQRNDSeed()
2. N can be any positive number except for very large numbers:
   * close to 2^31 on 32-bit systems
   * close to 2^62 on 64-bit systems
   An exception will be generated if N is too large.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::hqrnduniformi(
    hqrndstate state,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`hqrnduniformr` function

/*************************************************************************
This function generates random real number in (0,1),
not including interval boundaries

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::hqrnduniformr(
    hqrndstate state,
    const xparams _params = alglib::xdefault);

`hqrndunit2` function

/*************************************************************************
Random number generator: random X and Y such that X^2+Y^2=1

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::hqrndunit2(
    hqrndstate state,
    double& x,
    double& y,
    const xparams _params = alglib::xdefault);

`ibetaf` subpackage

Functions

incompletebeta
invincompletebeta

Examples

`incompletebeta` function

/*************************************************************************
Incomplete beta integral

Returns incomplete beta integral of the arguments, evaluated
from zero to x.  The function is defined as

                 x
    -            -
   | (a+b)      | |  a-1     b-1
 -----------    |   t   (1-t)   dt.
  -     -     | |
 | (a) | (b)   -
                0

The domain of definition is 0 <= x <= 1.  In this
implementation a and b are restricted to positive values.
The integral from x to 1 may be obtained by the symmetry
relation

   1 - incbet( a, b, x )  =  incbet( b, a, 1-x ).

The integral is evaluated by a continued fraction expansion
or, when b*x is small, by a power series.

ACCURACY:

Tested at uniformly distributed random points (a,b,x) with a and b
in "domain" and x between 0 and 1.
                                       Relative error
arithmetic   domain     # trials      peak         rms
   IEEE      0,5         10000       6.9e-15     4.5e-16
   IEEE      0,85       250000       2.2e-13     1.7e-14
   IEEE      0,1000      30000       5.3e-12     6.3e-13
   IEEE      0,10000    250000       9.3e-11     7.1e-12
   IEEE      0,100000    10000       8.7e-10     4.8e-11
Outputs smaller than the IEEE gradual underflow threshold
were excluded from these statistics.

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::incompletebeta(
    double a,
    double b,
    double x,
    const xparams _params = alglib::xdefault);

`invincompletebeta` function

/*************************************************************************
Inverse of imcomplete beta integral

Given y, the function finds x such that

 incbet( a, b, x ) = y .

The routine performs interval halving or Newton iterations to find the
root of incbet(a,b,x) - y = 0.


ACCURACY:

                     Relative error:
               x     a,b
arithmetic   domain  domain  # trials    peak       rms
   IEEE      0,1    .5,10000   50000    5.8e-12   1.3e-13
   IEEE      0,1   .25,100    100000    1.8e-13   3.9e-15
   IEEE      0,1     0,5       50000    1.1e-12   5.5e-15
With a and b constrained to half-integer or integer values:
   IEEE      0,1    .5,10000   50000    5.8e-12   1.1e-13
   IEEE      0,1    .5,100    100000    1.7e-14   7.9e-16
With a = .5, b constrained to half-integer or integer values:
   IEEE      0,1    .5,10000   10000    8.3e-11   1.0e-11

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1996, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::invincompletebeta(
    double a,
    double b,
    double y,
    const xparams _params = alglib::xdefault);

`idwint` subpackage

Classes

idwinterpolant

Functions

idwbuildmodifiedshepard
idwbuildmodifiedshepardr
idwbuildnoisy
idwcalc

Examples

`idwinterpolant` class

/*************************************************************************
IDW interpolant.
*************************************************************************/
class idwinterpolant
{
};

`idwbuildmodifiedshepard` function

/*************************************************************************
IDW interpolant using modified Shepard method for uniform point
distributions.

INPUT PARAMETERS:
    XY  -   X and Y values, array[0..N-1,0..NX].
            First NX columns contain X-values, last column contain
            Y-values.
    N   -   number of nodes, N>0.
    NX  -   space dimension, NX>=1.
    D   -   nodal function type, either:
            * 0     constant  model.  Just  for  demonstration only, worst
                    model ever.
            * 1     linear model, least squares fitting. Simpe  model  for
                    datasets too small for quadratic models
            * 2     quadratic  model,  least  squares  fitting. Best model
                    available (if your dataset is large enough).
            * -1    "fast"  linear  model,  use  with  caution!!!   It  is
                    significantly  faster than linear/quadratic and better
                    than constant model. But it is less robust (especially
                    in the presence of noise).
    NQ  -   number of points used to calculate  nodal  functions  (ignored
            for constant models). NQ should be LARGER than:
            * max(1.5*(1+NX),2^NX+1) for linear model,
            * max(3/4*(NX+2)*(NX+1),2^NX+1) for quadratic model.
            Values less than this threshold will be silently increased.
    NW  -   number of points used to calculate weights and to interpolate.
            Required: >=2^NX+1, values less than this  threshold  will  be
            silently increased.
            Recommended value: about 2*NQ

OUTPUT PARAMETERS:
    Z   -   IDW interpolant.

NOTES:
  * best results are obtained with quadratic models, worst - with constant
    models
  * when N is large, NQ and NW must be significantly smaller than  N  both
    to obtain optimal performance and to obtain optimal accuracy. In 2  or
    3-dimensional tasks NQ=15 and NW=25 are good values to start with.
  * NQ  and  NW  may  be  greater  than  N.  In  such  cases  they will be
    automatically decreased.
  * this subroutine is always succeeds (as long as correct parameters  are
    passed).
  * see  'Multivariate  Interpolation  of Large Sets of Scattered Data' by
    Robert J. Renka for more information on this algorithm.
  * this subroutine assumes that point distribution is uniform at the small
    scales.  If  it  isn't  -  for  example,  points are concentrated along
    "lines", but "lines" distribution is uniform at the larger scale - then
    you should use IDWBuildModifiedShepardR()


  -- ALGLIB PROJECT --
     Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::idwbuildmodifiedshepard(
    real_2d_array xy,
    ae_int_t n,
    ae_int_t nx,
    ae_int_t d,
    ae_int_t nq,
    ae_int_t nw,
    idwinterpolant& z,
    const xparams _params = alglib::xdefault);

`idwbuildmodifiedshepardr` function

/*************************************************************************
IDW interpolant using modified Shepard method for non-uniform datasets.

This type of model uses  constant  nodal  functions and interpolates using
all nodes which are closer than user-specified radius R. It  may  be  used
when points distribution is non-uniform at the small scale, but it  is  at
the distances as large as R.

INPUT PARAMETERS:
    XY  -   X and Y values, array[0..N-1,0..NX].
            First NX columns contain X-values, last column contain
            Y-values.
    N   -   number of nodes, N>0.
    NX  -   space dimension, NX>=1.
    R   -   radius, R>0

OUTPUT PARAMETERS:
    Z   -   IDW interpolant.

NOTES:
* if there is less than IDWKMin points within  R-ball,  algorithm  selects
  IDWKMin closest ones, so that continuity properties of  interpolant  are
  preserved even far from points.

  -- ALGLIB PROJECT --
     Copyright 11.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::idwbuildmodifiedshepardr(
    real_2d_array xy,
    ae_int_t n,
    ae_int_t nx,
    double r,
    idwinterpolant& z,
    const xparams _params = alglib::xdefault);

`idwbuildnoisy` function

/*************************************************************************
IDW model for noisy data.

This subroutine may be used to handle noisy data, i.e. data with noise  in
OUTPUT values.  It differs from IDWBuildModifiedShepard() in the following
aspects:
* nodal functions are not constrained to pass through  nodes:  Qi(xi)<>yi,
  i.e. we have fitting  instead  of  interpolation.
* weights which are used during least  squares fitting stage are all equal
  to 1.0 (independently of distance)
* "fast"-linear or constant nodal functions are not supported (either  not
  robust enough or too rigid)

This problem require far more complex tuning than interpolation  problems.
Below you can find some recommendations regarding this problem:
* focus on tuning NQ; it controls noise reduction. As for NW, you can just
  make it equal to 2*NQ.
* you can use cross-validation to determine optimal NQ.
* optimal NQ is a result of complex tradeoff  between  noise  level  (more
  noise = larger NQ required) and underlying  function  complexity  (given
  fixed N, larger NQ means smoothing of compex features in the data).  For
  example, NQ=N will reduce noise to the minimum level possible,  but  you
  will end up with just constant/linear/quadratic (depending on  D)  least
  squares model for the whole dataset.

INPUT PARAMETERS:
    XY  -   X and Y values, array[0..N-1,0..NX].
            First NX columns contain X-values, last column contain
            Y-values.
    N   -   number of nodes, N>0.
    NX  -   space dimension, NX>=1.
    D   -   nodal function degree, either:
            * 1     linear model, least squares fitting. Simpe  model  for
                    datasets too small for quadratic models (or  for  very
                    noisy problems).
            * 2     quadratic  model,  least  squares  fitting. Best model
                    available (if your dataset is large enough).
    NQ  -   number of points used to calculate nodal functions.  NQ should
            be  significantly   larger   than  1.5  times  the  number  of
            coefficients in a nodal function to overcome effects of noise:
            * larger than 1.5*(1+NX) for linear model,
            * larger than 3/4*(NX+2)*(NX+1) for quadratic model.
            Values less than this threshold will be silently increased.
    NW  -   number of points used to calculate weights and to interpolate.
            Required: >=2^NX+1, values less than this  threshold  will  be
            silently increased.
            Recommended value: about 2*NQ or larger

OUTPUT PARAMETERS:
    Z   -   IDW interpolant.

NOTES:
  * best results are obtained with quadratic models, linear models are not
    recommended to use unless you are pretty sure that it is what you want
  * this subroutine is always succeeds (as long as correct parameters  are
    passed).
  * see  'Multivariate  Interpolation  of Large Sets of Scattered Data' by
    Robert J. Renka for more information on this algorithm.


  -- ALGLIB PROJECT --
     Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::idwbuildnoisy(
    real_2d_array xy,
    ae_int_t n,
    ae_int_t nx,
    ae_int_t d,
    ae_int_t nq,
    ae_int_t nw,
    idwinterpolant& z,
    const xparams _params = alglib::xdefault);

`idwcalc` function

/*************************************************************************
IDW interpolation

INPUT PARAMETERS:
    Z   -   IDW interpolant built with one of model building
            subroutines.
    X   -   array[0..NX-1], interpolation point

Result:
    IDW interpolant Z(X)

  -- ALGLIB --
     Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
double alglib::idwcalc(
    idwinterpolant z,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

`igammaf` subpackage

Functions

incompletegamma
incompletegammac
invincompletegammac

Examples

`incompletegamma` function

/*************************************************************************
Incomplete gamma integral

The function is defined by

                          x
                           -
                  1       | |  -t  a-1
 igam(a,x)  =   -----     |   e   t   dt.
                 -      | |
                | (a)    -
                          0


In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30       200000       3.6e-14     2.9e-15
   IEEE      0,100      300000       9.9e-14     1.5e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::incompletegamma(
    double a,
    double x,
    const xparams _params = alglib::xdefault);

`incompletegammac` function

/*************************************************************************
Complemented incomplete gamma integral

The function is defined by


 igamc(a,x)   =   1 - igam(a,x)

                           inf.
                             -
                    1       | |  -t  a-1
              =   -----     |   e   t   dt.
                   -      | |
                  | (a)    -
                            x


In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.

ACCURACY:

Tested at random a, x.
               a         x                      Relative error:
arithmetic   domain   domain     # trials      peak         rms
   IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
   IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::incompletegammac(
    double a,
    double x,
    const xparams _params = alglib::xdefault);

`invincompletegammac` function

/*************************************************************************
Inverse of complemented imcomplete gamma integral

Given p, the function finds x such that

 igamc( a, x ) = p.

Starting with the approximate value

        3
 x = a t

 where

 t = 1 - d - ndtri(p) sqrt(d)

and

 d = 1/9a,

the routine performs up to 10 Newton iterations to find the
root of igamc(a,x) - p = 0.

ACCURACY:

Tested at random a, p in the intervals indicated.

               a        p                      Relative error:
arithmetic   domain   domain     # trials      peak         rms
   IEEE     0.5,100   0,0.5       100000       1.0e-14     1.7e-15
   IEEE     0.01,0.5  0,0.5       100000       9.0e-14     3.4e-15
   IEEE    0.5,10000  0,0.5        20000       2.3e-13     3.8e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::invincompletegammac(
    double a,
    double y0,
    const xparams _params = alglib::xdefault);

`intcomp` subpackage

Functions

nsfitspheremcc
nsfitspheremic
nsfitspheremzc
nsfitspherex

Examples

`nsfitspheremcc` function

/*************************************************************************
This function is left for backward compatibility.
Use fitspheremc() instead.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::nsfitspheremcc(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nx,
    real_1d_array& cx,
    double& rhi,
    const xparams _params = alglib::xdefault);

`nsfitspheremic` function

/*************************************************************************
This function is left for backward compatibility.
Use fitspheremi() instead.

  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::nsfitspheremic(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nx,
    real_1d_array& cx,
    double& rlo,
    const xparams _params = alglib::xdefault);

`nsfitspheremzc` function

/*************************************************************************
This function is left for backward compatibility.
Use fitspheremz() instead.

  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::nsfitspheremzc(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nx,
    real_1d_array& cx,
    double& rlo,
    double& rhi,
    const xparams _params = alglib::xdefault);

`nsfitspherex` function

/*************************************************************************
This function is left for backward compatibility.
Use fitspherex() instead.

  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::nsfitspherex(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nx,
    ae_int_t problemtype,
    double epsx,
    ae_int_t aulits,
    double penalty,
    real_1d_array& cx,
    double& rlo,
    double& rhi,
    const xparams _params = alglib::xdefault);

`inverseupdate` subpackage

Functions

rmatrixinvupdatecolumn
rmatrixinvupdaterow
rmatrixinvupdatesimple
rmatrixinvupdateuv

Examples

`rmatrixinvupdatecolumn` function

/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula

The algorithm updates matrix A^-1 when adding a vector to a column
of matrix A.

Input parameters:
    InvA        -   inverse of matrix A.
                    Array whose indexes range within [0..N-1, 0..N-1].
    N           -   size of matrix A.
    UpdColumn   -   the column of A whose vector U was added.
                    0 <= UpdColumn <= N-1
    U           -   the vector to be added to a column.
                    Array whose index ranges within [0..N-1].

Output parameters:
    InvA        -   inverse of modified matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixinvupdatecolumn(
    real_2d_array& inva,
    ae_int_t n,
    ae_int_t updcolumn,
    real_1d_array u,
    const xparams _params = alglib::xdefault);

`rmatrixinvupdaterow` function

/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula

The algorithm updates matrix A^-1 when adding a vector to a row
of matrix A.

Input parameters:
    InvA    -   inverse of matrix A.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    UpdRow  -   the row of A whose vector V was added.
                0 <= Row <= N-1
    V       -   the vector to be added to a row.
                Array whose index ranges within [0..N-1].

Output parameters:
    InvA    -   inverse of modified matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixinvupdaterow(
    real_2d_array& inva,
    ae_int_t n,
    ae_int_t updrow,
    real_1d_array v,
    const xparams _params = alglib::xdefault);

`rmatrixinvupdatesimple` function

/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula

The algorithm updates matrix A^-1 when adding a number to an element
of matrix A.

Input parameters:
    InvA    -   inverse of matrix A.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    UpdRow  -   row where the element to be updated is stored.
    UpdColumn - column where the element to be updated is stored.
    UpdVal  -   a number to be added to the element.


Output parameters:
    InvA    -   inverse of modified matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixinvupdatesimple(
    real_2d_array& inva,
    ae_int_t n,
    ae_int_t updrow,
    ae_int_t updcolumn,
    double updval,
    const xparams _params = alglib::xdefault);

`rmatrixinvupdateuv` function

/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula

The algorithm computes the inverse of matrix A+u*v' by using the given matrix
A^-1 and the vectors u and v.

Input parameters:
    InvA    -   inverse of matrix A.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    U       -   the vector modifying the matrix.
                Array whose index ranges within [0..N-1].
    V       -   the vector modifying the matrix.
                Array whose index ranges within [0..N-1].

Output parameters:
    InvA - inverse of matrix A + u*v'.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixinvupdateuv(
    real_2d_array& inva,
    ae_int_t n,
    real_1d_array u,
    real_1d_array v,
    const xparams _params = alglib::xdefault);

`jacobianelliptic` subpackage

Functions

jacobianellipticfunctions

Examples

`jacobianellipticfunctions` function

/*************************************************************************
Jacobian Elliptic Functions

Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
and dn(u|m) of parameter m between 0 and 1, and real
argument u.

These functions are periodic, with quarter-period on the
real axis equal to the complete elliptic integral
ellpk(1.0-m).

Relation to incomplete elliptic integral:
If u = ellik(phi,m), then sn(u|m) = sin(phi),
and cn(u|m) = cos(phi).  Phi is called the amplitude of u.

Computation is by means of the arithmetic-geometric mean
algorithm, except when m is within 1e-9 of 0 or 1.  In the
latter case with m close to 1, the approximation applies
only for phi < pi/2.

ACCURACY:

Tested at random points with u between 0 and 10, m between
0 and 1.

           Absolute error (* = relative error):
arithmetic   function   # trials      peak         rms
   IEEE      phi         10000       9.2e-16*    1.4e-16*
   IEEE      sn          50000       4.1e-15     4.6e-16
   IEEE      cn          40000       3.6e-15     4.4e-16
   IEEE      dn          10000       1.3e-12     1.8e-14

 Peak error observed in consistency check using addition
theorem for sn(u+v) was 4e-16 (absolute).  Also tested by
the above relation to the incomplete elliptic integral.
Accuracy deteriorates when u is large.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void alglib::jacobianellipticfunctions(
    double u,
    double m,
    double& sn,
    double& cn,
    double& dn,
    double& ph,
    const xparams _params = alglib::xdefault);

`jarquebera` subpackage

Functions

jarqueberatest

Examples

`jarqueberatest` function

/*************************************************************************
Jarque-Bera test

This test checks hypotheses about the fact that a  given  sample  X  is  a
sample of normal random variable.

Requirements:
    * the number of elements in the sample is not less than 5.

Input parameters:
    X   -   sample. Array whose index goes from 0 to N-1.
    N   -   size of the sample. N>=5

Output parameters:
    P           -   p-value for the test

Accuracy of the approximation used (5<=N<=1951):

p-value  	    relative error (5<=N<=1951)
[1, 0.1]            < 1%
[0.1, 0.01]         < 2%
[0.01, 0.001]       < 6%
[0.001, 0]          wasn't measured

For N>1951 accuracy wasn't measured but it shouldn't be sharply  different
from table values.

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::jarqueberatest(
    real_1d_array x,
    ae_int_t n,
    double& p,
    const xparams _params = alglib::xdefault);

`laguerre` subpackage

Functions

laguerrecalculate
laguerrecoefficients
laguerresum

Examples

`laguerrecalculate` function

/*************************************************************************
Calculation of the value of the Laguerre polynomial.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Laguerre polynomial Ln at x
*************************************************************************/
double alglib::laguerrecalculate(
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`laguerrecoefficients` function

/*************************************************************************
Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void alglib::laguerrecoefficients(
    ae_int_t n,
    real_1d_array& c,
    const xparams _params = alglib::xdefault);

`laguerresum` function

/*************************************************************************
Summation of Laguerre polynomials using Clenshaw's recurrence formula.

This routine calculates c[0]*L0(x) + c[1]*L1(x) + ... + c[N]*LN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Laguerre polynomial at x
*************************************************************************/
double alglib::laguerresum(
    real_1d_array c,
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`lda` subpackage

Functions

fisherlda
fisherldan

Examples

`fisherlda` function

/*************************************************************************
Multiclass Fisher LDA

Subroutine finds coefficients of linear combination which optimally separates
training set on classes.

COMMERCIAL EDITION OF ALGLIB:

  ! Commercial version of ALGLIB includes two important  improvements   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multithreading support
  !
  ! Intel MKL gives approximately constant  (with  respect  to  number  of
  ! worker threads) acceleration factor which depends on CPU  being  used,
  ! problem  size  and  "baseline"  ALGLIB  edition  which  is  used   for
  ! comparison. Best results are achieved  for  high-dimensional  problems
  ! (NVars is at least 256).
  !
  ! Multithreading is used to  accelerate  initial  phase  of  LDA,  which
  ! includes calculation of products of large matrices.  Again,  for  best
  ! efficiency problem must be high-dimensional.
  !
  ! Generally, commercial ALGLIB is several times faster than  open-source
  ! generic C edition, and many times faster than open-source C# edition.
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    XY          -   training set, array[0..NPoints-1,0..NVars].
                    First NVars columns store values of independent
                    variables, next column stores number of class (from 0
                    to NClasses-1) which dataset element belongs to. Fractional
                    values are rounded to nearest integer.
    NPoints     -   training set size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1
    NClasses    -   number of classes, NClasses>=2


OUTPUT PARAMETERS:
    Info        -   return code:
                    * -4, if internal EVD subroutine hasn't converged
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed (NPoints<0,
                          NVars<1, NClasses<2)
                    *  1, if task has been solved
                    *  2, if there was a multicollinearity in training set,
                          but task has been solved.
    W           -   linear combination coefficients, array[0..NVars-1]

  -- ALGLIB --
     Copyright 31.05.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::fisherlda(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t nclasses,
    ae_int_t& info,
    real_1d_array& w,
    const xparams _params = alglib::xdefault);

`fisherldan` function

/*************************************************************************
N-dimensional multiclass Fisher LDA

Subroutine finds coefficients of linear combinations which optimally separates
training set on classes. It returns N-dimensional basis whose vector are sorted
by quality of training set separation (in descending order).

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    XY          -   training set, array[0..NPoints-1,0..NVars].
                    First NVars columns store values of independent
                    variables, next column stores number of class (from 0
                    to NClasses-1) which dataset element belongs to. Fractional
                    values are rounded to nearest integer.
    NPoints     -   training set size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1
    NClasses    -   number of classes, NClasses>=2


OUTPUT PARAMETERS:
    Info        -   return code:
                    * -4, if internal EVD subroutine hasn't converged
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed (NPoints<0,
                          NVars<1, NClasses<2)
                    *  1, if task has been solved
                    *  2, if there was a multicollinearity in training set,
                          but task has been solved.
    W           -   basis, array[0..NVars-1,0..NVars-1]
                    columns of matrix stores basis vectors, sorted by
                    quality of training set separation (in descending order)

  -- ALGLIB --
     Copyright 31.05.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::fisherldan(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t nclasses,
    ae_int_t& info,
    real_2d_array& w,
    const xparams _params = alglib::xdefault);

`legendre` subpackage

Functions

legendrecalculate
legendrecoefficients
legendresum

Examples

`legendrecalculate` function

/*************************************************************************
Calculation of the value of the Legendre polynomial Pn.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Legendre polynomial Pn at x
*************************************************************************/
double alglib::legendrecalculate(
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`legendrecoefficients` function

/*************************************************************************
Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void alglib::legendrecoefficients(
    ae_int_t n,
    real_1d_array& c,
    const xparams _params = alglib::xdefault);

`legendresum` function

/*************************************************************************
Summation of Legendre polynomials using Clenshaw's recurrence formula.

This routine calculates
    c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Legendre polynomial at x
*************************************************************************/
double alglib::legendresum(
    real_1d_array c,
    ae_int_t n,
    double x,
    const xparams _params = alglib::xdefault);

`lincg` subpackage

Classes

lincgreport
lincgstate

Functions

lincgcreate
lincgresults
lincgsetcond
lincgsetprecdiag
lincgsetprecunit
lincgsetrestartfreq
lincgsetrupdatefreq
lincgsetstartingpoint
lincgsetxrep
lincgsolvesparse

Examples

lincg_d_1

Solution of sparse linear systems with CG

`lincgreport` class

/*************************************************************************

*************************************************************************/
class lincgreport
{
    ae_int_t             iterationscount;
    ae_int_t             nmv;
    ae_int_t             terminationtype;
    double               r2;
};

`lincgstate` class

/*************************************************************************
This object stores state of the linear CG method.

You should use ALGLIB functions to work with this object.
Never try to access its fields directly!
*************************************************************************/
class lincgstate
{
};

`lincgcreate` function

/*************************************************************************
This function initializes linear CG Solver. This solver is used  to  solve
symmetric positive definite problems. If you want  to  solve  nonsymmetric
(or non-positive definite) problem you may use LinLSQR solver provided  by
ALGLIB.

USAGE:
1. User initializes algorithm state with LinCGCreate() call
2. User tunes solver parameters with  LinCGSetCond() and other functions
3. Optionally, user sets starting point with LinCGSetStartingPoint()
4. User  calls LinCGSolveSparse() function which takes algorithm state and
   SparseMatrix object.
5. User calls LinCGResults() to get solution
6. Optionally, user may call LinCGSolveSparse()  again  to  solve  another
   problem  with different matrix and/or right part without reinitializing
   LinCGState structure.

INPUT PARAMETERS:
    N       -   problem dimension, N>0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgcreate(
    ae_int_t n,
    lincgstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lincgresults` function

/*************************************************************************
CG-solver: results.

This function must be called after LinCGSolve

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[N], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    * -5    input matrix is either not positive definite,
                            too large or too small
                    * -4    overflow/underflow during solution
                            (ill conditioned problem)
                    *  1    ||residual||<=EpsF*||b||
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            best point found is returned
                * Rep.IterationsCount contains iterations count
                * NMV countains number of matrix-vector calculations

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgresults(
    lincgstate state,
    real_1d_array& x,
    lincgreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lincgsetcond` function

/*************************************************************************
This function sets stopping criteria.

INPUT PARAMETERS:
    EpsF    -   algorithm will be stopped if norm of residual is less than
                EpsF*||b||.
    MaxIts  -   algorithm will be stopped if number of iterations is  more
                than MaxIts.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
If  both  EpsF  and  MaxIts  are  zero then small EpsF will be set to small
value.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgsetcond(
    lincgstate state,
    double epsf,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

`lincgsetprecdiag` function

/*************************************************************************
This  function  changes  preconditioning  settings  of  LinCGSolveSparse()
function.  LinCGSolveSparse() will use diagonal of the  system  matrix  as
preconditioner. This preconditioning mode is active by default.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgsetprecdiag(
    lincgstate state,
    const xparams _params = alglib::xdefault);

`lincgsetprecunit` function

/*************************************************************************
This  function  changes  preconditioning  settings  of  LinCGSolveSparse()
function. By default, SolveSparse() uses diagonal preconditioner,  but  if
you want to use solver without preconditioning, you can call this function
which forces solver to use unit matrix for preconditioning.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgsetprecunit(
    lincgstate state,
    const xparams _params = alglib::xdefault);

`lincgsetrestartfreq` function

/*************************************************************************
This function sets restart frequency. By default, algorithm  is  restarted
after N subsequent iterations.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgsetrestartfreq(
    lincgstate state,
    ae_int_t srf,
    const xparams _params = alglib::xdefault);

`lincgsetrupdatefreq` function

/*************************************************************************
This function sets frequency of residual recalculations.

Algorithm updates residual r_k using iterative formula,  but  recalculates
it from scratch after each 10 iterations. It is done to avoid accumulation
of numerical errors and to stop algorithm when r_k starts to grow.

Such low update frequence (1/10) gives very  little  overhead,  but  makes
algorithm a bit more robust against numerical errors. However, you may
change it

INPUT PARAMETERS:
    Freq    -   desired update frequency, Freq>=0.
                Zero value means that no updates will be done.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgsetrupdatefreq(
    lincgstate state,
    ae_int_t freq,
    const xparams _params = alglib::xdefault);

`lincgsetstartingpoint` function

/*************************************************************************
This function sets starting point.
By default, zero starting point is used.

INPUT PARAMETERS:
    X       -   starting point, array[N]

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgsetstartingpoint(
    lincgstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

`lincgsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinCGOptimize().

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgsetxrep(
    lincgstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

`lincgsolvesparse` function

/*************************************************************************
Procedure for solution of A*x=b with sparse A.

INPUT PARAMETERS:
    State   -   algorithm state
    A       -   sparse matrix in the CRS format (you MUST contvert  it  to
                CRS format by calling SparseConvertToCRS() function).
    IsUpper -   whether upper or lower triangle of A is used:
                * IsUpper=True  => only upper triangle is used and lower
                                   triangle is not referenced at all
                * IsUpper=False => only lower triangle is used and upper
                                   triangle is not referenced at all
    B       -   right part, array[N]

RESULT:
    This function returns no result.
    You can get solution by calling LinCGResults()

NOTE: this function uses lightweight preconditioning -  multiplication  by
      inverse of diag(A). If you want, you can turn preconditioning off by
      calling LinCGSetPrecUnit(). However, preconditioning cost is low and
      preconditioner  is  very  important  for  solution  of  badly scaled
      problems.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lincgsolvesparse(
    lincgstate state,
    sparsematrix a,
    bool isupper,
    real_1d_array b,
    const xparams _params = alglib::xdefault);

Examples: [1]

lincg_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example illustrates solution of sparse linear systems with
    // conjugate gradient method.
    // 
    // Suppose that we have linear system A*x=b with sparse symmetric
    // positive definite A (represented by sparsematrix object)
    //         [ 5 1       ]
    //         [ 1 7 2     ]
    //     A = [   2 8 1   ]
    //         [     1 4 1 ]
    //         [       1 4 ]
    // and right part b
    //     [  7 ]
    //     [ 17 ]
    // b = [ 14 ]
    //     [ 10 ]
    //     [  6 ]
    // and we want to solve this system using sparse linear CG. In order
    // to do so, we have to create left part (sparsematrix object) and
    // right part (dense array).
    //
    // Initially, sparse matrix is created in the Hash-Table format,
    // which allows easy initialization, but do not allow matrix to be
    // used in the linear solvers. So after construction you should convert
    // sparse matrix to CRS format (one suited for linear operations).
    //
    // It is important to note that in our example we initialize full
    // matrix A, both lower and upper triangles. However, it is symmetric
    // and sparse solver needs just one half of the matrix. So you may
    // save about half of the space by filling only one of the triangles.
    //
    sparsematrix a;
    sparsecreate(5, 5, a);
    sparseset(a, 0, 0, 5.0);
    sparseset(a, 0, 1, 1.0);
    sparseset(a, 1, 0, 1.0);
    sparseset(a, 1, 1, 7.0);
    sparseset(a, 1, 2, 2.0);
    sparseset(a, 2, 1, 2.0);
    sparseset(a, 2, 2, 8.0);
    sparseset(a, 2, 3, 1.0);
    sparseset(a, 3, 2, 1.0);
    sparseset(a, 3, 3, 4.0);
    sparseset(a, 3, 4, 1.0);
    sparseset(a, 4, 3, 1.0);
    sparseset(a, 4, 4, 4.0);

    //
    // Now our matrix is fully initialized, but we have to do one more
    // step - convert it from Hash-Table format to CRS format (see
    // documentation on sparse matrices for more information about these
    // formats).
    //
    // If you omit this call, ALGLIB will generate exception on the first
    // attempt to use A in linear operations. 
    //
    sparseconverttocrs(a);

    //
    // Initialization of the right part
    //
    real_1d_array b = "[7,17,14,10,6]";

    //
    // Now we have to create linear solver object and to use it for the
    // solution of the linear system.
    //
    // NOTE: lincgsolvesparse() accepts additional parameter which tells
    //       what triangle of the symmetric matrix should be used - upper
    //       or lower. Because we've filled both parts of the matrix, we
    //       can use any part - upper or lower.
    //
    lincgstate s;
    lincgreport rep;
    real_1d_array x;
    lincgcreate(5, s);
    lincgsolvesparse(s, a, true, b);
    lincgresults(s, x, rep);

    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [1.000,2.000,1.000,2.000,1.000]
    return 0;
}

`linlsqr` subpackage

Classes

linlsqrreport
linlsqrstate

Functions

linlsqrcreate
linlsqrcreatebuf
linlsqrresults
linlsqrsetcond
linlsqrsetlambdai
linlsqrsetprecdiag
linlsqrsetprecunit
linlsqrsetxrep
linlsqrsolvesparse

Examples

linlsqr_d_1

Solution of sparse linear systems with CG

`linlsqrreport` class

/*************************************************************************

*************************************************************************/
class linlsqrreport
{
    ae_int_t             iterationscount;
    ae_int_t             nmv;
    ae_int_t             terminationtype;
};

`linlsqrstate` class

/*************************************************************************
This object stores state of the LinLSQR method.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class linlsqrstate
{
};

`linlsqrcreate` function

/*************************************************************************
This function initializes linear LSQR Solver. This solver is used to solve
non-symmetric (and, possibly, non-square) problems. Least squares solution
is returned for non-compatible systems.

USAGE:
1. User initializes algorithm state with LinLSQRCreate() call
2. User tunes solver parameters with  LinLSQRSetCond() and other functions
3. User  calls  LinLSQRSolveSparse()  function which takes algorithm state
   and SparseMatrix object.
4. User calls LinLSQRResults() to get solution
5. Optionally, user may call LinLSQRSolveSparse() again to  solve  another
   problem  with different matrix and/or right part without reinitializing
   LinLSQRState structure.

INPUT PARAMETERS:
    M       -   number of rows in A
    N       -   number of variables, N>0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE: see also linlsqrcreatebuf()  for  version  which  reuses  previously
      allocated place as much as possible.

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::linlsqrcreate(
    ae_int_t m,
    ae_int_t n,
    linlsqrstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`linlsqrcreatebuf` function

/*************************************************************************
This function initializes linear LSQR Solver.  It  provides  exactly  same
functionality as linlsqrcreate(), but reuses  previously  allocated  space
as much as possible.

INPUT PARAMETERS:
    M       -   number of rows in A
    N       -   number of variables, N>0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 14.11.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::linlsqrcreatebuf(
    ae_int_t m,
    ae_int_t n,
    linlsqrstate state,
    const xparams _params = alglib::xdefault);

`linlsqrresults` function

/*************************************************************************
LSQR solver: results.

This function must be called after LinLSQRSolve

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[N], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    *  1    ||Rk||<=EpsB*||B||
                    *  4    ||A^T*Rk||/(||A||*||Rk||)<=EpsA
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            X contains best point found so far.
                            (sometimes returned on singular systems)
                * Rep.IterationsCount contains iterations count
                * NMV countains number of matrix-vector calculations

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::linlsqrresults(
    linlsqrstate state,
    real_1d_array& x,
    linlsqrreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`linlsqrsetcond` function

/*************************************************************************
This function sets stopping criteria.

INPUT PARAMETERS:
    EpsA    -   algorithm will be stopped if ||A^T*Rk||/(||A||*||Rk||)<=EpsA.
    EpsB    -   algorithm will be stopped if ||Rk||<=EpsB*||B||
    MaxIts  -   algorithm will be stopped if number of iterations
                more than MaxIts.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE: if EpsA,EpsB,EpsC and MaxIts are zero then these variables will
be setted as default values.

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::linlsqrsetcond(
    linlsqrstate state,
    double epsa,
    double epsb,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

`linlsqrsetlambdai` function

/*************************************************************************
This function sets optional Tikhonov regularization coefficient.
It is zero by default.

INPUT PARAMETERS:
    LambdaI -   regularization factor, LambdaI>=0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::linlsqrsetlambdai(
    linlsqrstate state,
    double lambdai,
    const xparams _params = alglib::xdefault);

`linlsqrsetprecdiag` function

/*************************************************************************
This  function  changes  preconditioning  settings  of  LinCGSolveSparse()
function.  LinCGSolveSparse() will use diagonal of the  system  matrix  as
preconditioner. This preconditioning mode is active by default.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::linlsqrsetprecdiag(
    linlsqrstate state,
    const xparams _params = alglib::xdefault);

`linlsqrsetprecunit` function

/*************************************************************************
This  function  changes  preconditioning  settings of LinLSQQSolveSparse()
function. By default, SolveSparse() uses diagonal preconditioner,  but  if
you want to use solver without preconditioning, you can call this function
which forces solver to use unit matrix for preconditioning.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::linlsqrsetprecunit(
    linlsqrstate state,
    const xparams _params = alglib::xdefault);

`linlsqrsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinCGOptimize().

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::linlsqrsetxrep(
    linlsqrstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

`linlsqrsolvesparse` function

/*************************************************************************
Procedure for solution of A*x=b with sparse A.

INPUT PARAMETERS:
    State   -   algorithm state
    A       -   sparse M*N matrix in the CRS format (you MUST contvert  it
                to CRS format  by  calling  SparseConvertToCRS()  function
                BEFORE you pass it to this function).
    B       -   right part, array[M]

RESULT:
    This function returns no result.
    You can get solution by calling LinCGResults()

NOTE: this function uses lightweight preconditioning -  multiplication  by
      inverse of diag(A). If you want, you can turn preconditioning off by
      calling LinLSQRSetPrecUnit(). However, preconditioning cost is   low
      and preconditioner is very important for solution  of  badly  scaled
      problems.

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::linlsqrsolvesparse(
    linlsqrstate state,
    sparsematrix a,
    real_1d_array b,
    const xparams _params = alglib::xdefault);

Examples: [1]

linlsqr_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example illustrates solution of sparse linear least squares problem
    // with LSQR algorithm.
    // 
    // Suppose that we have least squares problem min|A*x-b| with sparse A
    // represented by sparsematrix object
    //         [ 1 1 ]
    //         [ 1 1 ]
    //     A = [ 2 1 ]
    //         [ 1   ]
    //         [   1 ]
    // and right part b
    //     [ 4 ]
    //     [ 2 ]
    // b = [ 4 ]
    //     [ 1 ]
    //     [ 2 ]
    // and we want to solve this system in the least squares sense using
    // LSQR algorithm. In order to do so, we have to create left part
    // (sparsematrix object) and right part (dense array).
    //
    // Initially, sparse matrix is created in the Hash-Table format,
    // which allows easy initialization, but do not allow matrix to be
    // used in the linear solvers. So after construction you should convert
    // sparse matrix to CRS format (one suited for linear operations).
    //
    sparsematrix a;
    sparsecreate(5, 2, a);
    sparseset(a, 0, 0, 1.0);
    sparseset(a, 0, 1, 1.0);
    sparseset(a, 1, 0, 1.0);
    sparseset(a, 1, 1, 1.0);
    sparseset(a, 2, 0, 2.0);
    sparseset(a, 2, 1, 1.0);
    sparseset(a, 3, 0, 1.0);
    sparseset(a, 4, 1, 1.0);

    //
    // Now our matrix is fully initialized, but we have to do one more
    // step - convert it from Hash-Table format to CRS format (see
    // documentation on sparse matrices for more information about these
    // formats).
    //
    // If you omit this call, ALGLIB will generate exception on the first
    // attempt to use A in linear operations. 
    //
    sparseconverttocrs(a);

    //
    // Initialization of the right part
    //
    real_1d_array b = "[4,2,4,1,2]";

    //
    // Now we have to create linear solver object and to use it for the
    // solution of the linear system.
    //
    linlsqrstate s;
    linlsqrreport rep;
    real_1d_array x;
    linlsqrcreate(5, 2, s);
    linlsqrsolvesparse(s, a, b);
    linlsqrresults(s, x, rep);

    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [1.000,2.000]
    return 0;
}

`linreg` subpackage

Classes

linearmodel
lrreport

Functions

lravgerror
lravgrelerror
lrbuild
lrbuilds
lrbuildz
lrbuildzs
lrpack
lrprocess
lrrmserror
lrunpack

Examples

linreg_d_basic

Linear regression used to build the very basic model and unpack coefficients

`linearmodel` class

/*************************************************************************

*************************************************************************/
class linearmodel
{
};

`lrreport` class

/*************************************************************************
LRReport structure contains additional information about linear model:
* C             -   covariation matrix,  array[0..NVars,0..NVars].
                    C[i,j] = Cov(A[i],A[j])
* RMSError      -   root mean square error on a training set
* AvgError      -   average error on a training set
* AvgRelError   -   average relative error on a training set (excluding
                    observations with zero function value).
* CVRMSError    -   leave-one-out cross-validation estimate of
                    generalization error. Calculated using fast algorithm
                    with O(NVars*NPoints) complexity.
* CVAvgError    -   cross-validation estimate of average error
* CVAvgRelError -   cross-validation estimate of average relative error

All other fields of the structure are intended for internal use and should
not be used outside ALGLIB.
*************************************************************************/
class lrreport
{
    real_2d_array        c;
    double               rmserror;
    double               avgerror;
    double               avgrelerror;
    double               cvrmserror;
    double               cvavgerror;
    double               cvavgrelerror;
    ae_int_t             ncvdefects;
    integer_1d_array     cvdefects;
};

`lravgerror` function

/*************************************************************************
Average error on the test set

INPUT PARAMETERS:
    LM      -   linear model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average error.

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::lravgerror(
    linearmodel lm,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`lravgrelerror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    LM      -   linear model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average relative error.

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::lravgrelerror(
    linearmodel lm,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`lrbuild` function

/*************************************************************************
Linear regression

Subroutine builds model:

    Y = A(0)*X[0] + ... + A(N-1)*X[N-1] + A(N)

and model found in ALGLIB format, covariation matrix, training set  errors
(rms,  average,  average  relative)   and  leave-one-out  cross-validation
estimate of the generalization error. CV  estimate calculated  using  fast
algorithm with O(NPoints*NVars) complexity.

When  covariation  matrix  is  calculated  standard deviations of function
values are assumed to be equal to RMS error on the training set.

INPUT PARAMETERS:
    XY          -   training set, array [0..NPoints-1,0..NVars]:
                    * NVars columns - independent variables
                    * last column - dependent variable
    NPoints     -   training set size, NPoints>NVars+1
    NVars       -   number of independent variables

OUTPUT PARAMETERS:
    Info        -   return code:
                    * -255, in case of unknown internal error
                    * -4, if internal SVD subroutine haven't converged
                    * -1, if incorrect parameters was passed (NPoints<NVars+2, NVars<1).
                    *  1, if subroutine successfully finished
    LM          -   linear model in the ALGLIB format. Use subroutines of
                    this unit to work with the model.
    AR          -   additional results


  -- ALGLIB --
     Copyright 02.08.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::lrbuild(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t& info,
    linearmodel& lm,
    lrreport& ar,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lrbuilds` function

/*************************************************************************
Linear regression

Variant of LRBuild which uses vector of standatd deviations (errors in
function values).

INPUT PARAMETERS:
    XY          -   training set, array [0..NPoints-1,0..NVars]:
                    * NVars columns - independent variables
                    * last column - dependent variable
    S           -   standard deviations (errors in function values)
                    array[0..NPoints-1], S[i]>0.
    NPoints     -   training set size, NPoints>NVars+1
    NVars       -   number of independent variables

OUTPUT PARAMETERS:
    Info        -   return code:
                    * -255, in case of unknown internal error
                    * -4, if internal SVD subroutine haven't converged
                    * -1, if incorrect parameters was passed (NPoints<NVars+2, NVars<1).
                    * -2, if S[I]<=0
                    *  1, if subroutine successfully finished
    LM          -   linear model in the ALGLIB format. Use subroutines of
                    this unit to work with the model.
    AR          -   additional results


  -- ALGLIB --
     Copyright 02.08.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::lrbuilds(
    real_2d_array xy,
    real_1d_array s,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t& info,
    linearmodel& lm,
    lrreport& ar,
    const xparams _params = alglib::xdefault);

`lrbuildz` function

/*************************************************************************
Like LRBuild but builds model

    Y = A(0)*X[0] + ... + A(N-1)*X[N-1]

i.e. with zero constant term.

  -- ALGLIB --
     Copyright 30.10.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::lrbuildz(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t& info,
    linearmodel& lm,
    lrreport& ar,
    const xparams _params = alglib::xdefault);

`lrbuildzs` function

/*************************************************************************
Like LRBuildS, but builds model

    Y = A(0)*X[0] + ... + A(N-1)*X[N-1]

i.e. with zero constant term.

  -- ALGLIB --
     Copyright 30.10.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::lrbuildzs(
    real_2d_array xy,
    real_1d_array s,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t& info,
    linearmodel& lm,
    lrreport& ar,
    const xparams _params = alglib::xdefault);

`lrpack` function

/*************************************************************************
"Packs" coefficients and creates linear model in ALGLIB format (LRUnpack
reversed).

INPUT PARAMETERS:
    V           -   coefficients, array[0..NVars]
    NVars       -   number of independent variables

OUTPUT PAREMETERS:
    LM          -   linear model.

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::lrpack(
    real_1d_array v,
    ae_int_t nvars,
    linearmodel& lm,
    const xparams _params = alglib::xdefault);

`lrprocess` function

/*************************************************************************
Procesing

INPUT PARAMETERS:
    LM      -   linear model
    X       -   input vector,  array[0..NVars-1].

Result:
    value of linear model regression estimate

  -- ALGLIB --
     Copyright 03.09.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::lrprocess(
    linearmodel lm,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

`lrrmserror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    LM      -   linear model
    XY      -   test set
    NPoints -   test set size

RESULT:
    root mean square error.

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::lrrmserror(
    linearmodel lm,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`lrunpack` function

/*************************************************************************
Unpacks coefficients of linear model.

INPUT PARAMETERS:
    LM          -   linear model in ALGLIB format

OUTPUT PARAMETERS:
    V           -   coefficients, array[0..NVars]
                    constant term (intercept) is stored in the V[NVars].
    NVars       -   number of independent variables (one less than number
                    of coefficients)

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::lrunpack(
    linearmodel lm,
    real_1d_array& v,
    ae_int_t& nvars,
    const xparams _params = alglib::xdefault);

Examples: [1]

linreg_d_basic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // In this example we demonstrate linear fitting by f(x|a) = a*exp(0.5*x).
    //
    // We have:
    // * xy - matrix of basic function values (exp(0.5*x)) and expected values
    //
    real_2d_array xy = "[[0.606531,1.133719],[0.670320,1.306522],[0.740818,1.504604],[0.818731,1.554663],[0.904837,1.884638],[1.000000,2.072436],[1.105171,2.257285],[1.221403,2.534068],[1.349859,2.622017],[1.491825,2.897713],[1.648721,3.219371]]";
    ae_int_t info;
    ae_int_t nvars;
    linearmodel model;
    lrreport rep;
    real_1d_array c;

    lrbuildz(xy, 11, 1, info, model, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    lrunpack(model, c, nvars);
    printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [1.98650,0.00000]
    return 0;
}

`logit` subpackage

Classes

logitmodel
mnlreport

Functions

mnlavgce
mnlavgerror
mnlavgrelerror
mnlclserror
mnlpack
mnlprocess
mnlprocessi
mnlrelclserror
mnlrmserror
mnltrainh
mnlunpack

Examples

`logitmodel` class

/*************************************************************************

*************************************************************************/
class logitmodel
{
};

`mnlreport` class

/*************************************************************************
MNLReport structure contains information about training process:
* NGrad     -   number of gradient calculations
* NHess     -   number of Hessian calculations
*************************************************************************/
class mnlreport
{
    ae_int_t             ngrad;
    ae_int_t             nhess;
};

`mnlavgce` function

/*************************************************************************
Average cross-entropy (in bits per element) on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    CrossEntropy/(NPoints*ln(2)).

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::mnlavgce(
    logitmodel lm,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mnlavgerror` function

/*************************************************************************
Average error on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average error (error when estimating posterior probabilities).

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::mnlavgerror(
    logitmodel lm,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mnlavgrelerror` function

/*************************************************************************
Average relative error on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average relative error (error when estimating posterior probabilities).

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::mnlavgrelerror(
    logitmodel lm,
    real_2d_array xy,
    ae_int_t ssize,
    const xparams _params = alglib::xdefault);

`mnlclserror` function

/*************************************************************************
Classification error on test set = MNLRelClsError*NPoints

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::mnlclserror(
    logitmodel lm,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mnlpack` function

/*************************************************************************
"Packs" coefficients and creates logit model in ALGLIB format (MNLUnpack
reversed).

INPUT PARAMETERS:
    A           -   model (see MNLUnpack)
    NVars       -   number of independent variables
    NClasses    -   number of classes

OUTPUT PARAMETERS:
    LM          -   logit model.

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mnlpack(
    real_2d_array a,
    ae_int_t nvars,
    ae_int_t nclasses,
    logitmodel& lm,
    const xparams _params = alglib::xdefault);

`mnlprocess` function

/*************************************************************************
Procesing

INPUT PARAMETERS:
    LM      -   logit model, passed by non-constant reference
                (some fields of structure are used as temporaries
                when calculating model output).
    X       -   input vector,  array[0..NVars-1].
    Y       -   (possibly) preallocated buffer; if size of Y is less than
                NClasses, it will be reallocated.If it is large enough, it
                is NOT reallocated, so we can save some time on reallocation.

OUTPUT PARAMETERS:
    Y       -   result, array[0..NClasses-1]
                Vector of posterior probabilities for classification task.

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mnlprocess(
    logitmodel lm,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`mnlprocessi` function

/*************************************************************************
'interactive'  variant  of  MNLProcess  for  languages  like  Python which
support constructs like "Y = MNLProcess(LM,X)" and interactive mode of the
interpreter

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mnlprocessi(
    logitmodel lm,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`mnlrelclserror` function

/*************************************************************************
Relative classification error on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    percent of incorrectly classified cases.

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::mnlrelclserror(
    logitmodel lm,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mnlrmserror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    root mean square error (error when estimating posterior probabilities).

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::mnlrmserror(
    logitmodel lm,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mnltrainh` function

/*************************************************************************
This subroutine trains logit model.

INPUT PARAMETERS:
    XY          -   training set, array[0..NPoints-1,0..NVars]
                    First NVars columns store values of independent
                    variables, next column stores number of class (from 0
                    to NClasses-1) which dataset element belongs to. Fractional
                    values are rounded to nearest integer.
    NPoints     -   training set size, NPoints>=1
    NVars       -   number of independent variables, NVars>=1
    NClasses    -   number of classes, NClasses>=2

OUTPUT PARAMETERS:
    Info        -   return code:
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed
                          (NPoints<NVars+2, NVars<1, NClasses<2).
                    *  1, if task has been solved
    LM          -   model built
    Rep         -   training report

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mnltrainh(
    real_2d_array xy,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t nclasses,
    ae_int_t& info,
    logitmodel& lm,
    mnlreport& rep,
    const xparams _params = alglib::xdefault);

`mnlunpack` function

/*************************************************************************
Unpacks coefficients of logit model. Logit model have form:

    P(class=i) = S(i) / (S(0) + S(1) + ... +S(M-1))
          S(i) = Exp(A[i,0]*X[0] + ... + A[i,N-1]*X[N-1] + A[i,N]), when i<M-1
        S(M-1) = 1

INPUT PARAMETERS:
    LM          -   logit model in ALGLIB format

OUTPUT PARAMETERS:
    V           -   coefficients, array[0..NClasses-2,0..NVars]
    NVars       -   number of independent variables
    NClasses    -   number of classes

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mnlunpack(
    logitmodel lm,
    real_2d_array& a,
    ae_int_t& nvars,
    ae_int_t& nclasses,
    const xparams _params = alglib::xdefault);

`lsfit` subpackage

Classes

barycentricfitreport
lsfitreport
lsfitstate
polynomialfitreport

Functions

barycentricfitfloaterhormann
barycentricfitfloaterhormannwc
logisticcalc4
logisticcalc5
logisticfit4
logisticfit45x
logisticfit4ec
logisticfit5
logisticfit5ec
lsfitcreatef
lsfitcreatefg
lsfitcreatefgh
lsfitcreatewf
lsfitcreatewfg
lsfitcreatewfgh
lsfitfit
lsfitlinear
lsfitlinearc
lsfitlinearw
lsfitlinearwc
lsfitresults
lsfitsetbc
lsfitsetcond
lsfitsetgradientcheck
lsfitsetlc
lsfitsetscale
lsfitsetstpmax
lsfitsetxrep
lstfitpiecewiselinearrdp
lstfitpiecewiselinearrdpfixed
polynomialfit
polynomialfitwc
spline1dfitcubic
spline1dfitcubicwc
spline1dfithermite
spline1dfithermitewc

Examples

lsfit_d_lin		Unconstrained (general) linear least squares fitting with and without weights
lsfit_d_linc		Constrained (general) linear least squares fitting with and without weights
lsfit_d_nlf		Nonlinear fitting using function value only
lsfit_d_nlfb		Bound contstrained nonlinear fitting using function value only
lsfit_d_nlfg		Nonlinear fitting using gradient
lsfit_d_nlfgh		Nonlinear fitting using gradient and Hessian
lsfit_d_nlscale		Nonlinear fitting with custom scaling and bound constraints
lsfit_d_pol		Unconstrained polynomial fitting
lsfit_d_polc		Constrained polynomial fitting
lsfit_d_spline		Unconstrained fitting by penalized regression spline
lsfit_t_4pl		4-parameter logistic fitting
lsfit_t_5pl		5-parameter logistic fitting

`barycentricfitreport` class

/*************************************************************************
Barycentric fitting report:
    RMSError        RMS error
    AvgError        average error
    AvgRelError     average relative error (for non-zero Y[I])
    MaxError        maximum error
    TaskRCond       reciprocal of task's condition number
*************************************************************************/
class barycentricfitreport
{
    double               taskrcond;
    ae_int_t             dbest;
    double               rmserror;
    double               avgerror;
    double               avgrelerror;
    double               maxerror;
};

`lsfitreport` class

/*************************************************************************
Least squares fitting report. This structure contains informational fields
which are set by fitting functions provided by this unit.

Different functions initialize different sets of  fields,  so  you  should
read documentation on specific function you used in order  to  know  which
fields are initialized.

    TaskRCond       reciprocal of task's condition number
    IterationsCount number of internal iterations

    VarIdx          if user-supplied gradient contains errors  which  were
                    detected by nonlinear fitter, this  field  is  set  to
                    index  of  the  first  component  of gradient which is
                    suspected to be spoiled by bugs.

    RMSError        RMS error
    AvgError        average error
    AvgRelError     average relative error (for non-zero Y[I])
    MaxError        maximum error

    WRMSError       weighted RMS error

    CovPar          covariance matrix for parameters, filled by some solvers
    ErrPar          vector of errors in parameters, filled by some solvers
    ErrCurve        vector of fit errors -  variability  of  the  best-fit
                    curve, filled by some solvers.
    Noise           vector of per-point noise estimates, filled by
                    some solvers.
    R2              coefficient of determination (non-weighted, non-adjusted),
                    filled by some solvers.
*************************************************************************/
class lsfitreport
{
    double               taskrcond;
    ae_int_t             iterationscount;
    ae_int_t             varidx;
    double               rmserror;
    double               avgerror;
    double               avgrelerror;
    double               maxerror;
    double               wrmserror;
    real_2d_array        covpar;
    real_1d_array        errpar;
    real_1d_array        errcurve;
    real_1d_array        noise;
    double               r2;
};

`lsfitstate` class

/*************************************************************************
Nonlinear fitter.

You should use ALGLIB functions to work with fitter.
Never try to access its fields directly!
*************************************************************************/
class lsfitstate
{
};

`polynomialfitreport` class

/*************************************************************************
Polynomial fitting report:
    TaskRCond       reciprocal of task's condition number
    RMSError        RMS error
    AvgError        average error
    AvgRelError     average relative error (for non-zero Y[I])
    MaxError        maximum error
*************************************************************************/
class polynomialfitreport
{
    double               taskrcond;
    double               rmserror;
    double               avgerror;
    double               avgrelerror;
    double               maxerror;
};

`barycentricfitfloaterhormann` function

/*************************************************************************
Rational least squares fitting using  Floater-Hormann  rational  functions
with optimal D chosen from [0,9].

Equidistant  grid  with M node on [min(x),max(x)]  is  used to build basis
functions. Different values of D are tried, optimal  D  (least  root  mean
square error) is chosen.  Task  is  linear, so linear least squares solver
is used. Complexity  of  this  computational  scheme is  O(N*M^2)  (mostly
dominated by the least squares solver).

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    N   -   number of points, N>0.
    M   -   number of basis functions ( = number_of_nodes), M>=2.

OUTPUT PARAMETERS:
    Info-   same format as in LSFitLinearWC() subroutine.
            * Info>0    task is solved
            * Info<=0   an error occured:
                        -4 means inconvergence of internal SVD
                        -3 means inconsistent constraints
    B   -   barycentric interpolant.
    Rep -   report, same format as in LSFitLinearWC() subroutine.
            Following fields are set:
            * DBest         best value of the D parameter
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::barycentricfitfloaterhormann(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t m,
    ae_int_t& info,
    barycentricinterpolant& b,
    barycentricfitreport& rep,
    const xparams _params = alglib::xdefault);

`barycentricfitfloaterhormannwc` function

/*************************************************************************
Weghted rational least  squares  fitting  using  Floater-Hormann  rational
functions  with  optimal  D  chosen  from  [0,9],  with  constraints   and
individual weights.

Equidistant  grid  with M node on [min(x),max(x)]  is  used to build basis
functions. Different values of D are tried, optimal D (least WEIGHTED root
mean square error) is chosen.  Task  is  linear,  so  linear least squares
solver  is  used.  Complexity  of  this  computational  scheme is O(N*M^2)
(mostly dominated by the least squares solver).

SEE ALSO
* BarycentricFitFloaterHormann(), "lightweight" fitting without invididual
  weights and constraints.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    W   -   weights, array[0..N-1]
            Each summand in square  sum  of  approximation deviations from
            given  values  is  multiplied  by  the square of corresponding
            weight. Fill it by 1's if you don't  want  to  solve  weighted
            task.
    N   -   number of points, N>0.
    XC  -   points where function values/derivatives are constrained,
            array[0..K-1].
    YC  -   values of constraints, array[0..K-1]
    DC  -   array[0..K-1], types of constraints:
            * DC[i]=0   means that S(XC[i])=YC[i]
            * DC[i]=1   means that S'(XC[i])=YC[i]
            SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
    K   -   number of constraints, 0<=K<M.
            K=0 means no constraints (XC/YC/DC are not used in such cases)
    M   -   number of basis functions ( = number_of_nodes), M>=2.

OUTPUT PARAMETERS:
    Info-   same format as in LSFitLinearWC() subroutine.
            * Info>0    task is solved
            * Info<=0   an error occured:
                        -4 means inconvergence of internal SVD
                        -3 means inconsistent constraints
                        -1 means another errors in parameters passed
                           (N<=0, for example)
    B   -   barycentric interpolant.
    Rep -   report, same format as in LSFitLinearWC() subroutine.
            Following fields are set:
            * DBest         best value of the D parameter
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroutine doesn't calculate task's condition number for K<>0.

SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

Setting constraints can lead  to undesired  results,  like ill-conditioned
behavior, or inconsistency being detected. From the other side,  it allows
us to improve quality of the fit. Here we summarize  our  experience  with
constrained barycentric interpolants:
* excessive  constraints  can  be  inconsistent.   Floater-Hormann   basis
  functions aren't as flexible as splines (although they are very smooth).
* the more evenly constraints are spread across [min(x),max(x)],  the more
  chances that they will be consistent
* the  greater  is  M (given  fixed  constraints),  the  more chances that
  constraints will be consistent
* in the general case, consistency of constraints IS NOT GUARANTEED.
* in the several special cases, however, we CAN guarantee consistency.
* one of this cases is constraints on the function  VALUES at the interval
  boundaries. Note that consustency of the  constraints  on  the  function
  DERIVATIVES is NOT guaranteed (you can use in such cases  cubic  splines
  which are more flexible).
* another  special  case  is ONE constraint on the function value (OR, but
  not AND, derivative) anywhere in the interval

Our final recommendation is to use constraints  WHEN  AND  ONLY  WHEN  you
can't solve your task without them. Anything beyond  special  cases  given
above is not guaranteed and may result in inconsistency.

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::barycentricfitfloaterhormannwc(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    ae_int_t n,
    real_1d_array xc,
    real_1d_array yc,
    integer_1d_array dc,
    ae_int_t k,
    ae_int_t m,
    ae_int_t& info,
    barycentricinterpolant& b,
    barycentricfitreport& rep,
    const xparams _params = alglib::xdefault);

`logisticcalc4` function

/*************************************************************************
This function calculates value of four-parameter logistic (4PL)  model  at
specified point X. 4PL model has following form:

    F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B))

INPUT PARAMETERS:
    X       -   current point, X>=0:
                * zero X is correctly handled even for B<=0
                * negative X results in exception.
    A, B, C, D- parameters of 4PL model:
                * A is unconstrained
                * B is unconstrained; zero or negative values are handled
                  correctly.
                * C>0, non-positive value results in exception
                * D is unconstrained

RESULT:
    model value at X

NOTE: if B=0, denominator is assumed to be equal to 2.0 even  for  zero  X
      (strictly speaking, 0^0 is undefined).

NOTE: this function also throws exception  if  all  input  parameters  are
      correct, but overflow was detected during calculations.

NOTE: this function performs a lot of checks;  if  you  need  really  high
      performance, consider evaluating model  yourself,  without  checking
      for degenerate cases.


  -- ALGLIB PROJECT --
     Copyright 14.05.2014 by Bochkanov Sergey
*************************************************************************/
double alglib::logisticcalc4(
    double x,
    double a,
    double b,
    double c,
    double d,
    const xparams _params = alglib::xdefault);

Examples: [1]

`logisticcalc5` function

/*************************************************************************
This function calculates value of five-parameter logistic (5PL)  model  at
specified point X. 5PL model has following form:

    F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G)

INPUT PARAMETERS:
    X       -   current point, X>=0:
                * zero X is correctly handled even for B<=0
                * negative X results in exception.
    A, B, C, D, G- parameters of 5PL model:
                * A is unconstrained
                * B is unconstrained; zero or negative values are handled
                  correctly.
                * C>0, non-positive value results in exception
                * D is unconstrained
                * G>0, non-positive value results in exception

RESULT:
    model value at X

NOTE: if B=0, denominator is assumed to be equal to Power(2.0,G) even  for
      zero X (strictly speaking, 0^0 is undefined).

NOTE: this function also throws exception  if  all  input  parameters  are
      correct, but overflow was detected during calculations.

NOTE: this function performs a lot of checks;  if  you  need  really  high
      performance, consider evaluating model  yourself,  without  checking
      for degenerate cases.


  -- ALGLIB PROJECT --
     Copyright 14.05.2014 by Bochkanov Sergey
*************************************************************************/
double alglib::logisticcalc5(
    double x,
    double a,
    double b,
    double c,
    double d,
    double g,
    const xparams _params = alglib::xdefault);

Examples: [1]

`logisticfit4` function

/*************************************************************************
This function fits four-parameter logistic (4PL) model  to  data  provided
by user. 4PL model has following form:

    F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B))

Here:
    * A, D - unconstrained (see LogisticFit4EC() for constrained 4PL)
    * B>=0
    * C>0

IMPORTANT: output of this function is constrained in  such  way that  B>0.
           Because 4PL model is symmetric with respect to B, there  is  no
           need to explore  B<0.  Constraining  B  makes  algorithm easier
           to stabilize and debug.
           Users  who  for  some  reason  prefer to work with negative B's
           should transform output themselves (swap A and D, replace B  by
           -B).

4PL fitting is implemented as follows:
* we perform small number of restarts from random locations which helps to
  solve problem of bad local extrema. Locations are only partially  random
  - we use input data to determine good  initial  guess,  but  we  include
  controlled amount of randomness.
* we perform Levenberg-Marquardt fitting with very  tight  constraints  on
  parameters B and C - it allows us to find good  initial  guess  for  the
  second stage without risk of running into "flat spot".
* second  Levenberg-Marquardt  round  is   performed   without   excessive
  constraints. Results from the previous round are used as initial guess.
* after fitting is done, we compare results with best values found so far,
  rewrite "best solution" if needed, and move to next random location.

Overall algorithm is very stable and is not prone to  bad  local  extrema.
Furthermore, it automatically scales when input data have  very  large  or
very small range.

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.

OUTPUT PARAMETERS:
    A, B, C, D- parameters of 4PL model
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc4() function.

NOTE: if you need better control over fitting process than provided by this
      function, you may use LogisticFit45X().

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::logisticfit4(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    double& a,
    double& b,
    double& c,
    double& d,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`logisticfit45x` function

/*************************************************************************
This is "expert" 4PL/5PL fitting function, which can be used if  you  need
better control over fitting process than provided  by  LogisticFit4()  or
LogisticFit5().

This function fits model of the form

    F(x|A,B,C,D)   = D+(A-D)/(1+Power(x/C,B))           (4PL model)

or

    F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G)    (5PL model)

Here:
    * A, D - unconstrained
    * B>=0 for 4PL, unconstrained for 5PL
    * C>0
    * G>0 (if present)

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.
    CnstrLeft-  optional equality constraint for model value at the   left
                boundary (at X=0). Specify NAN (Not-a-Number)  if  you  do
                not need constraint on the model value at X=0 (in C++  you
                can pass alglib::fp_nan as parameter, in  C#  it  will  be
                Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.
    CnstrRight- optional equality constraint for model value at X=infinity.
                Specify NAN (Not-a-Number) if you do not  need  constraint
                on the model value (in C++  you can pass alglib::fp_nan as
                parameter, in  C# it will  be Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.
    Is4PL   -   whether 4PL or 5PL models are fitted
    LambdaV -   regularization coefficient, LambdaV>=0.
                Set it to zero unless you know what you are doing.
    EpsX    -   stopping condition (step size), EpsX>=0.
                Zero value means that small step is automatically chosen.
                See notes below for more information.
    RsCnt   -   number of repeated restarts from  random  points.  4PL/5PL
                models are prone to problem of bad local extrema. Utilizing
                multiple random restarts allows  us  to  improve algorithm
                convergence.
                RsCnt>=0.
                Zero value means that function automatically choose  small
                amount of restarts (recommended).

OUTPUT PARAMETERS:
    A, B, C, D- parameters of 4PL model
    G       -   parameter of 5PL model; for Is4PL=True, G=1 is returned.
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc5() function.

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.

EQUALITY CONSTRAINTS ON PARAMETERS

4PL/5PL solver supports equality constraints on model values at  the  left
boundary (X=0) and right  boundary  (X=infinity).  These  constraints  are
completely optional and you can specify both of them, only  one  -  or  no
constraints at all.

Parameter  CnstrLeft  contains  left  constraint (or NAN for unconstrained
fitting), and CnstrRight contains right  one.  For  4PL,  left  constraint
ALWAYS corresponds to parameter A, and right one is ALWAYS  constraint  on
D. That's because 4PL model is normalized in such way that B>=0.

For 5PL model things are different. Unlike  4PL  one,  5PL  model  is  NOT
symmetric with respect to  change  in  sign  of  B. Thus, negative B's are
possible, and left constraint may constrain parameter A (for positive B's)
- or parameter D (for negative B's). Similarly changes  meaning  of  right
constraint.

You do not have to decide what parameter to  constrain  -  algorithm  will
automatically determine correct parameters as fitting progresses. However,
question highlighted above is important when you interpret fitting results.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::logisticfit45x(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    double cnstrleft,
    double cnstrright,
    bool is4pl,
    double lambdav,
    double epsx,
    ae_int_t rscnt,
    double& a,
    double& b,
    double& c,
    double& d,
    double& g,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

`logisticfit4ec` function

/*************************************************************************
This function fits four-parameter logistic (4PL) model  to  data  provided
by user, with optional constraints on parameters A and D.  4PL  model  has
following form:

    F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B))

Here:
    * A, D - with optional equality constraints
    * B>=0
    * C>0

IMPORTANT: output of this function is constrained in  such  way that  B>0.
           Because 4PL model is symmetric with respect to B, there  is  no
           need to explore  B<0.  Constraining  B  makes  algorithm easier
           to stabilize and debug.
           Users  who  for  some  reason  prefer to work with negative B's
           should transform output themselves (swap A and D, replace B  by
           -B).

4PL fitting is implemented as follows:
* we perform small number of restarts from random locations which helps to
  solve problem of bad local extrema. Locations are only partially  random
  - we use input data to determine good  initial  guess,  but  we  include
  controlled amount of randomness.
* we perform Levenberg-Marquardt fitting with very  tight  constraints  on
  parameters B and C - it allows us to find good  initial  guess  for  the
  second stage without risk of running into "flat spot".
* second  Levenberg-Marquardt  round  is   performed   without   excessive
  constraints. Results from the previous round are used as initial guess.
* after fitting is done, we compare results with best values found so far,
  rewrite "best solution" if needed, and move to next random location.

Overall algorithm is very stable and is not prone to  bad  local  extrema.
Furthermore, it automatically scales when input data have  very  large  or
very small range.

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.
    CnstrLeft-  optional equality constraint for model value at the   left
                boundary (at X=0). Specify NAN (Not-a-Number)  if  you  do
                not need constraint on the model value at X=0 (in C++  you
                can pass alglib::fp_nan as parameter, in  C#  it  will  be
                Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.
    CnstrRight- optional equality constraint for model value at X=infinity.
                Specify NAN (Not-a-Number) if you do not  need  constraint
                on the model value (in C++  you can pass alglib::fp_nan as
                parameter, in  C# it will  be Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.

OUTPUT PARAMETERS:
    A, B, C, D- parameters of 4PL model
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc4() function.

NOTE: if you need better control over fitting process than provided by this
      function, you may use LogisticFit45X().

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.

EQUALITY CONSTRAINTS ON PARAMETERS

4PL/5PL solver supports equality constraints on model values at  the  left
boundary (X=0) and right  boundary  (X=infinity).  These  constraints  are
completely optional and you can specify both of them, only  one  -  or  no
constraints at all.

Parameter  CnstrLeft  contains  left  constraint (or NAN for unconstrained
fitting), and CnstrRight contains right  one.  For  4PL,  left  constraint
ALWAYS corresponds to parameter A, and right one is ALWAYS  constraint  on
D. That's because 4PL model is normalized in such way that B>=0.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::logisticfit4ec(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    double cnstrleft,
    double cnstrright,
    double& a,
    double& b,
    double& c,
    double& d,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

`logisticfit5` function

/*************************************************************************
This function fits five-parameter logistic (5PL) model  to  data  provided
by user. 5PL model has following form:

    F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G)

Here:
    * A, D - unconstrained
    * B - unconstrained
    * C>0
    * G>0

IMPORTANT: unlike in  4PL  fitting,  output  of  this  function   is   NOT
           constrained in  such  way that B is guaranteed to be  positive.
           Furthermore,  unlike  4PL,  5PL  model  is  NOT  symmetric with
           respect to B, so you can NOT transform model to equivalent one,
           with B having desired sign (>0 or <0).

5PL fitting is implemented as follows:
* we perform small number of restarts from random locations which helps to
  solve problem of bad local extrema. Locations are only partially  random
  - we use input data to determine good  initial  guess,  but  we  include
  controlled amount of randomness.
* we perform Levenberg-Marquardt fitting with very  tight  constraints  on
  parameters B and C - it allows us to find good  initial  guess  for  the
  second stage without risk of running into "flat spot".  Parameter  G  is
  fixed at G=1.
* second  Levenberg-Marquardt  round  is   performed   without   excessive
  constraints on B and C, but with G still equal to 1.  Results  from  the
  previous round are used as initial guess.
* third Levenberg-Marquardt round relaxes constraints on G  and  tries  two
  different models - one with B>0 and one with B<0.
* after fitting is done, we compare results with best values found so far,
  rewrite "best solution" if needed, and move to next random location.

Overall algorithm is very stable and is not prone to  bad  local  extrema.
Furthermore, it automatically scales when input data have  very  large  or
very small range.

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.

OUTPUT PARAMETERS:
    A,B,C,D,G-  parameters of 5PL model
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc5() function.

NOTE: if you need better control over fitting process than provided by this
      function, you may use LogisticFit45X().

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::logisticfit5(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    double& a,
    double& b,
    double& c,
    double& d,
    double& g,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`logisticfit5ec` function

/*************************************************************************
This function fits five-parameter logistic (5PL) model  to  data  provided
by user, subject to optional equality constraints on parameters A  and  D.
5PL model has following form:

    F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G)

Here:
    * A, D - with optional equality constraints
    * B - unconstrained
    * C>0
    * G>0

IMPORTANT: unlike in  4PL  fitting,  output  of  this  function   is   NOT
           constrained in  such  way that B is guaranteed to be  positive.
           Furthermore,  unlike  4PL,  5PL  model  is  NOT  symmetric with
           respect to B, so you can NOT transform model to equivalent one,
           with B having desired sign (>0 or <0).

5PL fitting is implemented as follows:
* we perform small number of restarts from random locations which helps to
  solve problem of bad local extrema. Locations are only partially  random
  - we use input data to determine good  initial  guess,  but  we  include
  controlled amount of randomness.
* we perform Levenberg-Marquardt fitting with very  tight  constraints  on
  parameters B and C - it allows us to find good  initial  guess  for  the
  second stage without risk of running into "flat spot".  Parameter  G  is
  fixed at G=1.
* second  Levenberg-Marquardt  round  is   performed   without   excessive
  constraints on B and C, but with G still equal to 1.  Results  from  the
  previous round are used as initial guess.
* third Levenberg-Marquardt round relaxes constraints on G  and  tries  two
  different models - one with B>0 and one with B<0.
* after fitting is done, we compare results with best values found so far,
  rewrite "best solution" if needed, and move to next random location.

Overall algorithm is very stable and is not prone to  bad  local  extrema.
Furthermore, it automatically scales when input data have  very  large  or
very small range.

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.
    CnstrLeft-  optional equality constraint for model value at the   left
                boundary (at X=0). Specify NAN (Not-a-Number)  if  you  do
                not need constraint on the model value at X=0 (in C++  you
                can pass alglib::fp_nan as parameter, in  C#  it  will  be
                Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.
    CnstrRight- optional equality constraint for model value at X=infinity.
                Specify NAN (Not-a-Number) if you do not  need  constraint
                on the model value (in C++  you can pass alglib::fp_nan as
                parameter, in  C# it will  be Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.

OUTPUT PARAMETERS:
    A,B,C,D,G-  parameters of 5PL model
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc5() function.

NOTE: if you need better control over fitting process than provided by this
      function, you may use LogisticFit45X().

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.

EQUALITY CONSTRAINTS ON PARAMETERS

5PL solver supports equality constraints on model  values  at   the   left
boundary (X=0) and right  boundary  (X=infinity).  These  constraints  are
completely optional and you can specify both of them, only  one  -  or  no
constraints at all.

Parameter  CnstrLeft  contains  left  constraint (or NAN for unconstrained
fitting), and CnstrRight contains right  one.

Unlike 4PL one, 5PL model is NOT symmetric with respect to  change in sign
of B. Thus, negative B's are possible, and left constraint  may  constrain
parameter A (for positive B's)  -  or  parameter  D  (for  negative  B's).
Similarly changes meaning of right constraint.

You do not have to decide what parameter to  constrain  -  algorithm  will
automatically determine correct parameters as fitting progresses. However,
question highlighted above is important when you interpret fitting results.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::logisticfit5ec(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    double cnstrleft,
    double cnstrright,
    double& a,
    double& b,
    double& c,
    double& d,
    double& g,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

`lsfitcreatef` function

/*************************************************************************
Nonlinear least squares fitting using function values only.

Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.

Nonlinear task min(F(c)) is solved, where

    F(c) = (f(c,x[0])-y[0])^2 + ... + (f(c,x[n-1])-y[n-1])^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * w is an N-dimensional vector of weight coefficients,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses only f(c,x[i]).

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted
    DiffStep-   numerical differentiation step;
                should not be very small or large;
                large = loss of accuracy
                small = growth of round-off errors

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitcreatef(
    real_2d_array x,
    real_1d_array y,
    real_1d_array c,
    double diffstep,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);
void alglib::lsfitcreatef(
    real_2d_array x,
    real_1d_array y,
    real_1d_array c,
    ae_int_t n,
    ae_int_t m,
    ae_int_t k,
    double diffstep,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`lsfitcreatefg` function

/*************************************************************************
Nonlinear least squares fitting using gradient only, without individual
weights.

Nonlinear task min(F(c)) is solved, where

    F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses only f(c,x[i]) and its gradient.

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted
    CheapFG -   boolean flag, which is:
                * True  if both function and gradient calculation complexity
                        are less than O(M^2).  An improved  algorithm  can
                        be  used  which corresponds  to  FGJ  scheme  from
                        MINLM unit.
                * False otherwise.
                        Standard Jacibian-bases  Levenberg-Marquardt  algo
                        will be used (FJ scheme).

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitcreatefg(
    real_2d_array x,
    real_1d_array y,
    real_1d_array c,
    bool cheapfg,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);
void alglib::lsfitcreatefg(
    real_2d_array x,
    real_1d_array y,
    real_1d_array c,
    ae_int_t n,
    ae_int_t m,
    ae_int_t k,
    bool cheapfg,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lsfitcreatefgh` function

/*************************************************************************
Nonlinear least squares fitting using gradient/Hessian, without individial
weights.

Nonlinear task min(F(c)) is solved, where

    F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses f(c,x[i]), its gradient and its Hessian.

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state


  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitcreatefgh(
    real_2d_array x,
    real_1d_array y,
    real_1d_array c,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);
void alglib::lsfitcreatefgh(
    real_2d_array x,
    real_1d_array y,
    real_1d_array c,
    ae_int_t n,
    ae_int_t m,
    ae_int_t k,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lsfitcreatewf` function

/*************************************************************************
Weighted nonlinear least squares fitting using function values only.

Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.

Nonlinear task min(F(c)) is solved, where

    F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * w is an N-dimensional vector of weight coefficients,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses only f(c,x[i]).

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    W       -   weights, array[0..N-1]
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted
    DiffStep-   numerical differentiation step;
                should not be very small or large;
                large = loss of accuracy
                small = growth of round-off errors

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitcreatewf(
    real_2d_array x,
    real_1d_array y,
    real_1d_array w,
    real_1d_array c,
    double diffstep,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);
void alglib::lsfitcreatewf(
    real_2d_array x,
    real_1d_array y,
    real_1d_array w,
    real_1d_array c,
    ae_int_t n,
    ae_int_t m,
    ae_int_t k,
    double diffstep,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`lsfitcreatewfg` function

/*************************************************************************
Weighted nonlinear least squares fitting using gradient only.

Nonlinear task min(F(c)) is solved, where

    F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * w is an N-dimensional vector of weight coefficients,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses only f(c,x[i]) and its gradient.

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    W       -   weights, array[0..N-1]
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted
    CheapFG -   boolean flag, which is:
                * True  if both function and gradient calculation complexity
                        are less than O(M^2).  An improved  algorithm  can
                        be  used  which corresponds  to  FGJ  scheme  from
                        MINLM unit.
                * False otherwise.
                        Standard Jacibian-bases  Levenberg-Marquardt  algo
                        will be used (FJ scheme).

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

See also:
    LSFitResults
    LSFitCreateFG (fitting without weights)
    LSFitCreateWFGH (fitting using Hessian)
    LSFitCreateFGH (fitting using Hessian, without weights)

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitcreatewfg(
    real_2d_array x,
    real_1d_array y,
    real_1d_array w,
    real_1d_array c,
    bool cheapfg,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);
void alglib::lsfitcreatewfg(
    real_2d_array x,
    real_1d_array y,
    real_1d_array w,
    real_1d_array c,
    ae_int_t n,
    ae_int_t m,
    ae_int_t k,
    bool cheapfg,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lsfitcreatewfgh` function

/*************************************************************************
Weighted nonlinear least squares fitting using gradient/Hessian.

Nonlinear task min(F(c)) is solved, where

    F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * w is an N-dimensional vector of weight coefficients,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses f(c,x[i]), its gradient and its Hessian.

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    W       -   weights, array[0..N-1]
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitcreatewfgh(
    real_2d_array x,
    real_1d_array y,
    real_1d_array w,
    real_1d_array c,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);
void alglib::lsfitcreatewfgh(
    real_2d_array x,
    real_1d_array y,
    real_1d_array w,
    real_1d_array c,
    ae_int_t n,
    ae_int_t m,
    ae_int_t k,
    lsfitstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lsfitfit` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear fitter

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    hess    -   callback which calculates function (or merit function)
                value func, gradient grad and Hessian hess at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

NOTES:

1. this algorithm is somewhat unusual because it works with  parameterized
   function f(C,X), where X is a function argument (we  have  many  points
   which are characterized by different  argument  values),  and  C  is  a
   parameter to fit.

   For example, if we want to do linear fit by f(c0,c1,x) = c0*x+c1,  then
   x will be argument, and {c0,c1} will be parameters.

   It is important to understand that this algorithm finds minimum in  the
   space of function PARAMETERS (not arguments), so it  needs  derivatives
   of f() with respect to C, not X.

   In the example above it will need f=c0*x+c1 and {df/dc0,df/dc1} = {x,1}
   instead of {df/dx} = {c0}.

2. Callback functions accept C as the first parameter, and X as the second

3. If  state  was  created  with  LSFitCreateFG(),  algorithm  needs  just
   function   and   its   gradient,   but   if   state   was  created with
   LSFitCreateFGH(), algorithm will need function, gradient and Hessian.

   According  to  the  said  above,  there  ase  several  versions of this
   function, which accept different sets of callbacks.

   This flexibility opens way to subtle errors - you may create state with
   LSFitCreateFGH() (optimization using Hessian), but call function  which
   does not accept Hessian. So when algorithm will request Hessian,  there
   will be no callback to call. In this case exception will be thrown.

   Be careful to avoid such errors because there is no way to find them at
   compile time - you can see them at runtime only.

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitfit(lsfitstate &state,
    void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &c, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void lsfitfit(lsfitstate &state,
    void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr),
    void (*grad)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &c, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void lsfitfit(lsfitstate &state,
    void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr),
    void (*grad)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void (*hess)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, real_2d_array &hess, void *ptr),
    void  (*rep)(const real_1d_array &c, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`lsfitlinear` function

/*************************************************************************
Linear least squares fitting.

QR decomposition is used to reduce task to MxM, then triangular solver  or
SVD-based solver is used depending on condition number of the  system.  It
allows to maximize speed and retain decent accuracy.

IMPORTANT: if you want to perform  polynomial  fitting,  it  may  be  more
           convenient to use PolynomialFit() function. This function gives
           best  results  on  polynomial  problems  and  solves  numerical
           stability  issues  which  arise  when   you   fit   high-degree
           polynomials to your data.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Y       -   array[0..N-1] Function values in  N  points.
    FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                FMatrix[I, J] - value of J-th basis function in I-th point.
    N       -   number of points used. N>=1.
    M       -   number of basis functions, M>=1.

OUTPUT PARAMETERS:
    Info    -   error code:
                * -4    internal SVD decomposition subroutine failed (very
                        rare and for degenerate systems only)
                *  1    task is solved
    C       -   decomposition coefficients, array[0..M-1]
    Rep     -   fitting report. Following fields are set:
                * Rep.TaskRCond     reciprocal of condition number
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(F*CovPar*F')),
                    where F is functions matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitlinear(
    real_1d_array y,
    real_2d_array fmatrix,
    ae_int_t& info,
    real_1d_array& c,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::lsfitlinear(
    real_1d_array y,
    real_2d_array fmatrix,
    ae_int_t n,
    ae_int_t m,
    ae_int_t& info,
    real_1d_array& c,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lsfitlinearc` function

/*************************************************************************
Constained linear least squares fitting.

This  is  variation  of LSFitLinear(),  which searchs for min|A*x=b| given
that  K  additional  constaints  C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints,  then  LSFitLinear()
is called.

IMPORTANT: if you want to perform  polynomial  fitting,  it  may  be  more
           convenient to use PolynomialFit() function. This function gives
           best  results  on  polynomial  problems  and  solves  numerical
           stability  issues  which  arise  when   you   fit   high-degree
           polynomials to your data.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Y       -   array[0..N-1] Function values in  N  points.
    FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                FMatrix[I,J] - value of J-th basis function in I-th point.
    CMatrix -   a table of constaints, array[0..K-1,0..M].
                I-th row of CMatrix corresponds to I-th linear constraint:
                CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
    N       -   number of points used. N>=1.
    M       -   number of basis functions, M>=1.
    K       -   number of constraints, 0 <= K < M
                K=0 corresponds to absence of constraints.

OUTPUT PARAMETERS:
    Info    -   error code:
                * -4    internal SVD decomposition subroutine failed (very
                        rare and for degenerate systems only)
                * -3    either   too   many  constraints  (M   or   more),
                        degenerate  constraints   (some   constraints  are
                        repetead twice) or inconsistent  constraints  were
                        specified.
                *  1    task is solved
    C       -   decomposition coefficients, array[0..M-1]
    Rep     -   fitting report. Following fields are set:
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(F*CovPar*F')),
                    where F is functions matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

IMPORTANT:  errors  in  parameters  are  calculated  without  taking  into
            account boundary/linear constraints! Presence  of  constraints
            changes distribution of errors, but there is no  easy  way  to
            account for constraints when you calculate covariance matrix.

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  -- ALGLIB --
     Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitlinearc(
    real_1d_array y,
    real_2d_array fmatrix,
    real_2d_array cmatrix,
    ae_int_t& info,
    real_1d_array& c,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::lsfitlinearc(
    real_1d_array y,
    real_2d_array fmatrix,
    real_2d_array cmatrix,
    ae_int_t n,
    ae_int_t m,
    ae_int_t k,
    ae_int_t& info,
    real_1d_array& c,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lsfitlinearw` function

/*************************************************************************
Weighted linear least squares fitting.

QR decomposition is used to reduce task to MxM, then triangular solver  or
SVD-based solver is used depending on condition number of the  system.  It
allows to maximize speed and retain decent accuracy.

IMPORTANT: if you want to perform  polynomial  fitting,  it  may  be  more
           convenient to use PolynomialFit() function. This function gives
           best  results  on  polynomial  problems  and  solves  numerical
           stability  issues  which  arise  when   you   fit   high-degree
           polynomials to your data.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Y       -   array[0..N-1] Function values in  N  points.
    W       -   array[0..N-1]  Weights  corresponding to function  values.
                Each summand in square  sum  of  approximation  deviations
                from  given  values  is  multiplied  by  the   square   of
                corresponding weight.
    FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                FMatrix[I, J] - value of J-th basis function in I-th point.
    N       -   number of points used. N>=1.
    M       -   number of basis functions, M>=1.

OUTPUT PARAMETERS:
    Info    -   error code:
                * -4    internal SVD decomposition subroutine failed (very
                        rare and for degenerate systems only)
                * -1    incorrect N/M were specified
                *  1    task is solved
    C       -   decomposition coefficients, array[0..M-1]
    Rep     -   fitting report. Following fields are set:
                * Rep.TaskRCond     reciprocal of condition number
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(F*CovPar*F')),
                    where F is functions matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitlinearw(
    real_1d_array y,
    real_1d_array w,
    real_2d_array fmatrix,
    ae_int_t& info,
    real_1d_array& c,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::lsfitlinearw(
    real_1d_array y,
    real_1d_array w,
    real_2d_array fmatrix,
    ae_int_t n,
    ae_int_t m,
    ae_int_t& info,
    real_1d_array& c,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lsfitlinearwc` function

/*************************************************************************
Weighted constained linear least squares fitting.

This  is  variation  of LSFitLinearW(), which searchs for min|A*x=b| given
that  K  additional  constaints  C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints,  then LSFitLinearW()
is called.

IMPORTANT: if you want to perform  polynomial  fitting,  it  may  be  more
           convenient to use PolynomialFit() function. This function gives
           best  results  on  polynomial  problems  and  solves  numerical
           stability  issues  which  arise  when   you   fit   high-degree
           polynomials to your data.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Y       -   array[0..N-1] Function values in  N  points.
    W       -   array[0..N-1]  Weights  corresponding to function  values.
                Each summand in square  sum  of  approximation  deviations
                from  given  values  is  multiplied  by  the   square   of
                corresponding weight.
    FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                FMatrix[I,J] - value of J-th basis function in I-th point.
    CMatrix -   a table of constaints, array[0..K-1,0..M].
                I-th row of CMatrix corresponds to I-th linear constraint:
                CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
    N       -   number of points used. N>=1.
    M       -   number of basis functions, M>=1.
    K       -   number of constraints, 0 <= K < M
                K=0 corresponds to absence of constraints.

OUTPUT PARAMETERS:
    Info    -   error code:
                * -4    internal SVD decomposition subroutine failed (very
                        rare and for degenerate systems only)
                * -3    either   too   many  constraints  (M   or   more),
                        degenerate  constraints   (some   constraints  are
                        repetead twice) or inconsistent  constraints  were
                        specified.
                *  1    task is solved
    C       -   decomposition coefficients, array[0..M-1]
    Rep     -   fitting report. Following fields are set:
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(F*CovPar*F')),
                    where F is functions matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

IMPORTANT:  errors  in  parameters  are  calculated  without  taking  into
            account boundary/linear constraints! Presence  of  constraints
            changes distribution of errors, but there is no  easy  way  to
            account for constraints when you calculate covariance matrix.

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  -- ALGLIB --
     Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitlinearwc(
    real_1d_array y,
    real_1d_array w,
    real_2d_array fmatrix,
    real_2d_array cmatrix,
    ae_int_t& info,
    real_1d_array& c,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::lsfitlinearwc(
    real_1d_array y,
    real_1d_array w,
    real_2d_array fmatrix,
    real_2d_array cmatrix,
    ae_int_t n,
    ae_int_t m,
    ae_int_t k,
    ae_int_t& info,
    real_1d_array& c,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lsfitresults` function

/*************************************************************************
Nonlinear least squares fitting results.

Called after return from LSFitFit().

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    Info    -   completion code:
                    * -8    optimizer   detected  NAN/INF  in  the  target
                            function and/or gradient
                    * -7    gradient verification failed.
                            See LSFitSetGradientCheck() for more information.
                    * -3    inconsistent constraints
                    *  2    relative step is no more than EpsX.
                    *  5    MaxIts steps was taken
                    *  7    stopping conditions are too stringent,
                            further improvement is impossible
    C       -   array[0..K-1], solution
    Rep     -   optimization report. On success following fields are set:
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED
                * WRMSError         weighted rms error on the (X,Y).

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(J*CovPar*J')),
                    where J is Jacobian matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

IMPORTANT:  errors  in  parameters  are  calculated  without  taking  into
            account boundary/linear constraints! Presence  of  constraints
            changes distribution of errors, but there is no  easy  way  to
            account for constraints when you calculate covariance matrix.

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitresults(
    lsfitstate state,
    ae_int_t& info,
    real_1d_array& c,
    lsfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`lsfitsetbc` function

/*************************************************************************
This function sets boundary constraints for underlying optimizer

Boundary constraints are inactive by default (after initial creation).
They are preserved until explicitly turned off with another SetBC() call.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[K].
                If some (all) variables are unbounded, you may specify
                very small number or -INF (latter is recommended because
                it will allow solver to use better algorithm).
    BndU    -   upper bounds, array[K].
                If some (all) variables are unbounded, you may specify
                very large number or +INF (latter is recommended because
                it will allow solver to use better algorithm).

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: unlike other constrained optimization algorithms, this solver  has
following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by bound constraints

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitsetbc(
    lsfitstate state,
    real_1d_array bndl,
    real_1d_array bndu,
    const xparams _params = alglib::xdefault);

`lsfitsetcond` function

/*************************************************************************
Stopping conditions for nonlinear least squares fitting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - ste pvector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by LSFitSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations   is    unlimited.   Only   Levenberg-Marquardt
                iterations  are  counted  (L-BFGS/CG  iterations  are  NOT
                counted because their cost is very low compared to that of
                LM).

NOTE

Passing EpsX=0  and  MaxIts=0  (simultaneously)  will  lead  to  automatic
stopping criterion selection (according to the scheme used by MINLM unit).


  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitsetcond(
    lsfitstate state,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`lsfitsetgradientcheck` function

/*************************************************************************
This  subroutine  turns  on  verification  of  the  user-supplied analytic
gradient:
* user calls this subroutine before fitting begins
* LSFitFit() is called
* prior to actual fitting, for  each  point  in  data  set  X_i  and  each
  component  of  parameters  being  fited C_j algorithm performs following
  steps:
  * two trial steps are made to C_j-TestStep*S[j] and C_j+TestStep*S[j],
    where C_j is j-th parameter and S[j] is a scale of j-th parameter
  * if needed, steps are bounded with respect to constraints on C[]
  * F(X_i|C) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point
  * in case difference between prediction and actual value is higher  than
    some predetermined threshold, algorithm stops with completion code -7;
    Rep.VarIdx is set to index of the parameter with incorrect derivative.
* after verification is over, algorithm proceeds to the actual optimization.

NOTE 1: verification needs N*K (points count * parameters count)  gradient
        evaluations. It is very costly and you should use it only for  low
        dimensional  problems,  when  you  want  to  be  sure  that you've
        correctly calculated analytic derivatives. You should not  use  it
        in the production code  (unless  you  want  to  check  derivatives
        provided by some third party).

NOTE 2: you  should  carefully  choose  TestStep. Value which is too large
        (so large that function behaviour is significantly non-cubic) will
        lead to false alarms. You may use  different  step  for  different
        parameters by means of setting scale with LSFitSetScale().

NOTE 3: this function may lead to false positives. In case it reports that
        I-th  derivative was calculated incorrectly, you may decrease test
        step  and  try  one  more  time  - maybe your function changes too
        sharply  and  your  step  is  too  large for such rapidly chanding
        function.

NOTE 4: this function works only for optimizers created with LSFitCreateWFG()
        or LSFitCreateFG() constructors.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification

  -- ALGLIB --
     Copyright 15.06.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitsetgradientcheck(
    lsfitstate state,
    double teststep,
    const xparams _params = alglib::xdefault);

`lsfitsetlc` function

/*************************************************************************
This function sets linear constraints for underlying optimizer

Linear constraints are inactive by default (after initial creation).
They are preserved until explicitly turned off with another SetLC() call.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

IMPORTANT: if you have linear constraints, it is strongly  recommended  to
           set scale of variables with lsfitsetscale(). QP solver which is
           used to calculate linearly constrained steps heavily relies  on
           good scaling of input problems.

NOTE: linear  (non-box)  constraints  are  satisfied only approximately  -
      there  always  exists some violation due  to  numerical  errors  and
      algorithmic limitations.

NOTE: general linear constraints  add  significant  overhead  to  solution
      process. Although solver performs roughly same amount of  iterations
      (when compared  with  similar  box-only  constrained  problem), each
      iteration   now    involves  solution  of  linearly  constrained  QP
      subproblem, which requires ~3-5 times more Cholesky  decompositions.
      Thus, if you can reformulate your problem in such way  this  it  has
      only box constraints, it may be beneficial to do so.

  -- ALGLIB --
     Copyright 29.04.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitsetlc(
    lsfitstate state,
    real_2d_array c,
    integer_1d_array ct,
    const xparams _params = alglib::xdefault);
void alglib::lsfitsetlc(
    lsfitstate state,
    real_2d_array c,
    integer_1d_array ct,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

`lsfitsetscale` function

/*************************************************************************
This function sets scaling coefficients for underlying optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Generally, scale is NOT considered to be a form of preconditioner.  But LM
optimizer is unique in that it uses scaling matrix both  in  the  stopping
condition tests and as Marquardt damping factor.

Proper scaling is very important for the algorithm performance. It is less
important for the quality of results, but still has some influence (it  is
easier  to  converge  when  variables  are  properly  scaled, so premature
stopping is possible when very badly scalled variables are  combined  with
relaxed stopping conditions).

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitsetscale(
    lsfitstate state,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`lsfitsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

NOTE: non-zero StpMax leads to moderate  performance  degradation  because
intermediate  step  of  preconditioned L-BFGS optimization is incompatible
with limits on step size.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitsetstpmax(
    lsfitstate state,
    double stpmax,
    const xparams _params = alglib::xdefault);

`lsfitsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

When reports are needed, State.C (current parameters) and State.F (current
value of fitting function) are reported.


  -- ALGLIB --
     Copyright 15.08.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::lsfitsetxrep(
    lsfitstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

`lstfitpiecewiselinearrdp` function

/*************************************************************************
This  subroutine fits piecewise linear curve to points with Ramer-Douglas-
Peucker algorithm, which stops after achieving desired precision.

IMPORTANT:
* it performs non-least-squares fitting; it builds curve, but  this  curve
  does not minimize some least squares  metric.  See  description  of  RDP
  algorithm (say, in Wikipedia) for more details on WHAT is performed.
* this function does NOT work with parametric curves  (i.e.  curves  which
  can be represented as {X(t),Y(t)}. It works with curves   which  can  be
  represented as Y(X). Thus, it is impossible to model figures like circles
  with this functions.
  If  you  want  to  work  with  parametric   curves,   you   should   use
  ParametricRDPFixed() function provided  by  "Parametric"  subpackage  of
  "Interpolation" package.

INPUT PARAMETERS:
    X       -   array of X-coordinates:
                * at least N elements
                * can be unordered (points are automatically sorted)
                * this function may accept non-distinct X (see below for
                  more information on handling of such inputs)
    Y       -   array of Y-coordinates:
                * at least N elements
    N       -   number of elements in X/Y
    Eps     -   positive number, desired precision.


OUTPUT PARAMETERS:
    X2      -   X-values of corner points for piecewise approximation,
                has length NSections+1 or zero (for NSections=0).
    Y2      -   Y-values of corner points,
                has length NSections+1 or zero (for NSections=0).
    NSections-  number of sections found by algorithm,
                NSections can be zero for degenerate datasets
                (N<=1 or all X[] are non-distinct).

NOTE: X2/Y2 are ordered arrays, i.e. (X2[0],Y2[0]) is  a  first  point  of
      curve, (X2[NSection-1],Y2[NSection-1]) is the last point.

  -- ALGLIB --
     Copyright 02.10.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::lstfitpiecewiselinearrdp(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    double eps,
    real_1d_array& x2,
    real_1d_array& y2,
    ae_int_t& nsections,
    const xparams _params = alglib::xdefault);

`lstfitpiecewiselinearrdpfixed` function

/*************************************************************************
This  subroutine fits piecewise linear curve to points with Ramer-Douglas-
Peucker algorithm, which stops after generating specified number of linear
sections.

IMPORTANT:
* it does NOT perform least-squares fitting; it  builds  curve,  but  this
  curve does not minimize some least squares metric.  See  description  of
  RDP algorithm (say, in Wikipedia) for more details on WHAT is performed.
* this function does NOT work with parametric curves  (i.e.  curves  which
  can be represented as {X(t),Y(t)}. It works with curves   which  can  be
  represented as Y(X). Thus,  it  is  impossible  to  model  figures  like
  circles  with  this  functions.
  If  you  want  to  work  with  parametric   curves,   you   should   use
  ParametricRDPFixed() function provided  by  "Parametric"  subpackage  of
  "Interpolation" package.

INPUT PARAMETERS:
    X       -   array of X-coordinates:
                * at least N elements
                * can be unordered (points are automatically sorted)
                * this function may accept non-distinct X (see below for
                  more information on handling of such inputs)
    Y       -   array of Y-coordinates:
                * at least N elements
    N       -   number of elements in X/Y
    M       -   desired number of sections:
                * at most M sections are generated by this function
                * less than M sections can be generated if we have N<M
                  (or some X are non-distinct).

OUTPUT PARAMETERS:
    X2      -   X-values of corner points for piecewise approximation,
                has length NSections+1 or zero (for NSections=0).
    Y2      -   Y-values of corner points,
                has length NSections+1 or zero (for NSections=0).
    NSections-  number of sections found by algorithm, NSections<=M,
                NSections can be zero for degenerate datasets
                (N<=1 or all X[] are non-distinct).

NOTE: X2/Y2 are ordered arrays, i.e. (X2[0],Y2[0]) is  a  first  point  of
      curve, (X2[NSection-1],Y2[NSection-1]) is the last point.

  -- ALGLIB --
     Copyright 02.10.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::lstfitpiecewiselinearrdpfixed(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t m,
    real_1d_array& x2,
    real_1d_array& y2,
    ae_int_t& nsections,
    const xparams _params = alglib::xdefault);

`polynomialfit` function

/*************************************************************************
Fitting by polynomials in barycentric form. This function provides  simple
unterface for unconstrained unweighted fitting. See  PolynomialFitWC()  if
you need constrained fitting.

Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver

SEE ALSO:
    PolynomialFitWC()

NOTES:
    you can convert P from barycentric form  to  the  power  or  Chebyshev
    basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions  from
    POLINT subpackage.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    N   -   number of points, N>0
            * if given, only leading N elements of X/Y are used
            * if not given, automatically determined from sizes of X/Y
    M   -   number of basis functions (= polynomial_degree + 1), M>=1

OUTPUT PARAMETERS:
    Info-   same format as in LSFitLinearW() subroutine:
            * Info>0    task is solved
            * Info<=0   an error occured:
                        -4 means inconvergence of internal SVD
    P   -   interpolant in barycentric form.
    Rep -   report, same format as in LSFitLinearW() subroutine.
            Following fields are set:
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

  -- ALGLIB PROJECT --
     Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialfit(
    real_1d_array x,
    real_1d_array y,
    ae_int_t m,
    ae_int_t& info,
    barycentricinterpolant& p,
    polynomialfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::polynomialfit(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t m,
    ae_int_t& info,
    barycentricinterpolant& p,
    polynomialfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`polynomialfitwc` function

/*************************************************************************
Weighted  fitting by polynomials in barycentric form, with constraints  on
function values or first derivatives.

Small regularizing term is used when solving constrained tasks (to improve
stability).

Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver

SEE ALSO:
    PolynomialFit()

NOTES:
    you can convert P from barycentric form  to  the  power  or  Chebyshev
    basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions  from
    POLINT subpackage.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    W   -   weights, array[0..N-1]
            Each summand in square  sum  of  approximation deviations from
            given  values  is  multiplied  by  the square of corresponding
            weight. Fill it by 1's if you don't  want  to  solve  weighted
            task.
    N   -   number of points, N>0.
            * if given, only leading N elements of X/Y/W are used
            * if not given, automatically determined from sizes of X/Y/W
    XC  -   points where polynomial values/derivatives are constrained,
            array[0..K-1].
    YC  -   values of constraints, array[0..K-1]
    DC  -   array[0..K-1], types of constraints:
            * DC[i]=0   means that P(XC[i])=YC[i]
            * DC[i]=1   means that P'(XC[i])=YC[i]
            SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
    K   -   number of constraints, 0<=K<M.
            K=0 means no constraints (XC/YC/DC are not used in such cases)
    M   -   number of basis functions (= polynomial_degree + 1), M>=1

OUTPUT PARAMETERS:
    Info-   same format as in LSFitLinearW() subroutine:
            * Info>0    task is solved
            * Info<=0   an error occured:
                        -4 means inconvergence of internal SVD
                        -3 means inconsistent constraints
    P   -   interpolant in barycentric form.
    Rep -   report, same format as in LSFitLinearW() subroutine.
            Following fields are set:
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

Setting constraints can lead  to undesired  results,  like ill-conditioned
behavior, or inconsistency being detected. From the other side,  it allows
us to improve quality of the fit. Here we summarize  our  experience  with
constrained regression splines:
* even simple constraints can be inconsistent, see  Wikipedia  article  on
  this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation
* the  greater  is  M (given  fixed  constraints),  the  more chances that
  constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the one special cases, however, we can  guarantee  consistency.  This
  case  is:  M>1  and constraints on the function values (NOT DERIVATIVES)

Our final recommendation is to use constraints  WHEN  AND  ONLY  when  you
can't solve your task without them. Anything beyond  special  cases  given
above is not guaranteed and may result in inconsistency.

  -- ALGLIB PROJECT --
     Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialfitwc(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    real_1d_array xc,
    real_1d_array yc,
    integer_1d_array dc,
    ae_int_t m,
    ae_int_t& info,
    barycentricinterpolant& p,
    polynomialfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::polynomialfitwc(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    ae_int_t n,
    real_1d_array xc,
    real_1d_array yc,
    integer_1d_array dc,
    ae_int_t k,
    ae_int_t m,
    ae_int_t& info,
    barycentricinterpolant& p,
    polynomialfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline1dfitcubic` function

/*************************************************************************
Least squares fitting by cubic spline.

This subroutine is "lightweight" alternative for more complex and feature-
rich Spline1DFitCubicWC().  See  Spline1DFitCubicWC() for more information
about subroutine parameters (we don't duplicate it here because of length)

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dfitcubic(
    real_1d_array x,
    real_1d_array y,
    ae_int_t m,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::spline1dfitcubic(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t m,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);

`spline1dfitcubicwc` function

/*************************************************************************
Weighted fitting by cubic  spline,  with constraints on function values or
derivatives.

Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is  used to build
basis functions. Basis functions are cubic splines with continuous  second
derivatives  and  non-fixed first  derivatives  at  interval  ends.  Small
regularizing term is used  when  solving  constrained  tasks  (to  improve
stability).

Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver

SEE ALSO
    Spline1DFitHermiteWC()  -   fitting by Hermite splines (more flexible,
                                less smooth)
    Spline1DFitCubic()      -   "lightweight" fitting  by  cubic  splines,
                                without invididual weights and constraints

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    W   -   weights, array[0..N-1]
            Each summand in square  sum  of  approximation deviations from
            given  values  is  multiplied  by  the square of corresponding
            weight. Fill it by 1's if you don't  want  to  solve  weighted
            task.
    N   -   number of points (optional):
            * N>0
            * if given, only first N elements of X/Y/W are processed
            * if not given, automatically determined from X/Y/W sizes
    XC  -   points where spline values/derivatives are constrained,
            array[0..K-1].
    YC  -   values of constraints, array[0..K-1]
    DC  -   array[0..K-1], types of constraints:
            * DC[i]=0   means that S(XC[i])=YC[i]
            * DC[i]=1   means that S'(XC[i])=YC[i]
            SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
    K   -   number of constraints (optional):
            * 0<=K<M.
            * K=0 means no constraints (XC/YC/DC are not used)
            * if given, only first K elements of XC/YC/DC are used
            * if not given, automatically determined from XC/YC/DC
    M   -   number of basis functions ( = number_of_nodes+2), M>=4.

OUTPUT PARAMETERS:
    Info-   same format as in LSFitLinearWC() subroutine.
            * Info>0    task is solved
            * Info<=0   an error occured:
                        -4 means inconvergence of internal SVD
                        -3 means inconsistent constraints
    S   -   spline interpolant.
    Rep -   report, same format as in LSFitLinearWC() subroutine.
            Following fields are set:
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

Setting constraints can lead  to undesired  results,  like ill-conditioned
behavior, or inconsistency being detected. From the other side,  it allows
us to improve quality of the fit. Here we summarize  our  experience  with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are  piecewise  cubic
  functions, and it is easy to create an example, where  large  number  of
  constraints  concentrated  in  small  area will result in inconsistency.
  Just because spline is not flexible enough to satisfy all of  them.  And
  same constraints spread across the  [min(x),max(x)]  will  be  perfectly
  consistent.
* the more evenly constraints are spread across [min(x),max(x)],  the more
  chances that they will be consistent
* the  greater  is  M (given  fixed  constraints),  the  more chances that
  constraints will be consistent
* in the general case, consistency of constraints IS NOT GUARANTEED.
* in the several special cases, however, we CAN guarantee consistency.
* one of this cases is constraints  on  the  function  values  AND/OR  its
  derivatives at the interval boundaries.
* another  special  case  is ONE constraint on the function value (OR, but
  not AND, derivative) anywhere in the interval

Our final recommendation is to use constraints  WHEN  AND  ONLY  WHEN  you
can't solve your task without them. Anything beyond  special  cases  given
above is not guaranteed and may result in inconsistency.


  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dfitcubicwc(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    real_1d_array xc,
    real_1d_array yc,
    integer_1d_array dc,
    ae_int_t m,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::spline1dfitcubicwc(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    ae_int_t n,
    real_1d_array xc,
    real_1d_array yc,
    integer_1d_array dc,
    ae_int_t k,
    ae_int_t m,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);

`spline1dfithermite` function

/*************************************************************************
Least squares fitting by Hermite spline.

This subroutine is "lightweight" alternative for more complex and feature-
rich Spline1DFitHermiteWC().  See Spline1DFitHermiteWC()  description  for
more information about subroutine parameters (we don't duplicate  it  here
because of length).

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dfithermite(
    real_1d_array x,
    real_1d_array y,
    ae_int_t m,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::spline1dfithermite(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t m,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);

`spline1dfithermitewc` function

/*************************************************************************
Weighted  fitting  by Hermite spline,  with constraints on function values
or first derivatives.

Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is  used  to  build
basis functions. Basis functions are Hermite splines.  Small  regularizing
term is used when solving constrained tasks (to improve stability).

Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver

SEE ALSO
    Spline1DFitCubicWC()    -   fitting by Cubic splines (less flexible,
                                more smooth)
    Spline1DFitHermite()    -   "lightweight" Hermite fitting, without
                                invididual weights and constraints

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    W   -   weights, array[0..N-1]
            Each summand in square  sum  of  approximation deviations from
            given  values  is  multiplied  by  the square of corresponding
            weight. Fill it by 1's if you don't  want  to  solve  weighted
            task.
    N   -   number of points (optional):
            * N>0
            * if given, only first N elements of X/Y/W are processed
            * if not given, automatically determined from X/Y/W sizes
    XC  -   points where spline values/derivatives are constrained,
            array[0..K-1].
    YC  -   values of constraints, array[0..K-1]
    DC  -   array[0..K-1], types of constraints:
            * DC[i]=0   means that S(XC[i])=YC[i]
            * DC[i]=1   means that S'(XC[i])=YC[i]
            SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
    K   -   number of constraints (optional):
            * 0<=K<M.
            * K=0 means no constraints (XC/YC/DC are not used)
            * if given, only first K elements of XC/YC/DC are used
            * if not given, automatically determined from XC/YC/DC
    M   -   number of basis functions (= 2 * number of nodes),
            M>=4,
            M IS EVEN!

OUTPUT PARAMETERS:
    Info-   same format as in LSFitLinearW() subroutine:
            * Info>0    task is solved
            * Info<=0   an error occured:
                        -4 means inconvergence of internal SVD
                        -3 means inconsistent constraints
                        -2 means odd M was passed (which is not supported)
                        -1 means another errors in parameters passed
                           (N<=0, for example)
    S   -   spline interpolant.
    Rep -   report, same format as in LSFitLinearW() subroutine.
            Following fields are set:
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

IMPORTANT:
    this subroitine supports only even M's


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

Setting constraints can lead  to undesired  results,  like ill-conditioned
behavior, or inconsistency being detected. From the other side,  it allows
us to improve quality of the fit. Here we summarize  our  experience  with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are  piecewise  cubic
  functions, and it is easy to create an example, where  large  number  of
  constraints  concentrated  in  small  area will result in inconsistency.
  Just because spline is not flexible enough to satisfy all of  them.  And
  same constraints spread across the  [min(x),max(x)]  will  be  perfectly
  consistent.
* the more evenly constraints are spread across [min(x),max(x)],  the more
  chances that they will be consistent
* the  greater  is  M (given  fixed  constraints),  the  more chances that
  constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the several special cases, however, we can guarantee consistency.
* one of this cases is  M>=4  and   constraints  on   the  function  value
  (AND/OR its derivative) at the interval boundaries.
* another special case is M>=4  and  ONE  constraint on the function value
  (OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)]

Our final recommendation is to use constraints  WHEN  AND  ONLY  when  you
can't solve your task without them. Anything beyond  special  cases  given
above is not guaranteed and may result in inconsistency.

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dfithermitewc(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    real_1d_array xc,
    real_1d_array yc,
    integer_1d_array dc,
    ae_int_t m,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::spline1dfithermitewc(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    ae_int_t n,
    real_1d_array xc,
    real_1d_array yc,
    integer_1d_array dc,
    ae_int_t k,
    ae_int_t m,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);

lsfit_d_lin example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // In this example we demonstrate linear fitting by f(x|a) = a*exp(0.5*x).
    //
    // We have:
    // * y - vector of experimental data
    // * fmatrix -  matrix of basis functions calculated at sample points
    //              Actually, we have only one basis function F0 = exp(0.5*x).
    //
    real_2d_array fmatrix = "[[0.606531],[0.670320],[0.740818],[0.818731],[0.904837],[1.000000],[1.105171],[1.221403],[1.349859],[1.491825],[1.648721]]";
    real_1d_array y = "[1.133719, 1.306522, 1.504604, 1.554663, 1.884638, 2.072436, 2.257285, 2.534068, 2.622017, 2.897713, 3.219371]";
    ae_int_t info;
    real_1d_array c;
    lsfitreport rep;

    //
    // Linear fitting without weights
    //
    lsfitlinear(y, fmatrix, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [1.98650]

    //
    // Linear fitting with individual weights.
    // Slightly different result is returned.
    //
    real_1d_array w = "[1.414213, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]";
    lsfitlinearw(y, w, fmatrix, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [1.983354]
    return 0;
}

lsfit_d_linc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // In this example we demonstrate linear fitting by f(x|a,b) = a*x+b
    // with simple constraint f(0)=0.
    //
    // We have:
    // * y - vector of experimental data
    // * fmatrix -  matrix of basis functions sampled at [0,1] with step 0.2:
    //                  [ 1.0   0.0 ]
    //                  [ 1.0   0.2 ]
    //                  [ 1.0   0.4 ]
    //                  [ 1.0   0.6 ]
    //                  [ 1.0   0.8 ]
    //                  [ 1.0   1.0 ]
    //              first column contains value of first basis function (constant term)
    //              second column contains second basis function (linear term)
    // * cmatrix -  matrix of linear constraints:
    //                  [ 1.0  0.0  0.0 ]
    //              first two columns contain coefficients before basis functions,
    //              last column contains desired value of their sum.
    //              So [1,0,0] means "1*constant_term + 0*linear_term = 0" 
    //
    real_1d_array y = "[0.072436,0.246944,0.491263,0.522300,0.714064,0.921929]";
    real_2d_array fmatrix = "[[1,0.0],[1,0.2],[1,0.4],[1,0.6],[1,0.8],[1,1.0]]";
    real_2d_array cmatrix = "[[1,0,0]]";
    ae_int_t info;
    real_1d_array c;
    lsfitreport rep;

    //
    // Constrained fitting without weights
    //
    lsfitlinearc(y, fmatrix, cmatrix, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [0,0.932933]

    //
    // Constrained fitting with individual weights
    //
    real_1d_array w = "[1, 1.414213, 1, 1, 1, 1]";
    lsfitlinearwc(y, w, fmatrix, cmatrix, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [0,0.938322]
    return 0;
}

lsfit_d_nlf example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0))
    // where x is a position on X-axis and c is adjustable parameter
    func = exp(-c[0]*pow(x[0],2));
}

int main(int argc, char **argv)
{
    //
    // In this example we demonstrate exponential fitting
    // by f(x) = exp(-c*x^2)
    // using function value only.
    //
    // Gradient is estimated using combination of numerical differences
    // and secant updates. diffstep variable stores differentiation step 
    // (we have to tell algorithm what step to use).
    //
    real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
    real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
    real_1d_array c = "[0.3]";
    double epsx = 0.000001;
    ae_int_t maxits = 0;
    ae_int_t info;
    lsfitstate state;
    lsfitreport rep;
    double diffstep = 0.0001;

    //
    // Fitting without weights
    //
    lsfitcreatef(x, y, c, diffstep, state);
    lsfitsetcond(state, epsx, maxits);
    alglib::lsfitfit(state, function_cx_1_func);
    lsfitresults(state, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 2
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]

    //
    // Fitting with weights
    // (you can change weights and see how it changes result)
    //
    real_1d_array w = "[1,1,1,1,1,1,1,1,1,1,1]";
    lsfitcreatewf(x, y, w, c, diffstep, state);
    lsfitsetcond(state, epsx, maxits);
    alglib::lsfitfit(state, function_cx_1_func);
    lsfitresults(state, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 2
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
    return 0;
}

lsfit_d_nlfb example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0))
    // where x is a position on X-axis and c is adjustable parameter
    func = exp(-c[0]*pow(x[0],2));
}

int main(int argc, char **argv)
{
    //
    // In this example we demonstrate exponential fitting by
    //     f(x) = exp(-c*x^2)
    // subject to bound constraints
    //     0.0 <= c <= 1.0
    // using function value only.
    //
    // Gradient is estimated using combination of numerical differences
    // and secant updates. diffstep variable stores differentiation step 
    // (we have to tell algorithm what step to use).
    //
    // Unconstrained solution is c=1.5, but because of constraints we should
    // get c=1.0 (at the boundary).
    //
    real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
    real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
    real_1d_array c = "[0.3]";
    real_1d_array bndl = "[0.0]";
    real_1d_array bndu = "[1.0]";
    double epsx = 0.000001;
    ae_int_t maxits = 0;
    ae_int_t info;
    lsfitstate state;
    lsfitreport rep;
    double diffstep = 0.0001;

    lsfitcreatef(x, y, c, diffstep, state);
    lsfitsetbc(state, bndl, bndu);
    lsfitsetcond(state, epsx, maxits);
    alglib::lsfitfit(state, function_cx_1_func);
    lsfitresults(state, info, c, rep);
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.0]
    return 0;
}

lsfit_d_nlfg example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0))
    // where x is a position on X-axis and c is adjustable parameter
    func = exp(-c[0]*pow(x[0],2));
}
void function_cx_1_grad(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0)) and gradient G={df/dc[i]}
    // where x is a position on X-axis and c is adjustable parameter.
    // IMPORTANT: gradient is calculated with respect to C, not to X
    func = exp(-c[0]*pow(x[0],2));
    grad[0] = -pow(x[0],2)*func;
}

int main(int argc, char **argv)
{
    //
    // In this example we demonstrate exponential fitting
    // by f(x) = exp(-c*x^2)
    // using function value and gradient (with respect to c).
    //
    real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
    real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
    real_1d_array c = "[0.3]";
    double epsx = 0.000001;
    ae_int_t maxits = 0;
    ae_int_t info;
    lsfitstate state;
    lsfitreport rep;

    //
    // Fitting without weights
    //
    lsfitcreatefg(x, y, c, true, state);
    lsfitsetcond(state, epsx, maxits);
    alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad);
    lsfitresults(state, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 2
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]

    //
    // Fitting with weights
    // (you can change weights and see how it changes result)
    //
    real_1d_array w = "[1,1,1,1,1,1,1,1,1,1,1]";
    lsfitcreatewfg(x, y, w, c, true, state);
    lsfitsetcond(state, epsx, maxits);
    alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad);
    lsfitresults(state, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 2
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
    return 0;
}

lsfit_d_nlfgh example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0))
    // where x is a position on X-axis and c is adjustable parameter
    func = exp(-c[0]*pow(x[0],2));
}
void function_cx_1_grad(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0)) and gradient G={df/dc[i]}
    // where x is a position on X-axis and c is adjustable parameter.
    // IMPORTANT: gradient is calculated with respect to C, not to X
    func = exp(-c[0]*pow(x[0],2));
    grad[0] = -pow(x[0],2)*func;
}
void function_cx_1_hess(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, real_2d_array &hess, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0)), gradient G={df/dc[i]} and Hessian H={d2f/(dc[i]*dc[j])}
    // where x is a position on X-axis and c is adjustable parameter.
    // IMPORTANT: gradient/Hessian are calculated with respect to C, not to X
    func = exp(-c[0]*pow(x[0],2));
    grad[0] = -pow(x[0],2)*func;
    hess[0][0] = pow(x[0],4)*func;
}

int main(int argc, char **argv)
{
    //
    // In this example we demonstrate exponential fitting
    // by f(x) = exp(-c*x^2)
    // using function value, gradient and Hessian (with respect to c)
    //
    real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
    real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
    real_1d_array c = "[0.3]";
    double epsx = 0.000001;
    ae_int_t maxits = 0;
    ae_int_t info;
    lsfitstate state;
    lsfitreport rep;

    //
    // Fitting without weights
    //
    lsfitcreatefgh(x, y, c, state);
    lsfitsetcond(state, epsx, maxits);
    alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad, function_cx_1_hess);
    lsfitresults(state, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 2
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]

    //
    // Fitting with weights
    // (you can change weights and see how it changes result)
    //
    real_1d_array w = "[1,1,1,1,1,1,1,1,1,1,1]";
    lsfitcreatewfgh(x, y, w, c, state);
    lsfitsetcond(state, epsx, maxits);
    alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad, function_cx_1_hess);
    lsfitresults(state, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 2
    printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
    return 0;
}

lsfit_d_nlscale example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;
void function_debt_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr) 
{
    //
    // this callback calculates f(c,x)=c[0]*(1+c[1]*(pow(x[0]-1999,c[2])-1))
    //
    func = c[0]*(1+c[1]*(pow(x[0]-1999,c[2])-1));
}

int main(int argc, char **argv)
{
    //
    // In this example we demonstrate fitting by
    //     f(x) = c[0]*(1+c[1]*((x-1999)^c[2]-1))
    // subject to bound constraints
    //     -INF  < c[0] < +INF
    //      -10 <= c[1] <= +10
    //      0.1 <= c[2] <= 2.0
    // Data we want to fit are time series of Japan national debt
    // collected from 2000 to 2008 measured in USD (dollars, not
    // millions of dollars).
    //
    // Our variables are:
    //     c[0] - debt value at initial moment (2000),
    //     c[1] - direction coefficient (growth or decrease),
    //     c[2] - curvature coefficient.
    // You may see that our variables are badly scaled - first one 
    // is order of 10^12, and next two are somewhere about 1 in 
    // magnitude. Such problem is difficult to solve without some
    // kind of scaling.
    // That is exactly where lsfitsetscale() function can be used.
    // We set scale of our variables to [1.0E12, 1, 1], which allows
    // us to easily solve this problem.
    //
    // You can try commenting out lsfitsetscale() call - and you will 
    // see that algorithm will fail to converge.
    //
    real_2d_array x = "[[2000],[2001],[2002],[2003],[2004],[2005],[2006],[2007],[2008]]";
    real_1d_array y = "[4323239600000.0, 4560913100000.0, 5564091500000.0, 6743189300000.0, 7284064600000.0, 7050129600000.0, 7092221500000.0, 8483907600000.0, 8625804400000.0]";
    real_1d_array c = "[1.0e+13, 1, 1]";
    double epsx = 1.0e-5;
    real_1d_array bndl = "[-inf, -10, 0.1]";
    real_1d_array bndu = "[+inf, +10, 2.0]";
    real_1d_array s = "[1.0e+12, 1, 1]";
    ae_int_t maxits = 0;
    ae_int_t info;
    lsfitstate state;
    lsfitreport rep;
    double diffstep = 1.0e-5;

    lsfitcreatef(x, y, c, diffstep, state);
    lsfitsetcond(state, epsx, maxits);
    lsfitsetbc(state, bndl, bndu);
    lsfitsetscale(state, s);
    alglib::lsfitfit(state, function_debt_func);
    lsfitresults(state, info, c, rep);
    printf("%d\n", int(info)); // EXPECTED: 2
    printf("%s\n", c.tostring(-2).c_str()); // EXPECTED: [4.142560E+12, 0.434240, 0.565376]
    return 0;
}

lsfit_d_pol example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates polynomial fitting.
    //
    // Fitting is done by two (M=2) functions from polynomial basis:
    //     f0 = 1
    //     f1 = x
    // Basically, it just a linear fit; more complex polynomials may be used
    // (e.g. parabolas with M=3, cubic with M=4), but even such simple fit allows
    // us to demonstrate polynomialfit() function in action.
    //
    // We have:
    // * x      set of abscissas
    // * y      experimental data
    //
    // Additionally we demonstrate weighted fitting, where second point has
    // more weight than other ones.
    //
    real_1d_array x = "[0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]";
    real_1d_array y = "[0.00,0.05,0.26,0.32,0.33,0.43,0.60,0.60,0.77,0.98,1.02]";
    ae_int_t m = 2;
    double t = 2;
    ae_int_t info;
    barycentricinterpolant p;
    polynomialfitreport rep;
    double v;

    //
    // Fitting without individual weights
    //
    // NOTE: result is returned as barycentricinterpolant structure.
    //       if you want to get representation in the power basis,
    //       you can use barycentricbar2pow() function to convert
    //       from barycentric to power representation (see docs for 
    //       POLINT subpackage for more info).
    //
    polynomialfit(x, y, m, info, p, rep);
    v = barycentriccalc(p, t);
    printf("%.2f\n", double(v)); // EXPECTED: 2.011

    //
    // Fitting with individual weights
    //
    // NOTE: slightly different result is returned
    //
    real_1d_array w = "[1,1.414213562,1,1,1,1,1,1,1,1,1]";
    real_1d_array xc = "[]";
    real_1d_array yc = "[]";
    integer_1d_array dc = "[]";
    polynomialfitwc(x, y, w, xc, yc, dc, m, info, p, rep);
    v = barycentriccalc(p, t);
    printf("%.2f\n", double(v)); // EXPECTED: 2.023
    return 0;
}

lsfit_d_polc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates polynomial fitting.
    //
    // Fitting is done by two (M=2) functions from polynomial basis:
    //     f0 = 1
    //     f1 = x
    // with simple constraint on function value
    //     f(0) = 0
    // Basically, it just a linear fit; more complex polynomials may be used
    // (e.g. parabolas with M=3, cubic with M=4), but even such simple fit allows
    // us to demonstrate polynomialfit() function in action.
    //
    // We have:
    // * x      set of abscissas
    // * y      experimental data
    // * xc     points where constraints are placed
    // * yc     constraints on derivatives
    // * dc     derivative indices
    //          (0 means function itself, 1 means first derivative)
    //
    real_1d_array x = "[1.0,1.0]";
    real_1d_array y = "[0.9,1.1]";
    real_1d_array w = "[1,1]";
    real_1d_array xc = "[0]";
    real_1d_array yc = "[0]";
    integer_1d_array dc = "[0]";
    double t = 2;
    ae_int_t m = 2;
    ae_int_t info;
    barycentricinterpolant p;
    polynomialfitreport rep;
    double v;

    polynomialfitwc(x, y, w, xc, yc, dc, m, info, p, rep);
    v = barycentriccalc(p, t);
    printf("%.2f\n", double(v)); // EXPECTED: 2.000
    return 0;
}

lsfit_d_spline example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // In this example we demonstrate penalized spline fitting of noisy data
    //
    // We have:
    // * x - abscissas
    // * y - vector of experimental data, straight line with small noise
    //
    real_1d_array x = "[0.00,0.10,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90]";
    real_1d_array y = "[0.10,0.00,0.30,0.40,0.30,0.40,0.62,0.68,0.75,0.95]";
    ae_int_t info;
    double v;
    spline1dinterpolant s;
    spline1dfitreport rep;
    double rho;

    //
    // Fit with VERY small amount of smoothing (rho = -5.0)
    // and large number of basis functions (M=50).
    //
    // With such small regularization penalized spline almost fully reproduces function values
    //
    rho = -5.0;
    spline1dfitpenalized(x, y, 50, rho, info, s, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    v = spline1dcalc(s, 0.0);
    printf("%.1f\n", double(v)); // EXPECTED: 0.10

    //
    // Fit with VERY large amount of smoothing (rho = 10.0)
    // and large number of basis functions (M=50).
    //
    // With such regularization our spline should become close to the straight line fit.
    // We will compare its value in x=1.0 with results obtained from such fit.
    //
    rho = +10.0;
    spline1dfitpenalized(x, y, 50, rho, info, s, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    v = spline1dcalc(s, 1.0);
    printf("%.2f\n", double(v)); // EXPECTED: 0.969

    //
    // In real life applications you may need some moderate degree of fitting,
    // so we try to fit once more with rho=3.0.
    //
    rho = +3.0;
    spline1dfitpenalized(x, y, 50, rho, info, s, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    return 0;
}

lsfit_t_4pl example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_1d_array x = "[1,2,3,4,5,6,7,8]";
    real_1d_array y = "[0.06313223,0.44552624,0.61838364,0.71385108,0.77345838,0.81383140,0.84280033,0.86449822]";
    ae_int_t n = 8;
    double a;
    double b;
    double c;
    double d;
    lsfitreport rep;

    //
    // Test logisticfit4() on carefully designed data with a priori known answer.
    //
    logisticfit4(x, y, n, a, b, c, d, rep);
    printf("%.1f\n", double(a)); // EXPECTED: -1.000
    printf("%.1f\n", double(b)); // EXPECTED: 1.200
    printf("%.1f\n", double(c)); // EXPECTED: 0.900
    printf("%.1f\n", double(d)); // EXPECTED: 1.000

    //
    // Evaluate model at point x=0.5
    //
    double v;
    v = logisticcalc4(0.5, a, b, c, d);
    printf("%.2f\n", double(v)); // EXPECTED: -0.33874308
    return 0;
}

lsfit_t_5pl example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_1d_array x = "[1,2,3,4,5,6,7,8]";
    real_1d_array y = "[0.1949776139,0.5710060208,0.726002637,0.8060434158,0.8534547965,0.8842071579,0.9054773317,0.9209088299]";
    ae_int_t n = 8;
    double a;
    double b;
    double c;
    double d;
    double g;
    lsfitreport rep;

    //
    // Test logisticfit5() on carefully designed data with a priori known answer.
    //
    logisticfit5(x, y, n, a, b, c, d, g, rep);
    printf("%.1f\n", double(a)); // EXPECTED: -1.000
    printf("%.1f\n", double(b)); // EXPECTED: 1.200
    printf("%.1f\n", double(c)); // EXPECTED: 0.900
    printf("%.1f\n", double(d)); // EXPECTED: 1.000
    printf("%.1f\n", double(g)); // EXPECTED: 1.200

    //
    // Evaluate model at point x=0.5
    //
    double v;
    v = logisticcalc5(0.5, a, b, c, d, g);
    printf("%.2f\n", double(v)); // EXPECTED: -0.2354656824
    return 0;
}

`mannwhitneyu` subpackage

Functions

mannwhitneyutest

Examples

`mannwhitneyutest` function

/*************************************************************************
Mann-Whitney U-test

This test checks hypotheses about whether X  and  Y  are  samples  of  two
continuous distributions of the same shape  and  same  median  or  whether
their medians are different.

The following tests are performed:
    * two-tailed test (null hypothesis - the medians are equal)
    * left-tailed test (null hypothesis - the median of the  first  sample
      is greater than or equal to the median of the second sample)
    * right-tailed test (null hypothesis - the median of the first  sample
      is less than or equal to the median of the second sample).

Requirements:
    * the samples are independent
    * X and Y are continuous distributions (or discrete distributions well-
      approximating continuous distributions)
    * distributions of X and Y have the  same  shape.  The  only  possible
      difference is their position (i.e. the value of the median)
    * the number of elements in each sample is not less than 5
    * the scale of measurement should be ordinal, interval or ratio  (i.e.
      the test could not be applied to nominal variables).

The test is non-parametric and doesn't require distributions to be normal.

Input parameters:
    X   -   sample 1. Array whose index goes from 0 to N-1.
    N   -   size of the sample. N>=5
    Y   -   sample 2. Array whose index goes from 0 to M-1.
    M   -   size of the sample. M>=5

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

To calculate p-values, special approximation is used. This method lets  us
calculate p-values with satisfactory  accuracy  in  interval  [0.0001, 1].
There is no approximation outside the [0.0001, 1] interval. Therefore,  if
the significance level outlies this interval, the test returns 0.0001.

Relative precision of approximation of p-value:

N          M          Max.err.   Rms.err.
5..10      N..10      1.4e-02    6.0e-04
5..10      N..100     2.2e-02    5.3e-06
10..15     N..15      1.0e-02    3.2e-04
10..15     N..100     1.0e-02    2.2e-05
15..100    N..100     6.1e-03    2.7e-06

For N,M>100 accuracy checks weren't put into  practice,  but  taking  into
account characteristics of asymptotic approximation used, precision should
not be sharply different from the values for interval [5, 100].

NOTE: P-value approximation was  optimized  for  0.0001<=p<=0.2500.  Thus,
      P's outside of this interval are enforced to these bounds. Say,  you
      may quite often get P equal to exactly 0.25 or 0.0001.

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mannwhitneyutest(
    real_1d_array x,
    ae_int_t n,
    real_1d_array y,
    ae_int_t m,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`matdet` subpackage

Functions

cmatrixdet
cmatrixludet
rmatrixdet
rmatrixludet
spdmatrixcholeskydet
spdmatrixdet

Examples

matdet_d_1		Determinant calculation, real matrix, short form
matdet_d_2		Determinant calculation, real matrix, full form
matdet_d_3		Determinant calculation, complex matrix, short form
matdet_d_4		Determinant calculation, complex matrix, full form
matdet_d_5		Determinant calculation, complex matrix with zero imaginary part, short form

`cmatrixdet` function

/*************************************************************************
Calculation of the determinant of a general matrix

Input parameters:
    A       -   matrix, array[0..N-1, 0..N-1]
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

Result: determinant of matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
alglib::complex alglib::cmatrixdet(
    complex_2d_array a,
    const xparams _params = alglib::xdefault);
alglib::complex alglib::cmatrixdet(
    complex_2d_array a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`cmatrixludet` function

/*************************************************************************
Determinant calculation of the matrix given by its LU decomposition.

Input parameters:
    A       -   LU decomposition of the matrix (output of
                RMatrixLU subroutine).
    Pivots  -   table of permutations which were made during
                the LU decomposition.
                Output of RMatrixLU subroutine.
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

Result: matrix determinant.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
alglib::complex alglib::cmatrixludet(
    complex_2d_array a,
    integer_1d_array pivots,
    const xparams _params = alglib::xdefault);
alglib::complex alglib::cmatrixludet(
    complex_2d_array a,
    integer_1d_array pivots,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`rmatrixdet` function

/*************************************************************************
Calculation of the determinant of a general matrix

Input parameters:
    A       -   matrix, array[0..N-1, 0..N-1]
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

Result: determinant of matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
double alglib::rmatrixdet(
    real_2d_array a,
    const xparams _params = alglib::xdefault);
double alglib::rmatrixdet(
    real_2d_array a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`rmatrixludet` function

/*************************************************************************
Determinant calculation of the matrix given by its LU decomposition.

Input parameters:
    A       -   LU decomposition of the matrix (output of
                RMatrixLU subroutine).
    Pivots  -   table of permutations which were made during
                the LU decomposition.
                Output of RMatrixLU subroutine.
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

Result: matrix determinant.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
double alglib::rmatrixludet(
    real_2d_array a,
    integer_1d_array pivots,
    const xparams _params = alglib::xdefault);
double alglib::rmatrixludet(
    real_2d_array a,
    integer_1d_array pivots,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskydet` function

/*************************************************************************
Determinant calculation of the matrix given by the Cholesky decomposition.

Input parameters:
    A       -   Cholesky decomposition,
                output of SMatrixCholesky subroutine.
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

As the determinant is equal to the product of squares of diagonal elements,
it's not necessary to specify which triangle - lower or upper - the matrix
is stored in.

Result:
    matrix determinant.

  -- ALGLIB --
     Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
double alglib::spdmatrixcholeskydet(
    real_2d_array a,
    const xparams _params = alglib::xdefault);
double alglib::spdmatrixcholeskydet(
    real_2d_array a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`spdmatrixdet` function

/*************************************************************************
Determinant calculation of the symmetric positive definite matrix.

Input parameters:
    A       -   matrix. Array with elements [0..N-1, 0..N-1].
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)
    IsUpper -   (optional) storage type:
                * if True, symmetric matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't used/changed  by
                  function
                * if False, symmetric matrix  A  is  given  by  its lower
                  triangle, and the upper triangle isn't used/changed  by
                  function
                * if not given, both lower and upper  triangles  must  be
                  filled.

Result:
    determinant of matrix A.
    If matrix A is not positive definite, exception is thrown.

  -- ALGLIB --
     Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
double alglib::spdmatrixdet(
    real_2d_array a,
    const xparams _params = alglib::xdefault);
double alglib::spdmatrixdet(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    const xparams _params = alglib::xdefault);

matdet_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_2d_array b = "[[1,2],[2,1]]";
    double a;
    a = rmatrixdet(b);
    printf("%.3f\n", double(a)); // EXPECTED: -3
    return 0;
}

matdet_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_2d_array b = "[[5,4],[4,5]]";
    double a;
    a = rmatrixdet(b, 2);
    printf("%.3f\n", double(a)); // EXPECTED: 9
    return 0;
}

matdet_d_3 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    complex_2d_array b = "[[1+1i,2],[2,1-1i]]";
    alglib::complex a;
    a = cmatrixdet(b);
    printf("%s\n", a.tostring(3).c_str()); // EXPECTED: -2
    return 0;
}

matdet_d_4 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    alglib::complex a;
    complex_2d_array b = "[[5i,4],[4i,5]]";
    a = cmatrixdet(b, 2);
    printf("%s\n", a.tostring(3).c_str()); // EXPECTED: 9i
    return 0;
}

matdet_d_5 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    alglib::complex a;
    complex_2d_array b = "[[9,1],[2,1]]";
    a = cmatrixdet(b);
    printf("%s\n", a.tostring(3).c_str()); // EXPECTED: 7
    return 0;
}

`matgen` subpackage

Functions

cmatrixrndcond
cmatrixrndorthogonal
cmatrixrndorthogonalfromtheleft
cmatrixrndorthogonalfromtheright
hmatrixrndcond
hmatrixrndmultiply
hpdmatrixrndcond
rmatrixrndcond
rmatrixrndorthogonal
rmatrixrndorthogonalfromtheleft
rmatrixrndorthogonalfromtheright
smatrixrndcond
smatrixrndmultiply
spdmatrixrndcond

Examples

`cmatrixrndcond` function

/*************************************************************************
Generation of random NxN complex matrix with given condition number C and
norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixrndcond(
    ae_int_t n,
    double c,
    complex_2d_array& a,
    const xparams _params = alglib::xdefault);

`cmatrixrndorthogonal` function

/*************************************************************************
Generation of a random Haar distributed orthogonal complex matrix

INPUT PARAMETERS:
    N   -   matrix size, N>=1

OUTPUT PARAMETERS:
    A   -   orthogonal NxN matrix, array[0..N-1,0..N-1]

NOTE: this function uses algorithm  described  in  Stewart, G. W.  (1980),
      "The Efficient Generation of  Random  Orthogonal  Matrices  with  an
      Application to Condition Estimators".

      Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it:
      * takes an NxN one
      * takes uniformly distributed unit vector of dimension N+1.
      * constructs a Householder reflection from the vector, then applies
        it to the smaller matrix (embedded in the larger size with a 1 at
        the bottom right corner).

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixrndorthogonal(
    ae_int_t n,
    complex_2d_array& a,
    const xparams _params = alglib::xdefault);

`cmatrixrndorthogonalfromtheleft` function

/*************************************************************************
Multiplication of MxN complex matrix by MxM random Haar distributed
complex orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..M-1, 0..N-1]
    M, N-   matrix size

OUTPUT PARAMETERS:
    A   -   Q*A, where Q is random MxM orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixrndorthogonalfromtheleft(
    complex_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`cmatrixrndorthogonalfromtheright` function

/*************************************************************************
Multiplication of MxN complex matrix by NxN random Haar distributed
complex orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..M-1, 0..N-1]
    M, N-   matrix size

OUTPUT PARAMETERS:
    A   -   A*Q, where Q is random NxN orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixrndorthogonalfromtheright(
    complex_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`hmatrixrndcond` function

/*************************************************************************
Generation of random NxN Hermitian matrix with given condition number  and
norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::hmatrixrndcond(
    ae_int_t n,
    double c,
    complex_2d_array& a,
    const xparams _params = alglib::xdefault);

`hmatrixrndmultiply` function

/*************************************************************************
Hermitian multiplication of NxN matrix by random Haar distributed
complex orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..N-1, 0..N-1]
    N   -   matrix size

OUTPUT PARAMETERS:
    A   -   Q^H*A*Q, where Q is random NxN orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::hmatrixrndmultiply(
    complex_2d_array& a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`hpdmatrixrndcond` function

/*************************************************************************
Generation of random NxN Hermitian positive definite matrix with given
condition number and norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random HPD matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixrndcond(
    ae_int_t n,
    double c,
    complex_2d_array& a,
    const xparams _params = alglib::xdefault);

`rmatrixrndcond` function

/*************************************************************************
Generation of random NxN matrix with given condition number and norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixrndcond(
    ae_int_t n,
    double c,
    real_2d_array& a,
    const xparams _params = alglib::xdefault);

`rmatrixrndorthogonal` function

/*************************************************************************
Generation of a random uniformly distributed (Haar) orthogonal matrix

INPUT PARAMETERS:
    N   -   matrix size, N>=1

OUTPUT PARAMETERS:
    A   -   orthogonal NxN matrix, array[0..N-1,0..N-1]

NOTE: this function uses algorithm  described  in  Stewart, G. W.  (1980),
      "The Efficient Generation of  Random  Orthogonal  Matrices  with  an
      Application to Condition Estimators".

      Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it:
      * takes an NxN one
      * takes uniformly distributed unit vector of dimension N+1.
      * constructs a Householder reflection from the vector, then applies
        it to the smaller matrix (embedded in the larger size with a 1 at
        the bottom right corner).

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixrndorthogonal(
    ae_int_t n,
    real_2d_array& a,
    const xparams _params = alglib::xdefault);

`rmatrixrndorthogonalfromtheleft` function

/*************************************************************************
Multiplication of MxN matrix by MxM random Haar distributed orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..M-1, 0..N-1]
    M, N-   matrix size

OUTPUT PARAMETERS:
    A   -   Q*A, where Q is random MxM orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixrndorthogonalfromtheleft(
    real_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`rmatrixrndorthogonalfromtheright` function

/*************************************************************************
Multiplication of MxN matrix by NxN random Haar distributed orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..M-1, 0..N-1]
    M, N-   matrix size

OUTPUT PARAMETERS:
    A   -   A*Q, where Q is random NxN orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixrndorthogonalfromtheright(
    real_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`smatrixrndcond` function

/*************************************************************************
Generation of random NxN symmetric matrix with given condition number  and
norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::smatrixrndcond(
    ae_int_t n,
    double c,
    real_2d_array& a,
    const xparams _params = alglib::xdefault);

`smatrixrndmultiply` function

/*************************************************************************
Symmetric multiplication of NxN matrix by random Haar distributed
orthogonal  matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..N-1, 0..N-1]
    N   -   matrix size

OUTPUT PARAMETERS:
    A   -   Q'*A*Q, where Q is random NxN orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::smatrixrndmultiply(
    real_2d_array& a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`spdmatrixrndcond` function

/*************************************************************************
Generation of random NxN symmetric positive definite matrix with given
condition number and norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random SPD matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixrndcond(
    ae_int_t n,
    double c,
    real_2d_array& a,
    const xparams _params = alglib::xdefault);

`matinv` subpackage

Classes

matinvreport

Functions

cmatrixinverse
cmatrixluinverse
cmatrixtrinverse
hpdmatrixcholeskyinverse
hpdmatrixinverse
rmatrixinverse
rmatrixluinverse
rmatrixtrinverse
spdmatrixcholeskyinverse
spdmatrixinverse

Examples

matinv_d_c1		Complex matrix inverse
matinv_d_hpd1		HPD matrix inverse
matinv_d_r1		Real matrix inverse
matinv_d_spd1		SPD matrix inverse

`matinvreport` class

/*************************************************************************
Matrix inverse report:
* R1    reciprocal of condition number in 1-norm
* RInf  reciprocal of condition number in inf-norm
*************************************************************************/
class matinvreport
{
    double               r1;
    double               rinf;
};

`cmatrixinverse` function

/*************************************************************************
Inversion of a general matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)

Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixinverse(
    complex_2d_array& a,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::cmatrixinverse(
    complex_2d_array& a,
    ae_int_t n,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`cmatrixluinverse` function

/*************************************************************************
Inversion of a matrix given by its LU decomposition.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    A       -   LU decomposition of the matrix
                (output of CMatrixLU subroutine).
    Pivots  -   table of permutations
                (the output of CMatrixLU subroutine).
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)

OUTPUT PARAMETERS:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse

  -- ALGLIB routine --
     05.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixluinverse(
    complex_2d_array& a,
    integer_1d_array pivots,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::cmatrixluinverse(
    complex_2d_array& a,
    integer_1d_array pivots,
    ae_int_t n,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

`cmatrixtrinverse` function

/*************************************************************************
Triangular matrix inverse (complex)

The subroutine inverts the following types of matrices:
    * upper triangular
    * upper triangular with unit diagonal
    * lower triangular
    * lower triangular with unit diagonal

In case of an upper (lower) triangular matrix,  the  inverse  matrix  will
also be upper (lower) triangular, and after the end of the algorithm,  the
inverse matrix replaces the source matrix. The elements  below (above) the
main diagonal are not changed by the algorithm.

If  the matrix  has a unit diagonal, the inverse matrix also  has  a  unit
diagonal, and the diagonal elements are not passed to the algorithm.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix, array[0..N-1, 0..N-1].
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   diagonal type (optional):
                * if True, matrix has unit diagonal (a[i,i] are NOT used)
                * if False, matrix diagonal is arbitrary
                * if not given, False is assumed

Output parameters:
    Info    -   same as for RMatrixLUInverse
    Rep     -   same as for RMatrixLUInverse
    A       -   same as for RMatrixLUInverse.

  -- ALGLIB --
     Copyright 05.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixtrinverse(
    complex_2d_array& a,
    bool isupper,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::cmatrixtrinverse(
    complex_2d_array& a,
    ae_int_t n,
    bool isupper,
    bool isunit,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

`hpdmatrixcholeskyinverse` function

/*************************************************************************
Inversion of a Hermitian positive definite matrix which is given
by Cholesky decomposition.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   Cholesky decomposition of the matrix to be inverted:
                A=U'*U or A = L*L'.
                Output of  HPDMatrixCholesky subroutine.
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   storage type (optional):
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function
                * if not given, lower half is used.

Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixcholeskyinverse(
    complex_2d_array& a,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::hpdmatrixcholeskyinverse(
    complex_2d_array& a,
    ae_int_t n,
    bool isupper,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

`hpdmatrixinverse` function

/*************************************************************************
Inversion of a Hermitian positive definite matrix.

Given an upper or lower triangle of a Hermitian positive definite matrix,
the algorithm generates matrix A^-1 and saves the upper or lower triangle
depending on the input.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix to be inverted (upper or lower triangle).
                Array with elements [0..N-1,0..N-1].
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   storage type (optional):
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function
                * if not given,  both lower and upper  triangles  must  be
                  filled.

Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::hpdmatrixinverse(
    complex_2d_array& a,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::hpdmatrixinverse(
    complex_2d_array& a,
    ae_int_t n,
    bool isupper,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rmatrixinverse` function

/*************************************************************************
Inversion of a general matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix.
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)

Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse

Result:
    True, if the matrix is not singular.
    False, if the matrix is singular.

  -- ALGLIB --
     Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixinverse(
    real_2d_array& a,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::rmatrixinverse(
    real_2d_array& a,
    ae_int_t n,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rmatrixluinverse` function

/*************************************************************************
Inversion of a matrix given by its LU decomposition.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    A       -   LU decomposition of the matrix
                (output of RMatrixLU subroutine).
    Pivots  -   table of permutations
                (the output of RMatrixLU subroutine).
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)

OUTPUT PARAMETERS:
    Info    -   return code:
                * -3    A is singular, or VERY close to singular.
                        it is filled by zeros in such cases.
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   solver report, see below for more info
    A       -   inverse of matrix A.
                Array whose indexes range within [0..N-1, 0..N-1].

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* R1        reciprocal of condition number: 1/cond(A), 1-norm.
* RInf      reciprocal of condition number: 1/cond(A), inf-norm.

  -- ALGLIB routine --
     05.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixluinverse(
    real_2d_array& a,
    integer_1d_array pivots,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::rmatrixluinverse(
    real_2d_array& a,
    integer_1d_array pivots,
    ae_int_t n,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

`rmatrixtrinverse` function

/*************************************************************************
Triangular matrix inverse (real)

The subroutine inverts the following types of matrices:
    * upper triangular
    * upper triangular with unit diagonal
    * lower triangular
    * lower triangular with unit diagonal

In case of an upper (lower) triangular matrix,  the  inverse  matrix  will
also be upper (lower) triangular, and after the end of the algorithm,  the
inverse matrix replaces the source matrix. The elements  below (above) the
main diagonal are not changed by the algorithm.

If  the matrix  has a unit diagonal, the inverse matrix also  has  a  unit
diagonal, and the diagonal elements are not passed to the algorithm.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix, array[0..N-1, 0..N-1].
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   diagonal type (optional):
                * if True, matrix has unit diagonal (a[i,i] are NOT used)
                * if False, matrix diagonal is arbitrary
                * if not given, False is assumed

Output parameters:
    Info    -   same as for RMatrixLUInverse
    Rep     -   same as for RMatrixLUInverse
    A       -   same as for RMatrixLUInverse.

  -- ALGLIB --
     Copyright 05.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixtrinverse(
    real_2d_array& a,
    bool isupper,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::rmatrixtrinverse(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    bool isunit,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskyinverse` function

/*************************************************************************
Inversion of a symmetric positive definite matrix which is given
by Cholesky decomposition.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   Cholesky decomposition of the matrix to be inverted:
                A=U'*U or A = L*L'.
                Output of  SPDMatrixCholesky subroutine.
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   storage type (optional):
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function
                * if not given, lower half is used.

Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixcholeskyinverse(
    real_2d_array& a,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::spdmatrixcholeskyinverse(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

`spdmatrixinverse` function

/*************************************************************************
Inversion of a symmetric positive definite matrix.

Given an upper or lower triangle of a symmetric positive definite matrix,
the algorithm generates matrix A^-1 and saves the upper or lower triangle
depending on the input.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix to be inverted (upper or lower triangle).
                Array with elements [0..N-1,0..N-1].
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   storage type (optional):
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function
                * if not given,  both lower and upper  triangles  must  be
                  filled.

Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::spdmatrixinverse(
    real_2d_array& a,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::spdmatrixinverse(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    ae_int_t& info,
    matinvreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

matinv_d_c1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    complex_2d_array a = "[[1i,-1],[1i,1]]";
    ae_int_t info;
    matinvreport rep;
    cmatrixinverse(a, info, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[-0.5i,-0.5i],[-0.5,0.5]]
    printf("%.4f\n", double(rep.r1)); // EXPECTED: 0.5
    printf("%.4f\n", double(rep.rinf)); // EXPECTED: 0.5
    return 0;
}

matinv_d_hpd1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    complex_2d_array a = "[[2,1],[1,2]]";
    ae_int_t info;
    matinvreport rep;
    hpdmatrixinverse(a, info, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[0.666666,-0.333333],[-0.333333,0.666666]]
    return 0;
}

matinv_d_r1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_2d_array a = "[[1,-1],[1,1]]";
    ae_int_t info;
    matinvreport rep;
    rmatrixinverse(a, info, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[0.5,0.5],[-0.5,0.5]]
    printf("%.4f\n", double(rep.r1)); // EXPECTED: 0.5
    printf("%.4f\n", double(rep.rinf)); // EXPECTED: 0.5
    return 0;
}

matinv_d_spd1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_2d_array a = "[[2,1],[1,2]]";
    ae_int_t info;
    matinvreport rep;
    spdmatrixinverse(a, info, rep);
    printf("%d\n", int(info)); // EXPECTED: 1
    printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[0.666666,-0.333333],[-0.333333,0.666666]]
    return 0;
}

`mcpd` subpackage

Classes

mcpdreport
mcpdstate

Functions

mcpdaddbc
mcpdaddec
mcpdaddtrack
mcpdcreate
mcpdcreateentry
mcpdcreateentryexit
mcpdcreateexit
mcpdresults
mcpdsetbc
mcpdsetec
mcpdsetlc
mcpdsetpredictionweights
mcpdsetprior
mcpdsettikhonovregularizer
mcpdsolve

Examples

mcpd_simple1		Simple unconstrained MCPD model (no entry/exit states)
mcpd_simple2		Simple MCPD model (no entry/exit states) with equality constraints

`mcpdreport` class

/*************************************************************************
This structure is a MCPD training report:
    InnerIterationsCount    -   number of inner iterations of the
                                underlying optimization algorithm
    OuterIterationsCount    -   number of outer iterations of the
                                underlying optimization algorithm
    NFEV                    -   number of merit function evaluations
    TerminationType         -   termination type
                                (same as for MinBLEIC optimizer, positive
                                values denote success, negative ones -
                                failure)

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
class mcpdreport
{
    ae_int_t             inneriterationscount;
    ae_int_t             outeriterationscount;
    ae_int_t             nfev;
    ae_int_t             terminationtype;
};

`mcpdstate` class

/*************************************************************************
This structure is a MCPD (Markov Chains for Population Data) solver.

You should use ALGLIB functions in order to work with this object.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
class mcpdstate
{
};

`mcpdaddbc` function

/*************************************************************************
This function is used to add bound constraints  on  the  elements  of  the
transition matrix P.

MCPD solver has four types of constraints which can be placed on P:
* user-specified equality constraints (optional)
* user-specified bound constraints (optional)
* user-specified general linear constraints (optional)
* basic constraints (always present):
  * non-negativity: P[i,j]>=0
  * consistency: every column of P sums to 1.0

Final  constraints  which  are  passed  to  the  underlying  optimizer are
calculated  as  intersection  of all present constraints. For example, you
may specify boundary constraint on P[0,0] and equality one:
    0.1<=P[0,0]<=0.9
    P[0,0]=0.5
Such  combination  of  constraints  will  be  silently  reduced  to  their
intersection, which is P[0,0]=0.5.

This  function  can  be  used to ADD bound constraint for one element of P
without changing constraints for other elements.

You  can  also  use  MCPDSetBC()  function  which  allows to  place  bound
constraints  on arbitrary subset of elements of P.   Set of constraints is
specified  by  BndL/BndU matrices, which may contain arbitrary combination
of finite numbers or infinities (like -INF<x<=0.5 or 0.1<=x<+INF).

These functions (MCPDSetBC and MCPDAddBC) interact as follows:
* there is internal matrix of bound constraints which is stored in the
  MCPD solver
* MCPDSetBC() replaces this matrix by another one (SET)
* MCPDAddBC() modifies one element of this matrix and  leaves  other  ones
  unchanged (ADD)
* thus  MCPDAddBC()  call  preserves  all  modifications  done by previous
  calls,  while  MCPDSetBC()  completely discards all changes  done to the
  equality constraints.

INPUT PARAMETERS:
    S       -   solver
    I       -   row index of element being constrained
    J       -   column index of element being constrained
    BndL    -   lower bound
    BndU    -   upper bound

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdaddbc(
    mcpdstate s,
    ae_int_t i,
    ae_int_t j,
    double bndl,
    double bndu,
    const xparams _params = alglib::xdefault);

`mcpdaddec` function

/*************************************************************************
This function is used to add equality constraints on the elements  of  the
transition matrix P.

MCPD solver has four types of constraints which can be placed on P:
* user-specified equality constraints (optional)
* user-specified bound constraints (optional)
* user-specified general linear constraints (optional)
* basic constraints (always present):
  * non-negativity: P[i,j]>=0
  * consistency: every column of P sums to 1.0

Final  constraints  which  are  passed  to  the  underlying  optimizer are
calculated  as  intersection  of all present constraints. For example, you
may specify boundary constraint on P[0,0] and equality one:
    0.1<=P[0,0]<=0.9
    P[0,0]=0.5
Such  combination  of  constraints  will  be  silently  reduced  to  their
intersection, which is P[0,0]=0.5.

This function can be used to ADD equality constraint for one element of  P
without changing constraints for other elements.

You  can  also  use  MCPDSetEC()  function  which  allows  you  to specify
arbitrary set of equality constraints in one call.

These functions (MCPDSetEC and MCPDAddEC) interact as follows:
* there is internal matrix of equality constraints which is stored in the
  MCPD solver
* MCPDSetEC() replaces this matrix by another one (SET)
* MCPDAddEC() modifies one element of this matrix and leaves  other  ones
  unchanged (ADD)
* thus  MCPDAddEC()  call  preserves  all  modifications done by previous
  calls,  while  MCPDSetEC()  completely discards all changes done to the
  equality constraints.

INPUT PARAMETERS:
    S       -   solver
    I       -   row index of element being constrained
    J       -   column index of element being constrained
    C       -   value (constraint for P[I,J]).  Can  be  either  NAN  (no
                constraint) or finite value from [0,1].

NOTES:

1. infinite values of C  will lead to exception being thrown. Values  less
than 0.0 or greater than 1.0 will lead to error code being returned  after
call to MCPDSolve().

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdaddec(
    mcpdstate s,
    ae_int_t i,
    ae_int_t j,
    double c,
    const xparams _params = alglib::xdefault);

Examples: [1]

`mcpdaddtrack` function

/*************************************************************************
This  function  is  used to add a track - sequence of system states at the
different moments of its evolution.

You  may  add  one  or several tracks to the MCPD solver. In case you have
several tracks, they won't overwrite each other. For example,  if you pass
two tracks, A1-A2-A3 (system at t=A+1, t=A+2 and t=A+3) and B1-B2-B3, then
solver will try to model transitions from t=A+1 to t=A+2, t=A+2 to  t=A+3,
t=B+1 to t=B+2, t=B+2 to t=B+3. But it WONT mix these two tracks - i.e. it
wont try to model transition from t=A+3 to t=B+1.

INPUT PARAMETERS:
    S       -   solver
    XY      -   track, array[K,N]:
                * I-th row is a state at t=I
                * elements of XY must be non-negative (exception will be
                  thrown on negative elements)
    K       -   number of points in a track
                * if given, only leading K rows of XY are used
                * if not given, automatically determined from size of XY

NOTES:

1. Track may contain either proportional or population data:
   * with proportional data all rows of XY must sum to 1.0, i.e. we have
     proportions instead of absolute population values
   * with population data rows of XY contain population counts and generally
     do not sum to 1.0 (although they still must be non-negative)

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdaddtrack(
    mcpdstate s,
    real_2d_array xy,
    const xparams _params = alglib::xdefault);
void alglib::mcpdaddtrack(
    mcpdstate s,
    real_2d_array xy,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`mcpdcreate` function

/*************************************************************************
DESCRIPTION:

This function creates MCPD (Markov Chains for Population Data) solver.

This  solver  can  be  used  to find transition matrix P for N-dimensional
prediction  problem  where transition from X[i] to X[i+1] is  modelled  as
    X[i+1] = P*X[i]
where X[i] and X[i+1] are N-dimensional population vectors (components  of
each X are non-negative), and P is a N*N transition matrix (elements of  P
are non-negative, each column sums to 1.0).

Such models arise when when:
* there is some population of individuals
* individuals can have different states
* individuals can transit from one state to another
* population size is constant, i.e. there is no new individuals and no one
  leaves population
* you want to model transitions of individuals from one state into another

USAGE:

Here we give very brief outline of the MCPD. We strongly recommend you  to
read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide
on data analysis which is available at http://www.alglib.net/dataanalysis/

1. User initializes algorithm state with MCPDCreate() call

2. User  adds  one  or  more  tracks -  sequences of states which describe
   evolution of a system being modelled from different starting conditions

3. User may add optional boundary, equality  and/or  linear constraints on
   the coefficients of P by calling one of the following functions:
   * MCPDSetEC() to set equality constraints
   * MCPDSetBC() to set bound constraints
   * MCPDSetLC() to set linear constraints

4. Optionally,  user  may  set  custom  weights  for prediction errors (by
   default, algorithm assigns non-equal, automatically chosen weights  for
   errors in the prediction of different components of X). It can be  done
   with a call of MCPDSetPredictionWeights() function.

5. User calls MCPDSolve() function which takes algorithm  state and
   pointer (delegate, etc.) to callback function which calculates F/G.

6. User calls MCPDResults() to get solution

INPUT PARAMETERS:
    N       -   problem dimension, N>=1

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdcreate(
    ae_int_t n,
    mcpdstate& s,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`mcpdcreateentry` function

/*************************************************************************
DESCRIPTION:

This function is a specialized version of MCPDCreate()  function,  and  we
recommend  you  to read comments for this function for general information
about MCPD solver.

This  function  creates  MCPD (Markov Chains for Population  Data)  solver
for "Entry-state" model,  i.e. model  where transition from X[i] to X[i+1]
is modelled as
    X[i+1] = P*X[i]
where
    X[i] and X[i+1] are N-dimensional state vectors
    P is a N*N transition matrix
and  one  selected component of X[] is called "entry" state and is treated
in a special way:
    system state always transits from "entry" state to some another state
    system state can not transit from any state into "entry" state
Such conditions basically mean that row of P which corresponds to  "entry"
state is zero.

Such models arise when:
* there is some population of individuals
* individuals can have different states
* individuals can transit from one state to another
* population size is NOT constant -  at every moment of time there is some
  (unpredictable) amount of "new" individuals, which can transit into  one
  of the states at the next turn, but still no one leaves population
* you want to model transitions of individuals from one state into another
* but you do NOT want to predict amount of "new"  individuals  because  it
  does not depends on individuals already present (hence  system  can  not
  transit INTO entry state - it can only transit FROM it).

This model is discussed  in  more  details  in  the ALGLIB User Guide (see
http://www.alglib.net/dataanalysis/ for more data).

INPUT PARAMETERS:
    N       -   problem dimension, N>=2
    EntryState- index of entry state, in 0..N-1

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdcreateentry(
    ae_int_t n,
    ae_int_t entrystate,
    mcpdstate& s,
    const xparams _params = alglib::xdefault);

`mcpdcreateentryexit` function

/*************************************************************************
DESCRIPTION:

This function is a specialized version of MCPDCreate()  function,  and  we
recommend  you  to read comments for this function for general information
about MCPD solver.

This  function  creates  MCPD (Markov Chains for Population  Data)  solver
for "Entry-Exit-states" model, i.e. model where  transition  from  X[i] to
X[i+1] is modelled as
    X[i+1] = P*X[i]
where
    X[i] and X[i+1] are N-dimensional state vectors
    P is a N*N transition matrix
one selected component of X[] is called "entry" state and is treated in  a
special way:
    system state always transits from "entry" state to some another state
    system state can not transit from any state into "entry" state
and another one component of X[] is called "exit" state and is treated  in
a special way too:
    system state can transit from any state into "exit" state
    system state can not transit from "exit" state into any other state
    transition operator discards "exit" state (makes it zero at each turn)
Such conditions basically mean that:
    row of P which corresponds to "entry" state is zero
    column of P which corresponds to "exit" state is zero
Multiplication by such P may decrease sum of vector components.

Such models arise when:
* there is some population of individuals
* individuals can have different states
* individuals can transit from one state to another
* population size is NOT constant
* at every moment of time there is some (unpredictable)  amount  of  "new"
  individuals, which can transit into one of the states at the next turn
* some  individuals  can  move  (predictably)  into "exit" state and leave
  population at the next turn
* you want to model transitions of individuals from one state into another,
  including transitions from the "entry" state and into the "exit" state.
* but you do NOT want to predict amount of "new"  individuals  because  it
  does not depends on individuals already present (hence  system  can  not
  transit INTO entry state - it can only transit FROM it).

This model is discussed  in  more  details  in  the ALGLIB User Guide (see
http://www.alglib.net/dataanalysis/ for more data).

INPUT PARAMETERS:
    N       -   problem dimension, N>=2
    EntryState- index of entry state, in 0..N-1
    ExitState-  index of exit state, in 0..N-1

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdcreateentryexit(
    ae_int_t n,
    ae_int_t entrystate,
    ae_int_t exitstate,
    mcpdstate& s,
    const xparams _params = alglib::xdefault);

`mcpdcreateexit` function

/*************************************************************************
DESCRIPTION:

This function is a specialized version of MCPDCreate()  function,  and  we
recommend  you  to read comments for this function for general information
about MCPD solver.

This  function  creates  MCPD (Markov Chains for Population  Data)  solver
for "Exit-state" model,  i.e. model  where  transition from X[i] to X[i+1]
is modelled as
    X[i+1] = P*X[i]
where
    X[i] and X[i+1] are N-dimensional state vectors
    P is a N*N transition matrix
and  one  selected component of X[] is called "exit"  state and is treated
in a special way:
    system state can transit from any state into "exit" state
    system state can not transit from "exit" state into any other state
    transition operator discards "exit" state (makes it zero at each turn)
Such  conditions  basically  mean  that  column  of P which corresponds to
"exit" state is zero. Multiplication by such P may decrease sum of  vector
components.

Such models arise when:
* there is some population of individuals
* individuals can have different states
* individuals can transit from one state to another
* population size is NOT constant - individuals can move into "exit" state
  and leave population at the next turn, but there are no new individuals
* amount of individuals which leave population can be predicted
* you want to model transitions of individuals from one state into another
  (including transitions into the "exit" state)

This model is discussed  in  more  details  in  the ALGLIB User Guide (see
http://www.alglib.net/dataanalysis/ for more data).

INPUT PARAMETERS:
    N       -   problem dimension, N>=2
    ExitState-  index of exit state, in 0..N-1

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdcreateexit(
    ae_int_t n,
    ae_int_t exitstate,
    mcpdstate& s,
    const xparams _params = alglib::xdefault);

`mcpdresults` function

/*************************************************************************
MCPD results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    P       -   array[N,N], transition matrix
    Rep     -   optimization report. You should check Rep.TerminationType
                in  order  to  distinguish  successful  termination  from
                unsuccessful one. Speaking short, positive values  denote
                success, negative ones are failures.
                More information about fields of this  structure  can  be
                found in the comments on MCPDReport datatype.


  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdresults(
    mcpdstate s,
    real_2d_array& p,
    mcpdreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`mcpdsetbc` function

/*************************************************************************
This function is used to add bound constraints  on  the  elements  of  the
transition matrix P.

MCPD solver has four types of constraints which can be placed on P:
* user-specified equality constraints (optional)
* user-specified bound constraints (optional)
* user-specified general linear constraints (optional)
* basic constraints (always present):
  * non-negativity: P[i,j]>=0
  * consistency: every column of P sums to 1.0

Final  constraints  which  are  passed  to  the  underlying  optimizer are
calculated  as  intersection  of all present constraints. For example, you
may specify boundary constraint on P[0,0] and equality one:
    0.1<=P[0,0]<=0.9
    P[0,0]=0.5
Such  combination  of  constraints  will  be  silently  reduced  to  their
intersection, which is P[0,0]=0.5.

This  function  can  be  used  to  place bound   constraints  on arbitrary
subset  of  elements  of  P.  Set of constraints is specified by BndL/BndU
matrices, which may contain arbitrary combination  of  finite  numbers  or
infinities (like -INF<x<=0.5 or 0.1<=x<+INF).

You can also use MCPDAddBC() function which allows to ADD bound constraint
for one element of P without changing constraints for other elements.

These functions (MCPDSetBC and MCPDAddBC) interact as follows:
* there is internal matrix of bound constraints which is stored in the
  MCPD solver
* MCPDSetBC() replaces this matrix by another one (SET)
* MCPDAddBC() modifies one element of this matrix and  leaves  other  ones
  unchanged (ADD)
* thus  MCPDAddBC()  call  preserves  all  modifications  done by previous
  calls,  while  MCPDSetBC()  completely discards all changes  done to the
  equality constraints.

INPUT PARAMETERS:
    S       -   solver
    BndL    -   lower bounds constraints, array[N,N]. Elements of BndL can
                be finite numbers or -INF.
    BndU    -   upper bounds constraints, array[N,N]. Elements of BndU can
                be finite numbers or +INF.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdsetbc(
    mcpdstate s,
    real_2d_array bndl,
    real_2d_array bndu,
    const xparams _params = alglib::xdefault);

`mcpdsetec` function

/*************************************************************************
This function is used to add equality constraints on the elements  of  the
transition matrix P.

MCPD solver has four types of constraints which can be placed on P:
* user-specified equality constraints (optional)
* user-specified bound constraints (optional)
* user-specified general linear constraints (optional)
* basic constraints (always present):
  * non-negativity: P[i,j]>=0
  * consistency: every column of P sums to 1.0

Final  constraints  which  are  passed  to  the  underlying  optimizer are
calculated  as  intersection  of all present constraints. For example, you
may specify boundary constraint on P[0,0] and equality one:
    0.1<=P[0,0]<=0.9
    P[0,0]=0.5
Such  combination  of  constraints  will  be  silently  reduced  to  their
intersection, which is P[0,0]=0.5.

This  function  can  be  used  to  place equality constraints on arbitrary
subset of elements of P. Set of constraints is specified by EC, which  may
contain either NAN's or finite numbers from [0,1]. NAN denotes absence  of
constraint, finite number denotes equality constraint on specific  element
of P.

You can also  use  MCPDAddEC()  function  which  allows  to  ADD  equality
constraint  for  one  element  of P without changing constraints for other
elements.

These functions (MCPDSetEC and MCPDAddEC) interact as follows:
* there is internal matrix of equality constraints which is stored in  the
  MCPD solver
* MCPDSetEC() replaces this matrix by another one (SET)
* MCPDAddEC() modifies one element of this matrix and  leaves  other  ones
  unchanged (ADD)
* thus  MCPDAddEC()  call  preserves  all  modifications  done by previous
  calls,  while  MCPDSetEC()  completely discards all changes  done to the
  equality constraints.

INPUT PARAMETERS:
    S       -   solver
    EC      -   equality constraints, array[N,N]. Elements of  EC  can  be
                either NAN's or finite  numbers from  [0,1].  NAN  denotes
                absence  of  constraints,  while  finite  value    denotes
                equality constraint on the corresponding element of P.

NOTES:

1. infinite values of EC will lead to exception being thrown. Values  less
than 0.0 or greater than 1.0 will lead to error code being returned  after
call to MCPDSolve().

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdsetec(
    mcpdstate s,
    real_2d_array ec,
    const xparams _params = alglib::xdefault);

`mcpdsetlc` function

/*************************************************************************
This function is used to set linear equality/inequality constraints on the
elements of the transition matrix P.

This function can be used to set one or several general linear constraints
on the elements of P. Two types of constraints are supported:
* equality constraints
* inequality constraints (both less-or-equal and greater-or-equal)

Coefficients  of  constraints  are  specified  by  matrix  C (one  of  the
parameters).  One  row  of  C  corresponds  to  one  constraint.   Because
transition  matrix P has N*N elements,  we  need  N*N columns to store all
coefficients  (they  are  stored row by row), and one more column to store
right part - hence C has N*N+1 columns.  Constraint  kind is stored in the
CT array.

Thus, I-th linear constraint is
    P[0,0]*C[I,0] + P[0,1]*C[I,1] + .. + P[0,N-1]*C[I,N-1] +
        + P[1,0]*C[I,N] + P[1,1]*C[I,N+1] + ... +
        + P[N-1,N-1]*C[I,N*N-1]  ?=?  C[I,N*N]
where ?=? can be either "=" (CT[i]=0), "<=" (CT[i]<0) or ">=" (CT[i]>0).

Your constraint may involve only some subset of P (less than N*N elements).
For example it can be something like
    P[0,0] + P[0,1] = 0.5
In this case you still should pass matrix  with N*N+1 columns, but all its
elements (except for C[0,0], C[0,1] and C[0,N*N-1]) will be zero.

INPUT PARAMETERS:
    S       -   solver
    C       -   array[K,N*N+1] - coefficients of constraints
                (see above for complete description)
    CT      -   array[K] - constraint types
                (see above for complete description)
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdsetlc(
    mcpdstate s,
    real_2d_array c,
    integer_1d_array ct,
    const xparams _params = alglib::xdefault);
void alglib::mcpdsetlc(
    mcpdstate s,
    real_2d_array c,
    integer_1d_array ct,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

`mcpdsetpredictionweights` function

/*************************************************************************
This function is used to change prediction weights

MCPD solver scales prediction errors as follows
    Error(P) = ||W*(y-P*x)||^2
where
    x is a system state at time t
    y is a system state at time t+1
    P is a transition matrix
    W is a diagonal scaling matrix

By default, weights are chosen in order  to  minimize  relative prediction
error instead of absolute one. For example, if one component of  state  is
about 0.5 in magnitude and another one is about 0.05, then algorithm  will
make corresponding weights equal to 2.0 and 20.0.

INPUT PARAMETERS:
    S       -   solver
    PW      -   array[N], weights:
                * must be non-negative values (exception will be thrown otherwise)
                * zero values will be replaced by automatically chosen values

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdsetpredictionweights(
    mcpdstate s,
    real_1d_array pw,
    const xparams _params = alglib::xdefault);

`mcpdsetprior` function

/*************************************************************************
This  function  allows to set prior values used for regularization of your
problem.

By default, regularizing term is equal to r*||P-prior_P||^2, where r is  a
small non-zero value,  P is transition matrix, prior_P is identity matrix,
||X||^2 is a sum of squared elements of X.

This  function  allows  you to change prior values prior_P. You  can  also
change r with MCPDSetTikhonovRegularizer() function.

INPUT PARAMETERS:
    S       -   solver
    PP      -   array[N,N], matrix of prior values:
                1. elements must be real numbers from [0,1]
                2. columns must sum to 1.0.
                First property is checked (exception is thrown otherwise),
                while second one is not checked/enforced.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdsetprior(
    mcpdstate s,
    real_2d_array pp,
    const xparams _params = alglib::xdefault);

`mcpdsettikhonovregularizer` function

/*************************************************************************
This function allows to  tune  amount  of  Tikhonov  regularization  being
applied to your problem.

By default, regularizing term is equal to r*||P-prior_P||^2, where r is  a
small non-zero value,  P is transition matrix, prior_P is identity matrix,
||X||^2 is a sum of squared elements of X.

This  function  allows  you to change coefficient r. You can  also  change
prior values with MCPDSetPrior() function.

INPUT PARAMETERS:
    S       -   solver
    V       -   regularization  coefficient, finite non-negative value. It
                is  not  recommended  to specify zero value unless you are
                pretty sure that you want it.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdsettikhonovregularizer(
    mcpdstate s,
    double v,
    const xparams _params = alglib::xdefault);

`mcpdsolve` function

/*************************************************************************
This function is used to start solution of the MCPD problem.

After return from this function, you can use MCPDResults() to get solution
and completion code.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mcpdsolve(
    mcpdstate s,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

mcpd_simple1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // The very simple MCPD example
    //
    // We have a loan portfolio. Our loans can be in one of two states:
    // * normal loans ("good" ones)
    // * past due loans ("bad" ones)
    //
    // We assume that:
    // * loans can transition from any state to any other state. In 
    //   particular, past due loan can become "good" one at any moment 
    //   with same (fixed) probability. Not realistic, but it is toy example :)
    // * portfolio size does not change over time
    //
    // Thus, we have following model
    //     state_new = P*state_old
    // where
    //         ( p00  p01 )
    //     P = (          )
    //         ( p10  p11 )
    //
    // We want to model transitions between these two states using MCPD
    // approach (Markov Chains for Proportional/Population Data), i.e.
    // to restore hidden transition matrix P using actual portfolio data.
    // We have:
    // * poportional data, i.e. proportion of loans in the normal and past 
    //   due states (not portfolio size measured in some currency, although 
    //   it is possible to work with population data too)
    // * two tracks, i.e. two sequences which describe portfolio
    //   evolution from two different starting states: [1,0] (all loans 
    //   are "good") and [0.8,0.2] (only 80% of portfolio is in the "good"
    //   state)
    //
    mcpdstate s;
    mcpdreport rep;
    real_2d_array p;
    real_2d_array track0 = "[[1.00000,0.00000],[0.95000,0.05000],[0.92750,0.07250],[0.91738,0.08263],[0.91282,0.08718]]";
    real_2d_array track1 = "[[0.80000,0.20000],[0.86000,0.14000],[0.88700,0.11300],[0.89915,0.10085]]";

    mcpdcreate(2, s);
    mcpdaddtrack(s, track0);
    mcpdaddtrack(s, track1);
    mcpdsolve(s);
    mcpdresults(s, p, rep);

    //
    // Hidden matrix P is equal to
    //         ( 0.95  0.50 )
    //         (            )
    //         ( 0.05  0.50 )
    // which means that "good" loans can become "bad" with 5% probability, 
    // while "bad" loans will return to good state with 50% probability.
    //
    printf("%s\n", p.tostring(2).c_str()); // EXPECTED: [[0.95,0.50],[0.05,0.50]]
    return 0;
}

mcpd_simple2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Simple MCPD example
    //
    // We have a loan portfolio. Our loans can be in one of three states:
    // * normal loans
    // * past due loans
    // * charged off loans
    //
    // We assume that:
    // * normal loan can stay normal or become past due (but not charged off)
    // * past due loan can stay past due, become normal or charged off
    // * charged off loan will stay charged off for the rest of eternity
    // * portfolio size does not change over time
    // Not realistic, but it is toy example :)
    //
    // Thus, we have following model
    //     state_new = P*state_old
    // where
    //         ( p00  p01    )
    //     P = ( p10  p11    )
    //         (      p21  1 )
    // i.e. four elements of P are known a priori.
    //
    // Although it is possible (given enough data) to In order to enforce 
    // this property we set equality constraints on these elements.
    //
    // We want to model transitions between these two states using MCPD
    // approach (Markov Chains for Proportional/Population Data), i.e.
    // to restore hidden transition matrix P using actual portfolio data.
    // We have:
    // * poportional data, i.e. proportion of loans in the current and past 
    //   due states (not portfolio size measured in some currency, although 
    //   it is possible to work with population data too)
    // * two tracks, i.e. two sequences which describe portfolio
    //   evolution from two different starting states: [1,0,0] (all loans 
    //   are "good") and [0.8,0.2,0.0] (only 80% of portfolio is in the "good"
    //   state)
    //
    mcpdstate s;
    mcpdreport rep;
    real_2d_array p;
    real_2d_array track0 = "[[1.000000,0.000000,0.000000],[0.950000,0.050000,0.000000],[0.927500,0.060000,0.012500],[0.911125,0.061375,0.027500],[0.896256,0.060900,0.042844]]";
    real_2d_array track1 = "[[0.800000,0.200000,0.000000],[0.860000,0.090000,0.050000],[0.862000,0.065500,0.072500],[0.851650,0.059475,0.088875],[0.838805,0.057451,0.103744]]";

    mcpdcreate(3, s);
    mcpdaddtrack(s, track0);
    mcpdaddtrack(s, track1);
    mcpdaddec(s, 0, 2, 0.0);
    mcpdaddec(s, 1, 2, 0.0);
    mcpdaddec(s, 2, 2, 1.0);
    mcpdaddec(s, 2, 0, 0.0);
    mcpdsolve(s);
    mcpdresults(s, p, rep);

    //
    // Hidden matrix P is equal to
    //         ( 0.95 0.50      )
    //         ( 0.05 0.25      )
    //         (      0.25 1.00 ) 
    // which means that "good" loans can become past due with 5% probability, 
    // while past due loans will become charged off with 25% probability or
    // return back to normal state with 50% probability.
    //
    printf("%s\n", p.tostring(2).c_str()); // EXPECTED: [[0.95,0.50,0.00],[0.05,0.25,0.00],[0.00,0.25,1.00]]
    return 0;
}

`minbc` subpackage

Classes

minbcreport
minbcstate

Functions

minbccreate
minbccreatef
minbcoptimize
minbcrequesttermination
minbcrestartfrom
minbcresults
minbcresultsbuf
minbcsetbc
minbcsetcond
minbcsetgradientcheck
minbcsetprecdefault
minbcsetprecdiag
minbcsetprecscale
minbcsetscale
minbcsetstpmax
minbcsetxrep

Examples

minbc_d_1		Nonlinear optimization with box constraints
minbc_numdiff		Nonlinear optimization with bound constraints and numerical differentiation

`minbcreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           number of iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -7    gradient verification failed.
        See MinBCSetGradientCheck() for more information.
  -3    inconsistent constraints.
   1    relative function improvement is no more than EpsF.
   2    relative step is no more than EpsX.
   4    gradient norm is no more than EpsG
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    terminated by user who called minbcrequesttermination(). X contains
        point which was "current accepted" when  termination  request  was
        submitted.

ADDITIONAL FIELDS

There are additional fields which can be used for debugging:
* DebugEqErr                error in the equality constraints (2-norm)
* DebugFS                   f, calculated at projection of initial point
                            to the feasible set
* DebugFF                   f, calculated at the final point
* DebugDX                   |X_start-X_final|
*************************************************************************/
class minbcreport
{
    ae_int_t             iterationscount;
    ae_int_t             nfev;
    ae_int_t             varidx;
    ae_int_t             terminationtype;
};

`minbcstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinBC subpackage to work with this
object
*************************************************************************/
class minbcstate
{
};

`minbccreate` function

/*************************************************************************
                     BOX CONSTRAINED OPTIMIZATION
          WITH FAST ACTIVATION OF MULTIPLE BOX CONSTRAINTS

DESCRIPTION:
The  subroutine  minimizes  function   F(x) of N arguments subject  to box
constraints (with some of box constraints actually being equality ones).

This optimizer uses algorithm similar to that of MinBLEIC (optimizer  with
general linear constraints), but presence of box-only  constraints  allows
us to use faster constraint activation strategies. On large-scale problems,
with multiple constraints active at the solution, this  optimizer  can  be
several times faster than BLEIC.

REQUIREMENTS:
* user must provide function value and gradient
* starting point X0 must be feasible or
  not too far away from the feasible set
* grad(f) must be Lipschitz continuous on a level set:
  L = { x : f(x)<=f(x0) }
* function must be defined everywhere on the feasible set F

USAGE:

Constrained optimization if far more complex than the unconstrained one.
Here we give very brief outline of the BC optimizer. We strongly recommend
you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide
on optimization, which is available at http://www.alglib.net/optimization/

1. User initializes algorithm state with MinBCCreate() call

2. USer adds box constraints by calling MinBCSetBC() function.

3. User sets stopping conditions with MinBCSetCond().

4. User calls MinBCOptimize() function which takes algorithm  state and
   pointer (delegate, etc.) to callback function which calculates F/G.

5. User calls MinBCResults() to get solution

6. Optionally user may call MinBCRestartFrom() to solve another problem
   with same N but another starting point.
   MinBCRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size ofX
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbccreate(
    real_1d_array x,
    minbcstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minbccreate(
    ae_int_t n,
    real_1d_array x,
    minbcstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minbccreatef` function

/*************************************************************************
The subroutine is finite difference variant of MinBCCreate().  It  uses
finite differences in order to differentiate target function.

Description below contains information which is specific to  this function
only. We recommend to read comments on MinBCCreate() in  order  to  get
more information about creation of BC optimizer.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[0..N-1].
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinBCSetScale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large truncation  errors, while too small
   step will result in too large numerical  errors.  1.0E-6  can  be  good
   value to start with.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is  less  robust and precise. CG needs exact gradient values. Imprecise
   gradient may slow  down  convergence, especially  on  highly  nonlinear
   problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 16.05.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minbccreatef(
    real_1d_array x,
    double diffstep,
    minbcstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minbccreatef(
    ae_int_t n,
    real_1d_array x,
    double diffstep,
    minbcstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minbcoptimize` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied gradient,  and one which uses function value
   only  and  numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   (either  MinBCCreate() for analytical gradient or  MinBCCreateF()
   for numerical differentiation) you should choose appropriate variant of
   MinBCOptimize() - one  which  accepts  function  AND gradient or one
   which accepts function ONLY.

   Be careful to choose variant of MinBCOptimize() which corresponds to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to MinBCOptimize()  and specific
   function used to create optimizer.


                     |         USER PASSED TO MinBCOptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   MinBCCreateF()    |     works               FAILS
   MinBCCreate()     |     FAILS               works

   Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
   function  and  MinBCOptimize()  version.   Attemps   to   use   such
   combination (for  example,  to  create optimizer with MinBCCreateF()
   and  to  pass  gradient  information  to  MinCGOptimize()) will lead to
   exception being thrown. Either  you  did  not pass gradient when it WAS
   needed or you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbcoptimize(minbcstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minbcoptimize(minbcstate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minbcrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcrequesttermination(
    minbcstate state,
    const xparams _params = alglib::xdefault);

`minbcrestartfrom` function

/*************************************************************************
This subroutine restarts algorithm from new point.
All optimization parameters (including constraints) are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have  same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinBCCreate call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcrestartfrom(
    minbcstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

`minbcresults` function

/*************************************************************************
BC results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report. You should check Rep.TerminationType
                in  order  to  distinguish  successful  termination  from
                unsuccessful one:
                * -8    internal integrity control  detected  infinite or
                        NAN   values   in   function/gradient.   Abnormal
                        termination signalled.
                * -7   gradient verification failed.
                       See MinBCSetGradientCheck() for more information.
                * -3   inconsistent constraints.
                *  1   relative function improvement is no more than EpsF.
                *  2   scaled step is no more than EpsX.
                *  4   scaled gradient norm is no more than EpsG.
                *  5   MaxIts steps was taken
                *  8   terminated by user who called minbcrequesttermination().
                       X contains point which was "current accepted"  when
                       termination request was submitted.
                More information about fields of this  structure  can  be
                found in the comments on MinBCReport datatype.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcresults(
    minbcstate state,
    real_1d_array& x,
    minbcreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`minbcresultsbuf` function

/*************************************************************************
BC results

Buffered implementation of MinBCResults() which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcresultsbuf(
    minbcstate state,
    real_1d_array& x,
    minbcreport& rep,
    const xparams _params = alglib::xdefault);

`minbcsetbc` function

/*************************************************************************
This function sets boundary constraints for BC optimizer.

Boundary constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with MinBCRestartFrom().

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF.

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: this solver has following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by  bound  constraints,
  even  when  numerical  differentiation is used (algorithm adjusts  nodes
  according to boundary constraints)

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcsetbc(
    minbcstate state,
    real_1d_array bndl,
    real_1d_array bndu,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`minbcsetcond` function

/*************************************************************************
This function sets stopping conditions for the optimizer.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinBCSetScale()
    EpsF    -   >=0
                The  subroutine  finishes  its work if on k+1-th iteration
                the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                is satisfied.
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinBCSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead
to automatic stopping criterion selection.

NOTE: when SetCond() called with non-zero MaxIts, BC solver may perform
      slightly more than MaxIts iterations. I.e., MaxIts  sets  non-strict
      limit on iterations count.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcsetcond(
    minbcstate state,
    double epsg,
    double epsf,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`minbcsetgradientcheck` function

/*************************************************************************
This  subroutine  turns  on  verification  of  the  user-supplied analytic
gradient:
* user calls this subroutine before optimization begins
* MinBCOptimize() is called
* prior to  actual  optimization, for each component  of  parameters being
  optimized X[i] algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * if needed, steps are bounded with respect to constraints on X[]
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point
  * in case difference between prediction and actual value is higher  than
    some predetermined threshold, algorithm stops with completion code -7;
    Rep.VarIdx is set to index of the parameter with incorrect derivative.
* after verification is over, algorithm proceeds to the actual optimization.

NOTE 1: verification  needs  N (parameters count) gradient evaluations. It
        is very costly and you should use  it  only  for  low  dimensional
        problems,  when  you  want  to  be  sure  that  you've   correctly
        calculated  analytic  derivatives.  You  should  not use it in the
        production code (unless you want to check derivatives provided  by
        some third party).

NOTE 2: you  should  carefully  choose  TestStep. Value which is too large
        (so large that function behaviour is significantly non-cubic) will
        lead to false alarms. You may use  different  step  for  different
        parameters by means of setting scale with MinBCSetScale().

NOTE 3: this function may lead to false positives. In case it reports that
        I-th  derivative was calculated incorrectly, you may decrease test
        step  and  try  one  more  time  - maybe your function changes too
        sharply  and  your  step  is  too  large for such rapidly chanding
        function.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification

  -- ALGLIB --
     Copyright 15.06.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcsetgradientcheck(
    minbcstate state,
    double teststep,
    const xparams _params = alglib::xdefault);

`minbcsetprecdefault` function

/*************************************************************************
Modification of the preconditioner: preconditioning is turned off.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcsetprecdefault(
    minbcstate state,
    const xparams _params = alglib::xdefault);

`minbcsetprecdiag` function

/*************************************************************************
Modification  of  the  preconditioner:  diagonal of approximate Hessian is
used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    D       -   diagonal of the approximate Hessian, array[0..N-1],
                (if larger, only leading N elements are used).

NOTE 1: D[i] should be positive. Exception will be thrown otherwise.

NOTE 2: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcsetprecdiag(
    minbcstate state,
    real_1d_array d,
    const xparams _params = alglib::xdefault);

`minbcsetprecscale` function

/*************************************************************************
Modification of the preconditioner: scale-based diagonal preconditioning.

This preconditioning mode can be useful when you  don't  have  approximate
diagonal of Hessian, but you know that your  variables  are  badly  scaled
(for  example,  one  variable is in [1,10], and another in [1000,100000]),
and most part of the ill-conditioning comes from different scales of vars.

In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
can greatly improve convergence.

IMPRTANT: you should set scale of your variables  with  MinBCSetScale()
call  (before  or after MinBCSetPrecScale() call). Without knowledge of
the scale of your variables scale-based preconditioner will be  just  unit
matrix.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcsetprecscale(
    minbcstate state,
    const xparams _params = alglib::xdefault);

`minbcsetscale` function

/*************************************************************************
This function sets scaling coefficients for BC optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

In  most  optimizers  (and  in  the  BC  too)  scaling is NOT a form of
preconditioning. It just  affects  stopping  conditions.  You  should  set
preconditioner  by  separate  call  to  one  of  the  MinBCSetPrec...()
functions.

There is a special  preconditioning  mode, however,  which  uses   scaling
coefficients to form diagonal preconditioning matrix. You  can  turn  this
mode on, if you want.   But  you should understand that scaling is not the
same thing as preconditioning - these are two different, although  related
forms of tuning solver.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcsetscale(
    minbcstate state,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

`minbcsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  lead   to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcsetstpmax(
    minbcstate state,
    double stpmax,
    const xparams _params = alglib::xdefault);

`minbcsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinBCOptimize().

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbcsetxrep(
    minbcstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

minbc_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // subject to bound constraints -1<=x<=+1, -1<=y<=+1, using MinBC optimizer.
    //
    real_1d_array x = "[0,0]";
    real_1d_array bndl = "[-1,-1]";
    real_1d_array bndu = "[+1,+1]";
    minbcstate state;
    minbcreport rep;

    //
    // These variables define stopping conditions for the optimizer.
    //
    // We use very simple condition - |g|<=epsg
    //
    double epsg = 0.000001;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;

    //
    // Now we are ready to actually optimize something:
    // * first we create optimizer
    // * we add boundary constraints
    // * we tune stopping conditions
    // * and, finally, optimize and obtain results...
    //
    minbccreate(x, state);
    minbcsetbc(state, bndl, bndu);
    minbcsetcond(state, epsg, epsf, epsx, maxits);
    alglib::minbcoptimize(state, function1_grad);
    minbcresults(state, x, rep);

    //
    // ...and evaluate these results
    //
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]
    return 0;
}

minbc_numdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // subject to bound constraints -1<=x<=+1, -1<=y<=+1, using MinBC optimizer.
    //
    real_1d_array x = "[0,0]";
    real_1d_array bndl = "[-1,-1]";
    real_1d_array bndu = "[+1,+1]";
    minbcstate state;
    minbcreport rep;

    //
    // These variables define stopping conditions for the optimizer.
    //
    // We use very simple condition - |g|<=epsg
    //
    double epsg = 0.000001;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;

    //
    // This variable contains differentiation step
    //
    double diffstep = 1.0e-6;

    //
    // Now we are ready to actually optimize something:
    // * first we create optimizer
    // * we add boundary constraints
    // * we tune stopping conditions
    // * and, finally, optimize and obtain results...
    //
    minbccreatef(x, diffstep, state);
    minbcsetbc(state, bndl, bndu);
    minbcsetcond(state, epsg, epsf, epsx, maxits);
    alglib::minbcoptimize(state, function1_func);
    minbcresults(state, x, rep);

    //
    // ...and evaluate these results
    //
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]
    return 0;
}

`minbleic` subpackage

Classes

minbleicreport
minbleicstate

Functions

minbleiccreate
minbleiccreatef
minbleicoptimize
minbleicrequesttermination
minbleicrestartfrom
minbleicresults
minbleicresultsbuf
minbleicsetbc
minbleicsetcond
minbleicsetgradientcheck
minbleicsetlc
minbleicsetprecdefault
minbleicsetprecdiag
minbleicsetprecscale
minbleicsetscale
minbleicsetstpmax
minbleicsetxrep

Examples

minbleic_d_1		Nonlinear optimization with bound constraints
minbleic_d_2		Nonlinear optimization with linear inequality constraints
minbleic_ftrim		Nonlinear optimization by BLEIC, function with singularities
minbleic_numdiff		Nonlinear optimization with bound constraints and numerical differentiation

`minbleicreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           number of iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -7    gradient verification failed.
        See MinBLEICSetGradientCheck() for more information.
  -3    inconsistent constraints. Feasible point is
        either nonexistent or too hard to find. Try to
        restart optimizer with better initial approximation
   1    relative function improvement is no more than EpsF.
   2    relative step is no more than EpsX.
   4    gradient norm is no more than EpsG
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    terminated by user who called minbleicrequesttermination(). X contains
        point which was "current accepted" when  termination  request  was
        submitted.

ADDITIONAL FIELDS

There are additional fields which can be used for debugging:
* DebugEqErr                error in the equality constraints (2-norm)
* DebugFS                   f, calculated at projection of initial point
                            to the feasible set
* DebugFF                   f, calculated at the final point
* DebugDX                   |X_start-X_final|
*************************************************************************/
class minbleicreport
{
    ae_int_t             iterationscount;
    ae_int_t             nfev;
    ae_int_t             varidx;
    ae_int_t             terminationtype;
    double               debugeqerr;
    double               debugfs;
    double               debugff;
    double               debugdx;
    ae_int_t             debugfeasqpits;
    ae_int_t             debugfeasgpaits;
    ae_int_t             inneriterationscount;
    ae_int_t             outeriterationscount;
};

`minbleicstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinBLEIC subpackage to work with this
object
*************************************************************************/
class minbleicstate
{
};

`minbleiccreate` function

/*************************************************************************
                     BOUND CONSTRAINED OPTIMIZATION
       WITH ADDITIONAL LINEAR EQUALITY AND INEQUALITY CONSTRAINTS

DESCRIPTION:
The  subroutine  minimizes  function   F(x)  of N arguments subject to any
combination of:
* bound constraints
* linear inequality constraints
* linear equality constraints

REQUIREMENTS:
* user must provide function value and gradient
* starting point X0 must be feasible or
  not too far away from the feasible set
* grad(f) must be Lipschitz continuous on a level set:
  L = { x : f(x)<=f(x0) }
* function must be defined everywhere on the feasible set F

USAGE:

Constrained optimization if far more complex than the unconstrained one.
Here we give very brief outline of the BLEIC optimizer. We strongly recommend
you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide
on optimization, which is available at http://www.alglib.net/optimization/

1. User initializes algorithm state with MinBLEICCreate() call

2. USer adds boundary and/or linear constraints by calling
   MinBLEICSetBC() and MinBLEICSetLC() functions.

3. User sets stopping conditions with MinBLEICSetCond().

4. User calls MinBLEICOptimize() function which takes algorithm  state and
   pointer (delegate, etc.) to callback function which calculates F/G.

5. User calls MinBLEICResults() to get solution

6. Optionally user may call MinBLEICRestartFrom() to solve another problem
   with same N but another starting point.
   MinBLEICRestartFrom() allows to reuse already initialized structure.

NOTE: if you have box-only constraints (no  general  linear  constraints),
      then MinBC optimizer can be better option. It uses  special,  faster
      constraint activation method, which performs better on problems with
      multiple constraints active at the solution.

      On small-scale problems performance of MinBC is similar to  that  of
      MinBLEIC, but on large-scale ones (hundreds and thousands of  active
      constraints) it can be several times faster than MinBLEIC.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size ofX
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleiccreate(
    real_1d_array x,
    minbleicstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minbleiccreate(
    ae_int_t n,
    real_1d_array x,
    minbleicstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minbleiccreatef` function

/*************************************************************************
The subroutine is finite difference variant of MinBLEICCreate().  It  uses
finite differences in order to differentiate target function.

Description below contains information which is specific to  this function
only. We recommend to read comments on MinBLEICCreate() in  order  to  get
more information about creation of BLEIC optimizer.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[0..N-1].
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinBLEICSetScale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large truncation  errors, while too small
   step will result in too large numerical  errors.  1.0E-6  can  be  good
   value to start with.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is  less  robust and precise. CG needs exact gradient values. Imprecise
   gradient may slow  down  convergence, especially  on  highly  nonlinear
   problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 16.05.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleiccreatef(
    real_1d_array x,
    double diffstep,
    minbleicstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minbleiccreatef(
    ae_int_t n,
    real_1d_array x,
    double diffstep,
    minbleicstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minbleicoptimize` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied gradient,  and one which uses function value
   only  and  numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   (either  MinBLEICCreate() for analytical gradient or  MinBLEICCreateF()
   for numerical differentiation) you should choose appropriate variant of
   MinBLEICOptimize() - one  which  accepts  function  AND gradient or one
   which accepts function ONLY.

   Be careful to choose variant of MinBLEICOptimize() which corresponds to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to MinBLEICOptimize()  and specific
   function used to create optimizer.


                     |         USER PASSED TO MinBLEICOptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   MinBLEICCreateF() |     work                FAIL
   MinBLEICCreate()  |     FAIL                work

   Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
   function  and  MinBLEICOptimize()  version.   Attemps   to   use   such
   combination (for  example,  to  create optimizer with MinBLEICCreateF()
   and  to  pass  gradient  information  to  MinCGOptimize()) will lead to
   exception being thrown. Either  you  did  not pass gradient when it WAS
   needed or you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicoptimize(minbleicstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minbleicoptimize(minbleicstate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minbleicrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicrequesttermination(
    minbleicstate state,
    const xparams _params = alglib::xdefault);

`minbleicrestartfrom` function

/*************************************************************************
This subroutine restarts algorithm from new point.
All optimization parameters (including constraints) are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have  same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinBLEICCreate call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicrestartfrom(
    minbleicstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

`minbleicresults` function

/*************************************************************************
BLEIC results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report. You should check Rep.TerminationType
                in  order  to  distinguish  successful  termination  from
                unsuccessful one:
                * -8    internal integrity control  detected  infinite or
                        NAN   values   in   function/gradient.   Abnormal
                        termination signalled.
                * -7   gradient verification failed.
                       See MinBLEICSetGradientCheck() for more information.
                * -3   inconsistent constraints. Feasible point is
                       either nonexistent or too hard to find. Try to
                       restart optimizer with better initial approximation
                *  1   relative function improvement is no more than EpsF.
                *  2   scaled step is no more than EpsX.
                *  4   scaled gradient norm is no more than EpsG.
                *  5   MaxIts steps was taken
                *  8   terminated by user who called minbleicrequesttermination().
                       X contains point which was "current accepted"  when
                       termination request was submitted.
                More information about fields of this  structure  can  be
                found in the comments on MinBLEICReport datatype.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicresults(
    minbleicstate state,
    real_1d_array& x,
    minbleicreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minbleicresultsbuf` function

/*************************************************************************
BLEIC results

Buffered implementation of MinBLEICResults() which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicresultsbuf(
    minbleicstate state,
    real_1d_array& x,
    minbleicreport& rep,
    const xparams _params = alglib::xdefault);

`minbleicsetbc` function

/*************************************************************************
This function sets boundary constraints for BLEIC optimizer.

Boundary constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with MinBLEICRestartFrom().

NOTE: if you have box-only constraints (no  general  linear  constraints),
      then MinBC optimizer can be better option. It uses  special,  faster
      constraint activation method, which performs better on problems with
      multiple constraints active at the solution.

      On small-scale problems performance of MinBC is similar to  that  of
      MinBLEIC, but on large-scale ones (hundreds and thousands of  active
      constraints) it can be several times faster than MinBLEIC.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF.

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: this solver has following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by  bound  constraints,
  even  when  numerical  differentiation is used (algorithm adjusts  nodes
  according to boundary constraints)

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetbc(
    minbleicstate state,
    real_1d_array bndl,
    real_1d_array bndu,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`minbleicsetcond` function

/*************************************************************************
This function sets stopping conditions for the optimizer.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinBLEICSetScale()
    EpsF    -   >=0
                The  subroutine  finishes  its work if on k+1-th iteration
                the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                is satisfied.
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinBLEICSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead
to automatic stopping criterion selection.

NOTE: when SetCond() called with non-zero MaxIts, BLEIC solver may perform
      slightly more than MaxIts iterations. I.e., MaxIts  sets  non-strict
      limit on iterations count.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetcond(
    minbleicstate state,
    double epsg,
    double epsf,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minbleicsetgradientcheck` function

/*************************************************************************
This  subroutine  turns  on  verification  of  the  user-supplied analytic
gradient:
* user calls this subroutine before optimization begins
* MinBLEICOptimize() is called
* prior to  actual  optimization, for each component  of  parameters being
  optimized X[i] algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * if needed, steps are bounded with respect to constraints on X[]
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point
  * in case difference between prediction and actual value is higher  than
    some predetermined threshold, algorithm stops with completion code -7;
    Rep.VarIdx is set to index of the parameter with incorrect derivative.
* after verification is over, algorithm proceeds to the actual optimization.

NOTE 1: verification  needs  N (parameters count) gradient evaluations. It
        is very costly and you should use  it  only  for  low  dimensional
        problems,  when  you  want  to  be  sure  that  you've   correctly
        calculated  analytic  derivatives.  You  should  not use it in the
        production code (unless you want to check derivatives provided  by
        some third party).

NOTE 2: you  should  carefully  choose  TestStep. Value which is too large
        (so large that function behaviour is significantly non-cubic) will
        lead to false alarms. You may use  different  step  for  different
        parameters by means of setting scale with MinBLEICSetScale().

NOTE 3: this function may lead to false positives. In case it reports that
        I-th  derivative was calculated incorrectly, you may decrease test
        step  and  try  one  more  time  - maybe your function changes too
        sharply  and  your  step  is  too  large for such rapidly chanding
        function.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification

  -- ALGLIB --
     Copyright 15.06.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetgradientcheck(
    minbleicstate state,
    double teststep,
    const xparams _params = alglib::xdefault);

`minbleicsetlc` function

/*************************************************************************
This function sets linear constraints for BLEIC optimizer.

Linear constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with MinBLEICRestartFrom().

INPUT PARAMETERS:
    State   -   structure previously allocated with MinBLEICCreate call.
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

NOTE 1: linear (non-bound) constraints are satisfied only approximately:
* there always exists some minor violation (about Epsilon in magnitude)
  due to rounding errors
* numerical differentiation, if used, may  lead  to  function  evaluations
  outside  of the feasible  area,   because   algorithm  does  NOT  change
  numerical differentiation formula according to linear constraints.
If you want constraints to be  satisfied  exactly, try to reformulate your
problem  in  such  manner  that  all constraints will become boundary ones
(this kind of constraints is always satisfied exactly, both in  the  final
solution and in all intermediate points).

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetlc(
    minbleicstate state,
    real_2d_array c,
    integer_1d_array ct,
    const xparams _params = alglib::xdefault);
void alglib::minbleicsetlc(
    minbleicstate state,
    real_2d_array c,
    integer_1d_array ct,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minbleicsetprecdefault` function

/*************************************************************************
Modification of the preconditioner: preconditioning is turned off.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetprecdefault(
    minbleicstate state,
    const xparams _params = alglib::xdefault);

`minbleicsetprecdiag` function

/*************************************************************************
Modification  of  the  preconditioner:  diagonal of approximate Hessian is
used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    D       -   diagonal of the approximate Hessian, array[0..N-1],
                (if larger, only leading N elements are used).

NOTE 1: D[i] should be positive. Exception will be thrown otherwise.

NOTE 2: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetprecdiag(
    minbleicstate state,
    real_1d_array d,
    const xparams _params = alglib::xdefault);

`minbleicsetprecscale` function

/*************************************************************************
Modification of the preconditioner: scale-based diagonal preconditioning.

This preconditioning mode can be useful when you  don't  have  approximate
diagonal of Hessian, but you know that your  variables  are  badly  scaled
(for  example,  one  variable is in [1,10], and another in [1000,100000]),
and most part of the ill-conditioning comes from different scales of vars.

In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
can greatly improve convergence.

IMPRTANT: you should set scale of your variables  with  MinBLEICSetScale()
call  (before  or after MinBLEICSetPrecScale() call). Without knowledge of
the scale of your variables scale-based preconditioner will be  just  unit
matrix.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetprecscale(
    minbleicstate state,
    const xparams _params = alglib::xdefault);

`minbleicsetscale` function

/*************************************************************************
This function sets scaling coefficients for BLEIC optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

In  most  optimizers  (and  in  the  BLEIC  too)  scaling is NOT a form of
preconditioning. It just  affects  stopping  conditions.  You  should  set
preconditioner  by  separate  call  to  one  of  the  MinBLEICSetPrec...()
functions.

There is a special  preconditioning  mode, however,  which  uses   scaling
coefficients to form diagonal preconditioning matrix. You  can  turn  this
mode on, if you want.   But  you should understand that scaling is not the
same thing as preconditioning - these are two different, although  related
forms of tuning solver.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetscale(
    minbleicstate state,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

`minbleicsetstpmax` function

/*************************************************************************
This function sets maximum step length

IMPORTANT: this feature is hard to combine with preconditioning. You can't
set upper limit on step length, when you solve optimization  problem  with
linear (non-boundary) constraints AND preconditioner turned on.

When  non-boundary  constraints  are  present,  you  have to either a) use
preconditioner, or b) use upper limit on step length.  YOU CAN'T USE BOTH!
In this case algorithm will terminate with appropriate error code.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  lead   to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetstpmax(
    minbleicstate state,
    double stpmax,
    const xparams _params = alglib::xdefault);

`minbleicsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinBLEICOptimize().

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetxrep(
    minbleicstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

minbleic_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // subject to bound constraints -1<=x<=+1, -1<=y<=+1, using BLEIC optimizer.
    //
    real_1d_array x = "[0,0]";
    real_1d_array bndl = "[-1,-1]";
    real_1d_array bndu = "[+1,+1]";
    minbleicstate state;
    minbleicreport rep;

    //
    // These variables define stopping conditions for the optimizer.
    //
    // We use very simple condition - |g|<=epsg
    //
    double epsg = 0.000001;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;

    //
    // Now we are ready to actually optimize something:
    // * first we create optimizer
    // * we add boundary constraints
    // * we tune stopping conditions
    // * and, finally, optimize and obtain results...
    //
    minbleiccreate(x, state);
    minbleicsetbc(state, bndl, bndu);
    minbleicsetcond(state, epsg, epsf, epsx, maxits);
    alglib::minbleicoptimize(state, function1_grad);
    minbleicresults(state, x, rep);

    //
    // ...and evaluate these results
    //
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]
    return 0;
}

minbleic_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // subject to inequality constraints:
    // * x>=2 (posed as general linear constraint),
    // * x+y>=6
    // using BLEIC optimizer.
    //
    real_1d_array x = "[5,5]";
    real_2d_array c = "[[1,0,2],[1,1,6]]";
    integer_1d_array ct = "[1,1]";
    minbleicstate state;
    minbleicreport rep;

    //
    // These variables define stopping conditions for the optimizer.
    //
    // We use very simple condition - |g|<=epsg
    //
    double epsg = 0.000001;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;

    //
    // Now we are ready to actually optimize something:
    // * first we create optimizer
    // * we add linear constraints
    // * we tune stopping conditions
    // * and, finally, optimize and obtain results...
    //
    minbleiccreate(x, state);
    minbleicsetlc(state, c, ct);
    minbleicsetcond(state, epsg, epsf, epsx, maxits);
    alglib::minbleicoptimize(state, function1_grad);
    minbleicresults(state, x, rep);

    //
    // ...and evaluate these results
    //
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2,4]
    return 0;
}

minbleic_ftrim example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void s1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
    //
    // this callback calculates f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x and its gradient.
    //
    // function is trimmed when we calculate it near the singular points or outside of the [-1,+1].
    // Note that we do NOT calculate gradient in this case.
    //
    if( (x[0]<=-0.999999999999) || (x[0]>=+0.999999999999) )
    {
        func = 1.0E+300;
        return;
    }
    func = pow(1+x[0],-0.2) + pow(1-x[0],-0.3) + 1000*x[0];
    grad[0] = -0.2*pow(1+x[0],-1.2) +0.3*pow(1-x[0],-1.3) + 1000;
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x.
    //
    // This function is undefined outside of (-1,+1) and has singularities at x=-1 and x=+1.
    // Special technique called "function trimming" allows us to solve this optimization problem 
    // - without using boundary constraints!
    //
    // See http://www.alglib.net/optimization/tipsandtricks.php#ftrimming for more information
    // on this subject.
    //
    real_1d_array x = "[0]";
    double epsg = 1.0e-6;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;
    minbleicstate state;
    minbleicreport rep;

    minbleiccreate(x, state);
    minbleicsetcond(state, epsg, epsf, epsx, maxits);
    alglib::minbleicoptimize(state, s1_grad);
    minbleicresults(state, x, rep);

    printf("%s\n", x.tostring(5).c_str()); // EXPECTED: [-0.99917305]
    return 0;
}

minbleic_numdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // subject to bound constraints -1<=x<=+1, -1<=y<=+1, using BLEIC optimizer.
    //
    real_1d_array x = "[0,0]";
    real_1d_array bndl = "[-1,-1]";
    real_1d_array bndu = "[+1,+1]";
    minbleicstate state;
    minbleicreport rep;

    //
    // These variables define stopping conditions for the optimizer.
    //
    // We use very simple condition - |g|<=epsg
    //
    double epsg = 0.000001;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;

    //
    // This variable contains differentiation step
    //
    double diffstep = 1.0e-6;

    //
    // Now we are ready to actually optimize something:
    // * first we create optimizer
    // * we add boundary constraints
    // * we tune stopping conditions
    // * and, finally, optimize and obtain results...
    //
    minbleiccreatef(x, diffstep, state);
    minbleicsetbc(state, bndl, bndu);
    minbleicsetcond(state, epsg, epsf, epsx, maxits);
    alglib::minbleicoptimize(state, function1_func);
    minbleicresults(state, x, rep);

    //
    // ...and evaluate these results
    //
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]
    return 0;
}

`mincg` subpackage

Classes

mincgreport
mincgstate

Functions

mincgcreate
mincgcreatef
mincgoptimize
mincgrequesttermination
mincgrestartfrom
mincgresults
mincgresultsbuf
mincgsetcgtype
mincgsetcond
mincgsetgradientcheck
mincgsetprecdefault
mincgsetprecdiag
mincgsetprecscale
mincgsetscale
mincgsetstpmax
mincgsetxrep
mincgsuggeststep

Examples

mincg_d_1		Nonlinear optimization by CG
mincg_d_2		Nonlinear optimization with additional settings and restarts
mincg_ftrim		Nonlinear optimization by CG, function with singularities
mincg_numdiff		Nonlinear optimization by CG with numerical differentiation

`mincgreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           total number of inner iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -7    gradient verification failed.
        See MinCGSetGradientCheck() for more information.
   1    relative function improvement is no more than EpsF.
   2    relative step is no more than EpsX.
   4    gradient norm is no more than EpsG
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    terminated by user who called mincgrequesttermination(). X contains
        point which was "current accepted" when  termination  request  was
        submitted.

Other fields of this structure are not documented and should not be used!
*************************************************************************/
class mincgreport
{
    ae_int_t             iterationscount;
    ae_int_t             nfev;
    ae_int_t             varidx;
    ae_int_t             terminationtype;
};

`mincgstate` class

/*************************************************************************
This object stores state of the nonlinear CG optimizer.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class mincgstate
{
};

`mincgcreate` function

/*************************************************************************
        NONLINEAR CONJUGATE GRADIENT METHOD

DESCRIPTION:
The subroutine minimizes function F(x) of N arguments by using one of  the
nonlinear conjugate gradient methods.

These CG methods are globally convergent (even on non-convex functions) as
long as grad(f) is Lipschitz continuous in  a  some  neighborhood  of  the
L = { x : f(x)<=f(x0) }.


REQUIREMENTS:
Algorithm will request following information during its operation:
* function value F and its gradient G (simultaneously) at given point X


USAGE:
1. User initializes algorithm state with MinCGCreate() call
2. User tunes solver parameters with MinCGSetCond(), MinCGSetStpMax() and
   other functions
3. User calls MinCGOptimize() function which takes algorithm  state   and
   pointer (delegate, etc.) to callback function which calculates F/G.
4. User calls MinCGResults() to get solution
5. Optionally, user may call MinCGRestartFrom() to solve another  problem
   with same N but another starting point and/or another function.
   MinCGRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[0..N-1].

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 25.03.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgcreate(
    real_1d_array x,
    mincgstate& state,
    const xparams _params = alglib::xdefault);
void alglib::mincgcreate(
    ae_int_t n,
    real_1d_array x,
    mincgstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`mincgcreatef` function

/*************************************************************************
The subroutine is finite difference variant of MinCGCreate(). It uses
finite differences in order to differentiate target function.

Description below contains information which is specific to this function
only. We recommend to read comments on MinCGCreate() in order to get more
information about creation of CG optimizer.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[0..N-1].
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinCGSetScale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large truncation  errors, while too small
   step will result in too large numerical  errors.  1.0E-6  can  be  good
   value to start with.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is  less  robust  and  precise.  L-BFGS  needs  exact  gradient values.
   Imprecise  gradient may slow down  convergence,  especially  on  highly
   nonlinear problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 16.05.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgcreatef(
    real_1d_array x,
    double diffstep,
    mincgstate& state,
    const xparams _params = alglib::xdefault);
void alglib::mincgcreatef(
    ae_int_t n,
    real_1d_array x,
    double diffstep,
    mincgstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`mincgoptimize` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied  gradient, and one which uses function value
   only  and  numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   (either MinCGCreate()  for analytical gradient  or  MinCGCreateF()  for
   numerical differentiation) you should  choose  appropriate  variant  of
   MinCGOptimize() - one which accepts function AND gradient or one  which
   accepts function ONLY.

   Be careful to choose variant of MinCGOptimize()  which  corresponds  to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed  to  MinCGOptimize()  and  specific
   function used to create optimizer.


                  |         USER PASSED TO MinCGOptimize()
   CREATED WITH   |  function only   |  function and gradient
   ------------------------------------------------------------
   MinCGCreateF() |     work                FAIL
   MinCGCreate()  |     FAIL                work

   Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
   function and MinCGOptimize() version. Attemps to use  such  combination
   (for  example,  to create optimizer with  MinCGCreateF()  and  to  pass
   gradient information to MinCGOptimize()) will lead to  exception  being
   thrown. Either  you  did  not  pass  gradient when it WAS needed or you
   passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 20.04.2009 by Bochkanov Sergey
*************************************************************************/
void mincgoptimize(mincgstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void mincgoptimize(mincgstate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`mincgrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgrequesttermination(
    mincgstate state,
    const xparams _params = alglib::xdefault);

`mincgrestartfrom` function

/*************************************************************************
This  subroutine  restarts  CG  algorithm from new point. All optimization
parameters are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure used to store algorithm state.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgrestartfrom(
    mincgstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

Examples: [1]

`mincgresults` function

/*************************************************************************
Conjugate gradient results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    * -8    internal integrity control  detected  infinite
                            or NAN values in  function/gradient.  Abnormal
                            termination signalled.
                    * -7    gradient verification failed.
                            See MinCGSetGradientCheck() for more information.
                    *  1    relative function improvement is no more than
                            EpsF.
                    *  2    relative step is no more than EpsX.
                    *  4    gradient norm is no more than EpsG
                    *  5    MaxIts steps was taken
                    *  7    stopping conditions are too stringent,
                            further improvement is impossible,
                            we return best X found so far
                    *  8    terminated by user
                * Rep.IterationsCount contains iterations count
                * NFEV countains number of function calculations

  -- ALGLIB --
     Copyright 20.04.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgresults(
    mincgstate state,
    real_1d_array& x,
    mincgreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4]

`mincgresultsbuf` function

/*************************************************************************
Conjugate gradient results

Buffered implementation of MinCGResults(), which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 20.04.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgresultsbuf(
    mincgstate state,
    real_1d_array& x,
    mincgreport& rep,
    const xparams _params = alglib::xdefault);

`mincgsetcgtype` function

/*************************************************************************
This function sets CG algorithm.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    CGType  -   algorithm type:
                * -1    automatic selection of the best algorithm
                * 0     DY (Dai and Yuan) algorithm
                * 1     Hybrid DY-HS algorithm

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsetcgtype(
    mincgstate state,
    ae_int_t cgtype,
    const xparams _params = alglib::xdefault);

`mincgsetcond` function

/*************************************************************************
This function sets stopping conditions for CG optimization algorithm.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinCGSetScale()
    EpsF    -   >=0
                The  subroutine  finishes  its work if on k+1-th iteration
                the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                is satisfied.
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - ste pvector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinCGSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to
automatic stopping criterion selection (small EpsX).

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsetcond(
    mincgstate state,
    double epsg,
    double epsf,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`mincgsetgradientcheck` function

/*************************************************************************

This  subroutine  turns  on  verification  of  the  user-supplied analytic
gradient:
* user calls this subroutine before optimization begins
* MinCGOptimize() is called
* prior to  actual  optimization, for each component  of  parameters being
  optimized X[i] algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point
  * in case difference between prediction and actual value is higher  than
    some predetermined threshold, algorithm stops with completion code -7;
    Rep.VarIdx is set to index of the parameter with incorrect derivative.
* after verification is over, algorithm proceeds to the actual optimization.

NOTE 1: verification  needs  N (parameters count) gradient evaluations. It
        is very costly and you should use  it  only  for  low  dimensional
        problems,  when  you  want  to  be  sure  that  you've   correctly
        calculated  analytic  derivatives.  You  should  not use it in the
        production code (unless you want to check derivatives provided  by
        some third party).

NOTE 2: you  should  carefully  choose  TestStep. Value which is too large
        (so large that function behaviour is significantly non-cubic) will
        lead to false alarms. You may use  different  step  for  different
        parameters by means of setting scale with MinCGSetScale().

NOTE 3: this function may lead to false positives. In case it reports that
        I-th  derivative was calculated incorrectly, you may decrease test
        step  and  try  one  more  time  - maybe your function changes too
        sharply  and  your  step  is  too  large for such rapidly chanding
        function.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification

  -- ALGLIB --
     Copyright 31.05.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsetgradientcheck(
    mincgstate state,
    double teststep,
    const xparams _params = alglib::xdefault);

`mincgsetprecdefault` function

/*************************************************************************
Modification of the preconditioner: preconditioning is turned off.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsetprecdefault(
    mincgstate state,
    const xparams _params = alglib::xdefault);

`mincgsetprecdiag` function

/*************************************************************************
Modification  of  the  preconditioner:  diagonal of approximate Hessian is
used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    D       -   diagonal of the approximate Hessian, array[0..N-1],
                (if larger, only leading N elements are used).

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

NOTE 2: D[i] should be positive. Exception will be thrown otherwise.

NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsetprecdiag(
    mincgstate state,
    real_1d_array d,
    const xparams _params = alglib::xdefault);

`mincgsetprecscale` function

/*************************************************************************
Modification of the preconditioner: scale-based diagonal preconditioning.

This preconditioning mode can be useful when you  don't  have  approximate
diagonal of Hessian, but you know that your  variables  are  badly  scaled
(for  example,  one  variable is in [1,10], and another in [1000,100000]),
and most part of the ill-conditioning comes from different scales of vars.

In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
can greatly improve convergence.

IMPRTANT: you should set scale of your variables with MinCGSetScale() call
(before or after MinCGSetPrecScale() call). Without knowledge of the scale
of your variables scale-based preconditioner will be just unit matrix.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsetprecscale(
    mincgstate state,
    const xparams _params = alglib::xdefault);

`mincgsetscale` function

/*************************************************************************
This function sets scaling coefficients for CG optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of CG optimizer  -  step
along I-th axis is equal to DiffStep*S[I].

In   most   optimizers  (and  in  the  CG  too)  scaling is NOT a form  of
preconditioning. It just  affects  stopping  conditions.  You  should  set
preconditioner by separate call to one of the MinCGSetPrec...() functions.

There  is  special  preconditioning  mode, however,  which  uses   scaling
coefficients to form diagonal preconditioning matrix. You  can  turn  this
mode on, if you want.   But  you should understand that scaling is not the
same thing as preconditioning - these are two different, although  related
forms of tuning solver.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsetscale(
    mincgstate state,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

`mincgsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsetstpmax(
    mincgstate state,
    double stpmax,
    const xparams _params = alglib::xdefault);

`mincgsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinCGOptimize().

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsetxrep(
    mincgstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

`mincgsuggeststep` function

/*************************************************************************
This function allows to suggest initial step length to the CG algorithm.

Suggested  step  length  is used as starting point for the line search. It
can be useful when you have  badly  scaled  problem,  i.e.  when  ||grad||
(which is used as initial estimate for the first step) is many  orders  of
magnitude different from the desired step.

Line search  may  fail  on  such problems without good estimate of initial
step length. Imagine, for example, problem with ||grad||=10^50 and desired
step equal to 0.1 Line  search function will use 10^50  as  initial  step,
then  it  will  decrease step length by 2 (up to 20 attempts) and will get
10^44, which is still too large.

This function allows us to tell than line search should  be  started  from
some moderate step length, like 1.0, so algorithm will be able  to  detect
desired step length in a several searches.

Default behavior (when no step is suggested) is to use preconditioner,  if
it is available, to generate initial estimate of step length.

This function influences only first iteration of algorithm. It  should  be
called between MinCGCreate/MinCGRestartFrom() call and MinCGOptimize call.
Suggested step is ignored if you have preconditioner.

INPUT PARAMETERS:
    State   -   structure used to store algorithm state.
    Stp     -   initial estimate of the step length.
                Can be zero (no estimate).

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mincgsuggeststep(
    mincgstate state,
    double stp,
    const xparams _params = alglib::xdefault);

mincg_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // with nonlinear conjugate gradient method.
    //
    real_1d_array x = "[0,0]";
    double epsg = 0.0000000001;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;
    mincgstate state;
    mincgreport rep;

    mincgcreate(x, state);
    mincgsetcond(state, epsg, epsf, epsx, maxits);
    alglib::mincgoptimize(state, function1_grad);
    mincgresults(state, x, rep);

    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
    return 0;
}

mincg_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // with nonlinear conjugate gradient method.
    //
    // Several advanced techniques are demonstrated:
    // * upper limit on step size
    // * restart from new point
    //
    real_1d_array x = "[0,0]";
    double epsg = 0.0000000001;
    double epsf = 0;
    double epsx = 0;
    double stpmax = 0.1;
    ae_int_t maxits = 0;
    mincgstate state;
    mincgreport rep;

    // first run
    mincgcreate(x, state);
    mincgsetcond(state, epsg, epsf, epsx, maxits);
    mincgsetstpmax(state, stpmax);
    alglib::mincgoptimize(state, function1_grad);
    mincgresults(state, x, rep);

    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]

    // second run - algorithm is restarted with mincgrestartfrom()
    x = "[10,10]";
    mincgrestartfrom(state, x);
    alglib::mincgoptimize(state, function1_grad);
    mincgresults(state, x, rep);

    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
    return 0;
}

mincg_ftrim example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void s1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
    //
    // this callback calculates f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x and its gradient.
    //
    // function is trimmed when we calculate it near the singular points or outside of the [-1,+1].
    // Note that we do NOT calculate gradient in this case.
    //
    if( (x[0]<=-0.999999999999) || (x[0]>=+0.999999999999) )
    {
        func = 1.0E+300;
        return;
    }
    func = pow(1+x[0],-0.2) + pow(1-x[0],-0.3) + 1000*x[0];
    grad[0] = -0.2*pow(1+x[0],-1.2) +0.3*pow(1-x[0],-1.3) + 1000;
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x.
    // This function has singularities at the boundary of the [-1,+1], but technique called
    // "function trimming" allows us to solve this optimization problem.
    //
    // See http://www.alglib.net/optimization/tipsandtricks.php#ftrimming for more information
    // on this subject.
    //
    real_1d_array x = "[0]";
    double epsg = 1.0e-6;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;
    mincgstate state;
    mincgreport rep;

    mincgcreate(x, state);
    mincgsetcond(state, epsg, epsf, epsx, maxits);
    alglib::mincgoptimize(state, s1_grad);
    mincgresults(state, x, rep);

    printf("%s\n", x.tostring(5).c_str()); // EXPECTED: [-0.99917305]
    return 0;
}

mincg_numdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // using numerical differentiation to calculate gradient.
    //
    real_1d_array x = "[0,0]";
    double epsg = 0.0000000001;
    double epsf = 0;
    double epsx = 0;
    double diffstep = 1.0e-6;
    ae_int_t maxits = 0;
    mincgstate state;
    mincgreport rep;

    mincgcreatef(x, diffstep, state);
    mincgsetcond(state, epsg, epsf, epsx, maxits);
    alglib::mincgoptimize(state, function1_func);
    mincgresults(state, x, rep);

    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
    return 0;
}

`mincomp` subpackage

Classes

minasareport
minasastate

Functions

minasacreate
minasaoptimize
minasarestartfrom
minasaresults
minasaresultsbuf
minasasetalgorithm
minasasetcond
minasasetstpmax
minasasetxrep
minbleicsetbarrierdecay
minbleicsetbarrierwidth
minlbfgssetcholeskypreconditioner
minlbfgssetdefaultpreconditioner

Examples

`minasareport` class

/*************************************************************************

*************************************************************************/
class minasareport
{
    ae_int_t             iterationscount;
    ae_int_t             nfev;
    ae_int_t             terminationtype;
    ae_int_t             activeconstraints;
};

`minasastate` class

/*************************************************************************

*************************************************************************/
class minasastate
{
};

`minasacreate` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 25.03.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minasacreate(
    real_1d_array x,
    real_1d_array bndl,
    real_1d_array bndu,
    minasastate& state,
    const xparams _params = alglib::xdefault);
void alglib::minasacreate(
    ae_int_t n,
    real_1d_array x,
    real_1d_array bndl,
    real_1d_array bndu,
    minasastate& state,
    const xparams _params = alglib::xdefault);

`minasaoptimize` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void minasaoptimize(minasastate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

`minasarestartfrom` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minasarestartfrom(
    minasastate state,
    real_1d_array x,
    real_1d_array bndl,
    real_1d_array bndu,
    const xparams _params = alglib::xdefault);

`minasaresults` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minasaresults(
    minasastate state,
    real_1d_array& x,
    minasareport& rep,
    const xparams _params = alglib::xdefault);

`minasaresultsbuf` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minasaresultsbuf(
    minasastate state,
    real_1d_array& x,
    minasareport& rep,
    const xparams _params = alglib::xdefault);

`minasasetalgorithm` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minasasetalgorithm(
    minasastate state,
    ae_int_t algotype,
    const xparams _params = alglib::xdefault);

`minasasetcond` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minasasetcond(
    minasastate state,
    double epsg,
    double epsf,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

`minasasetstpmax` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minasasetstpmax(
    minasastate state,
    double stpmax,
    const xparams _params = alglib::xdefault);

`minasasetxrep` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minasasetxrep(
    minasastate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

`minbleicsetbarrierdecay` function

/*************************************************************************
This is obsolete function which was used by previous version of the  BLEIC
optimizer. It does nothing in the current version of BLEIC.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetbarrierdecay(
    minbleicstate state,
    double mudecay,
    const xparams _params = alglib::xdefault);

`minbleicsetbarrierwidth` function

/*************************************************************************
This is obsolete function which was used by previous version of the  BLEIC
optimizer. It does nothing in the current version of BLEIC.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minbleicsetbarrierwidth(
    minbleicstate state,
    double mu,
    const xparams _params = alglib::xdefault);

`minlbfgssetcholeskypreconditioner` function

/*************************************************************************
Obsolete function, use MinLBFGSSetCholeskyPreconditioner() instead.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetcholeskypreconditioner(
    minlbfgsstate state,
    real_2d_array p,
    bool isupper,
    const xparams _params = alglib::xdefault);

`minlbfgssetdefaultpreconditioner` function

/*************************************************************************
Obsolete function, use MinLBFGSSetPrecDefault() instead.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetdefaultpreconditioner(
    minlbfgsstate state,
    const xparams _params = alglib::xdefault);

`minlbfgs` subpackage

Classes

minlbfgsreport
minlbfgsstate

Functions

minlbfgscreate
minlbfgscreatef
minlbfgsoptimize
minlbfgsrequesttermination
minlbfgsrestartfrom
minlbfgsresults
minlbfgsresultsbuf
minlbfgssetcond
minlbfgssetgradientcheck
minlbfgssetpreccholesky
minlbfgssetprecdefault
minlbfgssetprecdiag
minlbfgssetprecscale
minlbfgssetscale
minlbfgssetstpmax
minlbfgssetxrep

Examples

minlbfgs_d_1		Nonlinear optimization by L-BFGS
minlbfgs_d_2		Nonlinear optimization with additional settings and restarts
minlbfgs_ftrim		Nonlinear optimization by LBFGS, function with singularities
minlbfgs_numdiff		Nonlinear optimization by L-BFGS with numerical differentiation

`minlbfgsreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           total number of inner iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -7    gradient verification failed.
        See MinLBFGSSetGradientCheck() for more information.
   1    relative function improvement is no more than EpsF.
   2    relative step is no more than EpsX.
   4    gradient norm is no more than EpsG
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    terminated    by  user  who  called  minlbfgsrequesttermination().
        X contains point which was   "current accepted"  when  termination
        request was submitted.

Other fields of this structure are not documented and should not be used!
*************************************************************************/
class minlbfgsreport
{
    ae_int_t             iterationscount;
    ae_int_t             nfev;
    ae_int_t             varidx;
    ae_int_t             terminationtype;
};

`minlbfgsstate` class

/*************************************************************************

*************************************************************************/
class minlbfgsstate
{
};

`minlbfgscreate` function

/*************************************************************************
        LIMITED MEMORY BFGS METHOD FOR LARGE SCALE OPTIMIZATION

DESCRIPTION:
The subroutine minimizes function F(x) of N arguments by  using  a  quasi-
Newton method (LBFGS scheme) which is optimized to use  a  minimum  amount
of memory.
The subroutine generates the approximation of an inverse Hessian matrix by
using information about the last M steps of the algorithm  (instead of N).
It lessens a required amount of memory from a value  of  order  N^2  to  a
value of order 2*N*M.


REQUIREMENTS:
Algorithm will request following information during its operation:
* function value F and its gradient G (simultaneously) at given point X


USAGE:
1. User initializes algorithm state with MinLBFGSCreate() call
2. User tunes solver parameters with MinLBFGSSetCond() MinLBFGSSetStpMax()
   and other functions
3. User calls MinLBFGSOptimize() function which takes algorithm  state and
   pointer (delegate, etc.) to callback function which calculates F/G.
4. User calls MinLBFGSResults() to get solution
5. Optionally user may call MinLBFGSRestartFrom() to solve another problem
   with same N/M but another starting point and/or another function.
   MinLBFGSRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   problem dimension. N>0
    M       -   number of corrections in the BFGS scheme of Hessian
                approximation update. Recommended value:  3<=M<=7. The smaller
                value causes worse convergence, the bigger will  not  cause  a
                considerably better convergence, but will cause a fall in  the
                performance. M<=N.
    X       -   initial solution approximation, array[0..N-1].


OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state


NOTES:
1. you may tune stopping conditions with MinLBFGSSetCond() function
2. if target function contains exp() or other fast growing functions,  and
   optimization algorithm makes too large steps which leads  to  overflow,
   use MinLBFGSSetStpMax() function to bound algorithm's  steps.  However,
   L-BFGS rarely needs such a tuning.


  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgscreate(
    ae_int_t m,
    real_1d_array x,
    minlbfgsstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minlbfgscreate(
    ae_int_t n,
    ae_int_t m,
    real_1d_array x,
    minlbfgsstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minlbfgscreatef` function

/*************************************************************************
The subroutine is finite difference variant of MinLBFGSCreate().  It  uses
finite differences in order to differentiate target function.

Description below contains information which is specific to  this function
only. We recommend to read comments on MinLBFGSCreate() in  order  to  get
more information about creation of LBFGS optimizer.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    M       -   number of corrections in the BFGS scheme of Hessian
                approximation update. Recommended value:  3<=M<=7. The smaller
                value causes worse convergence, the bigger will  not  cause  a
                considerably better convergence, but will cause a fall in  the
                performance. M<=N.
    X       -   starting point, array[0..N-1].
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinLBFGSSetScale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large truncation  errors, while too small
   step will result in too large numerical  errors.  1.0E-6  can  be  good
   value to start with.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is   less  robust  and  precise.  LBFGS  needs  exact  gradient values.
   Imprecise gradient may slow  down  convergence,  especially  on  highly
   nonlinear problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 16.05.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgscreatef(
    ae_int_t m,
    real_1d_array x,
    double diffstep,
    minlbfgsstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minlbfgscreatef(
    ae_int_t n,
    ae_int_t m,
    real_1d_array x,
    double diffstep,
    minlbfgsstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minlbfgsoptimize` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied gradient,  and one which uses function value
   only  and  numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   (either MinLBFGSCreate() for analytical gradient  or  MinLBFGSCreateF()
   for numerical differentiation) you should choose appropriate variant of
   MinLBFGSOptimize() - one  which  accepts  function  AND gradient or one
   which accepts function ONLY.

   Be careful to choose variant of MinLBFGSOptimize() which corresponds to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to MinLBFGSOptimize()  and specific
   function used to create optimizer.


                     |         USER PASSED TO MinLBFGSOptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   MinLBFGSCreateF() |     work                FAIL
   MinLBFGSCreate()  |     FAIL                work

   Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
   function  and  MinLBFGSOptimize()  version.   Attemps   to   use   such
   combination (for example, to create optimizer with MinLBFGSCreateF() and
   to pass gradient information to MinCGOptimize()) will lead to exception
   being thrown. Either  you  did  not pass gradient when it WAS needed or
   you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void minlbfgsoptimize(minlbfgsstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minlbfgsoptimize(minlbfgsstate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minlbfgsrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgsrequesttermination(
    minlbfgsstate state,
    const xparams _params = alglib::xdefault);

`minlbfgsrestartfrom` function

/*************************************************************************
This  subroutine restarts LBFGS algorithm from new point. All optimization
parameters are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure used to store algorithm state
    X       -   new starting point.

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgsrestartfrom(
    minlbfgsstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minlbfgsresults` function

/*************************************************************************
L-BFGS algorithm results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    * -8    internal integrity control  detected  infinite
                            or NAN values in  function/gradient.  Abnormal
                            termination signalled.
                    * -7    gradient verification failed.
                            See MinLBFGSSetGradientCheck() for more information.
                    * -2    rounding errors prevent further improvement.
                            X contains best point found.
                    * -1    incorrect parameters were specified
                    *  1    relative function improvement is no more than
                            EpsF.
                    *  2    relative step is no more than EpsX.
                    *  4    gradient norm is no more than EpsG
                    *  5    MaxIts steps was taken
                    *  7    stopping conditions are too stringent,
                            further improvement is impossible
                    *  8    terminated by user who called minlbfgsrequesttermination().
                            X contains point which was "current accepted" when
                            termination request was submitted.
                * Rep.IterationsCount contains iterations count
                * NFEV countains number of function calculations

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgsresults(
    minlbfgsstate state,
    real_1d_array& x,
    minlbfgsreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minlbfgsresultsbuf` function

/*************************************************************************
L-BFGS algorithm results

Buffered implementation of MinLBFGSResults which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 20.08.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgsresultsbuf(
    minlbfgsstate state,
    real_1d_array& x,
    minlbfgsreport& rep,
    const xparams _params = alglib::xdefault);

`minlbfgssetcond` function

/*************************************************************************
This function sets stopping conditions for L-BFGS optimization algorithm.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinLBFGSSetScale()
    EpsF    -   >=0
                The  subroutine  finishes  its work if on k+1-th iteration
                the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                is satisfied.
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - ste pvector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinLBFGSSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to
automatic stopping criterion selection (small EpsX).

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetcond(
    minlbfgsstate state,
    double epsg,
    double epsf,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minlbfgssetgradientcheck` function

/*************************************************************************
This  subroutine  turns  on  verification  of  the  user-supplied analytic
gradient:
* user calls this subroutine before optimization begins
* MinLBFGSOptimize() is called
* prior to  actual  optimization, for each component  of  parameters being
  optimized X[i] algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * if needed, steps are bounded with respect to constraints on X[]
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point
  * in case difference between prediction and actual value is higher  than
    some predetermined threshold, algorithm stops with completion code -7;
    Rep.VarIdx is set to index of the parameter with incorrect derivative.
* after verification is over, algorithm proceeds to the actual optimization.

NOTE 1: verification  needs  N (parameters count) gradient evaluations. It
        is very costly and you should use  it  only  for  low  dimensional
        problems,  when  you  want  to  be  sure  that  you've   correctly
        calculated  analytic  derivatives.  You  should  not use it in the
        production code (unless you want to check derivatives provided  by
        some third party).

NOTE 2: you  should  carefully  choose  TestStep. Value which is too large
        (so large that function behaviour is significantly non-cubic) will
        lead to false alarms. You may use  different  step  for  different
        parameters by means of setting scale with MinLBFGSSetScale().

NOTE 3: this function may lead to false positives. In case it reports that
        I-th  derivative was calculated incorrectly, you may decrease test
        step  and  try  one  more  time  - maybe your function changes too
        sharply  and  your  step  is  too  large for such rapidly chanding
        function.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification

  -- ALGLIB --
     Copyright 24.05.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetgradientcheck(
    minlbfgsstate state,
    double teststep,
    const xparams _params = alglib::xdefault);

`minlbfgssetpreccholesky` function

/*************************************************************************
Modification of the preconditioner: Cholesky factorization of  approximate
Hessian is used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    P       -   triangular preconditioner, Cholesky factorization of
                the approximate Hessian. array[0..N-1,0..N-1],
                (if larger, only leading N elements are used).
    IsUpper -   whether upper or lower triangle of P is given
                (other triangle is not referenced)

After call to this function preconditioner is changed to P  (P  is  copied
into the internal buffer).

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

NOTE 2:  P  should  be nonsingular. Exception will be thrown otherwise.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetpreccholesky(
    minlbfgsstate state,
    real_2d_array p,
    bool isupper,
    const xparams _params = alglib::xdefault);

`minlbfgssetprecdefault` function

/*************************************************************************
Modification  of  the  preconditioner:  default  preconditioner    (simple
scaling, same for all elements of X) is used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetprecdefault(
    minlbfgsstate state,
    const xparams _params = alglib::xdefault);

`minlbfgssetprecdiag` function

/*************************************************************************
Modification  of  the  preconditioner:  diagonal of approximate Hessian is
used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    D       -   diagonal of the approximate Hessian, array[0..N-1],
                (if larger, only leading N elements are used).

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

NOTE 2: D[i] should be positive. Exception will be thrown otherwise.

NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetprecdiag(
    minlbfgsstate state,
    real_1d_array d,
    const xparams _params = alglib::xdefault);

`minlbfgssetprecscale` function

/*************************************************************************
Modification of the preconditioner: scale-based diagonal preconditioning.

This preconditioning mode can be useful when you  don't  have  approximate
diagonal of Hessian, but you know that your  variables  are  badly  scaled
(for  example,  one  variable is in [1,10], and another in [1000,100000]),
and most part of the ill-conditioning comes from different scales of vars.

In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
can greatly improve convergence.

IMPRTANT: you should set scale of your variables  with  MinLBFGSSetScale()
call  (before  or after MinLBFGSSetPrecScale() call). Without knowledge of
the scale of your variables scale-based preconditioner will be  just  unit
matrix.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetprecscale(
    minlbfgsstate state,
    const xparams _params = alglib::xdefault);

`minlbfgssetscale` function

/*************************************************************************
This function sets scaling coefficients for LBFGS optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

In  most  optimizers  (and  in  the  LBFGS  too)  scaling is NOT a form of
preconditioning. It just  affects  stopping  conditions.  You  should  set
preconditioner  by  separate  call  to  one  of  the  MinLBFGSSetPrec...()
functions.

There  is  special  preconditioning  mode, however,  which  uses   scaling
coefficients to form diagonal preconditioning matrix. You  can  turn  this
mode on, if you want.   But  you should understand that scaling is not the
same thing as preconditioning - these are two different, although  related
forms of tuning solver.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetscale(
    minlbfgsstate state,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

`minlbfgssetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0 (default),  if
                you don't want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetstpmax(
    minlbfgsstate state,
    double stpmax,
    const xparams _params = alglib::xdefault);

`minlbfgssetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinLBFGSOptimize().


  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlbfgssetxrep(
    minlbfgsstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

minlbfgs_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // using LBFGS method.
    //
    real_1d_array x = "[0,0]";
    double epsg = 0.0000000001;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;
    minlbfgsstate state;
    minlbfgsreport rep;

    minlbfgscreate(1, x, state);
    minlbfgssetcond(state, epsg, epsf, epsx, maxits);
    alglib::minlbfgsoptimize(state, function1_grad);
    minlbfgsresults(state, x, rep);

    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
    return 0;
}

minlbfgs_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // using LBFGS method.
    //
    // Several advanced techniques are demonstrated:
    // * upper limit on step size
    // * restart from new point
    //
    real_1d_array x = "[0,0]";
    double epsg = 0.0000000001;
    double epsf = 0;
    double epsx = 0;
    double stpmax = 0.1;
    ae_int_t maxits = 0;
    minlbfgsstate state;
    minlbfgsreport rep;

    // first run
    minlbfgscreate(1, x, state);
    minlbfgssetcond(state, epsg, epsf, epsx, maxits);
    minlbfgssetstpmax(state, stpmax);
    alglib::minlbfgsoptimize(state, function1_grad);
    minlbfgsresults(state, x, rep);

    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]

    // second run - algorithm is restarted
    x = "[10,10]";
    minlbfgsrestartfrom(state, x);
    alglib::minlbfgsoptimize(state, function1_grad);
    minlbfgsresults(state, x, rep);

    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
    return 0;
}

minlbfgs_ftrim example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void s1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
    //
    // this callback calculates f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x and its gradient.
    //
    // function is trimmed when we calculate it near the singular points or outside of the [-1,+1].
    // Note that we do NOT calculate gradient in this case.
    //
    if( (x[0]<=-0.999999999999) || (x[0]>=+0.999999999999) )
    {
        func = 1.0E+300;
        return;
    }
    func = pow(1+x[0],-0.2) + pow(1-x[0],-0.3) + 1000*x[0];
    grad[0] = -0.2*pow(1+x[0],-1.2) +0.3*pow(1-x[0],-1.3) + 1000;
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x.
    // This function has singularities at the boundary of the [-1,+1], but technique called
    // "function trimming" allows us to solve this optimization problem.
    //
    // See http://www.alglib.net/optimization/tipsandtricks.php#ftrimming for more information
    // on this subject.
    //
    real_1d_array x = "[0]";
    double epsg = 1.0e-6;
    double epsf = 0;
    double epsx = 0;
    ae_int_t maxits = 0;
    minlbfgsstate state;
    minlbfgsreport rep;

    minlbfgscreate(1, x, state);
    minlbfgssetcond(state, epsg, epsf, epsx, maxits);
    alglib::minlbfgsoptimize(state, s1_grad);
    minlbfgsresults(state, x, rep);

    printf("%s\n", x.tostring(5).c_str()); // EXPECTED: [-0.99917305]
    return 0;
}

minlbfgs_numdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
    // using numerical differentiation to calculate gradient.
    //
    real_1d_array x = "[0,0]";
    double epsg = 0.0000000001;
    double epsf = 0;
    double epsx = 0;
    double diffstep = 1.0e-6;
    ae_int_t maxits = 0;
    minlbfgsstate state;
    minlbfgsreport rep;

    minlbfgscreatef(1, x, diffstep, state);
    minlbfgssetcond(state, epsg, epsf, epsx, maxits);
    alglib::minlbfgsoptimize(state, function1_func);
    minlbfgsresults(state, x, rep);

    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
    return 0;
}

`minlm` subpackage

Classes

minlmreport
minlmstate

Functions

minlmcreatefgh
minlmcreatefgj
minlmcreatefj
minlmcreatev
minlmcreatevgj
minlmcreatevj
minlmoptimize
minlmrequesttermination
minlmrestartfrom
minlmresults
minlmresultsbuf
minlmsetacctype
minlmsetbc
minlmsetcond
minlmsetgradientcheck
minlmsetlc
minlmsetscale
minlmsetstpmax
minlmsetxrep

Examples

minlm_d_fgh		Nonlinear Hessian-based optimization for general functions
minlm_d_restarts		Efficient restarts of LM optimizer
minlm_d_v		Nonlinear least squares optimization using function vector only
minlm_d_vb		Bound constrained nonlinear least squares optimization
minlm_d_vj		Nonlinear least squares optimization using function vector and Jacobian

`minlmreport` class

/*************************************************************************
Optimization report, filled by MinLMResults() function

FIELDS:
* TerminationType, completetion code:
    * -8    optimizer detected NAN/INF values either in the function itself,
            or in its Jacobian
    * -7    derivative correctness check failed;
            see rep.funcidx, rep.varidx for
            more information.
    * -5    inappropriate solver was used:
            * solver created with minlmcreatefgh() used  on  problem  with
              general linear constraints (set with minlmsetlc() call).
    * -3    constraints are inconsistent
    *  2    relative step is no more than EpsX.
    *  5    MaxIts steps was taken
    *  7    stopping conditions are too stringent,
            further improvement is impossible
    *  8    terminated   by  user  who  called  MinLMRequestTermination().
            X contains point which was "current accepted" when termination
            request was submitted.
* IterationsCount, contains iterations count
* NFunc, number of function calculations
* NJac, number of Jacobi matrix calculations
* NGrad, number of gradient calculations
* NHess, number of Hessian calculations
* NCholesky, number of Cholesky decomposition calculations
*************************************************************************/
class minlmreport
{
    ae_int_t             iterationscount;
    ae_int_t             terminationtype;
    ae_int_t             funcidx;
    ae_int_t             varidx;
    ae_int_t             nfunc;
    ae_int_t             njac;
    ae_int_t             ngrad;
    ae_int_t             nhess;
    ae_int_t             ncholesky;
};

`minlmstate` class

/*************************************************************************
Levenberg-Marquardt optimizer.

This structure should be created using one of the MinLMCreate???()
functions. You should not access its fields directly; use ALGLIB functions
to work with it.
*************************************************************************/
class minlmstate
{
};

`minlmcreatefgh` function

/*************************************************************************
    LEVENBERG-MARQUARDT-LIKE METHOD FOR NON-LINEAR OPTIMIZATION

DESCRIPTION:
This  function  is  used  to  find  minimum  of general form (not "sum-of-
-squares") function
    F = F(x[0], ..., x[n-1])
using  its  gradient  and  Hessian.  Levenberg-Marquardt modification with
L-BFGS pre-optimization and internal pre-conditioned  L-BFGS  optimization
after each Levenberg-Marquardt step is used.


REQUIREMENTS:
This algorithm will request following information during its operation:

* function value F at given point X
* F and gradient G (simultaneously) at given point X
* F, G and Hessian H (simultaneously) at given point X

There are several overloaded versions of  MinLMOptimize()  function  which
correspond  to  different LM-like optimization algorithms provided by this
unit. You should choose version which accepts func(),  grad()  and  hess()
function pointers. First pointer is used to calculate F  at  given  point,
second  one  calculates  F(x)  and  grad F(x),  third one calculates F(x),
grad F(x), hess F(x).

You can try to initialize MinLMState structure with FGH-function and  then
use incorrect version of MinLMOptimize() (for example, version which  does
not provide Hessian matrix), but it will lead to  exception  being  thrown
after first attempt to calculate Hessian.


USAGE:
1. User initializes algorithm state with MinLMCreateFGH() call
2. User tunes solver parameters with MinLMSetCond(),  MinLMSetStpMax() and
   other functions
3. User calls MinLMOptimize() function which  takes algorithm  state   and
   pointers (delegates, etc.) to callback functions.
4. User calls MinLMResults() to get solution
5. Optionally, user may call MinLMRestartFrom() to solve  another  problem
   with same N but another starting point and/or another function.
   MinLMRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   dimension, N>1
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   initial solution, array[0..N-1]

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. you may tune stopping conditions with MinLMSetCond() function
2. if target function contains exp() or other fast growing functions,  and
   optimization algorithm makes too large steps which leads  to  overflow,
   use MinLMSetStpMax() function to bound algorithm's steps.

  -- ALGLIB --
     Copyright 30.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmcreatefgh(
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minlmcreatefgh(
    ae_int_t n,
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minlmcreatefgj` function

/*************************************************************************
This is obsolete function.

Since ALGLIB 3.3 it is equivalent to MinLMCreateFJ().

  -- ALGLIB --
     Copyright 30.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmcreatefgj(
    ae_int_t m,
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minlmcreatefgj(
    ae_int_t n,
    ae_int_t m,
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);

`minlmcreatefj` function

/*************************************************************************
This function is considered obsolete since ALGLIB 3.1.0 and is present for
backward  compatibility  only.  We  recommend  to use MinLMCreateVJ, which
provides similar, but more consistent and feature-rich interface.

  -- ALGLIB --
     Copyright 30.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmcreatefj(
    ae_int_t m,
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minlmcreatefj(
    ae_int_t n,
    ae_int_t m,
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);

`minlmcreatev` function

/*************************************************************************
                IMPROVED LEVENBERG-MARQUARDT METHOD FOR
                 NON-LINEAR LEAST SQUARES OPTIMIZATION

DESCRIPTION:
This function is used to find minimum of function which is represented  as
sum of squares:
    F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1])
using value of function vector f[] only. Finite differences  are  used  to
calculate Jacobian.


REQUIREMENTS:
This algorithm will request following information during its operation:
* function vector f[] at given point X

There are several overloaded versions of  MinLMOptimize()  function  which
correspond  to  different LM-like optimization algorithms provided by this
unit. You should choose version which accepts fvec() callback.

You can try to initialize MinLMState structure with VJ  function and  then
use incorrect version  of  MinLMOptimize()  (for  example,  version  which
works with general form function and does not accept function vector), but
it will  lead  to  exception being thrown after first attempt to calculate
Jacobian.


USAGE:
1. User initializes algorithm state with MinLMCreateV() call
2. User tunes solver parameters with MinLMSetCond(),  MinLMSetStpMax() and
   other functions
3. User calls MinLMOptimize() function which  takes algorithm  state   and
   callback functions.
4. User calls MinLMResults() to get solution
5. Optionally, user may call MinLMRestartFrom() to solve  another  problem
   with same N/M but another starting point and/or another function.
   MinLMRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   dimension, N>1
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    M       -   number of functions f[i]
    X       -   initial solution, array[0..N-1]
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

See also MinLMIteration, MinLMResults.

NOTES:
1. you may tune stopping conditions with MinLMSetCond() function
2. if target function contains exp() or other fast growing functions,  and
   optimization algorithm makes too large steps which leads  to  overflow,
   use MinLMSetStpMax() function to bound algorithm's steps.

  -- ALGLIB --
     Copyright 30.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmcreatev(
    ae_int_t m,
    real_1d_array x,
    double diffstep,
    minlmstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minlmcreatev(
    ae_int_t n,
    ae_int_t m,
    real_1d_array x,
    double diffstep,
    minlmstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minlmcreatevgj` function

/*************************************************************************
This is obsolete function.

Since ALGLIB 3.3 it is equivalent to MinLMCreateVJ().

  -- ALGLIB --
     Copyright 30.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmcreatevgj(
    ae_int_t m,
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minlmcreatevgj(
    ae_int_t n,
    ae_int_t m,
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);

`minlmcreatevj` function

/*************************************************************************
                IMPROVED LEVENBERG-MARQUARDT METHOD FOR
                 NON-LINEAR LEAST SQUARES OPTIMIZATION

DESCRIPTION:
This function is used to find minimum of function which is represented  as
sum of squares:
    F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1])
using value of function vector f[] and Jacobian of f[].


REQUIREMENTS:
This algorithm will request following information during its operation:

* function vector f[] at given point X
* function vector f[] and Jacobian of f[] (simultaneously) at given point

There are several overloaded versions of  MinLMOptimize()  function  which
correspond  to  different LM-like optimization algorithms provided by this
unit. You should choose version which accepts fvec()  and jac() callbacks.
First  one  is used to calculate f[] at given point, second one calculates
f[] and Jacobian df[i]/dx[j].

You can try to initialize MinLMState structure with VJ  function and  then
use incorrect version  of  MinLMOptimize()  (for  example,  version  which
works  with  general  form function and does not provide Jacobian), but it
will  lead  to  exception  being  thrown  after first attempt to calculate
Jacobian.


USAGE:
1. User initializes algorithm state with MinLMCreateVJ() call
2. User tunes solver parameters with MinLMSetCond(),  MinLMSetStpMax() and
   other functions
3. User calls MinLMOptimize() function which  takes algorithm  state   and
   callback functions.
4. User calls MinLMResults() to get solution
5. Optionally, user may call MinLMRestartFrom() to solve  another  problem
   with same N/M but another starting point and/or another function.
   MinLMRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   dimension, N>1
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    M       -   number of functions f[i]
    X       -   initial solution, array[0..N-1]

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. you may tune stopping conditions with MinLMSetCond() function
2. if target function contains exp() or other fast growing functions,  and
   optimization algorithm makes too large steps which leads  to  overflow,
   use MinLMSetStpMax() function to bound algorithm's steps.

  -- ALGLIB --
     Copyright 30.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmcreatevj(
    ae_int_t m,
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minlmcreatevj(
    ae_int_t n,
    ae_int_t m,
    real_1d_array x,
    minlmstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minlmoptimize` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    hess    -   callback which calculates function (or merit function)
                value func, gradient grad and Hessian hess at given point x
    fvec    -   callback which calculates function vector fi[]
                at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

NOTES:

1. Depending on function used to create state  structure,  this  algorithm
   may accept Jacobian and/or Hessian and/or gradient.  According  to  the
   said above, there ase several versions of this function,  which  accept
   different sets of callbacks.

   This flexibility opens way to subtle errors - you may create state with
   MinLMCreateFGH() (optimization using Hessian), but call function  which
   does not accept Hessian. So when algorithm will request Hessian,  there
   will be no callback to call. In this case exception will be thrown.

   Be careful to avoid such errors because there is no way to find them at
   compile time - you can see them at runtime only.

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void minlmoptimize(minlmstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minlmoptimize(minlmstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minlmoptimize(minlmstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void (*hess)(const real_1d_array &x, double &func, real_1d_array &grad, real_2d_array &hess, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minlmoptimize(minlmstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minlmoptimize(minlmstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`minlmrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmrequesttermination(
    minlmstate state,
    const xparams _params = alglib::xdefault);

`minlmrestartfrom` function

/*************************************************************************
This  subroutine  restarts  LM  algorithm from new point. All optimization
parameters are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure used for reverse communication previously
                allocated with MinLMCreateXXX call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmrestartfrom(
    minlmstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minlmresults` function

/*************************************************************************
Levenberg-Marquardt algorithm results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization  report;  includes  termination   codes   and
                additional information. Termination codes are listed below,
                see comments for this structure for more info.
                Termination code is stored in rep.terminationtype field:
                * -8    optimizer detected NAN/INF values either in the
                        function itself, or in its Jacobian
                * -7    derivative correctness check failed;
                        see rep.funcidx, rep.varidx for
                        more information.
                * -3    constraints are inconsistent
                *  2    relative step is no more than EpsX.
                *  5    MaxIts steps was taken
                *  7    stopping conditions are too stringent,
                        further improvement is impossible
                *  8    terminated by user who called minlmrequesttermination().
                        X contains point which was "current accepted" when
                        termination request was submitted.

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmresults(
    minlmstate state,
    real_1d_array& x,
    minlmreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`minlmresultsbuf` function

/*************************************************************************
Levenberg-Marquardt algorithm results

Buffered implementation of MinLMResults(), which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmresultsbuf(
    minlmstate state,
    real_1d_array& x,
    minlmreport& rep,
    const xparams _params = alglib::xdefault);

`minlmsetacctype` function

/*************************************************************************
This function is used to change acceleration settings

You can choose between three acceleration strategies:
* AccType=0, no acceleration.
* AccType=1, secant updates are used to update quadratic model after  each
  iteration. After fixed number of iterations (or after  model  breakdown)
  we  recalculate  quadratic  model  using  analytic  Jacobian  or  finite
  differences. Number of secant-based iterations depends  on  optimization
  settings: about 3 iterations - when we have analytic Jacobian, up to 2*N
  iterations - when we use finite differences to calculate Jacobian.

AccType=1 is recommended when Jacobian  calculation  cost is prohibitively
high (several Mx1 function vector calculations  followed  by  several  NxN
Cholesky factorizations are faster than calculation of one M*N  Jacobian).
It should also be used when we have no Jacobian, because finite difference
approximation takes too much time to compute.

Table below list  optimization  protocols  (XYZ  protocol  corresponds  to
MinLMCreateXYZ) and acceleration types they support (and use by  default).

ACCELERATION TYPES SUPPORTED BY OPTIMIZATION PROTOCOLS:

protocol    0   1   comment
V           +   +
VJ          +   +
FGH         +

DEFAULT VALUES:

protocol    0   1   comment
V               x   without acceleration it is so slooooooooow
VJ          x
FGH         x

NOTE: this  function should be called before optimization. Attempt to call
it during algorithm iterations may result in unexpected behavior.

NOTE: attempt to call this function with unsupported protocol/acceleration
combination will result in exception being thrown.

  -- ALGLIB --
     Copyright 14.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmsetacctype(
    minlmstate state,
    ae_int_t acctype,
    const xparams _params = alglib::xdefault);

`minlmsetbc` function

/*************************************************************************
This function sets boundary constraints for LM optimizer

Boundary constraints are inactive by default (after initial creation).
They are preserved until explicitly turned off with another SetBC() call.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF (latter is recommended because
                it will allow solver to use better algorithm).
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF (latter is recommended because
                it will allow solver to use better algorithm).

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: this solver has following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by bound constraints
  or at its boundary

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmsetbc(
    minlmstate state,
    real_1d_array bndl,
    real_1d_array bndu,
    const xparams _params = alglib::xdefault);

`minlmsetcond` function

/*************************************************************************
This function sets stopping conditions for Levenberg-Marquardt optimization
algorithm.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - ste pvector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinLMSetScale()
                Recommended values: 1E-9 ... 1E-12.
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations   is    unlimited.   Only   Levenberg-Marquardt
                iterations  are  counted  (L-BFGS/CG  iterations  are  NOT
                counted because their cost is very low compared to that of
                LM).

Passing  EpsX=0  and  MaxIts=0  (simultaneously)  will  lead  to automatic
stopping criterion selection (small EpsX).

NOTE: it is not recommended to set large EpsX (say, 0.001). Because LM  is
      a second-order method, it performs very precise steps anyway.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmsetcond(
    minlmstate state,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`minlmsetgradientcheck` function

/*************************************************************************
This  subroutine  turns  on  verification  of  the  user-supplied analytic
gradient:
* user calls this subroutine before optimization begins
* MinLMOptimize() is called
* prior to actual optimization, for  each  function Fi and each  component
  of parameters  being  optimized X[j] algorithm performs following steps:
  * two trial steps are made to X[j]-TestStep*S[j] and X[j]+TestStep*S[j],
    where X[j] is j-th parameter and S[j] is a scale of j-th parameter
  * if needed, steps are bounded with respect to constraints on X[]
  * Fi(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point
  * in case difference between prediction and actual value is higher  than
    some predetermined threshold, algorithm stops with completion code -7;
    Rep.VarIdx is set to index of the parameter with incorrect derivative,
    Rep.FuncIdx is set to index of the function.
* after verification is over, algorithm proceeds to the actual optimization.

NOTE 1: verification  needs  N (parameters count) Jacobian evaluations. It
        is  very  costly  and  you  should use it only for low dimensional
        problems,  when  you  want  to  be  sure  that  you've   correctly
        calculated  analytic  derivatives.  You should not  use  it in the
        production code  (unless  you  want  to check derivatives provided
        by some third party).

NOTE 2: you  should  carefully  choose  TestStep. Value which is too large
        (so large that function behaviour is significantly non-cubic) will
        lead to false alarms. You may use  different  step  for  different
        parameters by means of setting scale with MinLMSetScale().

NOTE 3: this function may lead to false positives. In case it reports that
        I-th  derivative was calculated incorrectly, you may decrease test
        step  and  try  one  more  time  - maybe your function changes too
        sharply  and  your  step  is  too  large for such rapidly chanding
        function.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification

  -- ALGLIB --
     Copyright 15.06.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmsetgradientcheck(
    minlmstate state,
    double teststep,
    const xparams _params = alglib::xdefault);

`minlmsetlc` function

/*************************************************************************
This function sets general linear constraints for LM optimizer

Linear constraints are inactive by default (after initial creation).  They
are preserved until explicitly turned off with another minlmsetlc() call.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

IMPORTANT: if you have linear constraints, it is strongly  recommended  to
           set scale of variables with minlmsetscale(). QP solver which is
           used to calculate linearly constrained steps heavily relies  on
           good scaling of input problems.

IMPORTANT: solvers created with minlmcreatefgh()  do  not  support  linear
           constraints.

NOTE: linear  (non-bound)  constraints are satisfied only approximately  -
      there  always  exists some violation due  to  numerical  errors  and
      algorithmic limitations.

NOTE: general linear constraints  add  significant  overhead  to  solution
      process. Although solver performs roughly same amount of  iterations
      (when compared  with  similar  box-only  constrained  problem), each
      iteration   now    involves  solution  of  linearly  constrained  QP
      subproblem, which requires ~3-5 times more Cholesky  decompositions.
      Thus, if you can reformulate your problem in such way  this  it  has
      only box constraints, it may be beneficial to do so.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmsetlc(
    minlmstate state,
    real_2d_array c,
    integer_1d_array ct,
    const xparams _params = alglib::xdefault);
void alglib::minlmsetlc(
    minlmstate state,
    real_2d_array c,
    integer_1d_array ct,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

`minlmsetscale` function

/*************************************************************************
This function sets scaling coefficients for LM optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Generally, scale is NOT considered to be a form of preconditioner.  But LM
optimizer is unique in that it uses scaling matrix both  in  the  stopping
condition tests and as Marquardt damping factor.

Proper scaling is very important for the algorithm performance. It is less
important for the quality of results, but still has some influence (it  is
easier  to  converge  when  variables  are  properly  scaled, so premature
stopping is possible when very badly scalled variables are  combined  with
relaxed stopping conditions).

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmsetscale(
    minlmstate state,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

`minlmsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

NOTE: non-zero StpMax leads to moderate  performance  degradation  because
intermediate  step  of  preconditioned L-BFGS optimization is incompatible
with limits on step size.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmsetstpmax(
    minlmstate state,
    double stpmax,
    const xparams _params = alglib::xdefault);

`minlmsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinLMOptimize(). Both Levenberg-Marquardt and internal  L-BFGS
iterations are reported.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minlmsetxrep(
    minlmstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

minlm_d_fgh example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}
void function1_hess(const real_1d_array &x, double &func, real_1d_array &grad, real_2d_array &hess, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // its derivatives df/d0 and df/dx1
    // and its Hessian.
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
    hess[0][0] = 1200*pow(x[0]+3,2);
    hess[0][1] = 0;
    hess[1][0] = 0;
    hess[1][1] = 12*pow(x[1]-3,2);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of F(x0,x1) = 100*(x0+3)^4+(x1-3)^4
    // using "FGH" mode of the Levenberg-Marquardt optimizer.
    //
    // F is treated like a monolitic function without internal structure,
    // i.e. we do NOT represent it as a sum of squares.
    //
    // Optimization algorithm uses:
    // * function value F(x0,x1)
    // * gradient G={dF/dxi}
    // * Hessian H={d2F/(dxi*dxj)}
    //
    real_1d_array x = "[0,0]";
    double epsx = 0.0000000001;
    ae_int_t maxits = 0;
    minlmstate state;
    minlmreport rep;

    minlmcreatefgh(x, state);
    minlmsetcond(state, epsx, maxits);
    alglib::minlmoptimize(state, function1_func, function1_grad, function1_hess);
    minlmresults(state, x, rep);

    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]
    return 0;
}

minlm_d_restarts example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
}
void  function2_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = x0^2+1
    // f1(x0,x1) = x1-1
    //
    fi[0] = x[0]*x[0]+1;
    fi[1] = x[1]-1;
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where 
    //
    //     f0(x0,x1) = 10*(x0+3)^2
    //     f1(x0,x1) = (x1-3)^2
    //
    // using several starting points and efficient restarts.
    //
    real_1d_array x;
    double epsx = 0.0000000001;
    ae_int_t maxits = 0;
    minlmstate state;
    minlmreport rep;

    //
    // create optimizer using minlmcreatev()
    //
    x = "[10,10]";
    minlmcreatev(2, x, 0.0001, state);
    minlmsetcond(state, epsx, maxits);
    alglib::minlmoptimize(state, function1_fvec);
    minlmresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]

    //
    // restart optimizer using minlmrestartfrom()
    //
    // we can use different starting point, different function,
    // different stopping conditions, but problem size
    // must remain unchanged.
    //
    x = "[4,4]";
    minlmrestartfrom(state, x);
    alglib::minlmoptimize(state, function2_fvec);
    minlmresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [0,1]
    return 0;
}

minlm_d_v example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where 
    //
    //     f0(x0,x1) = 10*(x0+3)^2
    //     f1(x0,x1) = (x1-3)^2
    //
    // using "V" mode of the Levenberg-Marquardt optimizer.
    //
    // Optimization algorithm uses:
    // * function vector f[] = {f1,f2}
    //
    // No other information (Jacobian, gradient, etc.) is needed.
    //
    real_1d_array x = "[0,0]";
    double epsx = 0.0000000001;
    ae_int_t maxits = 0;
    minlmstate state;
    minlmreport rep;

    minlmcreatev(2, x, 0.0001, state);
    minlmsetcond(state, epsx, maxits);
    alglib::minlmoptimize(state, function1_fvec);
    minlmresults(state, x, rep);

    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]
    return 0;
}

minlm_d_vb example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where 
    //
    //     f0(x0,x1) = 10*(x0+3)^2
    //     f1(x0,x1) = (x1-3)^2
    //
    // with boundary constraints
    //
    //     -1 <= x0 <= +1
    //     -1 <= x1 <= +1
    //
    // using "V" mode of the Levenberg-Marquardt optimizer.
    //
    // Optimization algorithm uses:
    // * function vector f[] = {f1,f2}
    //
    // No other information (Jacobian, gradient, etc.) is needed.
    //
    real_1d_array x = "[0,0]";
    real_1d_array bndl = "[-1,-1]";
    real_1d_array bndu = "[+1,+1]";
    double epsx = 0.0000000001;
    ae_int_t maxits = 0;
    minlmstate state;
    minlmreport rep;

    minlmcreatev(2, x, 0.0001, state);
    minlmsetbc(state, bndl, bndu);
    minlmsetcond(state, epsx, maxits);
    alglib::minlmoptimize(state, function1_fvec);
    minlmresults(state, x, rep);

    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,+1]
    return 0;
}

minlm_d_vj example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
}
void  function1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
    jac[0][0] = 20*(x[0]+3);
    jac[0][1] = 0;
    jac[1][0] = 0;
    jac[1][1] = 2*(x[1]-3);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where 
    //
    //     f0(x0,x1) = 10*(x0+3)^2
    //     f1(x0,x1) = (x1-3)^2
    //
    // using "VJ" mode of the Levenberg-Marquardt optimizer.
    //
    // Optimization algorithm uses:
    // * function vector f[] = {f1,f2}
    // * Jacobian matrix J = {dfi/dxj}.
    //
    real_1d_array x = "[0,0]";
    double epsx = 0.0000000001;
    ae_int_t maxits = 0;
    minlmstate state;
    minlmreport rep;

    minlmcreatevj(2, x, state);
    minlmsetcond(state, epsx, maxits);
    alglib::minlmoptimize(state, function1_fvec, function1_jac);
    minlmresults(state, x, rep);

    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]
    return 0;
}

`minnlc` subpackage

Classes

minnlcreport
minnlcstate

Functions

minnlccreate
minnlccreatef
minnlcoptimize
minnlcrestartfrom
minnlcresults
minnlcresultsbuf
minnlcsetalgoaul
minnlcsetalgoslp
minnlcsetbc
minnlcsetcond
minnlcsetgradientcheck
minnlcsetlc
minnlcsetnlc
minnlcsetprecexactlowrank
minnlcsetprecexactrobust
minnlcsetprecinexact
minnlcsetprecnone
minnlcsetscale
minnlcsetstpmax
minnlcsetxrep

Examples

minnlc_d_equality		Nonlinearly constrained optimization (equality constraints)
minnlc_d_inequality		Nonlinearly constrained optimization (inequality constraints)
minnlc_d_mixed		Nonlinearly constrained optimization with mixed equality/inequality constraints

`minnlcreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           total number of inner iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -7    gradient verification failed.
        See MinNLCSetGradientCheck() for more information.
   2    relative step is no more than EpsX.
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.

Other fields of this structure are not documented and should not be used!
*************************************************************************/
class minnlcreport
{
    ae_int_t             iterationscount;
    ae_int_t             nfev;
    ae_int_t             varidx;
    ae_int_t             funcidx;
    ae_int_t             terminationtype;
    ae_int_t             dbgphase0its;
};

`minnlcstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinNLC subpackage to work  with  this
object
*************************************************************************/
class minnlcstate
{
};

`minnlccreate` function

/*************************************************************************
                  NONLINEARLY  CONSTRAINED  OPTIMIZATION
            WITH PRECONDITIONED AUGMENTED LAGRANGIAN ALGORITHM

DESCRIPTION:
The  subroutine  minimizes  function   F(x)  of N arguments subject to any
combination of:
* bound constraints
* linear inequality constraints
* linear equality constraints
* nonlinear equality constraints Gi(x)=0
* nonlinear inequality constraints Hi(x)<=0

REQUIREMENTS:
* user must provide function value and gradient for F(), H(), G()
* starting point X0 must be feasible or not too far away from the feasible
  set
* F(), G(), H() are twice continuously differentiable on the feasible  set
  and its neighborhood
* nonlinear constraints G() and H() must have non-zero gradient at  G(x)=0
  and at H(x)=0. Say, constraint like x^2>=1 is supported, but x^2>=0   is
  NOT supported.

USAGE:

Constrained optimization if far more complex than the  unconstrained  one.
Nonlinearly constrained optimization is one of the most esoteric numerical
procedures.

Here we give very brief outline  of  the  MinNLC  optimizer.  We  strongly
recommend you to study examples in the ALGLIB Reference Manual and to read
ALGLIB User Guide on optimization, which is available at
http://www.alglib.net/optimization/

1. User initializes algorithm state with MinNLCCreate() call  and  chooses
   what NLC solver to use. There is some solver which is used by  default,
   with default settings, but you should NOT rely on  default  choice.  It
   may change in future releases of ALGLIB without notice, and no one  can
   guarantee that new solver will be  able  to  solve  your  problem  with
   default settings.

   From the other side, if you choose solver explicitly, you can be pretty
   sure that it will work with new ALGLIB releases.

   In the current release following solvers can be used:
   * SLP solver (activated with MinNLCSetAlgoSLP() function) -  successive
     linear programming, recommended option (default)
   * AUL solver (activated with MinNLCSetAlgoAUL() function)  -  augmented
     Lagrangian method with dense preconditioner

2. User adds boundary and/or linear and/or nonlinear constraints by  means
   of calling one of the following functions:
   a) MinNLCSetBC() for boundary constraints
   b) MinNLCSetLC() for linear constraints
   c) MinNLCSetNLC() for nonlinear constraints
   You may combine (a), (b) and (c) in one optimization problem.

3. User sets scale of the variables with MinNLCSetScale() function. It  is
   VERY important to set  scale  of  the  variables,  because  nonlinearly
   constrained problems are hard to solve when variables are badly scaled.

4. User sets  stopping  conditions  with  MinNLCSetCond(). If  NLC  solver
   uses  inner/outer  iteration  layout,  this  function   sets   stopping
   conditions for INNER iterations.

5. Finally, user calls MinNLCOptimize()  function  which  takes  algorithm
   state and pointer (delegate, etc.) to callback function which calculates
   F/G/H.

6. User calls MinNLCResults() to get solution

7. Optionally user may call MinNLCRestartFrom() to solve  another  problem
   with same N but another starting point. MinNLCRestartFrom()  allows  to
   reuse already initialized structure.


INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size ofX
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlccreate(
    real_1d_array x,
    minnlcstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minnlccreate(
    ae_int_t n,
    real_1d_array x,
    minnlcstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minnlccreatef` function

/*************************************************************************
This subroutine is a finite  difference variant of MinNLCCreate(). It uses
finite differences in order to differentiate target function.

Description below contains information which is specific to this  function
only. We recommend to read comments on MinNLCCreate() in order to get more
information about creation of NLC optimizer.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size ofX
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinNLCSetScale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large TRUNCATION  errors, while too small
   step will result in too large NUMERICAL  errors.  1.0E-4  can  be  good
   value to start from.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is  less   robust   and  precise.  Imprecise  gradient  may  slow  down
   convergence, especially on highly nonlinear problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlccreatef(
    real_1d_array x,
    double diffstep,
    minnlcstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minnlccreatef(
    ae_int_t n,
    real_1d_array x,
    double diffstep,
    minnlcstate& state,
    const xparams _params = alglib::xdefault);

`minnlcoptimize` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    fvec    -   callback which calculates function vector fi[]
                at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied Jacobian, and one which uses  only  function
   vector and numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   you should choose appropriate variant of MinNLCOptimize() -  one  which
   accepts function AND Jacobian or one which accepts ONLY function.

   Be careful to choose variant of MinNLCOptimize()  which  corresponds to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to MinNLCOptimize()   and  specific
   function used to create optimizer.


                     |         USER PASSED TO MinNLCOptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   MinNLCCreateF()   |     works               FAILS
   MinNLCCreate()    |     FAILS               works

   Here "FAILS" denotes inappropriate combinations  of  optimizer creation
   function  and  MinNLCOptimize()  version.   Attemps   to    use    such
   combination will lead to exception. Either  you  did  not pass gradient
   when it WAS needed or you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlcoptimize(minnlcstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minnlcoptimize(minnlcstate &state,
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`minnlcrestartfrom` function

/*************************************************************************
This subroutine restarts algorithm from new point.
All optimization parameters (including constraints) are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have  same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinNLCCreate call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcrestartfrom(
    minnlcstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

`minnlcresults` function

/*************************************************************************
MinNLC results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report. You should check Rep.TerminationType
                in  order  to  distinguish  successful  termination  from
                unsuccessful one:
                * -8    internal integrity control  detected  infinite or
                        NAN   values   in   function/gradient.   Abnormal
                        termination signalled.
                * -7   gradient verification failed.
                       See MinNLCSetGradientCheck() for more information.
                *  2   scaled step is no more than EpsX.
                *  5   MaxIts steps was taken
                More information about fields of this  structure  can  be
                found in the comments on MinNLCReport datatype.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcresults(
    minnlcstate state,
    real_1d_array& x,
    minnlcreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minnlcresultsbuf` function

/*************************************************************************
NLC results

Buffered implementation of MinNLCResults() which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcresultsbuf(
    minnlcstate state,
    real_1d_array& x,
    minnlcreport& rep,
    const xparams _params = alglib::xdefault);

`minnlcsetalgoaul` function

/*************************************************************************
This  function  tells MinNLC unit to use  Augmented  Lagrangian  algorithm
for nonlinearly constrained  optimization.  This  algorithm  is  a  slight
modification of one described in "A Modified Barrier-Augmented  Lagrangian
Method for  Constrained  Minimization  (1999)"  by  D.GOLDFARB,  R.POLYAK,
K. SCHEINBERG, I.YUZEFOVICH.

Augmented Lagrangian algorithm works by converting problem  of  minimizing
F(x) subject to equality/inequality constraints   to unconstrained problem
of the form

    min[ f(x) +
        + Rho*PENALTY_EQ(x)   + SHIFT_EQ(x,Nu1) +
        + Rho*PENALTY_INEQ(x) + SHIFT_INEQ(x,Nu2) ]

where:
* Rho is a fixed penalization coefficient
* PENALTY_EQ(x) is a penalty term, which is used to APPROXIMATELY  enforce
  equality constraints
* SHIFT_EQ(x) is a special "shift"  term  which  is  used  to  "fine-tune"
  equality constraints, greatly increasing precision
* PENALTY_INEQ(x) is a penalty term which is used to approximately enforce
  inequality constraints
* SHIFT_INEQ(x) is a special "shift"  term  which  is  used to "fine-tune"
  inequality constraints, greatly increasing precision
* Nu1/Nu2 are vectors of Lagrange coefficients which are fine-tuned during
  outer iterations of algorithm

This  version  of  AUL  algorithm  uses   preconditioner,  which   greatly
accelerates convergence. Because this  algorithm  is  similar  to  penalty
methods,  it  may  perform  steps  into  infeasible  area.  All  kinds  of
constraints (boundary, linear and nonlinear ones) may   be   violated   in
intermediate points - and in the solution.  However,  properly  configured
AUL method is significantly better at handling  constraints  than  barrier
and/or penalty methods.

The very basic outline of algorithm is given below:
1) first outer iteration is performed with "default"  values  of  Lagrange
   multipliers Nu1/Nu2. Solution quality is low (candidate  point  can  be
   too  far  away  from  true  solution; large violation of constraints is
   possible) and is comparable with that of penalty methods.
2) subsequent outer iterations  refine  Lagrange  multipliers  and improve
   quality of the solution.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    Rho     -   penalty coefficient, Rho>0:
                * large enough  that  algorithm  converges  with   desired
                  precision. Minimum value is 10*max(S'*diag(H)*S),  where
                  S is a scale matrix (set by MinNLCSetScale) and H  is  a
                  Hessian of the function being minimized. If you can  not
                  easily estimate Hessian norm,  see  our  recommendations
                  below.
                * not TOO large to prevent ill-conditioning
                * for unit-scale problems (variables and Hessian have unit
                  magnitude), Rho=100 or Rho=1000 can be used.
                * it is important to note that Rho is internally multiplied
                  by scaling matrix, i.e. optimum value of Rho depends  on
                  scale of variables specified  by  MinNLCSetScale().
    ItsCnt  -   number of outer iterations:
                * ItsCnt=0 means that small number of outer iterations  is
                  automatically chosen (10 iterations in current version).
                * ItsCnt=1 means that AUL algorithm performs just as usual
                  barrier method.
                * ItsCnt>1 means that  AUL  algorithm  performs  specified
                  number of outer iterations

HOW TO CHOOSE PARAMETERS

Nonlinear optimization is a tricky area and Augmented Lagrangian algorithm
is sometimes hard to tune. Good values of  Rho  and  ItsCnt  are  problem-
specific.  In  order  to  help  you   we   prepared   following   set   of
recommendations:

* for  unit-scale  problems  (variables  and Hessian have unit magnitude),
  Rho=100 or Rho=1000 can be used.

* start from  some  small  value of Rho and solve problem  with  just  one
  outer iteration (ItcCnt=1). In this case algorithm behaves like  penalty
  method. Increase Rho in 2x or 10x steps until you  see  that  one  outer
  iteration returns point which is "rough approximation to solution".

  It is very important to have Rho so  large  that  penalty  term  becomes
  constraining i.e. modified function becomes highly convex in constrained
  directions.

  From the other side, too large Rho may prevent you  from  converging  to
  the solution. You can diagnose it by studying number of inner iterations
  performed by algorithm: too few (5-10 on  1000-dimensional  problem)  or
  too many (orders of magnitude more than  dimensionality)  usually  means
  that Rho is too large.

* with just one outer iteration you  usually  have  low-quality  solution.
  Some constraints can be violated with very  large  margin,  while  other
  ones (which are NOT violated in the true solution) can push final  point
  too far in the inner area of the feasible set.

  For example, if you have constraint x0>=0 and true solution  x0=1,  then
  merely a presence of "x0>=0" will introduce a bias towards larger values
  of x0. Say, algorithm may stop at x0=1.5 instead of 1.0.

* after you found good Rho, you may increase number of  outer  iterations.
  ItsCnt=10 is a good value. Subsequent outer iteration will refine values
  of  Lagrange  multipliers.  Constraints  which  were  violated  will  be
  enforced, inactive constraints will be dropped (corresponding multipliers
  will be decreased). Ideally, you  should  see  10-1000x  improvement  in
  constraint handling (constraint violation is reduced).

* if  you  see  that  algorithm  converges  to  vicinity  of solution, but
  additional outer iterations do not refine solution,  it  may  mean  that
  algorithm is unstable - it wanders around true  solution,  but  can  not
  approach it. Sometimes algorithm may be stabilized by increasing Rho one
  more time, making it 5x or 10x larger.

SCALING OF CONSTRAINTS [IMPORTANT]

AUL optimizer scales   variables   according   to   scale   specified   by
MinNLCSetScale() function, so it can handle  problems  with  badly  scaled
variables (as long as we KNOW their scales).   However,  because  function
being optimized is a mix  of  original  function and  constraint-dependent
penalty  functions, it  is   important  to   rescale  both  variables  AND
constraints.

Say,  if  you  minimize f(x)=x^2 subject to 1000000*x>=0,  then  you  have
constraint whose scale is different from that of target  function (another
example is 0.000001*x>=0). It is also possible to have constraints   whose
scales  are   misaligned:   1000000*x0>=0, 0.000001*x1<=0.   Inappropriate
scaling may ruin convergence because minimizing x^2 subject to x>=0 is NOT
same as minimizing it subject to 1000000*x>=0.

Because we  know  coefficients  of  boundary/linear  constraints,  we  can
automatically rescale and normalize them. However,  there  is  no  way  to
automatically rescale nonlinear constraints Gi(x) and  Hi(x)  -  they  are
black boxes.

It means that YOU are the one who is  responsible  for  correct scaling of
nonlinear constraints  Gi(x)  and  Hi(x).  We  recommend  you  to  rescale
nonlinear constraints in such way that I-th component of dG/dX (or  dH/dx)
has magnitude approximately equal to 1/S[i] (where S  is  a  scale  set by
MinNLCSetScale() function).

WHAT IF IT DOES NOT CONVERGE?

It is possible that AUL algorithm fails to converge to precise  values  of
Lagrange multipliers. It stops somewhere around true solution, but candidate
point is still too far from solution, and some constraints  are  violated.
Such kind of failure is specific for Lagrangian algorithms -  technically,
they stop at some point, but this point is not constrained solution.

There are exist several reasons why algorithm may fail to converge:
a) too loose stopping criteria for inner iteration
b) degenerate, redundant constraints
c) target function has unconstrained extremum exactly at the  boundary  of
   some constraint
d) numerical noise in the target function

In all these cases algorithm is unstable - each outer iteration results in
large and almost random step which improves handling of some  constraints,
but violates other ones (ideally  outer iterations should form a  sequence
of progressively decreasing steps towards solution).

First reason possible is  that  too  loose  stopping  criteria  for  inner
iteration were specified. Augmented Lagrangian algorithm solves a sequence
of intermediate problems, and requries each of them to be solved with high
precision. Insufficient precision results in incorrect update of  Lagrange
multipliers.

Another reason is that you may have specified degenerate constraints: say,
some constraint was repeated twice. In most cases AUL algorithm gracefully
handles such situations, but sometimes it may spend too much time figuring
out subtle degeneracies in constraint matrix.

Third reason is tricky and hard to diagnose. Consider situation  when  you
minimize  f=x^2  subject to constraint x>=0.  Unconstrained   extremum  is
located  exactly  at  the  boundary  of  constrained  area.  In  this case
algorithm will tend to oscillate between negative  and  positive  x.  Each
time it stops at x<0 it "reinforces" constraint x>=0, and each time it  is
bounced to x>0 it "relaxes" constraint (and is  attracted  to  x<0).

Such situation  sometimes  happens  in  problems  with  hidden  symetries.
Algorithm  is  got  caught  in  a  loop with  Lagrange  multipliers  being
continuously increased/decreased. Luckily, such loop forms after at  least
three iterations, so this problem can be solved by  DECREASING  number  of
outer iterations down to 1-2 and increasing  penalty  coefficient  Rho  as
much as possible.

Final reason is numerical noise. AUL algorithm is robust against  moderate
noise (more robust than, say, active set methods),  but  large  noise  may
destabilize algorithm.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetalgoaul(
    minnlcstate state,
    double rho,
    ae_int_t itscnt,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minnlcsetalgoslp` function

/*************************************************************************
This   function  tells  MinNLC  optimizer  to  use  SLP (Successive Linear
Programming) algorithm for  nonlinearly  constrained   optimization.  This
algorithm  is  a  slight  modification  of  one  described  in  "A  Linear
programming-based optimization algorithm for solving nonlinear programming
problems" (2010) by Claus Still and Tapio Westerlund.

Despite its name ("linear" = "first order method") this algorithm performs
steps similar to that of conjugate gradients method;  internally  it  uses
orthogonality/conjugacy requirement for subsequent steps  which  makes  it
closer to second order methods in terms of convergence speed.

This algorithm has following nice properties:
* no parameters to tune
* no convexity requirements for target function or constraints
* initial point can be infeasible
* algorithm respects box constraints in all intermediate points  (it  does
  not even evaluate function outside of box constrained area)
* once linear constraints are enforced, algorithm will not violate them
* no such guarantees can be provided for nonlinear constraints,  but  once
  nonlinear constraints are enforced, algorithm will try  to  respect them
  as much as possible
* numerical differentiation does not  violate  box  constraints  (although
  general linear and nonlinear ones can be violated)

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 02.04.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetalgoslp(
    minnlcstate state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minnlcsetbc` function

/*************************************************************************
This function sets boundary constraints for NLC optimizer.

Boundary constraints are inactive by  default  (after  initial  creation).
They are preserved after algorithm restart with  MinNLCRestartFrom().

You may combine boundary constraints with  general  linear ones - and with
nonlinear ones! Boundary constraints are  handled  more  efficiently  than
other types.  Thus,  if  your  problem  has  mixed  constraints,  you  may
explicitly specify some of them as boundary and save some time/space.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF.

NOTE 1:  it is possible to specify  BndL[i]=BndU[i].  In  this  case  I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2:  when you solve your problem  with  augmented  Lagrangian  solver,
         boundary constraints are  satisfied  only  approximately!  It  is
         possible   that  algorithm  will  evaluate  function  outside  of
         feasible area!

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetbc(
    minnlcstate state,
    real_1d_array bndl,
    real_1d_array bndu,
    const xparams _params = alglib::xdefault);

`minnlcsetcond` function

/*************************************************************************
This function sets stopping conditions for inner iterations of  optimizer.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinNLCSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic
selection of the stopping condition.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetcond(
    minnlcstate state,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minnlcsetgradientcheck` function

/*************************************************************************
This  subroutine  turns  on  verification  of  the  user-supplied analytic
gradient:
* user calls this subroutine before optimization begins
* MinNLCOptimize() is called
* prior to  actual  optimization, for each component  of  parameters being
  optimized X[i] algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point
  * in case difference between prediction and actual value is higher  than
    some predetermined threshold, algorithm stops with completion code -7;
    Rep.VarIdx is set to index of the parameter with incorrect derivative,
    and Rep.FuncIdx is set to index of the function.
* after verification is over, algorithm proceeds to the actual optimization.

NOTE 1: verification  needs  N (parameters count) gradient evaluations. It
        is very costly and you should use  it  only  for  low  dimensional
        problems,  when  you  want  to  be  sure  that  you've   correctly
        calculated  analytic  derivatives.  You  should  not use it in the
        production code (unless you want to check derivatives provided  by
        some third party).

NOTE 2: you  should  carefully  choose  TestStep. Value which is too large
        (so large that function behaviour is significantly non-cubic) will
        lead to false alarms. You may use  different  step  for  different
        parameters by means of setting scale with MinNLCSetScale().

NOTE 3: this function may lead to false positives. In case it reports that
        I-th  derivative was calculated incorrectly, you may decrease test
        step  and  try  one  more  time  - maybe your function changes too
        sharply  and  your  step  is  too  large for such rapidly chanding
        function.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification

  -- ALGLIB --
     Copyright 15.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetgradientcheck(
    minnlcstate state,
    double teststep,
    const xparams _params = alglib::xdefault);

`minnlcsetlc` function

/*************************************************************************
This function sets linear constraints for MinNLC optimizer.

Linear constraints are inactive by default (after initial creation).  They
are preserved after algorithm restart with MinNLCRestartFrom().

You may combine linear constraints with boundary ones - and with nonlinear
ones! If your problem has mixed constraints, you  may  explicitly  specify
some of them as linear. It  may  help  optimizer   to   handle  them  more
efficiently.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinNLCCreate call.
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

NOTE 1: when you solve your problem  with  augmented  Lagrangian   solver,
        linear constraints are  satisfied  only   approximately!   It   is
        possible   that  algorithm  will  evaluate  function  outside   of
        feasible area!

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetlc(
    minnlcstate state,
    real_2d_array c,
    integer_1d_array ct,
    const xparams _params = alglib::xdefault);
void alglib::minnlcsetlc(
    minnlcstate state,
    real_2d_array c,
    integer_1d_array ct,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

`minnlcsetnlc` function

/*************************************************************************
This function sets nonlinear constraints for MinNLC optimizer.

In fact, this function sets NUMBER of nonlinear  constraints.  Constraints
itself (constraint functions) are passed to MinNLCOptimize() method.  This
method requires user-defined vector function F[]  and  its  Jacobian  J[],
where:
* first component of F[] and first row  of  Jacobian  J[]  corresponds  to
  function being minimized
* next NLEC components of F[] (and rows  of  J)  correspond  to  nonlinear
  equality constraints G_i(x)=0
* next NLIC components of F[] (and rows  of  J)  correspond  to  nonlinear
  inequality constraints H_i(x)<=0

NOTE: you may combine nonlinear constraints with linear/boundary ones.  If
      your problem has mixed constraints, you  may explicitly specify some
      of them as linear ones. It may help optimizer to  handle  them  more
      efficiently.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinNLCCreate call.
    NLEC    -   number of Non-Linear Equality Constraints (NLEC), >=0
    NLIC    -   number of Non-Linear Inquality Constraints (NLIC), >=0

NOTE 1: when you solve your problem  with  augmented  Lagrangian   solver,
        nonlinear constraints are satisfied only  approximately!   It   is
        possible   that  algorithm  will  evaluate  function  outside   of
        feasible area!

NOTE 2: algorithm scales variables  according  to   scale   specified   by
        MinNLCSetScale()  function,  so  it can handle problems with badly
        scaled variables (as long as we KNOW their scales).

        However,  there  is  no  way  to  automatically  scale   nonlinear
        constraints Gi(x) and Hi(x). Inappropriate scaling  of  Gi/Hi  may
        ruin convergence. Solving problem with  constraint  "1000*G0(x)=0"
        is NOT same as solving it with constraint "0.001*G0(x)=0".

        It  means  that  YOU  are  the  one who is responsible for correct
        scaling of nonlinear constraints Gi(x) and Hi(x). We recommend you
        to scale nonlinear constraints in such way that I-th component  of
        dG/dX (or dH/dx) has approximately unit  magnitude  (for  problems
        with unit scale)  or  has  magnitude approximately equal to 1/S[i]
        (where S is a scale set by MinNLCSetScale() function).


  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetnlc(
    minnlcstate state,
    ae_int_t nlec,
    ae_int_t nlic,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minnlcsetprecexactlowrank` function

/*************************************************************************
This function sets preconditioner to "exact low rank" mode.

Preconditioning is very important for convergence of  Augmented Lagrangian
algorithm because presence of penalty term makes problem  ill-conditioned.
Difference between  performance  of  preconditioned  and  unpreconditioned
methods can be as large as 100x!

MinNLC optimizer may use following preconditioners,  each  with   its  own
benefits and drawbacks:
    a) inexact LBFGS-based, with O(N*K) evaluation time
    b) exact low rank one,  with O(N*K^2) evaluation time
    c) exact robust one,    with O(N^3+K*N^2) evaluation time
where K is a total number of general linear and nonlinear constraints (box
ones are not counted).

It also provides special unpreconditioned mode of operation which  can  be
used for test purposes. Comments below discuss low rank preconditioner.

Exact low-rank preconditioner  uses  Woodbury  matrix  identity  to  build
quadratic model of the penalized function. It has following features:
* no special assumptions about orthogonality of constraints
* preconditioner evaluation is optimized for K<<N. Its cost  is  O(N*K^2),
  so it may become prohibitively slow for K>=N.
* finally, stability of the process is guaranteed only for K<<N.  Woodbury
  update often fail for K>=N due to degeneracy of  intermediate  matrices.
  That's why we recommend to use "exact robust"  preconditioner  for  such
  cases.

RECOMMENDATIONS

We  recommend  to  choose  between  "exact  low  rank"  and "exact robust"
preconditioners, with "low rank" version being chosen  when  you  know  in
advance that total count of non-box constraints won't exceed N, and "robust"
version being chosen when you need bulletproof solution.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    UpdateFreq- update frequency. Preconditioner is  rebuilt  after  every
                UpdateFreq iterations. Recommended value: 10 or higher.
                Zero value means that good default value will be used.

  -- ALGLIB --
     Copyright 26.09.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetprecexactlowrank(
    minnlcstate state,
    ae_int_t updatefreq,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minnlcsetprecexactrobust` function

/*************************************************************************
This function sets preconditioner to "exact robust" mode.

Preconditioning is very important for convergence of  Augmented Lagrangian
algorithm because presence of penalty term makes problem  ill-conditioned.
Difference between  performance  of  preconditioned  and  unpreconditioned
methods can be as large as 100x!

MinNLC optimizer may use following preconditioners,  each  with   its  own
benefits and drawbacks:
    a) inexact LBFGS-based, with O(N*K) evaluation time
    b) exact low rank one,  with O(N*K^2) evaluation time
    c) exact robust one,    with O(N^3+K*N^2) evaluation time
where K is a total number of general linear and nonlinear constraints (box
ones are not counted).

It also provides special unpreconditioned mode of operation which  can  be
used for test purposes. Comments below discuss robust preconditioner.

Exact  robust  preconditioner   uses   Cholesky  decomposition  to  invert
approximate Hessian matrix H=D+W'*C*W (where D stands for  diagonal  terms
of Hessian, combined result of initial scaling matrix and penalty from box
constraints; W stands for general linear constraints and linearization  of
nonlinear ones; C stands for diagonal matrix of penalty coefficients).

This preconditioner has following features:
* no special assumptions about constraint structure
* preconditioner is optimized  for  stability;  unlike  "exact  low  rank"
  version which fails for K>=N, this one works well for any value of K.
* the only drawback is that is takes O(N^3+K*N^2) time  to  build  it.  No
  economical  Woodbury update is applied even when it  makes  sense,  thus
  there  are  exist situations (K<<N) when "exact low rank" preconditioner
  outperforms this one.

RECOMMENDATIONS

We  recommend  to  choose  between  "exact  low  rank"  and "exact robust"
preconditioners, with "low rank" version being chosen  when  you  know  in
advance that total count of non-box constraints won't exceed N, and "robust"
version being chosen when you need bulletproof solution.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    UpdateFreq- update frequency. Preconditioner is  rebuilt  after  every
                UpdateFreq iterations. Recommended value: 10 or higher.
                Zero value means that good default value will be used.

  -- ALGLIB --
     Copyright 26.09.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetprecexactrobust(
    minnlcstate state,
    ae_int_t updatefreq,
    const xparams _params = alglib::xdefault);

`minnlcsetprecinexact` function

/*************************************************************************
This function sets preconditioner to "inexact LBFGS-based" mode.

Preconditioning is very important for convergence of  Augmented Lagrangian
algorithm because presence of penalty term makes problem  ill-conditioned.
Difference between  performance  of  preconditioned  and  unpreconditioned
methods can be as large as 100x!

MinNLC optimizer may use following preconditioners,  each  with   its  own
benefits and drawbacks:
    a) inexact LBFGS-based, with O(N*K) evaluation time
    b) exact low rank one,  with O(N*K^2) evaluation time
    c) exact robust one,    with O(N^3+K*N^2) evaluation time
where K is a total number of general linear and nonlinear constraints (box
ones are not counted).

Inexact  LBFGS-based  preconditioner  uses L-BFGS  formula  combined  with
orthogonality assumption to perform very fast updates. For a N-dimensional
problem with K general linear or nonlinear constraints (boundary ones  are
not counted) it has O(N*K) cost per iteration.  This   preconditioner  has
best  quality  (less  iterations)  when   general   linear  and  nonlinear
constraints are orthogonal to each other (orthogonality  with  respect  to
boundary constraints is not required). Number of iterations increases when
constraints  are  non-orthogonal, because algorithm assumes orthogonality,
but still it is better than no preconditioner at all.

INPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 26.09.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetprecinexact(
    minnlcstate state,
    const xparams _params = alglib::xdefault);

`minnlcsetprecnone` function

/*************************************************************************
This function sets preconditioner to "turned off" mode.

Preconditioning is very important for convergence of  Augmented Lagrangian
algorithm because presence of penalty term makes problem  ill-conditioned.
Difference between  performance  of  preconditioned  and  unpreconditioned
methods can be as large as 100x!

MinNLC optimizer may  utilize  two  preconditioners,  each  with  its  own
benefits and drawbacks: a) inexact LBFGS-based, and b) exact low rank one.
It also provides special unpreconditioned mode of operation which  can  be
used for test purposes.

This function activates this test mode. Do not use it in  production  code
to solve real-life problems.

INPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 26.09.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetprecnone(
    minnlcstate state,
    const xparams _params = alglib::xdefault);

`minnlcsetscale` function

/*************************************************************************
This function sets scaling coefficients for NLC optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetscale(
    minnlcstate state,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minnlcsetstpmax` function

/*************************************************************************
This function sets maximum step length (after scaling of step vector  with
respect to variable scales specified by minnlcsetscale() call).

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0 (default),  if
                you don't want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

NOTE: different solvers employed by MinNLC optimizer use  different  norms
      for step; AUL solver uses 2-norm, whilst SLP solver uses INF-norm.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetstpmax(
    minnlcstate state,
    double stpmax,
    const xparams _params = alglib::xdefault);

`minnlcsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinNLCOptimize().

NOTE: algorithm passes two parameters to rep() callback  -  current  point
      and penalized function value at current point. Important -  function
      value which is returned is NOT function being minimized. It  is  sum
      of the value of the function being minimized - and penalty term.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minnlcsetxrep(
    minnlcstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

minnlc_d_equality example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = -x0+x1
    //     f1(x0,x1) = x0^2+x1^2-1
    //
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = -x[0]+x[1];
    fi[1] = x[0]*x[0] + x[1]*x[1] - 1.0;
    jac[0][0] = -1.0;
    jac[0][1] = +1.0;
    jac[1][0] = 2*x[0];
    jac[1][1] = 2*x[1];
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of
    //
    //     f(x0,x1) = -x0+x1
    //
    // subject to nonlinear equality constraint
    //
    //    x0^2 + x1^2 - 1 = 0
    //
    real_1d_array x0 = "[0,0]";
    real_1d_array s = "[1,1]";
    double epsx = 0.000001;
    ae_int_t maxits = 0;
    minnlcstate state;
    minnlcreport rep;
    real_1d_array x1;

    //
    // Create optimizer object, choose AUL algorithm and tune its settings:
    // * epsx=0.000001  stopping condition for inner iterations
    // * s=[1,1]        all variables have unit scale
    //
    minnlccreate(2, x0, state);
    minnlcsetalgoslp(state);
    minnlcsetcond(state, epsx, maxits);
    minnlcsetscale(state, s);

    //
    // Set constraints:
    //
    // Nonlinear constraints are tricky - you can not "pack" general
    // nonlinear function into double precision array. That's why
    // minnlcsetnlc() does not accept constraints itself - only constraint
    // counts are passed: first parameter is number of equality constraints,
    // second one is number of inequality constraints.
    //
    // As for constraining functions - these functions are passed as part
    // of problem Jacobian (see below).
    //
    // NOTE: MinNLC optimizer supports arbitrary combination of boundary, general
    //       linear and general nonlinear constraints. This example does not
    //       show how to work with general linear constraints, but you can
    //       easily find it in documentation on minnlcsetbc() and
    //       minnlcsetlc() functions.
    //
    minnlcsetnlc(state, 1, 0);

    //
    // Optimize and test results.
    //
    // Optimizer object accepts vector function and its Jacobian, with first
    // component (Jacobian row) being target function, and next components
    // (Jacobian rows) being nonlinear equality and inequality constraints.
    //
    // So, our vector function has form
    //
    //     {f0,f1} = { -x0+x1 , x0^2+x1^2-1 }
    //
    // with Jacobian
    //
    //         [  -1    +1  ]
    //     J = [            ]
    //         [ 2*x0  2*x1 ]
    //
    // with f0 being target function, f1 being constraining function. Number
    // of equality/inequality constraints is specified by minnlcsetnlc(),
    // with equality ones always being first, inequality ones being last.
    //
    alglib::minnlcoptimize(state, nlcfunc1_jac);
    minnlcresults(state, x1, rep);
    printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0.70710,-0.70710]
    return 0;
}

minnlc_d_inequality example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = -x0+x1
    //     f1(x0,x1) = x0^2+x1^2-1
    //
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = -x[0]+x[1];
    fi[1] = x[0]*x[0] + x[1]*x[1] - 1.0;
    jac[0][0] = -1.0;
    jac[0][1] = +1.0;
    jac[1][0] = 2*x[0];
    jac[1][1] = 2*x[1];
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of
    //
    //     f(x0,x1) = -x0+x1
    //
    // subject to boundary constraints
    //
    //    x0>=0, x1>=0
    //
    // and nonlinear inequality constraint
    //
    //    x0^2 + x1^2 - 1 <= 0
    //
    real_1d_array x0 = "[0,0]";
    real_1d_array s = "[1,1]";
    double epsx = 0.000001;
    ae_int_t maxits = 0;
    real_1d_array bndl = "[0,0]";
    real_1d_array bndu = "[+inf,+inf]";
    minnlcstate state;
    minnlcreport rep;
    real_1d_array x1;

    //
    // Create optimizer object, choose SLP algorithm and tune its settings:
    // * epsx=0.000001  stopping condition for inner iterations
    // * s=[1,1]        all variables have unit scale
    //
    minnlccreate(2, x0, state);
    minnlcsetalgoslp(state);
    minnlcsetcond(state, epsx, maxits);
    minnlcsetscale(state, s);

    //
    // Set constraints:
    //
    // 1. boundary constraints are passed with minnlcsetbc() call
    //
    // 2. nonlinear constraints are more tricky - you can not "pack" general
    //    nonlinear function into double precision array. That's why
    //    minnlcsetnlc() does not accept constraints itself - only constraint
    //    counts are passed: first parameter is number of equality constraints,
    //    second one is number of inequality constraints.
    //
    //    As for constraining functions - these functions are passed as part
    //    of problem Jacobian (see below).
    //
    // NOTE: MinNLC optimizer supports arbitrary combination of boundary, general
    //       linear and general nonlinear constraints. This example does not
    //       show how to work with general linear constraints, but you can
    //       easily find it in documentation on minnlcsetlc() function.
    //
    minnlcsetbc(state, bndl, bndu);
    minnlcsetnlc(state, 0, 1);

    //
    // Optimize and test results.
    //
    // Optimizer object accepts vector function and its Jacobian, with first
    // component (Jacobian row) being target function, and next components
    // (Jacobian rows) being nonlinear equality and inequality constraints.
    //
    // So, our vector function has form
    //
    //     {f0,f1} = { -x0+x1 , x0^2+x1^2-1 }
    //
    // with Jacobian
    //
    //         [  -1    +1  ]
    //     J = [            ]
    //         [ 2*x0  2*x1 ]
    //
    // with f0 being target function, f1 being constraining function. Number
    // of equality/inequality constraints is specified by minnlcsetnlc(),
    // with equality ones always being first, inequality ones being last.
    //
    alglib::minnlcoptimize(state, nlcfunc1_jac);
    minnlcresults(state, x1, rep);
    printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [1.0000,0.0000]
    return 0;
}

minnlc_d_mixed example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc2_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1,x2) = x0+x1
    //     f1(x0,x1,x2) = x2-exp(x0)
    //     f2(x0,x1,x2) = x0^2+x1^2-1
    //
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = x[0]+x[1];
    fi[1] = x[2]-exp(x[0]);
    fi[2] = x[0]*x[0] + x[1]*x[1] - 1.0;
    jac[0][0] = 1.0;
    jac[0][1] = 1.0;
    jac[0][2] = 0.0;
    jac[1][0] = -exp(x[0]);
    jac[1][1] = 0.0;
    jac[1][2] = 1.0;
    jac[2][0] = 2*x[0];
    jac[2][1] = 2*x[1];
    jac[2][2] = 0.0;
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of
    //
    //     f(x0,x1) = x0+x1
    //
    // subject to nonlinear inequality constraint
    //
    //    x0^2 + x1^2 - 1 <= 0
    //
    // and nonlinear equality constraint
    //
    //    x2-exp(x0) = 0
    //
    real_1d_array x0 = "[0,0,0]";
    real_1d_array s = "[1,1,1]";
    double epsx = 0.000001;
    ae_int_t maxits = 0;
    minnlcstate state;
    minnlcreport rep;
    real_1d_array x1;

    //
    // Create optimizer object, choose AUL algorithm and tune its settings:
    // * rho=1000       penalty coefficient
    // * outerits=5     number of outer iterations to tune Lagrange coefficients
    // * epsx=0.000001  stopping condition for inner iterations
    // * s=[1,1]        all variables have unit scale
    // * exact low-rank preconditioner is used, updated after each 10 iterations
    // * upper limit on step length is specified (to avoid probing locations where exp() is large)
    //
    minnlccreate(3, x0, state);
    minnlcsetalgoslp(state);
    minnlcsetcond(state, epsx, maxits);
    minnlcsetscale(state, s);
    minnlcsetstpmax(state, 10.0);

    //
    // Set constraints:
    //
    // Nonlinear constraints are tricky - you can not "pack" general
    // nonlinear function into double precision array. That's why
    // minnlcsetnlc() does not accept constraints itself - only constraint
    // counts are passed: first parameter is number of equality constraints,
    // second one is number of inequality constraints.
    //
    // As for constraining functions - these functions are passed as part
    // of problem Jacobian (see below).
    //
    // NOTE: MinNLC optimizer supports arbitrary combination of boundary, general
    //       linear and general nonlinear constraints. This example does not
    //       show how to work with boundary or general linear constraints, but you
    //       can easily find it in documentation on minnlcsetbc() and
    //       minnlcsetlc() functions.
    //
    minnlcsetnlc(state, 1, 1);

    //
    // Optimize and test results.
    //
    // Optimizer object accepts vector function and its Jacobian, with first
    // component (Jacobian row) being target function, and next components
    // (Jacobian rows) being nonlinear equality and inequality constraints.
    //
    // So, our vector function has form
    //
    //     {f0,f1,f2} = { x0+x1 , x2-exp(x0) , x0^2+x1^2-1 }
    //
    // with Jacobian
    //
    //         [  +1      +1       0 ]
    //     J = [-exp(x0)  0        1 ]
    //         [ 2*x0    2*x1      0 ]
    //
    // with f0 being target function, f1 being equality constraint "f1=0",
    // f2 being inequality constraint "f2<=0". Number of equality/inequality
    // constraints is specified by minnlcsetnlc(), with equality ones always
    // being first, inequality ones being last.
    //
    alglib::minnlcoptimize(state, nlcfunc2_jac);
    minnlcresults(state, x1, rep);
    printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [-0.70710,-0.70710,0.49306]
    return 0;
}

`minns` subpackage

Classes

minnsreport
minnsstate

Functions

minnscreate
minnscreatef
minnsoptimize
minnsrequesttermination
minnsrestartfrom
minnsresults
minnsresultsbuf
minnssetalgoags
minnssetbc
minnssetcond
minnssetlc
minnssetnlc
minnssetscale
minnssetxrep

Examples

minns_d_bc		Nonsmooth box constrained optimization
minns_d_diff		Nonsmooth unconstrained optimization with numerical differentiation
minns_d_nlc		Nonsmooth nonlinearly constrained optimization
minns_d_unconstrained		Nonsmooth unconstrained optimization

`minnsreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           total number of inner iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)
* CErr                      maximum violation of all types of constraints
* LCErr                     maximum violation of linear constraints
* NLCErr                    maximum violation of nonlinear constraints

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -3    box constraints are inconsistent
  -1    inconsistent parameters were passed:
        * penalty parameter for minnssetalgoags() is zero,
          but we have nonlinear constraints set by minnssetnlc()
   2    sampling radius decreased below epsx
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    User requested termination via MinNSRequestTermination()

Other fields of this structure are not documented and should not be used!
*************************************************************************/
class minnsreport
{
    ae_int_t             iterationscount;
    ae_int_t             nfev;
    double               cerr;
    double               lcerr;
    double               nlcerr;
    ae_int_t             terminationtype;
    ae_int_t             varidx;
    ae_int_t             funcidx;
};

`minnsstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinNS subpackage to work  with  this
object
*************************************************************************/
class minnsstate
{
};

`minnscreate` function

/*************************************************************************
                  NONSMOOTH NONCONVEX OPTIMIZATION
            SUBJECT TO BOX/LINEAR/NONLINEAR-NONSMOOTH CONSTRAINTS

DESCRIPTION:

The  subroutine  minimizes  function   F(x)  of N arguments subject to any
combination of:
* bound constraints
* linear inequality constraints
* linear equality constraints
* nonlinear equality constraints Gi(x)=0
* nonlinear inequality constraints Hi(x)<=0

IMPORTANT: see MinNSSetAlgoAGS for important  information  on  performance
           restrictions of AGS solver.

REQUIREMENTS:
* starting point X0 must be feasible or not too far away from the feasible
  set
* F(), G(), H() are continuous, locally Lipschitz  and  continuously  (but
  not necessarily twice) differentiable in an open dense  subset  of  R^N.
  Functions F(), G() and H() may be nonsmooth and non-convex.
  Informally speaking, it means  that  functions  are  composed  of  large
  differentiable "patches" with nonsmoothness having  place  only  at  the
  boundaries between these "patches".
  Most real-life nonsmooth  functions  satisfy  these  requirements.  Say,
  anything which involves finite number of abs(), min() and max() is  very
  likely to pass the test.
  Say, it is possible to optimize anything of the following:
  * f=abs(x0)+2*abs(x1)
  * f=max(x0,x1)
  * f=sin(max(x0,x1)+abs(x2))
* for nonlinearly constrained problems: F()  must  be  bounded from  below
  without nonlinear constraints (this requirement is due to the fact that,
  contrary to box and linear constraints, nonlinear ones  require  special
  handling).
* user must provide function value and gradient for F(), H(), G()  at  all
  points where function/gradient can be calculated. If optimizer  requires
  value exactly at the boundary between "patches" (say, at x=0 for f=abs(x)),
  where gradient is not defined, user may resolve tie arbitrarily (in  our
  case - return +1 or -1 at its discretion).
* NS solver supports numerical differentiation, i.e. it may  differentiate
  your function for you,  but  it  results  in  2N  increase  of  function
  evaluations. Not recommended unless you solve really small problems. See
  minnscreatef() for more information on this functionality.

USAGE:

1. User initializes algorithm state with MinNSCreate() call  and   chooses
   what NLC solver to use. There is some solver which is used by  default,
   with default settings, but you should NOT rely on  default  choice.  It
   may change in future releases of ALGLIB without notice, and no one  can
   guarantee that new solver will be  able  to  solve  your  problem  with
   default settings.

   From the other side, if you choose solver explicitly, you can be pretty
   sure that it will work with new ALGLIB releases.

   In the current release following solvers can be used:
   * AGS solver (activated with MinNSSetAlgoAGS() function)

2. User adds boundary and/or linear and/or nonlinear constraints by  means
   of calling one of the following functions:
   a) MinNSSetBC() for boundary constraints
   b) MinNSSetLC() for linear constraints
   c) MinNSSetNLC() for nonlinear constraints
   You may combine (a), (b) and (c) in one optimization problem.

3. User sets scale of the variables with MinNSSetScale() function. It   is
   VERY important to set  scale  of  the  variables,  because  nonlinearly
   constrained problems are hard to solve when variables are badly scaled.

4. User sets stopping conditions with MinNSSetCond().

5. Finally, user calls MinNSOptimize()  function  which  takes   algorithm
   state and pointer (delegate, etc) to callback function which calculates
   F/G/H.

7. User calls MinNSResults() to get solution

8. Optionally user may call MinNSRestartFrom() to solve   another  problem
   with same N but another starting point. MinNSRestartFrom()  allows   to
   reuse already initialized structure.


INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

NOTE: minnscreatef() function may be used if  you  do  not  have  analytic
      gradient.   This   function  creates  solver  which  uses  numerical
      differentiation with user-specified step.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnscreate(
    real_1d_array x,
    minnsstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minnscreate(
    ae_int_t n,
    real_1d_array x,
    minnsstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`minnscreatef` function

/*************************************************************************
Version of minnscreatef() which uses numerical differentiation. I.e.,  you
do not have to calculate derivatives yourself. However, this version needs
2N times more function evaluations.

2-point differentiation formula is  used,  because  more  precise  4-point
formula is unstable when used on non-smooth functions.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.
    DiffStep-   differentiation  step,  DiffStep>0.   Algorithm   performs
                numerical differentiation  with  step  for  I-th  variable
                being equal to DiffStep*S[I] (here S[] is a  scale vector,
                set by minnssetscale() function).
                Do not use  too  small  steps,  because  it  may  lead  to
                catastrophic cancellation during intermediate calculations.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnscreatef(
    real_1d_array x,
    double diffstep,
    minnsstate& state,
    const xparams _params = alglib::xdefault);
void alglib::minnscreatef(
    ae_int_t n,
    real_1d_array x,
    double diffstep,
    minnsstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minnsoptimize` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    fvec    -   callback which calculates function vector fi[]
                at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied Jacobian, and one which uses  only  function
   vector and numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   you should choose appropriate variant of  minnsoptimize() -  one  which
   accepts function AND Jacobian or one which accepts ONLY function.

   Be careful to choose variant of minnsoptimize()  which  corresponds  to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to minnsoptimize()    and  specific
   function used to create optimizer.


                     |         USER PASSED TO minnsoptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   minnscreatef()    |     works               FAILS
   minnscreate()     |     FAILS               works

   Here "FAILS" denotes inappropriate combinations  of  optimizer creation
   function  and  minnsoptimize()  version.   Attemps   to    use     such
   combination will lead to exception. Either  you  did  not pass gradient
   when it WAS needed or you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnsoptimize(minnsstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minnsoptimize(minnsstate &state,
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnsrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnsrequesttermination(
    minnsstate state,
    const xparams _params = alglib::xdefault);

`minnsrestartfrom` function

/*************************************************************************
This subroutine restarts algorithm from new point.
All optimization parameters (including constraints) are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have  same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure previously allocated with minnscreate() call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnsrestartfrom(
    minnsstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

`minnsresults` function

/*************************************************************************
MinNS results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report. You should check Rep.TerminationType
                in  order  to  distinguish  successful  termination  from
                unsuccessful one:
                * -8   internal integrity control  detected  infinite  or
                       NAN   values   in   function/gradient.    Abnormal
                       termination signalled.
                * -3   box constraints are inconsistent
                * -1   inconsistent parameters were passed:
                       * penalty parameter for minnssetalgoags() is zero,
                         but we have nonlinear constraints set by minnssetnlc()
                *  2   sampling radius decreased below epsx
                *  7    stopping conditions are too stringent,
                        further improvement is impossible,
                        X contains best point found so far.
                *  8    User requested termination via minnsrequesttermination()

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnsresults(
    minnsstate state,
    real_1d_array& x,
    minnsreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnsresultsbuf` function

/*************************************************************************

Buffered implementation of minnsresults() which uses pre-allocated  buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnsresultsbuf(
    minnsstate state,
    real_1d_array& x,
    minnsreport& rep,
    const xparams _params = alglib::xdefault);

`minnssetalgoags` function

/*************************************************************************
This function tells MinNS unit to use  AGS  (adaptive  gradient  sampling)
algorithm for nonsmooth constrained  optimization.  This  algorithm  is  a
slight modification of one described in  "An  Adaptive  Gradient  Sampling
Algorithm for Nonsmooth Optimization" by Frank E. Curtisy and Xiaocun Quez.

This optimizer has following benefits and drawbacks:
+ robustness; it can be used with nonsmooth and nonconvex functions.
+ relatively easy tuning; most of the metaparameters are easy to select.
- it has convergence of steepest descent, slower than CG/LBFGS.
- each iteration involves evaluation of ~2N gradient values  and  solution
  of 2Nx2N quadratic programming problem, which  limits  applicability  of
  algorithm by small-scale problems (up to 50-100).

IMPORTANT: this  algorithm  has  convergence  guarantees,   i.e.  it  will
           steadily move towards some stationary point of the function.

           However, "stationary point" does not  always  mean  "solution".
           Nonsmooth problems often have "flat spots",  i.e.  areas  where
           function do not change at all. Such "flat spots" are stationary
           points by definition, and algorithm may be caught here.

           Nonsmooth CONVEX tasks are not prone to  this  problem. Say, if
           your function has form f()=MAX(f0,f1,...), and f_i are  convex,
           then f() is convex too and you have guaranteed  convergence  to
           solution.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    Radius  -   initial sampling radius, >=0.

                Internally multiplied  by  vector of  per-variable  scales
                specified by minnssetscale()).

                You should select relatively large sampling radius, roughly
                proportional to scaled length of the first  steps  of  the
                algorithm. Something close to 0.1 in magnitude  should  be
                good for most problems.

                AGS solver can automatically decrease radius, so too large
                radius is  not a problem (assuming that you  won't  choose
                so large radius that algorithm  will  sample  function  in
                too far away points, where gradient value is irrelevant).

                Too small radius won't cause algorithm to fail, but it may
                slow down algorithm (it may  have  to  perform  too  short
                steps).
    Penalty -   penalty coefficient for nonlinear constraints:
                * for problem with nonlinear constraints  should  be  some
                  problem-specific  positive   value,  large  enough  that
                  penalty term changes shape of the function.
                  Starting  from  some  problem-specific   value   penalty
                  coefficient becomes  large  enough  to  exactly  enforce
                  nonlinear constraints;  larger  values  do  not  improve
                  precision.
                  Increasing it too much may slow down convergence, so you
                  should choose it carefully.
                * can be zero for problems WITHOUT  nonlinear  constraints
                  (i.e. for unconstrained ones or ones with  just  box  or
                  linear constraints)
                * if you specify zero value for problem with at least  one
                  nonlinear  constraint,  algorithm  will  terminate  with
                  error code -1.

ALGORITHM OUTLINE

The very basic outline of unconstrained AGS algorithm is given below:

0. If sampling radius is below EpsX  or  we  performed  more  then  MaxIts
   iterations - STOP.
1. sample O(N) gradient values at random locations  around  current point;
   informally speaking, this sample is an implicit piecewise  linear model
   of the function, although algorithm formulation does  not  mention that
   explicitly
2. solve quadratic programming problem in order to find descent direction
3. if QP solver tells us that we  are  near  solution,  decrease  sampling
   radius and move to (0)
4. perform backtracking line search
5. after moving to new point, goto (0)

As for the constraints:
* box constraints are handled exactly  by  modification  of  the  function
  being minimized
* linear/nonlinear constraints are handled by adding L1  penalty.  Because
  our solver can handle nonsmoothness, we can  use  L1  penalty  function,
  which is an exact one  (i.e.  exact  solution  is  returned  under  such
  penalty).
* penalty coefficient for  linear  constraints  is  chosen  automatically;
  however, penalty coefficient for nonlinear constraints must be specified
  by user.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnssetalgoags(
    minnsstate state,
    double radius,
    double penalty,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnssetbc` function

/*************************************************************************
This function sets boundary constraints.

Boundary constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with minnsrestartfrom().

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF.

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: AGS solver has following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by  bound  constraints,
  even  when  numerical  differentiation is used (algorithm adjusts  nodes
  according to boundary constraints)

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnssetbc(
    minnsstate state,
    real_1d_array bndl,
    real_1d_array bndu,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minnssetcond` function

/*************************************************************************
This function sets stopping conditions for iterations of optimizer.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0
                The AGS solver finishes its work if  on  k+1-th  iteration
                sampling radius decreases below EpsX.
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsX=0  and  MaxIts=0  (simultaneously)  will  lead  to  automatic
stopping criterion selection. We do not recommend you to rely  on  default
choice in production code.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnssetcond(
    minnsstate state,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnssetlc` function

/*************************************************************************
This function sets linear constraints.

Linear constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with minnsrestartfrom().

INPUT PARAMETERS:
    State   -   structure previously allocated with minnscreate() call.
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

NOTE: linear (non-bound) constraints are satisfied only approximately:

* there always exists some minor violation (about current sampling  radius
  in magnitude during optimization, about EpsX in the solution) due to use
  of penalty method to handle constraints.
* numerical differentiation, if used, may  lead  to  function  evaluations
  outside  of the feasible  area,   because   algorithm  does  NOT  change
  numerical differentiation formula according to linear constraints.

If you want constraints to be  satisfied  exactly, try to reformulate your
problem  in  such  manner  that  all constraints will become boundary ones
(this kind of constraints is always satisfied exactly, both in  the  final
solution and in all intermediate points).

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnssetlc(
    minnsstate state,
    real_2d_array c,
    integer_1d_array ct,
    const xparams _params = alglib::xdefault);
void alglib::minnssetlc(
    minnsstate state,
    real_2d_array c,
    integer_1d_array ct,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

`minnssetnlc` function

/*************************************************************************
This function sets nonlinear constraints.

In fact, this function sets NUMBER of nonlinear  constraints.  Constraints
itself (constraint functions) are passed to minnsoptimize() method.   This
method requires user-defined vector function F[]  and  its  Jacobian  J[],
where:
* first component of F[] and first row  of  Jacobian  J[]  correspond   to
  function being minimized
* next NLEC components of F[] (and rows  of  J)  correspond  to  nonlinear
  equality constraints G_i(x)=0
* next NLIC components of F[] (and rows  of  J)  correspond  to  nonlinear
  inequality constraints H_i(x)<=0

NOTE: you may combine nonlinear constraints with linear/boundary ones.  If
      your problem has mixed constraints, you  may explicitly specify some
      of them as linear ones. It may help optimizer to  handle  them  more
      efficiently.

INPUT PARAMETERS:
    State   -   structure previously allocated with minnscreate() call.
    NLEC    -   number of Non-Linear Equality Constraints (NLEC), >=0
    NLIC    -   number of Non-Linear Inquality Constraints (NLIC), >=0

NOTE 1: nonlinear constraints are satisfied only  approximately!   It   is
        possible   that  algorithm  will  evaluate  function  outside   of
        the feasible area!

NOTE 2: algorithm scales variables  according  to   scale   specified   by
        minnssetscale()  function,  so  it can handle problems with  badly
        scaled variables (as long as we KNOW their scales).

        However,  there  is  no  way  to  automatically  scale   nonlinear
        constraints Gi(x) and Hi(x). Inappropriate scaling  of  Gi/Hi  may
        ruin convergence. Solving problem with  constraint  "1000*G0(x)=0"
        is NOT same as solving it with constraint "0.001*G0(x)=0".

        It  means  that  YOU  are  the  one who is responsible for correct
        scaling of nonlinear constraints Gi(x) and Hi(x). We recommend you
        to scale nonlinear constraints in such way that I-th component  of
        dG/dX (or dH/dx) has approximately unit  magnitude  (for  problems
        with unit scale)  or  has  magnitude approximately equal to 1/S[i]
        (where S is a scale set by minnssetscale() function).

NOTE 3: nonlinear constraints are always hard to handle,  no  matter  what
        algorithm you try to use. Even basic box/linear constraints modify
        function  curvature   by  adding   valleys  and  ridges.  However,
        nonlinear constraints add valleys which are very  hard  to  follow
        due to their "curved" nature.

        It means that optimization with single nonlinear constraint may be
        significantly slower than optimization with multiple linear  ones.
        It is normal situation, and we recommend you to  carefully  choose
        Rho parameter of minnssetalgoags(), because too  large  value  may
        slow down convergence.


  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnssetnlc(
    minnsstate state,
    ae_int_t nlec,
    ae_int_t nlic,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minnssetscale` function

/*************************************************************************
This function sets scaling coefficients for NLC optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void alglib::minnssetscale(
    minnsstate state,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnssetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to minnsoptimize().

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::minnssetxrep(
    minnsstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

minns_d_bc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nsfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = 2*|x0|+x1
    //
    // and Jacobian matrix J = [df0/dx0 df0/dx1]
    //
    fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
    jac[0][0] = 2*alglib::sign(x[0]);
    jac[0][1] = alglib::sign(x[1]);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of
    //
    //     f(x0,x1) = 2*|x0|+|x1|
    //
    // subject to box constraints
    //
    //        1 <= x0 < +INF
    //     -INF <= x1 < +INF
    //
    // using nonsmooth nonlinear optimizer.
    //
    real_1d_array x0 = "[1,1]";
    real_1d_array s = "[1,1]";
    real_1d_array bndl = "[1,-inf]";
    real_1d_array bndu = "[+inf,+inf]";
    double epsx = 0.00001;
    double radius = 0.1;
    double rho = 0.0;
    ae_int_t maxits = 0;
    minnsstate state;
    minnsreport rep;
    real_1d_array x1;

    //
    // Create optimizer object, choose AGS algorithm and tune its settings:
    // * radius=0.1     good initial value; will be automatically decreased later.
    // * rho=0.0        penalty coefficient for nonlinear constraints; can be zero
    //                  because we do not have such constraints
    // * epsx=0.000001  stopping conditions
    // * s=[1,1]        all variables have unit scale
    //
    minnscreate(2, x0, state);
    minnssetalgoags(state, radius, rho);
    minnssetcond(state, epsx, maxits);
    minnssetscale(state, s);

    //
    // Set box constraints.
    //
    // General linear constraints are set in similar way (see comments on
    // minnssetlc() function for more information).
    //
    // You may combine box, linear and nonlinear constraints in one optimization
    // problem.
    //
    minnssetbc(state, bndl, bndu);

    //
    // Optimize and test results.
    //
    // Optimizer object accepts vector function and its Jacobian, with first
    // component (Jacobian row) being target function, and next components
    // (Jacobian rows) being nonlinear equality and inequality constraints
    // (box/linear ones are passed separately by means of minnssetbc() and
    // minnssetlc() calls).
    //
    // If you do not have nonlinear constraints (exactly our situation), then
    // you will have one-component function vector and 1xN Jacobian matrix.
    //
    // So, our vector function has form
    //
    //     {f0} = { 2*|x0|+|x1| }
    //
    // with Jacobian
    //
    //         [                       ]
    //     J = [ 2*sign(x0)   sign(x1) ]
    //         [                       ]
    //
    // NOTE: nonsmooth optimizer requires considerably more function
    //       evaluations than smooth solver - about 2N times more. Using
    //       numerical differentiation introduces additional (multiplicative)
    //       2N speedup.
    //
    //       It means that if smooth optimizer WITH user-supplied gradient
    //       needs 100 function evaluations to solve 50-dimensional problem,
    //       then AGS solver with user-supplied gradient will need about 10.000
    //       function evaluations, and with numerical gradient about 1.000.000
    //       function evaluations will be performed.
    //
    // NOTE: AGS solver used by us can handle nonsmooth and nonconvex
    //       optimization problems. It has convergence guarantees, i.e. it will
    //       converge to stationary point of the function after running for some
    //       time.
    //
    //       However, it is important to remember that "stationary point" is not
    //       equal to "solution". If your problem is convex, everything is OK.
    //       But nonconvex optimization problems may have "flat spots" - large
    //       areas where gradient is exactly zero, but function value is far away
    //       from optimal. Such areas are stationary points too, and optimizer
    //       may be trapped here.
    //
    //       "Flat spots" are nonsmooth equivalent of the saddle points, but with
    //       orders of magnitude worse properties - they may be quite large and
    //       hard to avoid. All nonsmooth optimizers are prone to this kind of the
    //       problem, because it is impossible to automatically distinguish "flat
    //       spot" from true solution.
    //
    //       This note is here to warn you that you should be very careful when
    //       you solve nonsmooth optimization problems. Visual inspection of
    //       results is essential.
    //
    //
    alglib::minnsoptimize(state, nsfunc1_jac);
    minnsresults(state, x1, rep);
    printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [1.0000,0.0000]
    return 0;
}

minns_d_diff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nsfunc1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = 2*|x0|+x1
    //
    fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of
    //
    //     f(x0,x1) = 2*|x0|+|x1|
    //
    // using nonsmooth nonlinear optimizer with numerical
    // differentiation provided by ALGLIB.
    //
    // NOTE: nonsmooth optimizer requires considerably more function
    //       evaluations than smooth solver - about 2N times more. Using
    //       numerical differentiation introduces additional (multiplicative)
    //       2N speedup.
    //
    //       It means that if smooth optimizer WITH user-supplied gradient
    //       needs 100 function evaluations to solve 50-dimensional problem,
    //       then AGS solver with user-supplied gradient will need about 10.000
    //       function evaluations, and with numerical gradient about 1.000.000
    //       function evaluations will be performed.
    //
    real_1d_array x0 = "[1,1]";
    real_1d_array s = "[1,1]";
    double epsx = 0.00001;
    double diffstep = 0.000001;
    double radius = 0.1;
    double rho = 0.0;
    ae_int_t maxits = 0;
    minnsstate state;
    minnsreport rep;
    real_1d_array x1;

    //
    // Create optimizer object, choose AGS algorithm and tune its settings:
    // * radius=0.1     good initial value; will be automatically decreased later.
    // * rho=0.0        penalty coefficient for nonlinear constraints; can be zero
    //                  because we do not have such constraints
    // * epsx=0.000001  stopping conditions
    // * s=[1,1]        all variables have unit scale
    //
    minnscreatef(2, x0, diffstep, state);
    minnssetalgoags(state, radius, rho);
    minnssetcond(state, epsx, maxits);
    minnssetscale(state, s);

    //
    // Optimize and test results.
    //
    // Optimizer object accepts vector function, with first component
    // being target function, and next components being nonlinear equality
    // and inequality constraints (box/linear ones are passed separately
    // by means of minnssetbc() and minnssetlc() calls).
    //
    // If you do not have nonlinear constraints (exactly our situation), then
    // you will have one-component function vector.
    //
    // So, our vector function has form
    //
    //     {f0} = { 2*|x0|+|x1| }
    //
    alglib::minnsoptimize(state, nsfunc1_fvec);
    minnsresults(state, x1, rep);
    printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0.0000,0.0000]
    return 0;
}

minns_d_nlc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nsfunc2_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates function vector
    //
    //     f0(x0,x1) = 2*|x0|+x1
    //     f1(x0,x1) = x0-1
    //     f2(x0,x1) = -x1-1
    //
    // and Jacobian matrix J
    //
    //         [ df0/dx0   df0/dx1 ]
    //     J = [ df1/dx0   df1/dx1 ]
    //         [ df2/dx0   df2/dx1 ]
    //
    fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
    jac[0][0] = 2*alglib::sign(x[0]);
    jac[0][1] = alglib::sign(x[1]);
    fi[1] = x[0]-1;
    jac[1][0] = 1;
    jac[1][1] = 0;
    fi[2] = -x[1]-1;
    jac[2][0] = 0;
    jac[2][1] = -1;
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of
    //
    //     f(x0,x1) = 2*|x0|+|x1|
    //
    // subject to combination of equality and inequality constraints
    //
    //      x0  =  1
    //      x1 >= -1
    //
    // using nonsmooth nonlinear optimizer. Although these constraints
    // are linear, we treat them as general nonlinear ones in order to
    // demonstrate nonlinearly constrained optimization setup.
    //
    real_1d_array x0 = "[1,1]";
    real_1d_array s = "[1,1]";
    double epsx = 0.00001;
    double radius = 0.1;
    double rho = 50.0;
    ae_int_t maxits = 0;
    minnsstate state;
    minnsreport rep;
    real_1d_array x1;

    //
    // Create optimizer object, choose AGS algorithm and tune its settings:
    // * radius=0.1     good initial value; will be automatically decreased later.
    // * rho=50.0       penalty coefficient for nonlinear constraints. It is your
    //                  responsibility to choose good one - large enough that it
    //                  enforces constraints, but small enough in order to avoid
    //                  extreme slowdown due to ill-conditioning.
    // * epsx=0.000001  stopping conditions
    // * s=[1,1]        all variables have unit scale
    //
    minnscreate(2, x0, state);
    minnssetalgoags(state, radius, rho);
    minnssetcond(state, epsx, maxits);
    minnssetscale(state, s);

    //
    // Set general nonlinear constraints.
    //
    // This part is more tricky than working with box/linear constraints - you
    // can not "pack" general nonlinear function into double precision array.
    // That's why minnssetnlc() does not accept constraints itself - only
    // constraint COUNTS are passed: first parameter is number of equality
    // constraints, second one is number of inequality constraints.
    //
    // As for constraining functions - these functions are passed as part
    // of problem Jacobian (see below).
    //
    // NOTE: MinNS optimizer supports arbitrary combination of boundary, general
    //       linear and general nonlinear constraints. This example does not
    //       show how to work with general linear constraints, but you can
    //       easily find it in documentation on minnlcsetlc() function.
    //
    minnssetnlc(state, 1, 1);

    //
    // Optimize and test results.
    //
    // Optimizer object accepts vector function and its Jacobian, with first
    // component (Jacobian row) being target function, and next components
    // (Jacobian rows) being nonlinear equality and inequality constraints
    // (box/linear ones are passed separately by means of minnssetbc() and
    // minnssetlc() calls).
    //
    // Nonlinear equality constraints have form Gi(x)=0, inequality ones
    // have form Hi(x)<=0, so we may have to "normalize" constraints prior
    // to passing them to optimizer (right side is zero, constraints are
    // sorted, multiplied by -1 when needed).
    //
    // So, our vector function has form
    //
    //     {f0,f1,f2} = { 2*|x0|+|x1|,  x0-1, -x1-1 }
    //
    // with Jacobian
    //
    //         [ 2*sign(x0)   sign(x1) ]
    //     J = [     1           0     ]
    //         [     0          -1     ]
    //
    // which means that we have optimization problem
    //
    //     min{f0} subject to f1=0, f2<=0
    //
    // which is essentially same as
    //
    //     min { 2*|x0|+|x1| } subject to x0=1, x1>=-1
    //
    // NOTE: AGS solver used by us can handle nonsmooth and nonconvex
    //       optimization problems. It has convergence guarantees, i.e. it will
    //       converge to stationary point of the function after running for some
    //       time.
    //
    //       However, it is important to remember that "stationary point" is not
    //       equal to "solution". If your problem is convex, everything is OK.
    //       But nonconvex optimization problems may have "flat spots" - large
    //       areas where gradient is exactly zero, but function value is far away
    //       from optimal. Such areas are stationary points too, and optimizer
    //       may be trapped here.
    //
    //       "Flat spots" are nonsmooth equivalent of the saddle points, but with
    //       orders of magnitude worse properties - they may be quite large and
    //       hard to avoid. All nonsmooth optimizers are prone to this kind of the
    //       problem, because it is impossible to automatically distinguish "flat
    //       spot" from true solution.
    //
    //       This note is here to warn you that you should be very careful when
    //       you solve nonsmooth optimization problems. Visual inspection of
    //       results is essential.
    //
    alglib::minnsoptimize(state, nsfunc2_jac);
    minnsresults(state, x1, rep);
    printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [1.0000,0.0000]
    return 0;
}

minns_d_unconstrained example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nsfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = 2*|x0|+x1
    //
    // and Jacobian matrix J = [df0/dx0 df0/dx1]
    //
    fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
    jac[0][0] = 2*alglib::sign(x[0]);
    jac[0][1] = alglib::sign(x[1]);
}

int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of
    //
    //     f(x0,x1) = 2*|x0|+|x1|
    //
    // using nonsmooth nonlinear optimizer.
    //
    real_1d_array x0 = "[1,1]";
    real_1d_array s = "[1,1]";
    double epsx = 0.00001;
    double radius = 0.1;
    double rho = 0.0;
    ae_int_t maxits = 0;
    minnsstate state;
    minnsreport rep;
    real_1d_array x1;

    //
    // Create optimizer object, choose AGS algorithm and tune its settings:
    // * radius=0.1     good initial value; will be automatically decreased later.
    // * rho=0.0        penalty coefficient for nonlinear constraints; can be zero
    //                  because we do not have such constraints
    // * epsx=0.000001  stopping conditions
    // * s=[1,1]        all variables have unit scale
    //
    minnscreate(2, x0, state);
    minnssetalgoags(state, radius, rho);
    minnssetcond(state, epsx, maxits);
    minnssetscale(state, s);

    //
    // Optimize and test results.
    //
    // Optimizer object accepts vector function and its Jacobian, with first
    // component (Jacobian row) being target function, and next components
    // (Jacobian rows) being nonlinear equality and inequality constraints
    // (box/linear ones are passed separately by means of minnssetbc() and
    // minnssetlc() calls).
    //
    // If you do not have nonlinear constraints (exactly our situation), then
    // you will have one-component function vector and 1xN Jacobian matrix.
    //
    // So, our vector function has form
    //
    //     {f0} = { 2*|x0|+|x1| }
    //
    // with Jacobian
    //
    //         [                       ]
    //     J = [ 2*sign(x0)   sign(x1) ]
    //         [                       ]
    //
    // NOTE: nonsmooth optimizer requires considerably more function
    //       evaluations than smooth solver - about 2N times more. Using
    //       numerical differentiation introduces additional (multiplicative)
    //       2N speedup.
    //
    //       It means that if smooth optimizer WITH user-supplied gradient
    //       needs 100 function evaluations to solve 50-dimensional problem,
    //       then AGS solver with user-supplied gradient will need about 10.000
    //       function evaluations, and with numerical gradient about 1.000.000
    //       function evaluations will be performed.
    //
    // NOTE: AGS solver used by us can handle nonsmooth and nonconvex
    //       optimization problems. It has convergence guarantees, i.e. it will
    //       converge to stationary point of the function after running for some
    //       time.
    //
    //       However, it is important to remember that "stationary point" is not
    //       equal to "solution". If your problem is convex, everything is OK.
    //       But nonconvex optimization problems may have "flat spots" - large
    //       areas where gradient is exactly zero, but function value is far away
    //       from optimal. Such areas are stationary points too, and optimizer
    //       may be trapped here.
    //
    //       "Flat spots" are nonsmooth equivalent of the saddle points, but with
    //       orders of magnitude worse properties - they may be quite large and
    //       hard to avoid. All nonsmooth optimizers are prone to this kind of the
    //       problem, because it is impossible to automatically distinguish "flat
    //       spot" from true solution.
    //
    //       This note is here to warn you that you should be very careful when
    //       you solve nonsmooth optimization problems. Visual inspection of
    //       results is essential.
    //
    alglib::minnsoptimize(state, nsfunc1_jac);
    minnsresults(state, x1, rep);
    printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0.0000,0.0000]
    return 0;
}

`minqp` subpackage

Classes

minqpreport
minqpstate

Functions

minqpcreate
minqpoptimize
minqpresults
minqpresultsbuf
minqpsetalgobleic
minqpsetalgodenseaul
minqpsetalgoquickqp
minqpsetbc
minqpsetlc
minqpsetlcmixed
minqpsetlcsparse
minqpsetlinearterm
minqpsetorigin
minqpsetquadraticterm
minqpsetquadratictermsparse
minqpsetscale
minqpsetscaleautodiag
minqpsetstartingpoint

Examples

minqp_d_bc1		Bound constrained dense quadratic programming
minqp_d_lc1		Linearly constrained dense quadratic programming
minqp_d_nonconvex		Nonconvex quadratic programming
minqp_d_u1		Unconstrained dense quadratic programming
minqp_d_u2		Unconstrained sparse quadratic programming

`minqpreport` class

/*************************************************************************
This structure stores optimization report:
* InnerIterationsCount      number of inner iterations
* OuterIterationsCount      number of outer iterations
* NCholesky                 number of Cholesky decomposition
* NMV                       number of matrix-vector products
                            (only products calculated as part of iterative
                            process are counted)
* TerminationType           completion code (see below)

Completion codes:
* -9    failure of the automatic scale evaluation:  one  of  the  diagonal
        elements of the quadratic term is non-positive.  Specify  variable
        scales manually!
* -5    inappropriate solver was used:
        * QuickQP solver for problem with general linear constraints (dense/sparse)
* -4    BLEIC-QP or QuickQP solver found unconstrained direction
        of negative curvature (function is unbounded from
        below  even  under  constraints),  no  meaningful
        minimum can be found.
* -3    inconsistent constraints (or, maybe, feasible point is
        too hard to find). If you are sure that constraints are feasible,
        try to restart optimizer with better initial approximation.
* -1    solver error
*  1..4 successful completion
*  5    MaxIts steps was taken
*  7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
*************************************************************************/
class minqpreport
{
    ae_int_t             inneriterationscount;
    ae_int_t             outeriterationscount;
    ae_int_t             nmv;
    ae_int_t             ncholesky;
    ae_int_t             terminationtype;
};

`minqpstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinQP subpackage to work with this
object
*************************************************************************/
class minqpstate
{
};

`minqpcreate` function

/*************************************************************************
                    CONSTRAINED QUADRATIC PROGRAMMING

The subroutine creates QP optimizer. After initial creation,  it  contains
default optimization problem with zero quadratic and linear terms  and  no
constraints. You should set quadratic/linear terms with calls to functions
provided by MinQP subpackage.

You should also choose appropriate QP solver and set it  and  its stopping
criteria by means of MinQPSetAlgo??????() function. Then, you should start
solution process by means of MinQPOptimize() call. Solution itself can  be
obtained with MinQPResults() function.

Following solvers are recommended:
* QuickQP for dense problems with box-only constraints (or no constraints
  at all)
* QP-BLEIC for dense/sparse problems with moderate (up to 50) number of
  general linear constraints
* DENSE-AUL-QP for dense problems with any (small or large) number of
  general linear constraints

INPUT PARAMETERS:
    N       -   problem size

OUTPUT PARAMETERS:
    State   -   optimizer with zero quadratic/linear terms
                and no constraints

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpcreate(
    ae_int_t n,
    minqpstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`minqpoptimize` function

/*************************************************************************
This function solves quadratic programming problem.

Prior to calling this function you should choose solver by means of one of
the following functions:

* minqpsetalgoquickqp()     - for QuickQP solver
* minqpsetalgobleic()       - for BLEIC-QP solver
* minqpsetalgodenseaul()    - for Dense-AUL-QP solver

These functions also allow you to control stopping criteria of the solver.
If you did not set solver,  MinQP  subpackage  will  automatically  select
solver for your problem and will run it with default stopping criteria.

However, it is better to set explicitly solver and its stopping criteria.

INPUT PARAMETERS:
    State   -   algorithm state

You should use MinQPResults() function to access results after calls
to this function.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey.
     Special thanks to Elvira Illarionova  for  important  suggestions  on
     the linearly constrained QP algorithm.
*************************************************************************/
void alglib::minqpoptimize(
    minqpstate state,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`minqpresults` function

/*************************************************************************
QP solver results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution.
                This array is allocated and initialized only when
                Rep.TerminationType parameter is positive (success).
    Rep     -   optimization report. You should check Rep.TerminationType,
                which contains completion code, and you may check  another
                fields which contain another information  about  algorithm
                functioning.

                Failure codes returned by algorithm are:
                * -9    failure of the automatic scale evaluation:  one of
                        the diagonal elements of  the  quadratic  term  is
                        non-positive.  Specify variable scales manually!
                * -5    inappropriate solver was used:
                        * QuickQP solver for problem with  general  linear
                          constraints
                * -4    BLEIC-QP/QuickQP   solver    found   unconstrained
                        direction  of   negative  curvature  (function  is
                        unbounded from below even under constraints),   no
                        meaningful minimum can be found.
                * -3    inconsistent constraints (or maybe  feasible point
                        is too  hard  to  find).  If  you  are  sure  that
                        constraints are feasible, try to restart optimizer
                        with better initial approximation.

                Completion codes specific for Cholesky algorithm:
                *  4   successful completion

                Completion codes specific for BLEIC/QuickQP algorithms:
                *  1   relative function improvement is no more than EpsF.
                *  2   scaled step is no more than EpsX.
                *  4   scaled gradient norm is no more than EpsG.
                *  5   MaxIts steps was taken

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpresults(
    minqpstate state,
    real_1d_array& x,
    minqpreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`minqpresultsbuf` function

/*************************************************************************
QP results

Buffered implementation of MinQPResults() which uses pre-allocated  buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpresultsbuf(
    minqpstate state,
    real_1d_array& x,
    minqpreport& rep,
    const xparams _params = alglib::xdefault);

`minqpsetalgobleic` function

/*************************************************************************
This function tells solver to use BLEIC-based algorithm and sets  stopping
criteria for the algorithm.

This algorithm is fast  enough  for large-scale  problems  with  following
properties:
a) feasible initial point, moderate amount of general linear constraints
b) arbitrary (can be infeasible) initial point, small  amount  of  general
   linear constraints (say, hundred or less)

If you solve large-scale QP problem with many inequality  constraints  and
without initial feasibility guarantees, consider  using  DENSE-AUL  solver
instead. Initial feasibility detection stage by BLEIC may take too long on
such problems.

ALGORITHM FEATURES:

* supports dense and sparse QP problems
* supports box and general linear equality/inequality constraints
* can solve all types of problems  (convex,  semidefinite,  nonconvex)  as
  long as they are bounded from below under constraints.
  Say, it is possible to solve "min{-x^2} subject to -1<=x<=+1".
  Of course, global  minimum  is found only  for  positive  definite   and
  semidefinite  problems.  As  for indefinite ones - only local minimum is
  found.

ALGORITHM OUTLINE:

* BLEIC-QP solver is just a driver function for MinBLEIC solver; it solves
  quadratic  programming   problem   as   general   linearly   constrained
  optimization problem, which is solved by means of BLEIC solver  (part of
  ALGLIB, active set method).

ALGORITHM LIMITATIONS:
* This algorithm is inefficient on  problems with hundreds  and  thousands
  of general inequality constraints and infeasible initial point.  Initial
  feasibility detection stage may take too long on such constraint sets.
  Consider using DENSE-AUL instead.
* unlike QuickQP solver, this algorithm does not perform Newton steps  and
  does not use Level 3 BLAS. Being general-purpose active set  method,  it
  can activate constraints only one-by-one. Thus, its performance is lower
  than that of QuickQP.
* its precision is also a bit  inferior  to  that  of   QuickQP.  BLEIC-QP
  performs only LBFGS steps (no Newton steps), which are good at detecting
  neighborhood of the solution, buy needs many iterations to find solution
  with more than 6 digits of precision.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled constrained gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinQPSetScale()
    EpsF    -   >=0
                The  subroutine  finishes its work if exploratory steepest
                descent  step  on  k+1-th iteration  satisfies   following
                condition:  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
    EpsX    -   >=0
                The  subroutine  finishes its work if exploratory steepest
                descent  step  on  k+1-th iteration  satisfies   following
                condition:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinQPSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited. NOTE: this  algorithm uses  LBFGS
                iterations,  which  are  relatively  cheap,  but   improve
                function value only a bit. So you will need many iterations
                to converge - from 0.1*N to 10*N, depending  on  problem's
                condition number.

IT IS VERY IMPORTANT TO CALL MinQPSetScale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS STOPPING CRITERIA ARE SCALE-DEPENDENT!

Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead
to automatic stopping criterion selection (presently it is  small    step
length, but it may change in the future versions of ALGLIB).

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetalgobleic(
    minqpstate state,
    double epsg,
    double epsf,
    double epsx,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

`minqpsetalgodenseaul` function

/*************************************************************************
This function tells QP solver to use Dense-AUL algorithm and sets stopping
criteria for the algorithm.

ALGORITHM FEATURES:

* supports  box  and  dense/sparse  general   linear   equality/inequality
  constraints
* convergence is theoretically proved for positive-definite  (convex)   QP
  problems. Semidefinite and non-convex problems can be solved as long  as
  they  are   bounded  from  below  under  constraints,  although  without
  theoretical guarantees.
* this solver is better than QP-BLEIC on problems  with  large  number  of
  general linear constraints. It better handles infeasible initial points.

ALGORITHM OUTLINE:

* this  algorithm   is   an   augmented   Lagrangian   method  with  dense
  preconditioner (hence  its  name).  It  is  similar  to  barrier/penalty
  methods, but much more precise and faster.
* it performs several outer iterations in order to refine  values  of  the
  Lagrange multipliers. Single outer  iteration  is  a  solution  of  some
  unconstrained optimization problem: first  it  performs  dense  Cholesky
  factorization of the Hessian in order to build preconditioner  (adaptive
  regularization is applied to enforce positive  definiteness),  and  then
  it uses L-BFGS optimizer to solve optimization problem.
* typically you need about 5-10 outer iterations to converge to solution

ALGORITHM LIMITATIONS:

* because dense Cholesky driver is used, this algorithm has O(N^2)  memory
  requirements and O(OuterIterations*N^3) minimum running time.  From  the
  practical  point  of  view,  it  limits  its  applicability  by  several
  thousands of variables.
  From  the  other  side,  variables  count  is  the most limiting factor,
  and dependence on constraint count is  much  more  lower. Assuming  that
  constraint matrix is sparse, it may handle tens of thousands  of general
  linear constraints.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0, stopping criteria for inner optimizer.
                Inner  iterations  are  stopped  when  step  length  (with
                variable scaling being applied) is less than EpsX.
                See  minqpsetscale()  for  more  information  on  variable
                scaling.
    Rho     -   penalty coefficient, Rho>0:
                * large enough  that  algorithm  converges  with   desired
                  precision.
                * not TOO large to prevent ill-conditioning
                * recommended values are 100, 1000 or 10000
    ItsCnt  -   number of outer iterations:
                * recommended values: 10-15 (although  in  most  cases  it
                  converges within 5 iterations, you may need a  few  more
                  to be sure).
                * ItsCnt=0 means that small number of outer iterations  is
                  automatically chosen (10 iterations in current version).
                * ItsCnt=1 means that AUL algorithm performs just as usual
                  penalty method.
                * ItsCnt>1 means that  AUL  algorithm  performs  specified
                  number of outer iterations

IT IS VERY IMPORTANT TO CALL minqpsetscale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS CONVERGENCE PROPERTIES AND STOPPING CRITERIA ARE SCALE-DEPENDENT!

NOTE: Passing  EpsX=0  will  lead  to  automatic  step  length  selection
      (specific step length chosen may change in the future  versions  of
      ALGLIB, so it is better to specify step length explicitly).

  -- ALGLIB --
     Copyright 20.08.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetalgodenseaul(
    minqpstate state,
    double epsx,
    double rho,
    ae_int_t itscnt,
    const xparams _params = alglib::xdefault);

`minqpsetalgoquickqp` function

/*************************************************************************
This function tells solver to use QuickQP  algorithm:  special  extra-fast
algorithm for problems with box-only constrants. It may  solve  non-convex
problems as long as they are bounded from below under constraints.

ALGORITHM FEATURES:
* many times (from 5x to 50x!) faster than BLEIC-based QP solver; utilizes
  accelerated methods for activation of constraints.
* supports dense and sparse QP problems
* supports ONLY box constraints; general linear constraints are NOT
  supported by this solver
* can solve all types of problems  (convex,  semidefinite,  nonconvex)  as
  long as they are bounded from below under constraints.
  Say, it is possible to solve "min{-x^2} subject to -1<=x<=+1".
  In convex/semidefinite case global minimum  is  returned,  in  nonconvex
  case - algorithm returns one of the local minimums.

ALGORITHM OUTLINE:

* algorithm  performs  two kinds of iterations: constrained CG  iterations
  and constrained Newton iterations
* initially it performs small number of constrained CG  iterations,  which
  can efficiently activate/deactivate multiple constraints
* after CG phase algorithm tries to calculate Cholesky  decomposition  and
  to perform several constrained Newton steps. If  Cholesky  decomposition
  failed (matrix is indefinite even under constraints),  we  perform  more
  CG iterations until we converge to such set of constraints  that  system
  matrix becomes  positive  definite.  Constrained  Newton  steps  greatly
  increase convergence speed and precision.
* algorithm interleaves CG and Newton iterations which  allows  to  handle
  indefinite matrices (CG phase) and quickly converge after final  set  of
  constraints is found (Newton phase). Combination of CG and Newton phases
  is called "outer iteration".
* it is possible to turn off Newton  phase  (beneficial  for  semidefinite
  problems - Cholesky decomposition will fail too often)

ALGORITHM LIMITATIONS:

* algorithm does not support general  linear  constraints;  only  box ones
  are supported
* Cholesky decomposition for sparse problems  is  performed  with  Skyline
  Cholesky solver, which is intended for low-profile matrices. No profile-
  reducing reordering of variables is performed in this version of ALGLIB.
* problems with near-zero negative eigenvalues (or exacty zero  ones)  may
  experience about 2-3x performance penalty. The reason is  that  Cholesky
  decomposition can not be performed until we identify directions of  zero
  and negative curvature and activate corresponding boundary constraints -
  but we need a lot of trial and errors because these directions  are hard
  to notice in the matrix spectrum.
  In this case you may turn off Newton phase of algorithm.
  Large negative eigenvalues  are  not  an  issue,  so  highly  non-convex
  problems can be solved very efficiently.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled constrained gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinQPSetScale()
    EpsF    -   >=0
                The  subroutine  finishes its work if exploratory steepest
                descent  step  on  k+1-th iteration  satisfies   following
                condition:  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
    EpsX    -   >=0
                The  subroutine  finishes its work if exploratory steepest
                descent  step  on  k+1-th iteration  satisfies   following
                condition:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinQPSetScale()
    MaxOuterIts-maximum number of OUTER iterations.  One  outer  iteration
                includes some amount of CG iterations (from 5 to  ~N)  and
                one or several (usually small amount) Newton steps.  Thus,
                one outer iteration has high cost, but can greatly  reduce
                funcation value.
                Use 0 if you do not want to limit number of outer iterations.
    UseNewton-  use Newton phase or not:
                * Newton phase improves performance of  positive  definite
                  dense problems (about 2 times improvement can be observed)
                * can result in some performance penalty  on  semidefinite
                  or slightly negative definite  problems  -  each  Newton
                  phase will bring no improvement (Cholesky failure),  but
                  still will require computational time.
                * if you doubt, you can turn off this  phase  -  optimizer
                  will retain its most of its high speed.

IT IS VERY IMPORTANT TO CALL MinQPSetScale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS STOPPING CRITERIA ARE SCALE-DEPENDENT!

Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead
to automatic stopping criterion selection (presently it is  small    step
length, but it may change in the future versions of ALGLIB).

  -- ALGLIB --
     Copyright 22.05.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetalgoquickqp(
    minqpstate state,
    double epsg,
    double epsf,
    double epsx,
    ae_int_t maxouterits,
    bool usenewton,
    const xparams _params = alglib::xdefault);

`minqpsetbc` function

/*************************************************************************
This function sets box constraints for QP solver

Box constraints are inactive by default (after  initial  creation).  After
being  set,  they  are  preserved until explicitly turned off with another
SetBC() call.

All QP solvers may handle box constraints.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF (latter is recommended because
                it will allow solver to use better algorithm).
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF (latter is recommended because
                it will allow solver to use better algorithm).

NOTE: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetbc(
    minqpstate state,
    real_1d_array bndl,
    real_1d_array bndu,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minqpsetlc` function

/*************************************************************************
This function sets dense linear constraints for QP optimizer.

This  function  overrides  results  of  previous  calls  to  minqpsetlc(),
minqpsetlcsparse() and minqpsetlcmixed().  After  call  to  this  function
sparse constraints are dropped, and you have only those constraints  which
were specified in the present call.

If you want  to  specify  mixed  (with  dense  and  sparse  terms)  linear
constraints, you should call minqpsetlcmixed().

SUPPORT BY QP SOLVERS:

Following QP solvers can handle dense linear constraints:
* BLEIC-QP          -   handles them  with  high  precision,  but  may  be
                        inefficient for problems with hundreds of constraints
* Dense-AUL-QP      -   handles them with moderate precision (approx. 10^-6),
                        may efficiently handle thousands of constraints.

Following QP solvers can NOT handle dense linear constraints:
* QuickQP           -   can not handle general linear constraints

INPUT PARAMETERS:
    State   -   structure previously allocated with MinQPCreate call.
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

NOTE 1: linear (non-bound) constraints are satisfied only approximately  -
        there always exists some violation due  to  numerical  errors  and
        algorithmic limitations (BLEIC-QP solver is most  precise,  AUL-QP
        solver is less precise).

  -- ALGLIB --
     Copyright 19.06.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetlc(
    minqpstate state,
    real_2d_array c,
    integer_1d_array ct,
    const xparams _params = alglib::xdefault);
void alglib::minqpsetlc(
    minqpstate state,
    real_2d_array c,
    integer_1d_array ct,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minqpsetlcmixed` function

/*************************************************************************
This function sets mixed linear constraints, which include a set of  dense
rows, and a set of sparse rows.

This  function  overrides  results  of  previous  calls  to  minqpsetlc(),
minqpsetlcsparse() and minqpsetlcmixed().

This function may be useful if constraint matrix includes large number  of
both types of rows - dense and sparse. If you have just a few sparse rows,
you  may  represent  them  in  dense  format  without loosing performance.
Similarly, if you have just a few dense rows, you may store them in sparse
format with almost same performance.

SUPPORT BY QP SOLVERS:

Following QP solvers can handle mixed dense/sparse linear constraints:
* BLEIC-QP          -   handles them  with  high  precision,  but can  not
                        utilize their sparsity - sparse constraint  matrix
                        is silently converted to dense  format.  Thus,  it
                        may be inefficient for problems with  hundreds  of
                        constraints.
* Dense-AUL-QP      -   although this solver uses dense linear algebra  to
                        calculate   Cholesky   preconditioner,   it    may
                        efficiently  handle  sparse  constraints.  It  may
                        solve problems  with  hundreds  and  thousands  of
                        constraints. The only drawback is  that  precision
                        of constraint handling is typically within 1E-4...
                        ..1E-6 range.

Following QP solvers can NOT handle mixed linear constraints:
* QuickQP           -   can not handle general linear constraints at all

INPUT PARAMETERS:
    State   -   structure previously allocated with MinQPCreate call.
    DenseC  -   dense linear constraints, array[K,N+1].
                Each row of DenseC represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of DenseC (including right part) must be finite.
    DenseCT -   type of constraints, array[K]:
                * if DenseCT[i]>0, then I-th constraint is DenseC[i,*]*x >= DenseC[i,n+1]
                * if DenseCT[i]=0, then I-th constraint is DenseC[i,*]*x  = DenseC[i,n+1]
                * if DenseCT[i]<0, then I-th constraint is DenseC[i,*]*x <= DenseC[i,n+1]
    DenseK  -   number of equality/inequality constraints, DenseK>=0
    SparseC -   linear  constraints,  sparse  matrix  with  dimensions  at
                least [SparseK,N+1]. If matrix has  larger  size,  only  leading
                SPARSEKx(N+1) rectangle is used.
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    SparseCT-   type of sparse constraints, array[K]:
                * if SparseCT[i]>0, then I-th constraint is SparseC[i,*]*x >= SparseC[i,n+1]
                * if SparseCT[i]=0, then I-th constraint is SparseC[i,*]*x  = SparseC[i,n+1]
                * if SparseCT[i]<0, then I-th constraint is SparseC[i,*]*x <= SparseC[i,n+1]
    SparseK -   number of sparse equality/inequality constraints, K>=0

NOTE 1: linear (non-bound) constraints are satisfied only approximately  -
        there always exists some violation due  to  numerical  errors  and
        algorithmic limitations (BLEIC-QP solver is most  precise,  AUL-QP
        solver is less precise).

  -- ALGLIB --
     Copyright 22.08.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetlcmixed(
    minqpstate state,
    real_2d_array densec,
    integer_1d_array densect,
    ae_int_t densek,
    sparsematrix sparsec,
    integer_1d_array sparsect,
    ae_int_t sparsek,
    const xparams _params = alglib::xdefault);

`minqpsetlcsparse` function

/*************************************************************************
This function sets sparse linear constraints for QP optimizer.

This  function  overrides  results  of  previous  calls  to  minqpsetlc(),
minqpsetlcsparse() and minqpsetlcmixed().  After  call  to  this  function
dense constraints are dropped, and you have only those  constraints  which
were specified in the present call.

If you want  to  specify  mixed  (with  dense  and  sparse  terms)  linear
constraints, you should call minqpsetlcmixed().

SUPPORT BY QP SOLVERS:

Following QP solvers can handle sparse linear constraints:
* BLEIC-QP          -   handles them  with  high  precision,  but can  not
                        utilize their sparsity - sparse constraint  matrix
                        is silently converted to dense  format.  Thus,  it
                        may be inefficient for problems with  hundreds  of
                        constraints.
* Dense-AUL-QP      -   although this solver uses dense linear algebra  to
                        calculate   Cholesky   preconditioner,   it    may
                        efficiently  handle  sparse  constraints.  It  may
                        solve problems  with  hundreds  and  thousands  of
                        constraints. The only drawback is  that  precision
                        of constraint handling is typically within 1E-4...
                        ..1E-6 range.

Following QP solvers can NOT handle sparse linear constraints:
* QuickQP           -   can not handle general linear constraints

INPUT PARAMETERS:
    State   -   structure previously allocated with MinQPCreate call.
    C       -   linear  constraints,  sparse  matrix  with  dimensions  at
                least [K,N+1]. If matrix has  larger  size,  only  leading
                Kx(N+1) rectangle is used.
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0

NOTE 1: linear (non-bound) constraints are satisfied only approximately  -
        there always exists some violation due  to  numerical  errors  and
        algorithmic limitations (BLEIC-QP solver is most  precise,  AUL-QP
        solver is less precise).

  -- ALGLIB --
     Copyright 22.08.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetlcsparse(
    minqpstate state,
    sparsematrix c,
    integer_1d_array ct,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

`minqpsetlinearterm` function

/*************************************************************************
This function sets linear term for QP solver.

By default, linear term is zero.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    B       -   linear term, array[N].

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetlinearterm(
    minqpstate state,
    real_1d_array b,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`minqpsetorigin` function

/*************************************************************************
This  function sets origin for QP solver. By default, following QP program
is solved:

    min(0.5*x'*A*x+b'*x)

This function allows to solve different problem:

    min(0.5*(x-x_origin)'*A*(x-x_origin)+b'*(x-x_origin))

Specification of non-zero origin affects function being minimized, but not
constraints. Box and  linear  constraints  are  still  calculated  without
origin.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    XOrigin -   origin, array[N].

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetorigin(
    minqpstate state,
    real_1d_array xorigin,
    const xparams _params = alglib::xdefault);

`minqpsetquadraticterm` function

/*************************************************************************
This  function  sets  dense  quadratic  term  for  QP solver. By  default,
quadratic term is zero.

SUPPORT BY QP SOLVERS:

Dense quadratic term can be handled by following QP solvers:
* QuickQP
* BLEIC-QP
* Dense-AUL-QP

IMPORTANT:

This solver minimizes following  function:
    f(x) = 0.5*x'*A*x + b'*x.
Note that quadratic term has 0.5 before it. So if  you  want  to  minimize
    f(x) = x^2 + x
you should rewrite your problem as follows:
    f(x) = 0.5*(2*x^2) + x
and your matrix A will be equal to [[2.0]], not to [[1.0]]

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    A       -   matrix, array[N,N]
    IsUpper -   (optional) storage type:
                * if True, symmetric matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't used
                * if False, symmetric matrix  A  is  given  by  its lower
                  triangle, and the upper triangle isn't used
                * if not given, both lower and upper  triangles  must  be
                  filled.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetquadraticterm(
    minqpstate state,
    real_2d_array a,
    const xparams _params = alglib::xdefault);
void alglib::minqpsetquadraticterm(
    minqpstate state,
    real_2d_array a,
    bool isupper,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minqpsetquadratictermsparse` function

/*************************************************************************
This  function  sets  sparse  quadratic  term  for  QP solver. By default,
quadratic  term  is  zero.  This  function  overrides  previous  calls  to
minqpsetquadraticterm() or minqpsetquadratictermsparse().

SUPPORT BY QP SOLVERS:

Sparse quadratic term can be handled by following QP solvers:
* QuickQP
* BLEIC-QP
* Dense-AUL-QP (internally converts sparse matrix to dense format)

IMPORTANT:

This solver minimizes following  function:
    f(x) = 0.5*x'*A*x + b'*x.
Note that quadratic term has 0.5 before it. So if  you  want  to  minimize
    f(x) = x^2 + x
you should rewrite your problem as follows:
    f(x) = 0.5*(2*x^2) + x
and your matrix A will be equal to [[2.0]], not to [[1.0]]

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    A       -   matrix, array[N,N]
    IsUpper -   (optional) storage type:
                * if True, symmetric matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't used
                * if False, symmetric matrix  A  is  given  by  its lower
                  triangle, and the upper triangle isn't used
                * if not given, both lower and upper  triangles  must  be
                  filled.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetquadratictermsparse(
    minqpstate state,
    sparsematrix a,
    bool isupper,
    const xparams _params = alglib::xdefault);

Examples: [1]

`minqpsetscale` function

/*************************************************************************
This function sets scaling coefficients.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison  with  tolerances)  and  as
preconditioner.

Scale of the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the
   function

If you do not know how to choose scales of your variables, you can:
* read www.alglib.net/optimization/scaling.php article
* use minqpsetscaleautodiag(), which calculates scale  using  diagonal  of
  the  quadratic  term:  S  is  set to 1/sqrt(diag(A)), which works well
  sometimes.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetscale(
    minqpstate state,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

`minqpsetscaleautodiag` function

/*************************************************************************
This function sets automatic evaluation of variable scaling.

IMPORTANT: this function works only for  matrices  with positive  diagonal
           elements! Zero or negative elements will  result  in  -9  error
           code  being  returned.  Specify  scale  vector  manually   with
           minqpsetscale() in such cases.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison  with  tolerances)  and  as
preconditioner.

The  best  way  to  set  scaling  is  to manually specify variable scales.
However, sometimes you just need quick-and-dirty solution  -  either  when
you perform fast prototyping, or when you know your problem well  and  you
are 100% sure that this quick solution is robust enough in your case.

One such solution is to evaluate scale of I-th variable as 1/Sqrt(A[i,i]),
where A[i,i] is an I-th diagonal element of the quadratic term.

Such approach works well sometimes, but you have to be careful here.

INPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 26.12.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetscaleautodiag(
    minqpstate state,
    const xparams _params = alglib::xdefault);

`minqpsetstartingpoint` function

/*************************************************************************
This function sets starting point for QP solver. It is useful to have
good initial approximation to the solution, because it will increase
speed of convergence and identification of active constraints.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    X       -   starting point, array[N].

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::minqpsetstartingpoint(
    minqpstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

minqp_d_bc1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
    // subject to bound constraints 0<=x0<=2.5, 0<=x1<=2.5
    //
    // Exact solution is [x0,x1] = [2.5,2]
    //
    // We provide algorithm with starting point. With such small problem good starting
    // point is not really necessary, but with high-dimensional problem it can save us
    // a lot of time.
    //
    // Several QP solvers are tried: QuickQP, BLEIC, DENSE-AUL.
    //
    // IMPORTANT: this solver minimizes  following  function:
    //     f(x) = 0.5*x'*A*x + b'*x.
    // Note that quadratic term has 0.5 before it. So if you want to minimize
    // quadratic function, you should rewrite it in such way that quadratic term
    // is multiplied by 0.5 too.
    // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as 
    //     f(x) = 0.5*(2*x0^2+2*x1^2) + ....
    // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
    //
    real_2d_array a = "[[2,0],[0,2]]";
    real_1d_array b = "[-6,-4]";
    real_1d_array x0 = "[0,1]";
    real_1d_array s = "[1,1]";
    real_1d_array bndl = "[0.0,0.0]";
    real_1d_array bndu = "[2.5,2.5]";
    real_1d_array x;
    minqpstate state;
    minqpreport rep;

    // create solver, set quadratic/linear terms
    minqpcreate(2, state);
    minqpsetquadraticterm(state, a);
    minqpsetlinearterm(state, b);
    minqpsetstartingpoint(state, x0);
    minqpsetbc(state, bndl, bndu);

    // Set scale of the parameters.
    // It is strongly recommended that you set scale of your variables.
    // Knowing their scales is essential for evaluation of stopping criteria
    // and for preconditioning of the algorithm steps.
    // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
    //
    // NOTE: for convex problems you may try using minqpsetscaleautodiag()
    //       which automatically determines variable scales.
    minqpsetscale(state, s);

    //
    // Solve problem with QuickQP solver.
    //
    // This solver is intended for medium and large-scale problems with box
    // constraints (general linear constraints are not supported).
    //
    // Default stopping criteria are used, Newton phase is active.
    //
    minqpsetalgoquickqp(state, 0.0, 0.0, 0.0, 0, true);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2.5,2]

    //
    // Solve problem with BLEIC-based QP solver.
    //
    // This solver is intended for problems with moderate (up to 50) number
    // of general linear constraints and unlimited number of box constraints.
    //
    // Default stopping criteria are used.
    //
    minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2.5,2]

    //
    // Solve problem with DENSE-AUL solver.
    //
    // This solver is optimized for problems with up to several thousands of
    // variables and large amount of general linear constraints. Problems with
    // less than 50 general linear constraints can be efficiently solved with
    // BLEIC, problems with box-only constraints can be solved with QuickQP.
    // However, DENSE-AUL will work in any (including unconstrained) case.
    //
    // Default stopping criteria are used.
    //
    minqpsetalgodenseaul(state, 1.0e-9, 1.0e+4, 5);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2.5,2]
    return 0;
}

minqp_d_lc1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
    // subject to linear constraint x0+x1<=2
    //
    // Exact solution is [x0,x1] = [1.5,0.5]
    //
    // IMPORTANT: this solver minimizes  following  function:
    //     f(x) = 0.5*x'*A*x + b'*x.
    // Note that quadratic term has 0.5 before it. So if you want to minimize
    // quadratic function, you should rewrite it in such way that quadratic term
    // is multiplied by 0.5 too.
    // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as 
    //     f(x) = 0.5*(2*x0^2+2*x1^2) + ....
    // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
    //
    real_2d_array a = "[[2,0],[0,2]]";
    real_1d_array b = "[-6,-4]";
    real_1d_array s = "[1,1]";
    real_2d_array c = "[[1.0,1.0,2.0]]";
    integer_1d_array ct = "[-1]";
    real_1d_array x;
    minqpstate state;
    minqpreport rep;

    // create solver, set quadratic/linear terms
    minqpcreate(2, state);
    minqpsetquadraticterm(state, a);
    minqpsetlinearterm(state, b);
    minqpsetlc(state, c, ct);

    // Set scale of the parameters.
    // It is strongly recommended that you set scale of your variables.
    // Knowing their scales is essential for evaluation of stopping criteria
    // and for preconditioning of the algorithm steps.
    // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
    //
    // NOTE: for convex problems you may try using minqpsetscaleautodiag()
    //       which automatically determines variable scales.
    minqpsetscale(state, s);

    //
    // Solve problem with BLEIC-based QP solver.
    //
    // This solver is intended for problems with moderate (up to 50) number
    // of general linear constraints and unlimited number of box constraints.
    //
    // Default stopping criteria are used.
    //
    minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(1).c_str()); // EXPECTED: [1.500,0.500]

    //
    // Solve problem with DENSE-AUL solver.
    //
    // This solver is optimized for problems with up to several thousands of
    // variables and large amount of general linear constraints. Problems with
    // less than 50 general linear constraints can be efficiently solved with
    // BLEIC, problems with box-only constraints can be solved with QuickQP.
    // However, DENSE-AUL will work in any (including unconstrained) case.
    //
    // Default stopping criteria are used.
    //
    minqpsetalgodenseaul(state, 1.0e-9, 1.0e+4, 5);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(1).c_str()); // EXPECTED: [1.500,0.500]

    //
    // Solve problem with QuickQP solver.
    //
    // This solver is intended for medium and large-scale problems with box
    // constraints, and...
    //
    // ...Oops! It does not support general linear constraints, -5 returned as completion code!
    //
    minqpsetalgoquickqp(state, 0.0, 0.0, 0.0, 0, true);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: -5
    return 0;
}

minqp_d_nonconvex example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of nonconvex function
    //     F(x0,x1) = -(x0^2+x1^2)
    // subject to constraints x0,x1 in [1.0,2.0]
    // Exact solution is [x0,x1] = [2,2].
    //
    // Non-convex problems are harded to solve than convex ones, and they
    // may have more than one local minimum. However, ALGLIB solves may deal
    // with such problems (altough they do not guarantee convergence to
    // global minimum).
    //
    // IMPORTANT: this solver minimizes  following  function:
    //     f(x) = 0.5*x'*A*x + b'*x.
    // Note that quadratic term has 0.5 before it. So if you want to minimize
    // quadratic function, you should rewrite it in such way that quadratic term
    // is multiplied by 0.5 too.
    //
    // For example, our function is f(x)=-(x0^2+x1^2), but we rewrite it as 
    //     f(x) = 0.5*(-2*x0^2-2*x1^2)
    // and pass diag(-2,-2) as quadratic term - NOT diag(-1,-1)!
    //
    real_2d_array a = "[[-2,0],[0,-2]]";
    real_1d_array x0 = "[1,1]";
    real_1d_array s = "[1,1]";
    real_1d_array bndl = "[1.0,1.0]";
    real_1d_array bndu = "[2.0,2.0]";
    real_1d_array x;
    minqpstate state;
    minqpreport rep;

    // create solver, set quadratic/linear terms, constraints
    minqpcreate(2, state);
    minqpsetquadraticterm(state, a);
    minqpsetstartingpoint(state, x0);
    minqpsetbc(state, bndl, bndu);

    // Set scale of the parameters.
    // It is strongly recommended that you set scale of your variables.
    // Knowing their scales is essential for evaluation of stopping criteria
    // and for preconditioning of the algorithm steps.
    // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
    //
    // NOTE: there also exists minqpsetscaleautodiag() function
    //       which automatically determines variable scales; however,
    //       it does NOT work for non-convex problems.
    minqpsetscale(state, s);

    //
    // Solve problem with BLEIC-based QP solver.
    //
    // This solver is intended for problems with moderate (up to 50) number
    // of general linear constraints and unlimited number of box constraints.
    //
    // It may solve non-convex problems as long as they are bounded from
    // below under constraints.
    //
    // Default stopping criteria are used.
    //
    minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2,2]

    //
    // Solve problem with DENSE-AUL solver.
    //
    // This solver is optimized for problems with up to several thousands of
    // variables and large amount of general linear constraints. Problems with
    // less than 50 general linear constraints can be efficiently solved with
    // BLEIC, problems with box-only constraints can be solved with QuickQP.
    // However, DENSE-AUL will work in any (including unconstrained) case.
    //
    // Algorithm convergence is guaranteed only for convex case, but you may
    // expect that it will work for non-convex problems too (because near the
    // solution they are locally convex).
    //
    // Default stopping criteria are used.
    //
    minqpsetalgodenseaul(state, 1.0e-9, 1.0e+4, 5);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2,2]

    // Hmm... this problem is bounded from below (has solution) only under constraints.
    // What it we remove them?
    //
    // You may see that BLEIC algorithm detects unboundedness of the problem, 
    // -4 is returned as completion code. However, DENSE-AUL is unable to detect
    // such situation and it will cycle forever (we do not test it here).
    real_1d_array nobndl = "[-inf,-inf]";
    real_1d_array nobndu = "[+inf,+inf]";
    minqpsetbc(state, nobndl, nobndu);
    minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: -4
    return 0;
}

minqp_d_u1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
    //
    // Exact solution is [x0,x1] = [3,2]
    //
    // We provide algorithm with starting point, although in this case
    // (dense matrix, no constraints) it can work without such information.
    //
    // Several QP solvers are tried: QuickQP, BLEIC, DENSE-AUL.
    //
    // IMPORTANT: this solver minimizes  following  function:
    //     f(x) = 0.5*x'*A*x + b'*x.
    // Note that quadratic term has 0.5 before it. So if you want to minimize
    // quadratic function, you should rewrite it in such way that quadratic term
    // is multiplied by 0.5 too.
    //
    // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as 
    //     f(x) = 0.5*(2*x0^2+2*x1^2) + .... 
    // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
    //
    real_2d_array a = "[[2,0],[0,2]]";
    real_1d_array b = "[-6,-4]";
    real_1d_array x0 = "[0,1]";
    real_1d_array s = "[1,1]";
    real_1d_array x;
    minqpstate state;
    minqpreport rep;

    // create solver, set quadratic/linear terms
    minqpcreate(2, state);
    minqpsetquadraticterm(state, a);
    minqpsetlinearterm(state, b);
    minqpsetstartingpoint(state, x0);

    // Set scale of the parameters.
    // It is strongly recommended that you set scale of your variables.
    // Knowing their scales is essential for evaluation of stopping criteria
    // and for preconditioning of the algorithm steps.
    // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
    //
    // NOTE: for convex problems you may try using minqpsetscaleautodiag()
    //       which automatically determines variable scales.
    minqpsetscale(state, s);

    //
    // Solve problem with QuickQP solver.
    //
    // This solver is intended for medium and large-scale problems with box
    // constraints (general linear constraints are not supported), but it can
    // also be efficiently used on unconstrained problems.
    //
    // Default stopping criteria are used, Newton phase is active.
    //
    minqpsetalgoquickqp(state, 0.0, 0.0, 0.0, 0, true);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]

    //
    // Solve problem with BLEIC-based QP solver.
    //
    // This solver is intended for problems with moderate (up to 50) number
    // of general linear constraints and unlimited number of box constraints.
    // Of course, unconstrained problems can be solved too.
    //
    // Default stopping criteria are used.
    //
    minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]

    //
    // Solve problem with DENSE-AUL solver.
    //
    // This solver is optimized for problems with up to several thousands of
    // variables and large amount of general linear constraints. Problems with
    // less than 50 general linear constraints can be efficiently solved with
    // BLEIC, problems with box-only constraints can be solved with QuickQP.
    // However, DENSE-AUL will work in any (including unconstrained) case.
    //
    // Default stopping criteria are used.
    //
    minqpsetalgodenseaul(state, 1.0e-9, 1.0e+4, 5);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]
    return 0;
}

minqp_d_u2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1,
    // with quadratic term given by sparse matrix structure.
    //
    // Exact solution is [x0,x1] = [3,2]
    //
    // We provide algorithm with starting point, although in this case
    // (dense matrix, no constraints) it can work without such information.
    //
    // IMPORTANT: this solver minimizes  following  function:
    //     f(x) = 0.5*x'*A*x + b'*x.
    // Note that quadratic term has 0.5 before it. So if you want to minimize
    // quadratic function, you should rewrite it in such way that quadratic term
    // is multiplied by 0.5 too.
    //
    // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as 
    //     f(x) = 0.5*(2*x0^2+2*x1^2) + ....
    // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
    //
    sparsematrix a;
    real_1d_array b = "[-6,-4]";
    real_1d_array x0 = "[0,1]";
    real_1d_array s = "[1,1]";
    real_1d_array x;
    minqpstate state;
    minqpreport rep;

    // initialize sparsematrix structure
    sparsecreate(2, 2, 0, a);
    sparseset(a, 0, 0, 2.0);
    sparseset(a, 1, 1, 2.0);

    // create solver, set quadratic/linear terms
    minqpcreate(2, state);
    minqpsetquadratictermsparse(state, a, true);
    minqpsetlinearterm(state, b);
    minqpsetstartingpoint(state, x0);

    // Set scale of the parameters.
    // It is strongly recommended that you set scale of your variables.
    // Knowing their scales is essential for evaluation of stopping criteria
    // and for preconditioning of the algorithm steps.
    // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
    //
    // NOTE: for convex problems you may try using minqpsetscaleautodiag()
    //       which automatically determines variable scales.
    minqpsetscale(state, s);

    //
    // Solve problem with BLEIC-based QP solver.
    //
    // This solver is intended for problems with moderate (up to 50) number
    // of general linear constraints and unlimited number of box constraints.
    // It also supports sparse problems.
    //
    // Default stopping criteria are used.
    //
    minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
    minqpoptimize(state);
    minqpresults(state, x, rep);
    printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]
    return 0;
}

`mlpbase` subpackage

Classes

modelerrors
multilayerperceptron

Functions

mlpactivationfunction
mlpallerrorssparsesubset
mlpallerrorssubset
mlpavgce
mlpavgcesparse
mlpavgerror
mlpavgerrorsparse
mlpavgrelerror
mlpavgrelerrorsparse
mlpclserror
mlpcopy
mlpcopytunableparameters
mlpcreate0
mlpcreate1
mlpcreate2
mlpcreateb0
mlpcreateb1
mlpcreateb2
mlpcreatec0
mlpcreatec1
mlpcreatec2
mlpcreater0
mlpcreater1
mlpcreater2
mlperror
mlperrorn
mlperrorsparse
mlperrorsparsesubset
mlperrorsubset
mlpgetinputscaling
mlpgetinputscount
mlpgetlayerscount
mlpgetlayersize
mlpgetneuroninfo
mlpgetoutputscaling
mlpgetoutputscount
mlpgetweight
mlpgetweightscount
mlpgrad
mlpgradbatch
mlpgradbatchsparse
mlpgradbatchsparsesubset
mlpgradbatchsubset
mlpgradn
mlpgradnbatch
mlphessianbatch
mlphessiannbatch
mlpinitpreprocessor
mlpissoftmax
mlpprocess
mlpprocessi
mlpproperties
mlprandomize
mlprandomizefull
mlprelclserror
mlprelclserrorsparse
mlprmserror
mlprmserrorsparse
mlpserialize
mlpsetinputscaling
mlpsetneuroninfo
mlpsetoutputscaling
mlpsetweight
mlpunserialize

Examples

`modelerrors` class

/*************************************************************************
Model's errors:
    * RelCLSError   -   fraction of misclassified cases.
    * AvgCE         -   acerage cross-entropy
    * RMSError      -   root-mean-square error
    * AvgError      -   average error
    * AvgRelError   -   average relative error

NOTE 1: RelCLSError/AvgCE are zero on regression problems.

NOTE 2: on classification problems  RMSError/AvgError/AvgRelError  contain
        errors in prediction of posterior probabilities
*************************************************************************/
class modelerrors
{
    double               relclserror;
    double               avgce;
    double               rmserror;
    double               avgerror;
    double               avgrelerror;
};

`multilayerperceptron` class

/*************************************************************************

*************************************************************************/
class multilayerperceptron
{
};

`mlpactivationfunction` function

/*************************************************************************
Neural network activation function

INPUT PARAMETERS:
    NET         -   neuron input
    K           -   function index (zero for linear function)

OUTPUT PARAMETERS:
    F           -   function
    DF          -   its derivative
    D2F         -   its second derivative

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpactivationfunction(
    double net,
    ae_int_t k,
    double& f,
    double& df,
    double& d2f,
    const xparams _params = alglib::xdefault);

`mlpallerrorssparsesubset` function

/*************************************************************************
Calculation of all types of errors on subset of dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset given by sparse matrix;
                one sample = one row;
                first NIn columns contain inputs,
                next NOut columns - desired outputs.
    SetSize -   real size of XY, SetSize>=0;
    Subset  -   subset of SubsetSize elements, array[SubsetSize];
    SubsetSize- number of elements in Subset[] array:
                * if SubsetSize>0, rows of XY with indices Subset[0]...
                  ...Subset[SubsetSize-1] are processed
                * if SubsetSize=0, zeros are returned
                * if SubsetSize<0, entire dataset is  processed;  Subset[]
                  array is ignored in this case.

OUTPUT PARAMETERS:
    Rep     -   it contains all type of errors.


  -- ALGLIB --
     Copyright 04.09.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpallerrorssparsesubset(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t setsize,
    integer_1d_array subset,
    ae_int_t subsetsize,
    modelerrors& rep,
    const xparams _params = alglib::xdefault);

`mlpallerrorssubset` function

/*************************************************************************
Calculation of all types of errors on subset of dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset; one sample = one row;
                first NIn columns contain inputs,
                next NOut columns - desired outputs.
    SetSize -   real size of XY, SetSize>=0;
    Subset  -   subset of SubsetSize elements, array[SubsetSize];
    SubsetSize- number of elements in Subset[] array:
                * if SubsetSize>0, rows of XY with indices Subset[0]...
                  ...Subset[SubsetSize-1] are processed
                * if SubsetSize=0, zeros are returned
                * if SubsetSize<0, entire dataset is  processed;  Subset[]
                  array is ignored in this case.

OUTPUT PARAMETERS:
    Rep     -   it contains all type of errors.

  -- ALGLIB --
     Copyright 04.09.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpallerrorssubset(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t setsize,
    integer_1d_array subset,
    ae_int_t subsetsize,
    modelerrors& rep,
    const xparams _params = alglib::xdefault);

`mlpavgce` function

/*************************************************************************
Average cross-entropy  (in bits  per element) on the test set.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
CrossEntropy/(NPoints*LN(2)).
Zero if network solves regression task.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 08.01.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpavgce(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpavgcesparse` function

/*************************************************************************
Average  cross-entropy  (in bits  per element)  on the  test set  given by
sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0.

RESULT:
CrossEntropy/(NPoints*LN(2)).
Zero if network solves regression task.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 9.08.2012 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpavgcesparse(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpavgerror` function

/*************************************************************************
Average absolute error on the test set.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
Its meaning for regression task is obvious. As for classification task, it
means average error when estimating posterior probabilities.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 11.03.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpavgerror(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpavgerrorsparse` function

/*************************************************************************
Average absolute error on the test set given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0.

RESULT:
Its meaning for regression task is obvious. As for classification task, it
means average error when estimating posterior probabilities.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 09.08.2012 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpavgerrorsparse(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpavgrelerror` function

/*************************************************************************
Average relative error on the test set.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
Its meaning for regression task is obvious. As for classification task, it
means  average  relative  error  when  estimating posterior probability of
belonging to the correct class.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 11.03.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpavgrelerror(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpavgrelerrorsparse` function

/*************************************************************************
Average relative error on the test set given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0.

RESULT:
Its meaning for regression task is obvious. As for classification task, it
means  average  relative  error  when  estimating posterior probability of
belonging to the correct class.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 09.08.2012 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpavgrelerrorsparse(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpclserror` function

/*************************************************************************
Classification error of the neural network on dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
    classification error (number of misclassified cases)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::mlpclserror(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpcopy` function

/*************************************************************************
Copying of neural network

INPUT PARAMETERS:
    Network1 -   original

OUTPUT PARAMETERS:
    Network2 -   copy

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcopy(
    multilayerperceptron network1,
    multilayerperceptron& network2,
    const xparams _params = alglib::xdefault);

`mlpcopytunableparameters` function

/*************************************************************************
This function copies tunable  parameters (weights/means/sigmas)  from  one
network to another with same architecture. It  performs  some  rudimentary
checks that architectures are same, and throws exception if check fails.

It is intended for fast copying of states between two  network  which  are
known to have same geometry.

INPUT PARAMETERS:
    Network1 -   source, must be correctly initialized
    Network2 -   target, must have same architecture

OUTPUT PARAMETERS:
    Network2 -   network state is copied from source to target

  -- ALGLIB --
     Copyright 20.06.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcopytunableparameters(
    multilayerperceptron network1,
    multilayerperceptron network2,
    const xparams _params = alglib::xdefault);

`mlpcreate0` function

/*************************************************************************
Creates  neural  network  with  NIn  inputs,  NOut outputs, without hidden
layers, with linear output layer. Network weights are  filled  with  small
random values.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreate0(
    ae_int_t nin,
    ae_int_t nout,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreate1` function

/*************************************************************************
Same  as  MLPCreate0,  but  with  one  hidden  layer  (NHid  neurons) with
non-linear activation function. Output layer is linear.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreate1(
    ae_int_t nin,
    ae_int_t nhid,
    ae_int_t nout,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreate2` function

/*************************************************************************
Same as MLPCreate0, but with two hidden layers (NHid1 and  NHid2  neurons)
with non-linear activation function. Output layer is linear.
 $ALL

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreate2(
    ae_int_t nin,
    ae_int_t nhid1,
    ae_int_t nhid2,
    ae_int_t nout,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreateb0` function

/*************************************************************************
Creates  neural  network  with  NIn  inputs,  NOut outputs, without hidden
layers with non-linear output layer. Network weights are filled with small
random values.

Activation function of the output layer takes values:

    (B, +INF), if D>=0

or

    (-INF, B), if D<0.


  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreateb0(
    ae_int_t nin,
    ae_int_t nout,
    double b,
    double d,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreateb1` function

/*************************************************************************
Same as MLPCreateB0 but with non-linear hidden layer.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreateb1(
    ae_int_t nin,
    ae_int_t nhid,
    ae_int_t nout,
    double b,
    double d,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreateb2` function

/*************************************************************************
Same as MLPCreateB0 but with two non-linear hidden layers.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreateb2(
    ae_int_t nin,
    ae_int_t nhid1,
    ae_int_t nhid2,
    ae_int_t nout,
    double b,
    double d,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreatec0` function

/*************************************************************************
Creates classifier network with NIn  inputs  and  NOut  possible  classes.
Network contains no hidden layers and linear output  layer  with  SOFTMAX-
normalization  (so  outputs  sums  up  to  1.0  and  converge to posterior
probabilities).

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreatec0(
    ae_int_t nin,
    ae_int_t nout,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreatec1` function

/*************************************************************************
Same as MLPCreateC0, but with one non-linear hidden layer.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreatec1(
    ae_int_t nin,
    ae_int_t nhid,
    ae_int_t nout,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreatec2` function

/*************************************************************************
Same as MLPCreateC0, but with two non-linear hidden layers.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreatec2(
    ae_int_t nin,
    ae_int_t nhid1,
    ae_int_t nhid2,
    ae_int_t nout,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreater0` function

/*************************************************************************
Creates  neural  network  with  NIn  inputs,  NOut outputs, without hidden
layers with non-linear output layer. Network weights are filled with small
random values. Activation function of the output layer takes values [A,B].

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreater0(
    ae_int_t nin,
    ae_int_t nout,
    double a,
    double b,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreater1` function

/*************************************************************************
Same as MLPCreateR0, but with non-linear hidden layer.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreater1(
    ae_int_t nin,
    ae_int_t nhid,
    ae_int_t nout,
    double a,
    double b,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlpcreater2` function

/*************************************************************************
Same as MLPCreateR0, but with two non-linear hidden layers.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreater2(
    ae_int_t nin,
    ae_int_t nhid1,
    ae_int_t nhid2,
    ae_int_t nout,
    double a,
    double b,
    multilayerperceptron& network,
    const xparams _params = alglib::xdefault);

`mlperror` function

/*************************************************************************
Error of the neural network on dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
    sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::mlperror(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlperrorn` function

/*************************************************************************
Natural error function for neural network, internal subroutine.

NOTE: this function is single-threaded. Unlike other  error  function,  it
receives no speed-up from being executed in SMP mode.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::mlperrorn(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t ssize,
    const xparams _params = alglib::xdefault);

`mlperrorsparse` function

/*************************************************************************
Error of the neural network on dataset given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0

RESULT:
    sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
double alglib::mlperrorsparse(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlperrorsparsesubset` function

/*************************************************************************
Error of the neural network on subset of sparse dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network   -     neural network;
    XY        -     training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    SetSize   -     real size of XY, SetSize>=0;
                    it is used when SubsetSize<0;
    Subset    -     subset of SubsetSize elements, array[SubsetSize];
    SubsetSize-     number of elements in Subset[] array:
                    * if SubsetSize>0, rows of XY with indices Subset[0]...
                      ...Subset[SubsetSize-1] are processed
                    * if SubsetSize=0, zeros are returned
                    * if SubsetSize<0, entire dataset is  processed;  Subset[]
                      array is ignored in this case.

RESULT:
    sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.09.2012 by Bochkanov Sergey
*************************************************************************/
double alglib::mlperrorsparsesubset(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t setsize,
    integer_1d_array subset,
    ae_int_t subsetsize,
    const xparams _params = alglib::xdefault);

`mlperrorsubset` function

/*************************************************************************
Error of the neural network on subset of dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network   -     neural network;
    XY        -     training  set,  see  below  for  information  on   the
                    training set format;
    SetSize   -     real size of XY, SetSize>=0;
    Subset    -     subset of SubsetSize elements, array[SubsetSize];
    SubsetSize-     number of elements in Subset[] array:
                    * if SubsetSize>0, rows of XY with indices Subset[0]...
                      ...Subset[SubsetSize-1] are processed
                    * if SubsetSize=0, zeros are returned
                    * if SubsetSize<0, entire dataset is  processed;  Subset[]
                      array is ignored in this case.

RESULT:
    sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.09.2012 by Bochkanov Sergey
*************************************************************************/
double alglib::mlperrorsubset(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t setsize,
    integer_1d_array subset,
    ae_int_t subsetsize,
    const xparams _params = alglib::xdefault);

`mlpgetinputscaling` function

/*************************************************************************
This function returns offset/scaling coefficients for I-th input of the
network.

INPUT PARAMETERS:
    Network     -   network
    I           -   input index

OUTPUT PARAMETERS:
    Mean        -   mean term
    Sigma       -   sigma term, guaranteed to be nonzero.

I-th input is passed through linear transformation
    IN[i] = (IN[i]-Mean)/Sigma
before feeding to the network

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgetinputscaling(
    multilayerperceptron network,
    ae_int_t i,
    double& mean,
    double& sigma,
    const xparams _params = alglib::xdefault);

`mlpgetinputscount` function

/*************************************************************************
Returns number of inputs.

  -- ALGLIB --
     Copyright 19.10.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::mlpgetinputscount(
    multilayerperceptron network,
    const xparams _params = alglib::xdefault);

`mlpgetlayerscount` function

/*************************************************************************
This function returns total number of layers (including input, hidden and
output layers).

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::mlpgetlayerscount(
    multilayerperceptron network,
    const xparams _params = alglib::xdefault);

`mlpgetlayersize` function

/*************************************************************************
This function returns size of K-th layer.

K=0 corresponds to input layer, K=CNT-1 corresponds to output layer.

Size of the output layer is always equal to the number of outputs, although
when we have softmax-normalized network, last neuron doesn't have any
connections - it is just zero.

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::mlpgetlayersize(
    multilayerperceptron network,
    ae_int_t k,
    const xparams _params = alglib::xdefault);

`mlpgetneuroninfo` function

/*************************************************************************
This function returns information about Ith neuron of Kth layer

INPUT PARAMETERS:
    Network     -   network
    K           -   layer index
    I           -   neuron index (within layer)

OUTPUT PARAMETERS:
    FKind       -   activation function type (used by MLPActivationFunction())
                    this value is zero for input or linear neurons
    Threshold   -   also called offset, bias
                    zero for input neurons

NOTE: this function throws exception if layer or neuron with  given  index
do not exists.

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgetneuroninfo(
    multilayerperceptron network,
    ae_int_t k,
    ae_int_t i,
    ae_int_t& fkind,
    double& threshold,
    const xparams _params = alglib::xdefault);

`mlpgetoutputscaling` function

/*************************************************************************
This function returns offset/scaling coefficients for I-th output of the
network.

INPUT PARAMETERS:
    Network     -   network
    I           -   input index

OUTPUT PARAMETERS:
    Mean        -   mean term
    Sigma       -   sigma term, guaranteed to be nonzero.

I-th output is passed through linear transformation
    OUT[i] = OUT[i]*Sigma+Mean
before returning it to user. In case we have SOFTMAX-normalized network,
we return (Mean,Sigma)=(0.0,1.0).

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgetoutputscaling(
    multilayerperceptron network,
    ae_int_t i,
    double& mean,
    double& sigma,
    const xparams _params = alglib::xdefault);

`mlpgetoutputscount` function

/*************************************************************************
Returns number of outputs.

  -- ALGLIB --
     Copyright 19.10.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::mlpgetoutputscount(
    multilayerperceptron network,
    const xparams _params = alglib::xdefault);

`mlpgetweight` function

/*************************************************************************
This function returns information about connection from I0-th neuron of
K0-th layer to I1-th neuron of K1-th layer.

INPUT PARAMETERS:
    Network     -   network
    K0          -   layer index
    I0          -   neuron index (within layer)
    K1          -   layer index
    I1          -   neuron index (within layer)

RESULT:
    connection weight (zero for non-existent connections)

This function:
1. throws exception if layer or neuron with given index do not exists.
2. returns zero if neurons exist, but there is no connection between them

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpgetweight(
    multilayerperceptron network,
    ae_int_t k0,
    ae_int_t i0,
    ae_int_t k1,
    ae_int_t i1,
    const xparams _params = alglib::xdefault);

`mlpgetweightscount` function

/*************************************************************************
Returns number of weights.

  -- ALGLIB --
     Copyright 19.10.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::mlpgetweightscount(
    multilayerperceptron network,
    const xparams _params = alglib::xdefault);

`mlpgrad` function

/*************************************************************************
Gradient calculation

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    X       -   input vector, length of array must be at least NIn
    DesiredY-   desired outputs, length of array must be at least NOut
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgrad(
    multilayerperceptron network,
    real_1d_array x,
    real_1d_array desiredy,
    double& e,
    real_1d_array& grad,
    const xparams _params = alglib::xdefault);

`mlpgradbatch` function

/*************************************************************************
Batch gradient calculation for a set of inputs/outputs

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset in dense format; one sample = one row:
                * first NIn columns contain inputs,
                * for regression problem, next NOut columns store
                  desired outputs.
                * for classification problem, next column (just one!)
                  stores class number.
    SSize   -   number of elements in XY
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgradbatch(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t ssize,
    double& e,
    real_1d_array& grad,
    const xparams _params = alglib::xdefault);

`mlpgradbatchsparse` function

/*************************************************************************
Batch gradient calculation for a set  of inputs/outputs  given  by  sparse
matrices

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset in sparse format; one sample = one row:
                * MATRIX MUST BE STORED IN CRS FORMAT
                * first NIn columns contain inputs.
                * for regression problem, next NOut columns store
                  desired outputs.
                * for classification problem, next column (just one!)
                  stores class number.
    SSize   -   number of elements in XY
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 26.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgradbatchsparse(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t ssize,
    double& e,
    real_1d_array& grad,
    const xparams _params = alglib::xdefault);

`mlpgradbatchsparsesubset` function

/*************************************************************************
Batch gradient calculation for a set of inputs/outputs  for  a  subset  of
dataset given by set of indexes.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset in sparse format; one sample = one row:
                * MATRIX MUST BE STORED IN CRS FORMAT
                * first NIn columns contain inputs,
                * for regression problem, next NOut columns store
                  desired outputs.
                * for classification problem, next column (just one!)
                  stores class number.
    SetSize -   real size of XY, SetSize>=0;
    Idx     -   subset of SubsetSize elements, array[SubsetSize]:
                * Idx[I] stores row index in the original dataset which is
                  given by XY. Gradient is calculated with respect to rows
                  whose indexes are stored in Idx[].
                * Idx[]  must store correct indexes; this function  throws
                  an  exception  in  case  incorrect index (less than 0 or
                  larger than rows(XY)) is given
                * Idx[]  may  store  indexes  in  any  order and even with
                  repetitions.
    SubsetSize- number of elements in Idx[] array:
                * positive value means that subset given by Idx[] is processed
                * zero value results in zero gradient
                * negative value means that full dataset is processed
    Grad      - possibly  preallocated array. If size of array is  smaller
                than WCount, it will be reallocated. It is  recommended to
                reuse  previously  allocated  array  to  reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad    -   gradient  of  E  with  respect   to  weights  of  network,
                array[WCount]

NOTE: when  SubsetSize<0 is used full dataset by call MLPGradBatchSparse
      function.

  -- ALGLIB --
     Copyright 26.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgradbatchsparsesubset(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t setsize,
    integer_1d_array idx,
    ae_int_t subsetsize,
    double& e,
    real_1d_array& grad,
    const xparams _params = alglib::xdefault);

`mlpgradbatchsubset` function

/*************************************************************************
Batch gradient calculation for a subset of dataset

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset in dense format; one sample = one row:
                * first NIn columns contain inputs,
                * for regression problem, next NOut columns store
                  desired outputs.
                * for classification problem, next column (just one!)
                  stores class number.
    SetSize -   real size of XY, SetSize>=0;
    Idx     -   subset of SubsetSize elements, array[SubsetSize]:
                * Idx[I] stores row index in the original dataset which is
                  given by XY. Gradient is calculated with respect to rows
                  whose indexes are stored in Idx[].
                * Idx[]  must store correct indexes; this function  throws
                  an  exception  in  case  incorrect index (less than 0 or
                  larger than rows(XY)) is given
                * Idx[]  may  store  indexes  in  any  order and even with
                  repetitions.
    SubsetSize- number of elements in Idx[] array:
                * positive value means that subset given by Idx[] is processed
                * zero value results in zero gradient
                * negative value means that full dataset is processed
    Grad      - possibly  preallocated array. If size of array is  smaller
                than WCount, it will be reallocated. It is  recommended to
                reuse  previously  allocated  array  to  reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E         - error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad      - gradient  of  E  with  respect   to  weights  of  network,
                array[WCount]

  -- ALGLIB --
     Copyright 26.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgradbatchsubset(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t setsize,
    integer_1d_array idx,
    ae_int_t subsetsize,
    double& e,
    real_1d_array& grad,
    const xparams _params = alglib::xdefault);

`mlpgradn` function

/*************************************************************************
Gradient calculation (natural error function is used)

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    X       -   input vector, length of array must be at least NIn
    DesiredY-   desired outputs, length of array must be at least NOut
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, sum-of-squares for regression networks,
                cross-entropy for classification networks.
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgradn(
    multilayerperceptron network,
    real_1d_array x,
    real_1d_array desiredy,
    double& e,
    real_1d_array& grad,
    const xparams _params = alglib::xdefault);

`mlpgradnbatch` function

/*************************************************************************
Batch gradient calculation for a set of inputs/outputs
(natural error function is used)

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   set of inputs/outputs; one sample = one row;
                first NIn columns contain inputs,
                next NOut columns - desired outputs.
    SSize   -   number of elements in XY
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, sum-of-squares for regression networks,
                cross-entropy for classification networks.
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpgradnbatch(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t ssize,
    double& e,
    real_1d_array& grad,
    const xparams _params = alglib::xdefault);

`mlphessianbatch` function

/*************************************************************************
Batch Hessian calculation using R-algorithm.
Internal subroutine.

  -- ALGLIB --
     Copyright 26.01.2008 by Bochkanov Sergey.

     Hessian calculation based on R-algorithm described in
     "Fast Exact Multiplication by the Hessian",
     B. A. Pearlmutter,
     Neural Computation, 1994.
*************************************************************************/
void alglib::mlphessianbatch(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t ssize,
    double& e,
    real_1d_array& grad,
    real_2d_array& h,
    const xparams _params = alglib::xdefault);

`mlphessiannbatch` function

/*************************************************************************
Batch Hessian calculation (natural error function) using R-algorithm.
Internal subroutine.

  -- ALGLIB --
     Copyright 26.01.2008 by Bochkanov Sergey.

     Hessian calculation based on R-algorithm described in
     "Fast Exact Multiplication by the Hessian",
     B. A. Pearlmutter,
     Neural Computation, 1994.
*************************************************************************/
void alglib::mlphessiannbatch(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t ssize,
    double& e,
    real_1d_array& grad,
    real_2d_array& h,
    const xparams _params = alglib::xdefault);

`mlpinitpreprocessor` function

/*************************************************************************
Internal subroutine.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpinitpreprocessor(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t ssize,
    const xparams _params = alglib::xdefault);

`mlpissoftmax` function

/*************************************************************************
Tells whether network is SOFTMAX-normalized (i.e. classifier) or not.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
bool alglib::mlpissoftmax(
    multilayerperceptron network,
    const xparams _params = alglib::xdefault);

`mlpprocess` function

/*************************************************************************
Procesing

INPUT PARAMETERS:
    Network -   neural network
    X       -   input vector,  array[0..NIn-1].

OUTPUT PARAMETERS:
    Y       -   result. Regression estimate when solving regression  task,
                vector of posterior probabilities for classification task.

See also MLPProcessI

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpprocess(
    multilayerperceptron network,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`mlpprocessi` function

/*************************************************************************
'interactive'  variant  of  MLPProcess  for  languages  like  Python which
support constructs like "Y = MLPProcess(NN,X)" and interactive mode of the
interpreter

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 21.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpprocessi(
    multilayerperceptron network,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`mlpproperties` function

/*************************************************************************
Returns information about initialized network: number of inputs, outputs,
weights.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpproperties(
    multilayerperceptron network,
    ae_int_t& nin,
    ae_int_t& nout,
    ae_int_t& wcount,
    const xparams _params = alglib::xdefault);

`mlprandomize` function

/*************************************************************************
Randomization of neural network weights

  -- ALGLIB --
     Copyright 06.11.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlprandomize(
    multilayerperceptron network,
    const xparams _params = alglib::xdefault);

`mlprandomizefull` function

/*************************************************************************
Randomization of neural network weights and standartisator

  -- ALGLIB --
     Copyright 10.03.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::mlprandomizefull(
    multilayerperceptron network,
    const xparams _params = alglib::xdefault);

`mlprelclserror` function

/*************************************************************************
Relative classification error on the test set.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
Percent   of incorrectly   classified  cases.  Works  both  for classifier
networks and general purpose networks used as classifiers.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 25.12.2008 by Bochkanov Sergey
*************************************************************************/
double alglib::mlprelclserror(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlprelclserrorsparse` function

/*************************************************************************
Relative classification error on the test set given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. Sparse matrix must use CRS format
                    for storage.
    NPoints     -   points count, >=0.

RESULT:
Percent   of incorrectly   classified  cases.  Works  both  for classifier
networks and general purpose networks used as classifiers.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 09.08.2012 by Bochkanov Sergey
*************************************************************************/
double alglib::mlprelclserrorsparse(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlprmserror` function

/*************************************************************************
RMS error on the test set given.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
Root mean  square error. Its meaning for regression task is obvious. As for
classification  task,  RMS  error  means  error  when estimating  posterior
probabilities.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::mlprmserror(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlprmserrorsparse` function

/*************************************************************************
RMS error on the test set given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0.

RESULT:
Root mean  square error. Its meaning for regression task is obvious. As for
classification  task,  RMS  error  means  error  when estimating  posterior
probabilities.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 09.08.2012 by Bochkanov Sergey
*************************************************************************/
double alglib::mlprmserrorsparse(
    multilayerperceptron network,
    sparsematrix xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpserialize` function

/*************************************************************************
This function serializes data structure to string.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string  in  case you want to include it into text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor you should change case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize  it  in  C++ version of ALGLIB and unserialize in C# one, 
  and vice versa.
*************************************************************************/
void mlpserialize(multilayerperceptron &obj, std::string &s_out);
void mlpserialize(multilayerperceptron &obj, std::ostream &s_out);

`mlpsetinputscaling` function

/*************************************************************************
This function sets offset/scaling coefficients for I-th input of the
network.

INPUT PARAMETERS:
    Network     -   network
    I           -   input index
    Mean        -   mean term
    Sigma       -   sigma term (if zero, will be replaced by 1.0)

NTE: I-th input is passed through linear transformation
    IN[i] = (IN[i]-Mean)/Sigma
before feeding to the network. This function sets Mean and Sigma.

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpsetinputscaling(
    multilayerperceptron network,
    ae_int_t i,
    double mean,
    double sigma,
    const xparams _params = alglib::xdefault);

`mlpsetneuroninfo` function

/*************************************************************************
This function modifies information about Ith neuron of Kth layer

INPUT PARAMETERS:
    Network     -   network
    K           -   layer index
    I           -   neuron index (within layer)
    FKind       -   activation function type (used by MLPActivationFunction())
                    this value must be zero for input neurons
                    (you can not set activation function for input neurons)
    Threshold   -   also called offset, bias
                    this value must be zero for input neurons
                    (you can not set threshold for input neurons)

NOTES:
1. this function throws exception if layer or neuron with given index do
   not exists.
2. this function also throws exception when you try to set non-linear
   activation function for input neurons (any kind of network) or for output
   neurons of classifier network.
3. this function throws exception when you try to set non-zero threshold for
   input neurons (any kind of network).

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpsetneuroninfo(
    multilayerperceptron network,
    ae_int_t k,
    ae_int_t i,
    ae_int_t fkind,
    double threshold,
    const xparams _params = alglib::xdefault);

`mlpsetoutputscaling` function

/*************************************************************************
This function sets offset/scaling coefficients for I-th output of the
network.

INPUT PARAMETERS:
    Network     -   network
    I           -   input index
    Mean        -   mean term
    Sigma       -   sigma term (if zero, will be replaced by 1.0)

OUTPUT PARAMETERS:

NOTE: I-th output is passed through linear transformation
    OUT[i] = OUT[i]*Sigma+Mean
before returning it to user. This function sets Sigma/Mean. In case we
have SOFTMAX-normalized network, you can not set (Sigma,Mean) to anything
other than(0.0,1.0) - this function will throw exception.

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpsetoutputscaling(
    multilayerperceptron network,
    ae_int_t i,
    double mean,
    double sigma,
    const xparams _params = alglib::xdefault);

`mlpsetweight` function

/*************************************************************************
This function modifies information about connection from I0-th neuron of
K0-th layer to I1-th neuron of K1-th layer.

INPUT PARAMETERS:
    Network     -   network
    K0          -   layer index
    I0          -   neuron index (within layer)
    K1          -   layer index
    I1          -   neuron index (within layer)
    W           -   connection weight (must be zero for non-existent
                    connections)

This function:
1. throws exception if layer or neuron with given index do not exists.
2. throws exception if you try to set non-zero weight for non-existent
   connection

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpsetweight(
    multilayerperceptron network,
    ae_int_t k0,
    ae_int_t i0,
    ae_int_t k1,
    ae_int_t i1,
    double w,
    const xparams _params = alglib::xdefault);

`mlpunserialize` function

/*************************************************************************
This function unserializes data structure from string.
*************************************************************************/
void mlpunserialize(const std::string &s_in, multilayerperceptron &obj);
void mlpunserialize(const std::istream &s_in, multilayerperceptron &obj);

`mlpe` subpackage

Classes

mlpensemble

Functions

mlpeavgce
mlpeavgerror
mlpeavgrelerror
mlpecreate0
mlpecreate1
mlpecreate2
mlpecreateb0
mlpecreateb1
mlpecreateb2
mlpecreatec0
mlpecreatec1
mlpecreatec2
mlpecreatefromnetwork
mlpecreater0
mlpecreater1
mlpecreater2
mlpeissoftmax
mlpeprocess
mlpeprocessi
mlpeproperties
mlperandomize
mlperelclserror
mlpermserror
mlpeserialize
mlpeunserialize

Examples

`mlpensemble` class

/*************************************************************************
Neural networks ensemble
*************************************************************************/
class mlpensemble
{
};

`mlpeavgce` function

/*************************************************************************
Average cross-entropy (in bits per element) on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    CrossEntropy/(NPoints*LN(2)).
    Zero if ensemble solves regression task.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpeavgce(
    mlpensemble ensemble,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpeavgerror` function

/*************************************************************************
Average error on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    Its meaning for regression task is obvious. As for classification task
it means average error when estimating posterior probabilities.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpeavgerror(
    mlpensemble ensemble,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpeavgrelerror` function

/*************************************************************************
Average relative error on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    Its meaning for regression task is obvious. As for classification task
it means average relative error when estimating posterior probabilities.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpeavgrelerror(
    mlpensemble ensemble,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpecreate0` function

/*************************************************************************
Like MLPCreate0, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreate0(
    ae_int_t nin,
    ae_int_t nout,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreate1` function

/*************************************************************************
Like MLPCreate1, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreate1(
    ae_int_t nin,
    ae_int_t nhid,
    ae_int_t nout,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreate2` function

/*************************************************************************
Like MLPCreate2, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreate2(
    ae_int_t nin,
    ae_int_t nhid1,
    ae_int_t nhid2,
    ae_int_t nout,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreateb0` function

/*************************************************************************
Like MLPCreateB0, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreateb0(
    ae_int_t nin,
    ae_int_t nout,
    double b,
    double d,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreateb1` function

/*************************************************************************
Like MLPCreateB1, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreateb1(
    ae_int_t nin,
    ae_int_t nhid,
    ae_int_t nout,
    double b,
    double d,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreateb2` function

/*************************************************************************
Like MLPCreateB2, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreateb2(
    ae_int_t nin,
    ae_int_t nhid1,
    ae_int_t nhid2,
    ae_int_t nout,
    double b,
    double d,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreatec0` function

/*************************************************************************
Like MLPCreateC0, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreatec0(
    ae_int_t nin,
    ae_int_t nout,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreatec1` function

/*************************************************************************
Like MLPCreateC1, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreatec1(
    ae_int_t nin,
    ae_int_t nhid,
    ae_int_t nout,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreatec2` function

/*************************************************************************
Like MLPCreateC2, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreatec2(
    ae_int_t nin,
    ae_int_t nhid1,
    ae_int_t nhid2,
    ae_int_t nout,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreatefromnetwork` function

/*************************************************************************
Creates ensemble from network. Only network geometry is copied.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreatefromnetwork(
    multilayerperceptron network,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreater0` function

/*************************************************************************
Like MLPCreateR0, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreater0(
    ae_int_t nin,
    ae_int_t nout,
    double a,
    double b,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreater1` function

/*************************************************************************
Like MLPCreateR1, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreater1(
    ae_int_t nin,
    ae_int_t nhid,
    ae_int_t nout,
    double a,
    double b,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpecreater2` function

/*************************************************************************
Like MLPCreateR2, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpecreater2(
    ae_int_t nin,
    ae_int_t nhid1,
    ae_int_t nhid2,
    ae_int_t nout,
    double a,
    double b,
    ae_int_t ensemblesize,
    mlpensemble& ensemble,
    const xparams _params = alglib::xdefault);

`mlpeissoftmax` function

/*************************************************************************
Return normalization type (whether ensemble is SOFTMAX-normalized or not).

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
bool alglib::mlpeissoftmax(
    mlpensemble ensemble,
    const xparams _params = alglib::xdefault);

`mlpeprocess` function

/*************************************************************************
Procesing

INPUT PARAMETERS:
    Ensemble-   neural networks ensemble
    X       -   input vector,  array[0..NIn-1].
    Y       -   (possibly) preallocated buffer; if size of Y is less than
                NOut, it will be reallocated. If it is large enough, it
                is NOT reallocated, so we can save some time on reallocation.


OUTPUT PARAMETERS:
    Y       -   result. Regression estimate when solving regression  task,
                vector of posterior probabilities for classification task.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpeprocess(
    mlpensemble ensemble,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`mlpeprocessi` function

/*************************************************************************
'interactive'  variant  of  MLPEProcess  for  languages  like Python which
support constructs like "Y = MLPEProcess(LM,X)" and interactive mode of the
interpreter

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpeprocessi(
    mlpensemble ensemble,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`mlpeproperties` function

/*************************************************************************
Return ensemble properties (number of inputs and outputs).

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpeproperties(
    mlpensemble ensemble,
    ae_int_t& nin,
    ae_int_t& nout,
    const xparams _params = alglib::xdefault);

`mlperandomize` function

/*************************************************************************
Randomization of MLP ensemble

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlperandomize(
    mlpensemble ensemble,
    const xparams _params = alglib::xdefault);

`mlperelclserror` function

/*************************************************************************
Relative classification error on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    percent of incorrectly classified cases.
    Works both for classifier betwork and for regression networks which
are used as classifiers.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::mlperelclserror(
    mlpensemble ensemble,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpermserror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    root mean square error.
    Its meaning for regression task is obvious. As for classification task
RMS error means error when estimating posterior probabilities.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::mlpermserror(
    mlpensemble ensemble,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpeserialize` function

/*************************************************************************
This function serializes data structure to string.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string  in  case you want to include it into text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor you should change case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize  it  in  C++ version of ALGLIB and unserialize in C# one, 
  and vice versa.
*************************************************************************/
void mlpeserialize(mlpensemble &obj, std::string &s_out);
void mlpeserialize(mlpensemble &obj, std::ostream &s_out);

`mlpeunserialize` function

/*************************************************************************
This function unserializes data structure from string.
*************************************************************************/
void mlpeunserialize(const std::string &s_in, mlpensemble &obj);
void mlpeunserialize(const std::istream &s_in, mlpensemble &obj);

`mlptrain` subpackage

Classes

mlpcvreport
mlpreport
mlptrainer

Functions

mlpcontinuetraining
mlpcreatetrainer
mlpcreatetrainercls
mlpebagginglbfgs
mlpebagginglm
mlpetraines
mlpkfoldcv
mlpkfoldcvlbfgs
mlpkfoldcvlm
mlpsetalgobatch
mlpsetcond
mlpsetdataset
mlpsetdecay
mlpsetsparsedataset
mlpstarttraining
mlptrainensemblees
mlptraines
mlptrainlbfgs
mlptrainlm
mlptrainnetwork

Examples

nn_cls2		Binary classification problem
nn_cls3		Multiclass classification problem
nn_crossvalidation		Cross-validation
nn_ensembles_es		Early stopping ensembles
nn_parallel		Parallel training
nn_regr		Regression problem with one output (2=>1)
nn_regr_n		Regression problem with multiple outputs (2=>2)
nn_trainerobject		Advanced example on trainer object

`mlpcvreport` class

/*************************************************************************
Cross-validation estimates of generalization error
*************************************************************************/
class mlpcvreport
{
    double               relclserror;
    double               avgce;
    double               rmserror;
    double               avgerror;
    double               avgrelerror;
};

`mlpreport` class

/*************************************************************************
Training report:
    * RelCLSError   -   fraction of misclassified cases.
    * AvgCE         -   acerage cross-entropy
    * RMSError      -   root-mean-square error
    * AvgError      -   average error
    * AvgRelError   -   average relative error
    * NGrad         -   number of gradient calculations
    * NHess         -   number of Hessian calculations
    * NCholesky     -   number of Cholesky decompositions

NOTE 1: RelCLSError/AvgCE are zero on regression problems.

NOTE 2: on classification problems  RMSError/AvgError/AvgRelError  contain
        errors in prediction of posterior probabilities
*************************************************************************/
class mlpreport
{
    double               relclserror;
    double               avgce;
    double               rmserror;
    double               avgerror;
    double               avgrelerror;
    ae_int_t             ngrad;
    ae_int_t             nhess;
    ae_int_t             ncholesky;
};

`mlptrainer` class

/*************************************************************************
Trainer object for neural network.

You should not try to access fields of this object directly -  use  ALGLIB
functions to work with this object.
*************************************************************************/
class mlptrainer
{
};

`mlpcontinuetraining` function

/*************************************************************************
IMPORTANT: this is an "expert" version of the MLPTrain() function.  We  do
           not recommend you to use it unless you are pretty sure that you
           need ability to monitor training progress.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

This function performs step-by-step training of the neural  network.  Here
"step-by-step" means that training starts  with  MLPStartTraining()  call,
and then user subsequently calls MLPContinueTraining() to perform one more
iteration of the training.

This  function  performs  one  more  iteration of the training and returns
either True (training continues) or False (training stopped). In case True
was returned, Network weights are updated according to the  current  state
of the optimization progress. In case False was  returned,  no  additional
updates is performed (previous update of  the  network weights moved us to
the final point, and no additional updates is needed).

EXAMPLE:
    >
    > [initialize network and trainer object]
    >
    > MLPStartTraining(Trainer, Network, True)
    > while MLPContinueTraining(Trainer, Network) do
    >     [visualize training progress]
    >

INPUT PARAMETERS:
    S           -   trainer object
    Network     -   neural  network  structure,  which  is  used to  store
                    current state of the training process.

OUTPUT PARAMETERS:
    Network     -   weights of the neural network  are  rewritten  by  the
                    current approximation.

NOTE: this method uses sum-of-squares error function for training.

NOTE: it is expected that trainer object settings are NOT  changed  during
      step-by-step training, i.e. no  one  changes  stopping  criteria  or
      training set during training. It is possible and there is no defense
      against  such  actions,  but  algorithm  behavior  in  such cases is
      undefined and can be unpredictable.

NOTE: It  is  expected that Network is the same one which  was  passed  to
      MLPStartTraining() function.  However,  THIS  function  checks  only
      following:
      * that number of network inputs is consistent with trainer object
        settings
      * that number of network outputs/classes is consistent with  trainer
        object settings
      * that number of network weights is the same as number of weights in
        the network passed to MLPStartTraining() function
      Exception is thrown when these conditions are violated.

      It is also expected that you do not change state of the  network  on
      your own - the only party who has right to change network during its
      training is a trainer object. Any attempt to interfere with  trainer
      may lead to unpredictable results.


  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
bool alglib::mlpcontinuetraining(
    mlptrainer s,
    multilayerperceptron network,
    const xparams _params = alglib::xdefault);

`mlpcreatetrainer` function

/*************************************************************************
Creation of the network trainer object for regression networks

INPUT PARAMETERS:
    NIn         -   number of inputs, NIn>=1
    NOut        -   number of outputs, NOut>=1

OUTPUT PARAMETERS:
    S           -   neural network trainer object.
                    This structure can be used to train any regression
                    network with NIn inputs and NOut outputs.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreatetrainer(
    ae_int_t nin,
    ae_int_t nout,
    mlptrainer& s,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`mlpcreatetrainercls` function

/*************************************************************************
Creation of the network trainer object for classification networks

INPUT PARAMETERS:
    NIn         -   number of inputs, NIn>=1
    NClasses    -   number of classes, NClasses>=2

OUTPUT PARAMETERS:
    S           -   neural network trainer object.
                    This structure can be used to train any classification
                    network with NIn inputs and NOut outputs.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpcreatetrainercls(
    ae_int_t nin,
    ae_int_t nclasses,
    mlptrainer& s,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`mlpebagginglbfgs` function

/*************************************************************************
Training neural networks ensemble using  bootstrap  aggregating (bagging).
L-BFGS algorithm is used as base training method.

INPUT PARAMETERS:
    Ensemble    -   model with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay coefficient, >=0.001
    Restarts    -   restarts, >0.
    WStep       -   stopping criterion, same as in MLPTrainLBFGS
    MaxIts      -   stopping criterion, same as in MLPTrainLBFGS

OUTPUT PARAMETERS:
    Ensemble    -   trained model
    Info        -   return code:
                    * -8, if both WStep=0 and MaxIts=0
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed
                          (NPoints<0, Restarts<1).
                    *  2, if task has been solved.
    Rep         -   training report.
    OOBErrors   -   out-of-bag generalization error estimate

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpebagginglbfgs(
    mlpensemble ensemble,
    real_2d_array xy,
    ae_int_t npoints,
    double decay,
    ae_int_t restarts,
    double wstep,
    ae_int_t maxits,
    ae_int_t& info,
    mlpreport& rep,
    mlpcvreport& ooberrors,
    const xparams _params = alglib::xdefault);

`mlpebagginglm` function

/*************************************************************************
Training neural networks ensemble using  bootstrap  aggregating (bagging).
Modified Levenberg-Marquardt algorithm is used as base training method.

INPUT PARAMETERS:
    Ensemble    -   model with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay coefficient, >=0.001
    Restarts    -   restarts, >0.

OUTPUT PARAMETERS:
    Ensemble    -   trained model
    Info        -   return code:
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed
                          (NPoints<0, Restarts<1).
                    *  2, if task has been solved.
    Rep         -   training report.
    OOBErrors   -   out-of-bag generalization error estimate

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpebagginglm(
    mlpensemble ensemble,
    real_2d_array xy,
    ae_int_t npoints,
    double decay,
    ae_int_t restarts,
    ae_int_t& info,
    mlpreport& rep,
    mlpcvreport& ooberrors,
    const xparams _params = alglib::xdefault);

`mlpetraines` function

/*************************************************************************
Training neural networks ensemble using early stopping.

INPUT PARAMETERS:
    Ensemble    -   model with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay coefficient, >=0.001
    Restarts    -   restarts, >0.

OUTPUT PARAMETERS:
    Ensemble    -   trained model
    Info        -   return code:
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed
                          (NPoints<0, Restarts<1).
                    *  6, if task has been solved.
    Rep         -   training report.
    OOBErrors   -   out-of-bag generalization error estimate

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpetraines(
    mlpensemble ensemble,
    real_2d_array xy,
    ae_int_t npoints,
    double decay,
    ae_int_t restarts,
    ae_int_t& info,
    mlpreport& rep,
    const xparams _params = alglib::xdefault);

`mlpkfoldcv` function

/*************************************************************************
This function estimates generalization error using cross-validation on the
current dataset with current training settings.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S           -   trainer object
    Network     -   neural network. It must have same number of inputs and
                    output/classes as was specified during creation of the
                    trainer object. Network is not changed  during  cross-
                    validation and is not trained - it  is  used  only  as
                    representative of its architecture. I.e., we  estimate
                    generalization properties of  ARCHITECTURE,  not  some
                    specific network.
    NRestarts   -   number of restarts, >=0:
                    * NRestarts>0  means  that  for  each cross-validation
                      round   specified  number   of  random  restarts  is
                      performed,  with  best  network  being  chosen after
                      training.
                    * NRestarts=0 is same as NRestarts=1
    FoldsCount  -   number of folds in k-fold cross-validation:
                    * 2<=FoldsCount<=size of dataset
                    * recommended value: 10.
                    * values larger than dataset size will be silently
                      truncated down to dataset size

OUTPUT PARAMETERS:
    Rep         -   structure which contains cross-validation estimates:
                    * Rep.RelCLSError - fraction of misclassified cases.
                    * Rep.AvgCE - acerage cross-entropy
                    * Rep.RMSError - root-mean-square error
                    * Rep.AvgError - average error
                    * Rep.AvgRelError - average relative error

NOTE: when no dataset was specified with MLPSetDataset/SetSparseDataset(),
      or subset with only one point  was  given,  zeros  are  returned  as
      estimates.

NOTE: this method performs FoldsCount cross-validation  rounds,  each  one
      with NRestarts random starts.  Thus,  FoldsCount*NRestarts  networks
      are trained in total.

NOTE: Rep.RelCLSError/Rep.AvgCE are zero on regression problems.

NOTE: on classification problems Rep.RMSError/Rep.AvgError/Rep.AvgRelError
      contain errors in prediction of posterior probabilities.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpkfoldcv(
    mlptrainer s,
    multilayerperceptron network,
    ae_int_t nrestarts,
    ae_int_t foldscount,
    mlpreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`mlpkfoldcvlbfgs` function

/*************************************************************************
Cross-validation estimate of generalization error.

Base algorithm - L-BFGS.

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry.   Network is
                    not changed during cross-validation -  it is used only
                    as a representative of its architecture.
    XY          -   training set.
    SSize       -   training set size
    Decay       -   weight  decay, same as in MLPTrainLBFGS
    Restarts    -   number of restarts, >0.
                    restarts are counted for each partition separately, so
                    total number of restarts will be Restarts*FoldsCount.
    WStep       -   stopping criterion, same as in MLPTrainLBFGS
    MaxIts      -   stopping criterion, same as in MLPTrainLBFGS
    FoldsCount  -   number of folds in k-fold cross-validation,
                    2<=FoldsCount<=SSize.
                    recommended value: 10.

OUTPUT PARAMETERS:
    Info        -   return code, same as in MLPTrainLBFGS
    Rep         -   report, same as in MLPTrainLM/MLPTrainLBFGS
    CVRep       -   generalization error estimates

  -- ALGLIB --
     Copyright 09.12.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpkfoldcvlbfgs(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    double decay,
    ae_int_t restarts,
    double wstep,
    ae_int_t maxits,
    ae_int_t foldscount,
    ae_int_t& info,
    mlpreport& rep,
    mlpcvreport& cvrep,
    const xparams _params = alglib::xdefault);

`mlpkfoldcvlm` function

/*************************************************************************
Cross-validation estimate of generalization error.

Base algorithm - Levenberg-Marquardt.

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry.   Network is
                    not changed during cross-validation -  it is used only
                    as a representative of its architecture.
    XY          -   training set.
    SSize       -   training set size
    Decay       -   weight  decay, same as in MLPTrainLBFGS
    Restarts    -   number of restarts, >0.
                    restarts are counted for each partition separately, so
                    total number of restarts will be Restarts*FoldsCount.
    FoldsCount  -   number of folds in k-fold cross-validation,
                    2<=FoldsCount<=SSize.
                    recommended value: 10.

OUTPUT PARAMETERS:
    Info        -   return code, same as in MLPTrainLBFGS
    Rep         -   report, same as in MLPTrainLM/MLPTrainLBFGS
    CVRep       -   generalization error estimates

  -- ALGLIB --
     Copyright 09.12.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpkfoldcvlm(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    double decay,
    ae_int_t restarts,
    ae_int_t foldscount,
    ae_int_t& info,
    mlpreport& rep,
    mlpcvreport& cvrep,
    const xparams _params = alglib::xdefault);

`mlpsetalgobatch` function

/*************************************************************************
This function sets training algorithm: batch training using L-BFGS will be
used.

This algorithm:
* the most robust for small-scale problems, but may be too slow for  large
  scale ones.
* perfoms full pass through the dataset before performing step
* uses conditions specified by MLPSetCond() for stopping
* is default one used by trainer object

INPUT PARAMETERS:
    S           -   trainer object

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpsetalgobatch(
    mlptrainer s,
    const xparams _params = alglib::xdefault);

`mlpsetcond` function

/*************************************************************************
This function sets stopping criteria for the optimizer.

INPUT PARAMETERS:
    S           -   trainer object
    WStep       -   stopping criterion. Algorithm stops if  step  size  is
                    less than WStep. Recommended value - 0.01.  Zero  step
                    size means stopping after MaxIts iterations.
                    WStep>=0.
    MaxIts      -   stopping   criterion.  Algorithm  stops  after  MaxIts
                    epochs (full passes over entire dataset).  Zero MaxIts
                    means stopping when step is sufficiently small.
                    MaxIts>=0.

NOTE: by default, WStep=0.005 and MaxIts=0 are used. These values are also
      used when MLPSetCond() is called with WStep=0 and MaxIts=0.

NOTE: these stopping criteria are used for all kinds of neural training  -
      from "conventional" networks to early stopping ensembles. When  used
      for "conventional" networks, they are  used  as  the  only  stopping
      criteria. When combined with early stopping, they used as ADDITIONAL
      stopping criteria which can terminate early stopping algorithm.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpsetcond(
    mlptrainer s,
    double wstep,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

`mlpsetdataset` function

/*************************************************************************
This function sets "current dataset" of the trainer object to  one  passed
by user.

INPUT PARAMETERS:
    S           -   trainer object
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.
    NPoints     -   points count, >=0.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
datasetformat is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpsetdataset(
    mlptrainer s,
    real_2d_array xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6] [7] [8]

`mlpsetdecay` function

/*************************************************************************
This function sets weight decay coefficient which is used for training.

INPUT PARAMETERS:
    S           -   trainer object
    Decay       -   weight  decay  coefficient,  >=0.  Weight  decay  term
                    'Decay*||Weights||^2' is added to error  function.  If
                    you don't know what Decay to choose, use 1.0E-3.
                    Weight decay can be set to zero,  in this case network
                    is trained without weight decay.

NOTE: by default network uses some small nonzero value for weight decay.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpsetdecay(
    mlptrainer s,
    double decay,
    const xparams _params = alglib::xdefault);

`mlpsetsparsedataset` function

/*************************************************************************
This function sets "current dataset" of the trainer object to  one  passed
by user (sparse matrix is used to store dataset).

INPUT PARAMETERS:
    S           -   trainer object
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed. Any  sparse  storage  format  can be  used:
                    Hash-table, CRS...
    NPoints     -   points count, >=0

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
datasetformat is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpsetsparsedataset(
    mlptrainer s,
    sparsematrix xy,
    ae_int_t npoints,
    const xparams _params = alglib::xdefault);

`mlpstarttraining` function

/*************************************************************************
IMPORTANT: this is an "expert" version of the MLPTrain() function.  We  do
           not recommend you to use it unless you are pretty sure that you
           need ability to monitor training progress.

This function performs step-by-step training of the neural  network.  Here
"step-by-step" means that training  starts  with  MLPStartTraining() call,
and then user subsequently calls MLPContinueTraining() to perform one more
iteration of the training.

After call to this function trainer object remembers network and  is ready
to  train  it.  However,  no  training  is  performed  until first call to
MLPContinueTraining() function. Subsequent calls  to MLPContinueTraining()
will advance training progress one iteration further.

EXAMPLE:
    >
    > ...initialize network and trainer object....
    >
    > MLPStartTraining(Trainer, Network, True)
    > while MLPContinueTraining(Trainer, Network) do
    >     ...visualize training progress...
    >

INPUT PARAMETERS:
    S           -   trainer object
    Network     -   neural network. It must have same number of inputs and
                    output/classes as was specified during creation of the
                    trainer object.
    RandomStart -   randomize network before training or not:
                    * True  means  that  network  is  randomized  and  its
                      initial state (one which was passed to  the  trainer
                      object) is lost.
                    * False  means  that  training  is  started  from  the
                      current state of the network

OUTPUT PARAMETERS:
    Network     -   neural network which is ready to training (weights are
                    initialized, preprocessor is initialized using current
                    training set)

NOTE: this method uses sum-of-squares error function for training.

NOTE: it is expected that trainer object settings are NOT  changed  during
      step-by-step training, i.e. no  one  changes  stopping  criteria  or
      training set during training. It is possible and there is no defense
      against  such  actions,  but  algorithm  behavior  in  such cases is
      undefined and can be unpredictable.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlpstarttraining(
    mlptrainer s,
    multilayerperceptron network,
    bool randomstart,
    const xparams _params = alglib::xdefault);

`mlptrainensemblees` function

/*************************************************************************
This function trains neural network ensemble passed to this function using
current dataset and early stopping training algorithm. Each early stopping
round performs NRestarts  random  restarts  (thus,  EnsembleSize*NRestarts
training rounds is performed in total).

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S           -   trainer object;
    Ensemble    -   neural network ensemble. It must have same  number  of
                    inputs and outputs/classes  as  was  specified  during
                    creation of the trainer object.
    NRestarts   -   number of restarts, >=0:
                    * NRestarts>0 means that specified  number  of  random
                      restarts are performed during each ES round;
                    * NRestarts=0 is silently replaced by 1.

OUTPUT PARAMETERS:
    Ensemble    -   trained ensemble;
    Rep         -   it contains all type of errors.

NOTE: this training method uses BOTH early stopping and weight decay!  So,
      you should select weight decay before starting training just as  you
      select it before training "conventional" networks.

NOTE: when no dataset was specified with MLPSetDataset/SetSparseDataset(),
      or  single-point  dataset  was  passed,  ensemble  is filled by zero
      values.

NOTE: this method uses sum-of-squares error function for training.

  -- ALGLIB --
     Copyright 22.08.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlptrainensemblees(
    mlptrainer s,
    mlpensemble ensemble,
    ae_int_t nrestarts,
    mlpreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`mlptraines` function

/*************************************************************************
Neural network training using early stopping (base algorithm - L-BFGS with
regularization).

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry
    TrnXY       -   training set
    TrnSize     -   training set size, TrnSize>0
    ValXY       -   validation set
    ValSize     -   validation set size, ValSize>0
    Decay       -   weight decay constant, >=0.001
                    Decay term 'Decay*||Weights||^2' is added to error
                    function.
                    If you don't know what Decay to choose, use 0.001.
    Restarts    -   number of restarts, either:
                    * strictly positive number - algorithm make specified
                      number of restarts from random position.
                    * -1, in which case algorithm makes exactly one run
                      from the initial state of the network (no randomization).
                    If you don't know what Restarts to choose, choose one
                    one the following:
                    * -1 (deterministic start)
                    * +1 (one random restart)
                    * +5 (moderate amount of random restarts)

OUTPUT PARAMETERS:
    Network     -   trained neural network.
    Info        -   return code:
                    * -2, if there is a point with class number
                          outside of [0..NOut-1].
                    * -1, if wrong parameters specified
                          (NPoints<0, Restarts<1, ...).
                    *  2, task has been solved, stopping  criterion  met -
                          sufficiently small step size.  Not expected  (we
                          use  EARLY  stopping)  but  possible  and not an
                          error.
                    *  6, task has been solved, stopping  criterion  met -
                          increasing of validation set error.
    Rep         -   training report

NOTE:

Algorithm stops if validation set error increases for  a  long  enough  or
step size is small enought  (there  are  task  where  validation  set  may
decrease for eternity). In any case solution returned corresponds  to  the
minimum of validation set error.

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlptraines(
    multilayerperceptron network,
    real_2d_array trnxy,
    ae_int_t trnsize,
    real_2d_array valxy,
    ae_int_t valsize,
    double decay,
    ae_int_t restarts,
    ae_int_t& info,
    mlpreport& rep,
    const xparams _params = alglib::xdefault);

`mlptrainlbfgs` function

/*************************************************************************
Neural  network  training  using  L-BFGS  algorithm  with  regularization.
Subroutine  trains  neural  network  with  restarts from random positions.
Algorithm  is  well  suited  for  problems  of  any dimensionality (memory
requirements and step complexity are linear by weights number).

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay constant, >=0.001
                    Decay term 'Decay*||Weights||^2' is added to error
                    function.
                    If you don't know what Decay to choose, use 0.001.
    Restarts    -   number of restarts from random position, >0.
                    If you don't know what Restarts to choose, use 2.
    WStep       -   stopping criterion. Algorithm stops if  step  size  is
                    less than WStep. Recommended value - 0.01.  Zero  step
                    size means stopping after MaxIts iterations.
    MaxIts      -   stopping   criterion.  Algorithm  stops  after  MaxIts
                    iterations (NOT gradient  calculations).  Zero  MaxIts
                    means stopping when step is sufficiently small.

OUTPUT PARAMETERS:
    Network     -   trained neural network.
    Info        -   return code:
                    * -8, if both WStep=0 and MaxIts=0
                    * -2, if there is a point with class number
                          outside of [0..NOut-1].
                    * -1, if wrong parameters specified
                          (NPoints<0, Restarts<1).
                    *  2, if task has been solved.
    Rep         -   training report

  -- ALGLIB --
     Copyright 09.12.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::mlptrainlbfgs(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    double decay,
    ae_int_t restarts,
    double wstep,
    ae_int_t maxits,
    ae_int_t& info,
    mlpreport& rep,
    const xparams _params = alglib::xdefault);

`mlptrainlm` function

/*************************************************************************
Neural network training  using  modified  Levenberg-Marquardt  with  exact
Hessian calculation and regularization. Subroutine trains  neural  network
with restarts from random positions. Algorithm is well  suited  for  small
and medium scale problems (hundreds of weights).

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay constant, >=0.001
                    Decay term 'Decay*||Weights||^2' is added to error
                    function.
                    If you don't know what Decay to choose, use 0.001.
    Restarts    -   number of restarts from random position, >0.
                    If you don't know what Restarts to choose, use 2.

OUTPUT PARAMETERS:
    Network     -   trained neural network.
    Info        -   return code:
                    * -9, if internal matrix inverse subroutine failed
                    * -2, if there is a point with class number
                          outside of [0..NOut-1].
                    * -1, if wrong parameters specified
                          (NPoints<0, Restarts<1).
                    *  2, if task has been solved.
    Rep         -   training report

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::mlptrainlm(
    multilayerperceptron network,
    real_2d_array xy,
    ae_int_t npoints,
    double decay,
    ae_int_t restarts,
    ae_int_t& info,
    mlpreport& rep,
    const xparams _params = alglib::xdefault);

`mlptrainnetwork` function

/*************************************************************************
This function trains neural network passed to this function, using current
dataset (one which was passed to MLPSetDataset() or MLPSetSparseDataset())
and current training settings. Training  from  NRestarts  random  starting
positions is performed, best network is chosen.

Training is performed using current training algorithm.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S           -   trainer object
    Network     -   neural network. It must have same number of inputs and
                    output/classes as was specified during creation of the
                    trainer object.
    NRestarts   -   number of restarts, >=0:
                    * NRestarts>0 means that specified  number  of  random
                      restarts are performed, best network is chosen after
                      training
                    * NRestarts=0 means that current state of the  network
                      is used for training.

OUTPUT PARAMETERS:
    Network     -   trained network

NOTE: when no dataset was specified with MLPSetDataset/SetSparseDataset(),
      network  is  filled  by zero  values.  Same  behavior  for functions
      MLPStartTraining and MLPContinueTraining.

NOTE: this method uses sum-of-squares error function for training.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::mlptrainnetwork(
    mlptrainer s,
    multilayerperceptron network,
    ae_int_t nrestarts,
    mlpreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

nn_cls2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Suppose that we want to classify numbers as positive (class 0) and negative
    // (class 1). We have training set which includes several strictly positive
    // or negative numbers - and zero.
    //
    // The problem is that we are not sure how to classify zero, so from time to
    // time we mark it as positive or negative (with equal probability). Other
    // numbers are marked in pure deterministic setting. How will neural network
    // cope with such classification task?
    //
    // NOTE: we use network with excessive amount of neurons, which guarantees
    //       almost exact reproduction of the training set. Generalization ability
    //       of such network is rather low, but we are not concerned with such
    //       questions in this basic demo.
    //
    mlptrainer trn;
    multilayerperceptron network;
    mlpreport rep;
    real_1d_array x = "[0]";
    real_1d_array y = "[0,0]";

    //
    // Training set. One row corresponds to one record [A => class(A)].
    //
    // Classes are denoted by numbers from 0 to 1, where 0 corresponds to positive
    // numbers and 1 to negative numbers.
    //
    // [ +1  0]
    // [ +2  0]
    // [ -1  1]
    // [ -2  1]
    // [  0  0]   !! sometimes we classify 0 as positive, sometimes as negative
    // [  0  1]   !!
    //
    real_2d_array xy = "[[+1,0],[+2,0],[-1,1],[-2,1],[0,0],[0,1]]";

    //
    //
    // When we solve classification problems, everything is slightly different from
    // the regression ones:
    //
    // 1. Network is created. Because we solve classification problem, we use
    //    mlpcreatec1() function instead of mlpcreate1(). This function creates
    //    classifier network with SOFTMAX-normalized outputs. This network returns
    //    vector of class membership probabilities which are normalized to be
    //    non-negative and sum to 1.0
    //
    // 2. We use mlpcreatetrainercls() function instead of mlpcreatetrainer() to
    //    create trainer object. Trainer object process dataset and neural network
    //    slightly differently to account for specifics of the classification
    //    problems.
    //
    // 3. Dataset is attached to trainer object. Note that dataset format is slightly
    //    different from one used for regression.
    //
    mlpcreatetrainercls(1, 2, trn);
    mlpcreatec1(1, 5, 2, network);
    mlpsetdataset(trn, xy, 6);

    //
    // Network is trained with 5 restarts from random positions
    //
    mlptrainnetwork(trn, network, 5, rep);

    //
    // Test our neural network on strictly positive and strictly negative numbers.
    //
    // IMPORTANT! Classifier network returns class membership probabilities instead
    // of class indexes. Network returns two values (probabilities) instead of one
    // (class index).
    //
    // Thus, for +1 we expect to get [P0,P1] = [1,0], where P0 is probability that
    // number is positive (belongs to class 0), and P1 is probability that number
    // is negative (belongs to class 1).
    //
    // For -1 we expect to get [P0,P1] = [0,1]
    //
    // Following properties are guaranteed by network architecture:
    // * P0>=0, P1>=0   non-negativity
    // * P0+P1=1        normalization
    //
    x = "[1]";
    mlpprocess(network, x, y);
    printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [1.000,0.000]
    x = "[-1]";
    mlpprocess(network, x, y);
    printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.000,1.000]

    //
    // But what our network will return for 0, which is between classes 0 and 1?
    //
    // In our dataset it has two different marks assigned (class 0 AND class 1).
    // So network will return something average between class 0 and class 1:
    //     0 => [0.5, 0.5]
    //
    x = "[0]";
    mlpprocess(network, x, y);
    printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.500,0.500]
    return 0;
}

nn_cls3 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Suppose that we want to classify numbers as positive (class 0) and negative
    // (class 1). We also have one more class for zero (class 2).
    //
    // NOTE: we use network with excessive amount of neurons, which guarantees
    //       almost exact reproduction of the training set. Generalization ability
    //       of such network is rather low, but we are not concerned with such
    //       questions in this basic demo.
    //
    mlptrainer trn;
    multilayerperceptron network;
    mlpreport rep;
    real_1d_array x = "[0]";
    real_1d_array y = "[0,0,0]";

    //
    // Training set. One row corresponds to one record [A => class(A)].
    //
    // Classes are denoted by numbers from 0 to 2, where 0 corresponds to positive
    // numbers, 1 to negative numbers, 2 to zero
    //
    // [ +1  0]
    // [ +2  0]
    // [ -1  1]
    // [ -2  1]
    // [  0  2]
    //
    real_2d_array xy = "[[+1,0],[+2,0],[-1,1],[-2,1],[0,2]]";

    //
    //
    // When we solve classification problems, everything is slightly different from
    // the regression ones:
    //
    // 1. Network is created. Because we solve classification problem, we use
    //    mlpcreatec1() function instead of mlpcreate1(). This function creates
    //    classifier network with SOFTMAX-normalized outputs. This network returns
    //    vector of class membership probabilities which are normalized to be
    //    non-negative and sum to 1.0
    //
    // 2. We use mlpcreatetrainercls() function instead of mlpcreatetrainer() to
    //    create trainer object. Trainer object process dataset and neural network
    //    slightly differently to account for specifics of the classification
    //    problems.
    //
    // 3. Dataset is attached to trainer object. Note that dataset format is slightly
    //    different from one used for regression.
    //
    mlpcreatetrainercls(1, 3, trn);
    mlpcreatec1(1, 5, 3, network);
    mlpsetdataset(trn, xy, 5);

    //
    // Network is trained with 5 restarts from random positions
    //
    mlptrainnetwork(trn, network, 5, rep);

    //
    // Test our neural network on strictly positive and strictly negative numbers.
    //
    // IMPORTANT! Classifier network returns class membership probabilities instead
    // of class indexes. Network returns three values (probabilities) instead of one
    // (class index).
    //
    // Thus, for +1 we expect to get [P0,P1,P2] = [1,0,0],
    // for -1 we expect to get [P0,P1,P2] = [0,1,0],
    // and for 0 we will get [P0,P1,P2] = [0,0,1].
    //
    // Following properties are guaranteed by network architecture:
    // * P0>=0, P1>=0, P2>=0    non-negativity
    // * P0+P1+P2=1             normalization
    //
    x = "[1]";
    mlpprocess(network, x, y);
    printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [1.000,0.000,0.000]
    x = "[-1]";
    mlpprocess(network, x, y);
    printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.000,1.000,0.000]
    x = "[0]";
    mlpprocess(network, x, y);
    printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.000,0.000,1.000]
    return 0;
}

nn_crossvalidation example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example shows how to perform cross-validation with ALGLIB
    //
    mlptrainer trn;
    multilayerperceptron network;
    mlpreport rep;

    //
    // Training set: f(x)=1/(x^2+1)
    // One row corresponds to one record [x,f(x)]
    //
    real_2d_array xy = "[[-2.0,0.2],[-1.6,0.3],[-1.3,0.4],[-1,0.5],[-0.6,0.7],[-0.3,0.9],[0,1],[2.0,0.2],[1.6,0.3],[1.3,0.4],[1,0.5],[0.6,0.7],[0.3,0.9]]";

    //
    // Trainer object is created.
    // Dataset is attached to trainer object.
    //
    // NOTE: it is not good idea to perform cross-validation on sample
    //       as small as ours (13 examples). It is done for demonstration
    //       purposes only. Generalization error estimates won't be
    //       precise enough for practical purposes.
    //
    mlpcreatetrainer(1, 1, trn);
    mlpsetdataset(trn, xy, 13);

    //
    // The key property of the cross-validation is that it estimates
    // generalization properties of neural ARCHITECTURE. It does NOT
    // estimates generalization error of some specific network which
    // is passed to the k-fold CV routine.
    //
    // In our example we create 1x4x1 neural network and pass it to
    // CV routine without training it. Original state of the network
    // is not used for cross-validation - each round is restarted from
    // random initial state. Only geometry of network matters.
    //
    // We perform 5 restarts from different random positions for each
    // of the 10 cross-validation rounds.
    //
    mlpcreate1(1, 4, 1, network);
    mlpkfoldcv(trn, network, 5, 10, rep);

    //
    // Cross-validation routine stores estimates of the generalization
    // error to MLP report structure. You may examine its fields and
    // see estimates of different errors (RMS, CE, Avg).
    //
    // Because cross-validation is non-deterministic, in our manual we
    // can not say what values will be stored to rep after call to
    // mlpkfoldcv(). Every CV round will return slightly different
    // estimates.
    //
    return 0;
}

nn_ensembles_es example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example shows how to train early stopping ensebles.
    //
    mlptrainer trn;
    mlpensemble ensemble;
    mlpreport rep;

    //
    // Training set: f(x)=1/(x^2+1)
    // One row corresponds to one record [x,f(x)]
    //
    real_2d_array xy = "[[-2.0,0.2],[-1.6,0.3],[-1.3,0.4],[-1,0.5],[-0.6,0.7],[-0.3,0.9],[0,1],[2.0,0.2],[1.6,0.3],[1.3,0.4],[1,0.5],[0.6,0.7],[0.3,0.9]]";

    //
    // Trainer object is created.
    // Dataset is attached to trainer object.
    //
    // NOTE: it is not good idea to use early stopping ensemble on sample
    //       as small as ours (13 examples). It is done for demonstration
    //       purposes only. Ensemble training algorithm won't find good
    //       solution on such small sample.
    //
    mlpcreatetrainer(1, 1, trn);
    mlpsetdataset(trn, xy, 13);

    //
    // Ensemble is created and trained. Each of 50 network is trained
    // with 5 restarts.
    //
    mlpecreate1(1, 4, 1, 50, ensemble);
    mlptrainensemblees(trn, ensemble, 5, rep);
    return 0;
}

nn_parallel example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example shows how to use parallel functionality of ALGLIB.
    // We generate simple 1-dimensional regression problem and show how
    // to use parallel training, parallel cross-validation, parallel
    // training of neural ensembles.
    //
    // We assume that you already know how to use ALGLIB in serial mode
    // and concentrate on its parallel capabilities.
    //
    // NOTE: it is not good idea to use parallel features on sample as small
    //       as ours (13 examples). It is done only for demonstration purposes.
    //
    mlptrainer trn;
    multilayerperceptron network;
    mlpensemble ensemble;
    mlpreport rep;
    real_2d_array xy = "[[-2.0,0.2],[-1.6,0.3],[-1.3,0.4],[-1,0.5],[-0.6,0.7],[-0.3,0.9],[0,1],[2.0,0.2],[1.6,0.3],[1.3,0.4],[1,0.5],[0.6,0.7],[0.3,0.9]]";
    mlpcreatetrainer(1, 1, trn);
    mlpsetdataset(trn, xy, 13);
    mlpcreate1(1, 4, 1, network);
    mlpecreate1(1, 4, 1, 50, ensemble);

    //
    // Below we demonstrate how to perform:
    // * parallel training of individual networks
    // * parallel cross-validation
    // * parallel training of neural ensembles
    //
    // In order to use multithreading, you have to:
    // 1) Install SMP edition of ALGLIB.
    // 2) This step is specific for C++ users: you should activate OS-specific
    //    capabilities of ALGLIB by defining AE_OS=AE_POSIX (for *nix systems)
    //    or AE_OS=AE_WINDOWS (for Windows systems).
    //    C# users do not have to perform this step because C# programs are
    //    portable across different systems without OS-specific tuning.
    // 3) Tell ALGLIB that you want it to use multithreading by means of
    //    setnworkers() call:
    //          * alglib::setnworkers(0)  = use all cores
    //          * alglib::setnworkers(-1) = leave one core unused
    //          * alglib::setnworkers(-2) = leave two cores unused
    //          * alglib::setnworkers(+2) = use 2 cores (even if you have more)
    //    During runtime ALGLIB will automatically determine whether it is
    //    feasible to start worker threads and split your task between cores.
    //
    alglib::setnworkers(+2);

    //
    // First, we perform parallel training of individual network with 5
    // restarts from random positions. These 5 rounds of  training  are
    // executed in parallel manner,  with  best  network  chosen  after
    // training.
    //
    // ALGLIB can use additional way to speed up computations -  divide
    // dataset   into   smaller   subsets   and   process these subsets
    // simultaneously. It allows us  to  efficiently  parallelize  even
    // single training round. This operation is performed automatically
    // for large datasets, but our toy dataset is too small.
    //
    mlptrainnetwork(trn, network, 5, rep);

    //
    // Then, we perform parallel 10-fold cross-validation, with 5 random
    // restarts per each CV round. I.e., 5*10=50  networks  are trained
    // in total. All these operations can be parallelized.
    //
    // NOTE: again, ALGLIB can parallelize  calculation   of   gradient
    //       over entire dataset - but our dataset is too small.
    //
    mlpkfoldcv(trn, network, 5, 10, rep);

    //
    // Finally, we train early stopping ensemble of 50 neural networks,
    // each  of them is trained with 5 random restarts. I.e.,  5*50=250
    // networks aretrained in total.
    //
    mlptrainensemblees(trn, ensemble, 5, rep);
    return 0;
}

nn_regr example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // The very simple example on neural network: network is trained to reproduce
    // small 2x2 multiplication table.
    //
    // NOTE: we use network with excessive amount of neurons, which guarantees
    //       almost exact reproduction of the training set. Generalization ability
    //       of such network is rather low, but we are not concerned with such
    //       questions in this basic demo.
    //
    mlptrainer trn;
    multilayerperceptron network;
    mlpreport rep;

    //
    // Training set:
    // * one row corresponds to one record A*B=C in the multiplication table
    // * first two columns store A and B, last column stores C
    //
    // [1 * 1 = 1]
    // [1 * 2 = 2]
    // [2 * 1 = 2]
    // [2 * 2 = 4]
    //
    real_2d_array xy = "[[1,1,1],[1,2,2],[2,1,2],[2,2,4]]";

    //
    // Network is created.
    // Trainer object is created.
    // Dataset is attached to trainer object.
    //
    mlpcreatetrainer(2, 1, trn);
    mlpcreate1(2, 5, 1, network);
    mlpsetdataset(trn, xy, 4);

    //
    // Network is trained with 5 restarts from random positions
    //
    mlptrainnetwork(trn, network, 5, rep);

    //
    // 2*2=?
    //
    real_1d_array x = "[2,2]";
    real_1d_array y = "[0]";
    mlpprocess(network, x, y);
    printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [4.000]
    return 0;
}

nn_regr_n example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Network with 2 inputs and 2 outputs is trained to reproduce vector function:
    //     (x0,x1) => (x0+x1, x0*x1)
    //
    // Informally speaking, we want neural network to simultaneously calculate
    // both sum of two numbers and their product.
    //
    // NOTE: we use network with excessive amount of neurons, which guarantees
    //       almost exact reproduction of the training set. Generalization ability
    //       of such network is rather low, but we are not concerned with such
    //       questions in this basic demo.
    //
    mlptrainer trn;
    multilayerperceptron network;
    mlpreport rep;

    //
    // Training set. One row corresponds to one record [A,B,A+B,A*B].
    //
    // [ 1   1  1+1  1*1 ]
    // [ 1   2  1+2  1*2 ]
    // [ 2   1  2+1  2*1 ]
    // [ 2   2  2+2  2*2 ]
    //
    real_2d_array xy = "[[1,1,2,1],[1,2,3,2],[2,1,3,2],[2,2,4,4]]";

    //
    // Network is created.
    // Trainer object is created.
    // Dataset is attached to trainer object.
    //
    mlpcreatetrainer(2, 2, trn);
    mlpcreate1(2, 5, 2, network);
    mlpsetdataset(trn, xy, 4);

    //
    // Network is trained with 5 restarts from random positions
    //
    mlptrainnetwork(trn, network, 5, rep);

    //
    // 2+1=?
    // 2*1=?
    //
    real_1d_array x = "[2,1]";
    real_1d_array y = "[0,0]";
    mlpprocess(network, x, y);
    printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [3.000,2.000]
    return 0;
}

nn_trainerobject example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Trainer object is used to train network. It stores dataset, training settings,
    // and other information which is NOT part of neural network. You should use
    // trainer object as follows:
    // (1) you create trainer object and specify task type (classification/regression)
    //     and number of inputs/outputs
    // (2) you add dataset to the trainer object
    // (3) you may change training settings (stopping criteria or weight decay)
    // (4) finally, you may train one or more networks
    //
    // You may interleave stages 2...4 and repeat them many times. Trainer object
    // remembers its internal state and can be used several times after its creation
    // and initialization.
    //
    mlptrainer trn;

    //
    // Stage 1: object creation.
    //
    // We have to specify number of inputs and outputs. Trainer object can be used
    // only for problems with same number of inputs/outputs as was specified during
    // its creation.
    //
    // In case you want to train SOFTMAX-normalized network which solves classification
    // problems,  you  must  use  another  function  to  create  trainer  object:
    // mlpcreatetrainercls().
    //
    // Below we create trainer object which can be used to train regression networks
    // with 2 inputs and 1 output.
    //
    mlpcreatetrainer(2, 1, trn);

    //
    // Stage 2: specification of the training set
    //
    // By default trainer object stores empty dataset. So to solve your non-empty problem
    // you have to set dataset by passing to trainer dense or sparse matrix.
    //
    // One row of the matrix corresponds to one record A*B=C in the multiplication table.
    // First two columns store A and B, last column stores C
    //
    //     [1 * 1 = 1]   [ 1 1 1 ]
    //     [1 * 2 = 2]   [ 1 2 2 ]
    //     [2 * 1 = 2] = [ 2 1 2 ]
    //     [2 * 2 = 4]   [ 2 2 4 ]
    //
    real_2d_array xy = "[[1,1,1],[1,2,2],[2,1,2],[2,2,4]]";
    mlpsetdataset(trn, xy, 4);

    //
    // Stage 3: modification of the training parameters.
    //
    // You may modify parameters like weights decay or stopping criteria:
    // * we set moderate weight decay
    // * we choose iterations limit as stopping condition (another condition - step size -
    //   is zero, which means than this condition is not active)
    //
    double wstep = 0.000;
    ae_int_t maxits = 100;
    mlpsetdecay(trn, 0.01);
    mlpsetcond(trn, wstep, maxits);

    //
    // Stage 4: training.
    //
    // We will train several networks with different architecture using same trainer object.
    // We may change training parameters or even dataset, so different networks are trained
    // differently. But in this simple example we will train all networks with same settings.
    //
    // We create and train three networks:
    // * network 1 has 2x1 architecture     (2 inputs, no hidden neurons, 1 output)
    // * network 2 has 2x5x1 architecture   (2 inputs, 5 hidden neurons, 1 output)
    // * network 3 has 2x5x5x1 architecture (2 inputs, two hidden layers, 1 output)
    //
    // NOTE: these networks solve regression problems. For classification problems you
    //       should use mlpcreatec0/c1/c2 to create neural networks which have SOFTMAX-
    //       normalized outputs.
    //
    multilayerperceptron net1;
    multilayerperceptron net2;
    multilayerperceptron net3;
    mlpreport rep;

    mlpcreate0(2, 1, net1);
    mlpcreate1(2, 5, 1, net2);
    mlpcreate2(2, 5, 5, 1, net3);

    mlptrainnetwork(trn, net1, 5, rep);
    mlptrainnetwork(trn, net2, 5, rep);
    mlptrainnetwork(trn, net3, 5, rep);
    return 0;
}

`nearestneighbor` subpackage

Classes

kdtree
kdtreerequestbuffer

Functions

kdtreebuild
kdtreebuildtagged
kdtreecreaterequestbuffer
kdtreequeryaknn
kdtreequerybox
kdtreequeryknn
kdtreequeryresultsdistances
kdtreequeryresultsdistancesi
kdtreequeryresultstags
kdtreequeryresultstagsi
kdtreequeryresultsx
kdtreequeryresultsxi
kdtreequeryresultsxy
kdtreequeryresultsxyi
kdtreequeryrnn
kdtreeserialize
kdtreetsqueryaknn
kdtreetsquerybox
kdtreetsqueryknn
kdtreetsqueryresultsdistances
kdtreetsqueryresultstags
kdtreetsqueryresultsx
kdtreetsqueryresultsxy
kdtreetsqueryrnn
kdtreeunserialize

Examples

nneighbor_d_1		Nearest neighbor search, KNN queries
nneighbor_d_2		Serialization of KD-trees

`kdtree` class

/*************************************************************************
KD-tree object.
*************************************************************************/
class kdtree
{
};

`kdtreerequestbuffer` class

/*************************************************************************
Buffer object which is used to perform nearest neighbor  requests  in  the
multithreaded mode (multiple threads working with same KD-tree object).

This object should be created with KDTreeCreateBuffer().
*************************************************************************/
class kdtreerequestbuffer
{
};

`kdtreebuild` function

/*************************************************************************
KD-tree creation

This subroutine creates KD-tree from set of X-values and optional Y-values

INPUT PARAMETERS
    XY      -   dataset, array[0..N-1,0..NX+NY-1].
                one row corresponds to one point.
                first NX columns contain X-values, next NY (NY may be zero)
                columns may contain associated Y-values
    N       -   number of points, N>=0.
    NX      -   space dimension, NX>=1.
    NY      -   number of optional Y-values, NY>=0.
    NormType-   norm type:
                * 0 denotes infinity-norm
                * 1 denotes 1-norm
                * 2 denotes 2-norm (Euclidean norm)

OUTPUT PARAMETERS
    KDT     -   KD-tree


NOTES

1. KD-tree  creation  have O(N*logN) complexity and O(N*(2*NX+NY))  memory
   requirements.
2. Although KD-trees may be used with any combination of N  and  NX,  they
   are more efficient than brute-force search only when N >> 4^NX. So they
   are most useful in low-dimensional tasks (NX=2, NX=3). NX=1  is another
   inefficient case, because  simple  binary  search  (without  additional
   structures) is much more efficient in such tasks than KD-trees.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreebuild(
    real_2d_array xy,
    ae_int_t nx,
    ae_int_t ny,
    ae_int_t normtype,
    kdtree& kdt,
    const xparams _params = alglib::xdefault);
void alglib::kdtreebuild(
    real_2d_array xy,
    ae_int_t n,
    ae_int_t nx,
    ae_int_t ny,
    ae_int_t normtype,
    kdtree& kdt,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`kdtreebuildtagged` function

/*************************************************************************
KD-tree creation

This  subroutine  creates  KD-tree  from set of X-values, integer tags and
optional Y-values

INPUT PARAMETERS
    XY      -   dataset, array[0..N-1,0..NX+NY-1].
                one row corresponds to one point.
                first NX columns contain X-values, next NY (NY may be zero)
                columns may contain associated Y-values
    Tags    -   tags, array[0..N-1], contains integer tags associated
                with points.
    N       -   number of points, N>=0
    NX      -   space dimension, NX>=1.
    NY      -   number of optional Y-values, NY>=0.
    NormType-   norm type:
                * 0 denotes infinity-norm
                * 1 denotes 1-norm
                * 2 denotes 2-norm (Euclidean norm)

OUTPUT PARAMETERS
    KDT     -   KD-tree

NOTES

1. KD-tree  creation  have O(N*logN) complexity and O(N*(2*NX+NY))  memory
   requirements.
2. Although KD-trees may be used with any combination of N  and  NX,  they
   are more efficient than brute-force search only when N >> 4^NX. So they
   are most useful in low-dimensional tasks (NX=2, NX=3). NX=1  is another
   inefficient case, because  simple  binary  search  (without  additional
   structures) is much more efficient in such tasks than KD-trees.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreebuildtagged(
    real_2d_array xy,
    integer_1d_array tags,
    ae_int_t nx,
    ae_int_t ny,
    ae_int_t normtype,
    kdtree& kdt,
    const xparams _params = alglib::xdefault);
void alglib::kdtreebuildtagged(
    real_2d_array xy,
    integer_1d_array tags,
    ae_int_t n,
    ae_int_t nx,
    ae_int_t ny,
    ae_int_t normtype,
    kdtree& kdt,
    const xparams _params = alglib::xdefault);

Examples: [1]

`kdtreecreaterequestbuffer` function

/*************************************************************************
This function creates buffer  structure  which  can  be  used  to  perform
parallel KD-tree requests.

KD-tree subpackage provides two sets of request functions - ones which use
internal buffer of KD-tree object  (these  functions  are  single-threaded
because they use same buffer, which can not shared between  threads),  and
ones which use external buffer.

This function is used to initialize external buffer.

INPUT PARAMETERS
    KDT         -   KD-tree which is associated with newly created buffer

OUTPUT PARAMETERS
    Buf         -   external buffer.


IMPORTANT: KD-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 18.03.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreecreaterequestbuffer(
    kdtree kdt,
    kdtreerequestbuffer& buf,
    const xparams _params = alglib::xdefault);

`kdtreequeryaknn` function

/*************************************************************************
K-NN query: approximate K nearest neighbors

IMPORTANT: this function can not be used in multithreaded code because  it
           uses internal temporary buffer of kd-tree object, which can not
           be shared between multiple threads.  If  you  want  to  perform
           parallel requests, use function  which  uses  external  request
           buffer: KDTreeTsQueryAKNN() ("Ts" stands for "thread-safe").

INPUT PARAMETERS
    KDT         -   KD-tree
    X           -   point, array[0..NX-1].
    K           -   number of neighbors to return, K>=1
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True
    Eps         -   approximation factor, Eps>=0. eps-approximate  nearest
                    neighbor  is  a  neighbor  whose distance from X is at
                    most (1+eps) times distance of true nearest neighbor.

RESULT
    number of actual neighbors found (either K or N, if K>N).

NOTES
    significant performance gain may be achieved only when Eps  is  is  on
    the order of magnitude of 1 or larger.

This  subroutine  performs  query  and  stores  its result in the internal
structures of the KD-tree. You can use  following  subroutines  to  obtain
these results:
* KDTreeQueryResultsX() to get X-values
* KDTreeQueryResultsXY() to get X- and Y-values
* KDTreeQueryResultsTags() to get tag values
* KDTreeQueryResultsDistances() to get distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::kdtreequeryaknn(
    kdtree kdt,
    real_1d_array x,
    ae_int_t k,
    double eps,
    const xparams _params = alglib::xdefault);
ae_int_t alglib::kdtreequeryaknn(
    kdtree kdt,
    real_1d_array x,
    ae_int_t k,
    bool selfmatch,
    double eps,
    const xparams _params = alglib::xdefault);

Examples: [1]

`kdtreequerybox` function

/*************************************************************************
Box query: all points within user-specified box.

IMPORTANT: this function can not be used in multithreaded code because  it
           uses internal temporary buffer of kd-tree object, which can not
           be shared between multiple threads.  If  you  want  to  perform
           parallel requests, use function  which  uses  external  request
           buffer: KDTreeTsQueryBox() ("Ts" stands for "thread-safe").

INPUT PARAMETERS
    KDT         -   KD-tree
    BoxMin      -   lower bounds, array[0..NX-1].
    BoxMax      -   upper bounds, array[0..NX-1].


RESULT
    number of actual neighbors found (in [0,N]).

This  subroutine  performs  query  and  stores  its result in the internal
structures of the KD-tree. You can use  following  subroutines  to  obtain
these results:
* KDTreeQueryResultsX() to get X-values
* KDTreeQueryResultsXY() to get X- and Y-values
* KDTreeQueryResultsTags() to get tag values
* KDTreeQueryResultsDistances() returns zeros for this request

NOTE: this particular query returns unordered results, because there is no
      meaningful way of  ordering  points.  Furthermore,  no 'distance' is
      associated with points - it is either INSIDE  or OUTSIDE (so request
      for distances will return zeros).

  -- ALGLIB --
     Copyright 14.05.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::kdtreequerybox(
    kdtree kdt,
    real_1d_array boxmin,
    real_1d_array boxmax,
    const xparams _params = alglib::xdefault);

`kdtreequeryknn` function

/*************************************************************************
K-NN query: K nearest neighbors

IMPORTANT: this function can not be used in multithreaded code because  it
           uses internal temporary buffer of kd-tree object, which can not
           be shared between multiple threads.  If  you  want  to  perform
           parallel requests, use function  which  uses  external  request
           buffer: KDTreeTsQueryKNN() ("Ts" stands for "thread-safe").

INPUT PARAMETERS
    KDT         -   KD-tree
    X           -   point, array[0..NX-1].
    K           -   number of neighbors to return, K>=1
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of actual neighbors found (either K or N, if K>N).

This  subroutine  performs  query  and  stores  its result in the internal
structures of the KD-tree. You can use  following  subroutines  to  obtain
these results:
* KDTreeQueryResultsX() to get X-values
* KDTreeQueryResultsXY() to get X- and Y-values
* KDTreeQueryResultsTags() to get tag values
* KDTreeQueryResultsDistances() to get distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::kdtreequeryknn(
    kdtree kdt,
    real_1d_array x,
    ae_int_t k,
    const xparams _params = alglib::xdefault);
ae_int_t alglib::kdtreequeryknn(
    kdtree kdt,
    real_1d_array x,
    ae_int_t k,
    bool selfmatch,
    const xparams _params = alglib::xdefault);

Examples: [1]

`kdtreequeryresultsdistances` function

/*************************************************************************
Distances from last query

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - kdtreetsqueryresultsdistances().

INPUT PARAMETERS
    KDT     -   KD-tree
    R       -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    R       -   filled with distances (in corresponding norm)

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsTags()          tag values

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreequeryresultsdistances(
    kdtree kdt,
    real_1d_array& r,
    const xparams _params = alglib::xdefault);

Examples: [1]

`kdtreequeryresultsdistancesi` function

/*************************************************************************
Distances from last query; 'interactive' variant for languages like Python
which  support  constructs   like  "R = KDTreeQueryResultsDistancesI(KDT)"
and interactive mode of interpreter.

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreequeryresultsdistancesi(
    kdtree kdt,
    real_1d_array& r,
    const xparams _params = alglib::xdefault);

`kdtreequeryresultstags` function

/*************************************************************************
Tags from last query

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - kdtreetsqueryresultstags().

INPUT PARAMETERS
    KDT     -   KD-tree
    Tags    -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    Tags    -   filled with tags associated with points,
                or, when no tags were supplied, with zeros

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreequeryresultstags(
    kdtree kdt,
    integer_1d_array& tags,
    const xparams _params = alglib::xdefault);

Examples: [1]

`kdtreequeryresultstagsi` function

/*************************************************************************
Tags  from  last  query;  'interactive' variant for languages like  Python
which  support  constructs  like "Tags = KDTreeQueryResultsTagsI(KDT)" and
interactive mode of interpreter.

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreequeryresultstagsi(
    kdtree kdt,
    integer_1d_array& tags,
    const xparams _params = alglib::xdefault);

`kdtreequeryresultsx` function

/*************************************************************************
X-values from last query.

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - kdtreetsqueryresultsx().

INPUT PARAMETERS
    KDT     -   KD-tree
    X       -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    X       -   rows are filled with X-values

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsTags()          tag values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreequeryresultsx(
    kdtree kdt,
    real_2d_array& x,
    const xparams _params = alglib::xdefault);

Examples: [1]

`kdtreequeryresultsxi` function

/*************************************************************************
X-values from last query; 'interactive' variant for languages like  Python
which   support    constructs   like  "X = KDTreeQueryResultsXI(KDT)"  and
interactive mode of interpreter.

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreequeryresultsxi(
    kdtree kdt,
    real_2d_array& x,
    const xparams _params = alglib::xdefault);

`kdtreequeryresultsxy` function

/*************************************************************************
X- and Y-values from last query

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - kdtreetsqueryresultsxy().

INPUT PARAMETERS
    KDT     -   KD-tree
    XY      -   possibly pre-allocated buffer. If XY is too small to store
                result, it is resized. If size(XY) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    XY      -   rows are filled with points: first NX columns with
                X-values, next NY columns - with Y-values.

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsTags()          tag values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreequeryresultsxy(
    kdtree kdt,
    real_2d_array& xy,
    const xparams _params = alglib::xdefault);

Examples: [1]

`kdtreequeryresultsxyi` function

/*************************************************************************
XY-values from last query; 'interactive' variant for languages like Python
which   support    constructs   like "XY = KDTreeQueryResultsXYI(KDT)" and
interactive mode of interpreter.

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreequeryresultsxyi(
    kdtree kdt,
    real_2d_array& xy,
    const xparams _params = alglib::xdefault);

`kdtreequeryrnn` function

/*************************************************************************
R-NN query: all points within R-sphere centered at X

IMPORTANT: this function can not be used in multithreaded code because  it
           uses internal temporary buffer of kd-tree object, which can not
           be shared between multiple threads.  If  you  want  to  perform
           parallel requests, use function  which  uses  external  request
           buffer: KDTreeTsQueryRNN() ("Ts" stands for "thread-safe").

INPUT PARAMETERS
    KDT         -   KD-tree
    X           -   point, array[0..NX-1].
    R           -   radius of sphere (in corresponding norm), R>0
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of neighbors found, >=0

This  subroutine  performs  query  and  stores  its result in the internal
structures of the KD-tree. You can use  following  subroutines  to  obtain
actual results:
* KDTreeQueryResultsX() to get X-values
* KDTreeQueryResultsXY() to get X- and Y-values
* KDTreeQueryResultsTags() to get tag values
* KDTreeQueryResultsDistances() to get distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::kdtreequeryrnn(
    kdtree kdt,
    real_1d_array x,
    double r,
    const xparams _params = alglib::xdefault);
ae_int_t alglib::kdtreequeryrnn(
    kdtree kdt,
    real_1d_array x,
    double r,
    bool selfmatch,
    const xparams _params = alglib::xdefault);

Examples: [1]

`kdtreeserialize` function

/*************************************************************************
This function serializes data structure to string.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string  in  case you want to include it into text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor you should change case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize  it  in  C++ version of ALGLIB and unserialize in C# one, 
  and vice versa.
*************************************************************************/
void kdtreeserialize(kdtree &obj, std::string &s_out);
void kdtreeserialize(kdtree &obj, std::ostream &s_out);

`kdtreetsqueryaknn` function

/*************************************************************************
K-NN query: approximate K nearest neighbors, using thread-local buffer.

You can call this function from multiple threads for same kd-tree instance,
assuming that different instances of buffer object are passed to different
threads.

INPUT PARAMETERS
    KDT         -   KD-tree
    Buf         -   request buffer  object  created  for  this  particular
                    instance of kd-tree structure with kdtreecreaterequestbuffer()
                    function.
    X           -   point, array[0..NX-1].
    K           -   number of neighbors to return, K>=1
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True
    Eps         -   approximation factor, Eps>=0. eps-approximate  nearest
                    neighbor  is  a  neighbor  whose distance from X is at
                    most (1+eps) times distance of true nearest neighbor.

RESULT
    number of actual neighbors found (either K or N, if K>N).

NOTES
    significant performance gain may be achieved only when Eps  is  is  on
    the order of magnitude of 1 or larger.

This  subroutine  performs  query  and  stores  its result in the internal
structures  of  the  buffer object. You can use following  subroutines  to
obtain these results (pay attention to "buf" in their names):
* KDTreeTsQueryResultsX() to get X-values
* KDTreeTsQueryResultsXY() to get X- and Y-values
* KDTreeTsQueryResultsTags() to get tag values
* KDTreeTsQueryResultsDistances() to get distances

IMPORTANT: kd-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 18.03.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::kdtreetsqueryaknn(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_1d_array x,
    ae_int_t k,
    double eps,
    const xparams _params = alglib::xdefault);
ae_int_t alglib::kdtreetsqueryaknn(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_1d_array x,
    ae_int_t k,
    bool selfmatch,
    double eps,
    const xparams _params = alglib::xdefault);

`kdtreetsquerybox` function

/*************************************************************************
Box query: all points within user-specified box, using thread-local buffer.

You can call this function from multiple threads for same kd-tree instance,
assuming that different instances of buffer object are passed to different
threads.

INPUT PARAMETERS
    KDT         -   KD-tree
    Buf         -   request buffer  object  created  for  this  particular
                    instance of kd-tree structure with kdtreecreaterequestbuffer()
                    function.
    BoxMin      -   lower bounds, array[0..NX-1].
    BoxMax      -   upper bounds, array[0..NX-1].

RESULT
    number of actual neighbors found (in [0,N]).

This  subroutine  performs  query  and  stores  its result in the internal
structures  of  the  buffer object. You can use following  subroutines  to
obtain these results (pay attention to "ts" in their names):
* KDTreeTsQueryResultsX() to get X-values
* KDTreeTsQueryResultsXY() to get X- and Y-values
* KDTreeTsQueryResultsTags() to get tag values
* KDTreeTsQueryResultsDistances() returns zeros for this query

NOTE: this particular query returns unordered results, because there is no
      meaningful way of  ordering  points.  Furthermore,  no 'distance' is
      associated with points - it is either INSIDE  or OUTSIDE (so request
      for distances will return zeros).

IMPORTANT: kd-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 14.05.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::kdtreetsquerybox(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_1d_array boxmin,
    real_1d_array boxmax,
    const xparams _params = alglib::xdefault);

`kdtreetsqueryknn` function

/*************************************************************************
K-NN query: K nearest neighbors, using external thread-local buffer.

You can call this function from multiple threads for same kd-tree instance,
assuming that different instances of buffer object are passed to different
threads.

INPUT PARAMETERS
    KDT         -   kd-tree
    Buf         -   request buffer  object  created  for  this  particular
                    instance of kd-tree structure with kdtreecreaterequestbuffer()
                    function.
    X           -   point, array[0..NX-1].
    K           -   number of neighbors to return, K>=1
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of actual neighbors found (either K or N, if K>N).

This  subroutine  performs  query  and  stores  its result in the internal
structures  of  the  buffer object. You can use following  subroutines  to
obtain these results (pay attention to "buf" in their names):
* KDTreeTsQueryResultsX() to get X-values
* KDTreeTsQueryResultsXY() to get X- and Y-values
* KDTreeTsQueryResultsTags() to get tag values
* KDTreeTsQueryResultsDistances() to get distances

IMPORTANT: kd-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 18.03.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::kdtreetsqueryknn(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_1d_array x,
    ae_int_t k,
    const xparams _params = alglib::xdefault);
ae_int_t alglib::kdtreetsqueryknn(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_1d_array x,
    ae_int_t k,
    bool selfmatch,
    const xparams _params = alglib::xdefault);

`kdtreetsqueryresultsdistances` function

/*************************************************************************
Distances from last query associated with kdtreerequestbuffer object.

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - KDTreeTsqueryresultsdistances().

INPUT PARAMETERS
    KDT     -   KD-tree
    Buf     -   request  buffer  object  created   for   this   particular
                instance of kd-tree structure.
    R       -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    R       -   filled with distances (in corresponding norm)

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsTags()          tag values

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreetsqueryresultsdistances(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_1d_array& r,
    const xparams _params = alglib::xdefault);

`kdtreetsqueryresultstags` function

/*************************************************************************
Tags from last query associated with kdtreerequestbuffer object.

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - KDTreeTsqueryresultstags().

INPUT PARAMETERS
    KDT     -   KD-tree
    Buf     -   request  buffer  object  created   for   this   particular
                instance of kd-tree structure.
    Tags    -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    Tags    -   filled with tags associated with points,
                or, when no tags were supplied, with zeros

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreetsqueryresultstags(
    kdtree kdt,
    kdtreerequestbuffer buf,
    integer_1d_array& tags,
    const xparams _params = alglib::xdefault);

`kdtreetsqueryresultsx` function

/*************************************************************************
X-values from last query associated with kdtreerequestbuffer object.

INPUT PARAMETERS
    KDT     -   KD-tree
    Buf     -   request  buffer  object  created   for   this   particular
                instance of kd-tree structure.
    X       -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    X       -   rows are filled with X-values

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsTags()          tag values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreetsqueryresultsx(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_2d_array& x,
    const xparams _params = alglib::xdefault);

`kdtreetsqueryresultsxy` function

/*************************************************************************
X- and Y-values from last query associated with kdtreerequestbuffer object.

INPUT PARAMETERS
    KDT     -   KD-tree
    Buf     -   request  buffer  object  created   for   this   particular
                instance of kd-tree structure.
    XY      -   possibly pre-allocated buffer. If XY is too small to store
                result, it is resized. If size(XY) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    XY      -   rows are filled with points: first NX columns with
                X-values, next NY columns - with Y-values.

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsTags()          tag values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::kdtreetsqueryresultsxy(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_2d_array& xy,
    const xparams _params = alglib::xdefault);

`kdtreetsqueryrnn` function

/*************************************************************************
R-NN query: all points within  R-sphere  centered  at  X,  using  external
thread-local buffer.

You can call this function from multiple threads for same kd-tree instance,
assuming that different instances of buffer object are passed to different
threads.

INPUT PARAMETERS
    KDT         -   KD-tree
    Buf         -   request buffer  object  created  for  this  particular
                    instance of kd-tree structure with kdtreecreaterequestbuffer()
                    function.
    X           -   point, array[0..NX-1].
    R           -   radius of sphere (in corresponding norm), R>0
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of neighbors found, >=0

This  subroutine  performs  query  and  stores  its result in the internal
structures  of  the  buffer object. You can use following  subroutines  to
obtain these results (pay attention to "buf" in their names):
* KDTreeTsQueryResultsX() to get X-values
* KDTreeTsQueryResultsXY() to get X- and Y-values
* KDTreeTsQueryResultsTags() to get tag values
* KDTreeTsQueryResultsDistances() to get distances

IMPORTANT: kd-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 18.03.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::kdtreetsqueryrnn(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_1d_array x,
    double r,
    const xparams _params = alglib::xdefault);
ae_int_t alglib::kdtreetsqueryrnn(
    kdtree kdt,
    kdtreerequestbuffer buf,
    real_1d_array x,
    double r,
    bool selfmatch,
    const xparams _params = alglib::xdefault);

`kdtreeunserialize` function

/*************************************************************************
This function unserializes data structure from string.
*************************************************************************/
void kdtreeunserialize(const std::string &s_in, kdtree &obj);
void kdtreeunserialize(const std::istream &s_in, kdtree &obj);

nneighbor_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "alglibmisc.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_2d_array a = "[[0,0],[0,1],[1,0],[1,1]]";
    ae_int_t nx = 2;
    ae_int_t ny = 0;
    ae_int_t normtype = 2;
    kdtree kdt;
    real_1d_array x;
    real_2d_array r = "[[]]";
    ae_int_t k;
    kdtreebuild(a, nx, ny, normtype, kdt);
    x = "[-1,0]";
    k = kdtreequeryknn(kdt, x, 1);
    printf("%d\n", int(k)); // EXPECTED: 1
    kdtreequeryresultsx(kdt, r);
    printf("%s\n", r.tostring(1).c_str()); // EXPECTED: [[0,0]]
    return 0;
}

nneighbor_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "alglibmisc.h"

using namespace alglib;


int main(int argc, char **argv)
{
    real_2d_array a = "[[0,0],[0,1],[1,0],[1,1]]";
    ae_int_t nx = 2;
    ae_int_t ny = 0;
    ae_int_t normtype = 2;
    kdtree kdt0;
    kdtree kdt1;
    std::string s;
    real_1d_array x;
    real_2d_array r0 = "[[]]";
    real_2d_array r1 = "[[]]";

    //
    // Build tree and serialize it
    //
    kdtreebuild(a, nx, ny, normtype, kdt0);
    alglib::kdtreeserialize(kdt0, s);
    alglib::kdtreeunserialize(s, kdt1);

    //
    // Compare results from KNN queries
    //
    x = "[-1,0]";
    kdtreequeryknn(kdt0, x, 1);
    kdtreequeryresultsx(kdt0, r0);
    kdtreequeryknn(kdt1, x, 1);
    kdtreequeryresultsx(kdt1, r1);
    printf("%s\n", r0.tostring(1).c_str()); // EXPECTED: [[0,0]]
    printf("%s\n", r1.tostring(1).c_str()); // EXPECTED: [[0,0]]
    return 0;
}

`nleq` subpackage

Classes

nleqreport
nleqstate

Functions

nleqcreatelm
nleqrestartfrom
nleqresults
nleqresultsbuf
nleqsetcond
nleqsetstpmax
nleqsetxrep
nleqsolve

Examples

`nleqreport` class

/*************************************************************************

*************************************************************************/
class nleqreport
{
    ae_int_t             iterationscount;
    ae_int_t             nfunc;
    ae_int_t             njac;
    ae_int_t             terminationtype;
};

`nleqstate` class

/*************************************************************************

*************************************************************************/
class nleqstate
{
};

`nleqcreatelm` function

/*************************************************************************
                LEVENBERG-MARQUARDT-LIKE NONLINEAR SOLVER

DESCRIPTION:
This algorithm solves system of nonlinear equations
    F[0](x[0], ..., x[n-1])   = 0
    F[1](x[0], ..., x[n-1])   = 0
    ...
    F[M-1](x[0], ..., x[n-1]) = 0
with M/N do not necessarily coincide.  Algorithm  converges  quadratically
under following conditions:
    * the solution set XS is nonempty
    * for some xs in XS there exist such neighbourhood N(xs) that:
      * vector function F(x) and its Jacobian J(x) are continuously
        differentiable on N
      * ||F(x)|| provides local error bound on N, i.e. there  exists  such
        c1, that ||F(x)||>c1*distance(x,XS)
Note that these conditions are much more weaker than usual non-singularity
conditions. For example, algorithm will converge for any  affine  function
F (whether its Jacobian singular or not).


REQUIREMENTS:
Algorithm will request following information during its operation:
* function vector F[] and Jacobian matrix at given point X
* value of merit function f(x)=F[0]^2(x)+...+F[M-1]^2(x) at given point X


USAGE:
1. User initializes algorithm state with NLEQCreateLM() call
2. User tunes solver parameters with  NLEQSetCond(),  NLEQSetStpMax()  and
   other functions
3. User  calls  NLEQSolve()  function  which  takes  algorithm  state  and
   pointers (delegates, etc.) to callback functions which calculate  merit
   function value and Jacobian.
4. User calls NLEQResults() to get solution
5. Optionally, user may call NLEQRestartFrom() to  solve  another  problem
   with same parameters (N/M) but another starting  point  and/or  another
   function vector. NLEQRestartFrom() allows to reuse already  initialized
   structure.


INPUT PARAMETERS:
    N       -   space dimension, N>1:
                * if provided, only leading N elements of X are used
                * if not provided, determined automatically from size of X
    M       -   system size
    X       -   starting point


OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state


NOTES:
1. you may tune stopping conditions with NLEQSetCond() function
2. if target function contains exp() or other fast growing functions,  and
   optimization algorithm makes too large steps which leads  to  overflow,
   use NLEQSetStpMax() function to bound algorithm's steps.
3. this  algorithm  is  a  slightly  modified implementation of the method
   described  in  'Levenberg-Marquardt  method  for constrained  nonlinear
   equations with strong local convergence properties' by Christian Kanzow
   Nobuo Yamashita and Masao Fukushima and further  developed  in  'On the
   convergence of a New Levenberg-Marquardt Method'  by  Jin-yan  Fan  and
   Ya-Xiang Yuan.


  -- ALGLIB --
     Copyright 20.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::nleqcreatelm(
    ae_int_t m,
    real_1d_array x,
    nleqstate& state,
    const xparams _params = alglib::xdefault);
void alglib::nleqcreatelm(
    ae_int_t n,
    ae_int_t m,
    real_1d_array x,
    nleqstate& state,
    const xparams _params = alglib::xdefault);

`nleqrestartfrom` function

/*************************************************************************
This  subroutine  restarts  CG  algorithm from new point. All optimization
parameters are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure used for reverse communication previously
                allocated with MinCGCreate call.
    X       -   new starting point.
    BndL    -   new lower bounds
    BndU    -   new upper bounds

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::nleqrestartfrom(
    nleqstate state,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

`nleqresults` function

/*************************************************************************
NLEQ solver results

INPUT PARAMETERS:
    State   -   algorithm state.

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    * -4    ERROR:  algorithm   has   converged   to   the
                            stationary point Xf which is local minimum  of
                            f=F[0]^2+...+F[m-1]^2, but is not solution  of
                            nonlinear system.
                    *  1    sqrt(f)<=EpsF.
                    *  5    MaxIts steps was taken
                    *  7    stopping conditions are too stringent,
                            further improvement is impossible
                * Rep.IterationsCount contains iterations count
                * NFEV countains number of function calculations
                * ActiveConstraints contains number of active constraints

  -- ALGLIB --
     Copyright 20.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::nleqresults(
    nleqstate state,
    real_1d_array& x,
    nleqreport& rep,
    const xparams _params = alglib::xdefault);

`nleqresultsbuf` function

/*************************************************************************
NLEQ solver results

Buffered implementation of NLEQResults(), which uses pre-allocated  buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 20.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::nleqresultsbuf(
    nleqstate state,
    real_1d_array& x,
    nleqreport& rep,
    const xparams _params = alglib::xdefault);

`nleqsetcond` function

/*************************************************************************
This function sets stopping conditions for the nonlinear solver

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsF    -   >=0
                The subroutine finishes  its work if on k+1-th iteration
                the condition ||F||<=EpsF is satisfied
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsF=0 and MaxIts=0 simultaneously will lead to  automatic
stopping criterion selection (small EpsF).

NOTES:

  -- ALGLIB --
     Copyright 20.08.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::nleqsetcond(
    nleqstate state,
    double epsf,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

`nleqsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when target function  contains  exp()  or  other  fast
growing functions, and algorithm makes  too  large  steps  which  lead  to
overflow. This function allows us to reject steps that are too large  (and
therefore expose us to the possible overflow) without actually calculating
function value at the x+stp*d.

  -- ALGLIB --
     Copyright 20.08.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::nleqsetstpmax(
    nleqstate state,
    double stpmax,
    const xparams _params = alglib::xdefault);

`nleqsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to NLEQSolve().

  -- ALGLIB --
     Copyright 20.08.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::nleqsetxrep(
    nleqstate state,
    bool needxrep,
    const xparams _params = alglib::xdefault);

`nleqsolve` function

/*************************************************************************
This family of functions is used to launcn iterations of nonlinear solver

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void nleqsolve(nleqstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

`normaldistr` subpackage

Functions

errorfunction
errorfunctionc
inverf
invnormaldistribution
normaldistribution

Examples

`errorfunction` function

/*************************************************************************
Error function

The integral is

                          x
                           -
                2         | |          2
  erf(x)  =  --------     |    exp( - t  ) dt.
             sqrt(pi)   | |
                         -
                          0

For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
erf(x) = 1 - erfc(x).


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,1         30000       3.7e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::errorfunction(
    double x,
    const xparams _params = alglib::xdefault);

`errorfunctionc` function

/*************************************************************************
Complementary error function

 1 - erf(x) =

                          inf.
                            -
                 2         | |          2
  erfc(x)  =  --------     |    exp( - t  ) dt
              sqrt(pi)   | |
                          -
                           x


For small x, erfc(x) = 1 - erf(x); otherwise rational
approximations are computed.


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,26.6417   30000       5.7e-14     1.5e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::errorfunctionc(
    double x,
    const xparams _params = alglib::xdefault);

`inverf` function

/*************************************************************************
Inverse of the error function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::inverf(double e, const xparams _params = alglib::xdefault);

`invnormaldistribution` function

/*************************************************************************
Inverse of Normal distribution function

Returns the argument, x, for which the area under the
Gaussian probability density function (integrated from
minus infinity to x) is equal to y.


For small arguments 0 < y < exp(-2), the program computes
z = sqrt( -2.0 * log(y) );  then the approximation is
x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
There are two rational functions P/Q, one for 0 < y < exp(-32)
and the other for y up to exp(-2).  For larger arguments,
w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).

ACCURACY:

                     Relative error:
arithmetic   domain        # trials      peak         rms
   IEEE     0.125, 1        20000       7.2e-16     1.3e-16
   IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::invnormaldistribution(
    double y0,
    const xparams _params = alglib::xdefault);

`normaldistribution` function

/*************************************************************************
Normal distribution function

Returns the area under the Gaussian probability density
function, integrated from minus infinity to x:

                           x
                            -
                  1        | |          2
   ndtr(x)  = ---------    |    exp( - t /2 ) dt
              sqrt(2pi)  | |
                          -
                         -inf.

            =  ( 1 + erf(z) ) / 2
            =  erfc(z) / 2

where z = x/sqrt(2). Computation is via the functions
erf and erfc.


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -13,0        30000       3.4e-14     6.7e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::normaldistribution(
    double x,
    const xparams _params = alglib::xdefault);

`normestimator` subpackage

Classes

normestimatorstate

Functions

normestimatorcreate
normestimatorestimatesparse
normestimatorresults
normestimatorsetseed

Examples

`normestimatorstate` class

/*************************************************************************
This object stores state of the iterative norm estimation algorithm.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class normestimatorstate
{
};

`normestimatorcreate` function

/*************************************************************************
This procedure initializes matrix norm estimator.

USAGE:
1. User initializes algorithm state with NormEstimatorCreate() call
2. User calls NormEstimatorEstimateSparse() (or NormEstimatorIteration())
3. User calls NormEstimatorResults() to get solution.

INPUT PARAMETERS:
    M       -   number of rows in the matrix being estimated, M>0
    N       -   number of columns in the matrix being estimated, N>0
    NStart  -   number of random starting vectors
                recommended value - at least 5.
    NIts    -   number of iterations to do with best starting vector
                recommended value - at least 5.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state


NOTE: this algorithm is effectively deterministic, i.e. it always  returns
same result when repeatedly called for the same matrix. In fact, algorithm
uses randomized starting vectors, but internal  random  numbers  generator
always generates same sequence of the random values (it is a  feature, not
bug).

Algorithm can be made non-deterministic with NormEstimatorSetSeed(0) call.

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::normestimatorcreate(
    ae_int_t m,
    ae_int_t n,
    ae_int_t nstart,
    ae_int_t nits,
    normestimatorstate& state,
    const xparams _params = alglib::xdefault);

`normestimatorestimatesparse` function

/*************************************************************************
This function estimates norm of the sparse M*N matrix A.

INPUT PARAMETERS:
    State       -   norm estimator state, must be initialized with a  call
                    to NormEstimatorCreate()
    A           -   sparse M*N matrix, must be converted to CRS format
                    prior to calling this function.

After this function  is  over  you can call NormEstimatorResults() to get
estimate of the norm(A).

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::normestimatorestimatesparse(
    normestimatorstate state,
    sparsematrix a,
    const xparams _params = alglib::xdefault);

`normestimatorresults` function

/*************************************************************************
Matrix norm estimation results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    Nrm     -   estimate of the matrix norm, Nrm>=0

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::normestimatorresults(
    normestimatorstate state,
    double& nrm,
    const xparams _params = alglib::xdefault);

`normestimatorsetseed` function

/*************************************************************************
This function changes seed value used by algorithm. In some cases we  need
deterministic processing, i.e. subsequent calls must return equal results,
in other cases we need non-deterministic algorithm which returns different
results for the same matrix on every pass.

Setting zero seed will lead to non-deterministic algorithm, while non-zero
value will make our algorithm deterministic.

INPUT PARAMETERS:
    State       -   norm estimator state, must be initialized with a  call
                    to NormEstimatorCreate()
    SeedVal     -   seed value, >=0. Zero value = non-deterministic algo.

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::normestimatorsetseed(
    normestimatorstate state,
    ae_int_t seedval,
    const xparams _params = alglib::xdefault);

`odesolver` subpackage

Classes

odesolverreport
odesolverstate

Functions

odesolverresults
odesolverrkck
odesolversolve

Examples

odesolver_d1

Solving y'=-y with ODE solver

`odesolverreport` class

/*************************************************************************

*************************************************************************/
class odesolverreport
{
    ae_int_t             nfev;
    ae_int_t             terminationtype;
};

`odesolverstate` class

/*************************************************************************

*************************************************************************/
class odesolverstate
{
};

`odesolverresults` function

/*************************************************************************
ODE solver results

Called after OdeSolverIteration returned False.

INPUT PARAMETERS:
    State   -   algorithm state (used by OdeSolverIteration).

OUTPUT PARAMETERS:
    M       -   number of tabulated values, M>=1
    XTbl    -   array[0..M-1], values of X
    YTbl    -   array[0..M-1,0..N-1], values of Y in X[i]
    Rep     -   solver report:
                * Rep.TerminationType completetion code:
                    * -2    X is not ordered  by  ascending/descending  or
                            there are non-distinct X[],  i.e.  X[i]=X[i+1]
                    * -1    incorrect parameters were specified
                    *  1    task has been solved
                * Rep.NFEV contains number of function calculations

  -- ALGLIB --
     Copyright 01.09.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::odesolverresults(
    odesolverstate state,
    ae_int_t& m,
    real_1d_array& xtbl,
    real_2d_array& ytbl,
    odesolverreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`odesolverrkck` function

/*************************************************************************
Cash-Karp adaptive ODE solver.

This subroutine solves ODE  Y'=f(Y,x)  with  initial  conditions  Y(xs)=Ys
(here Y may be single variable or vector of N variables).

INPUT PARAMETERS:
    Y       -   initial conditions, array[0..N-1].
                contains values of Y[] at X[0]
    N       -   system size
    X       -   points at which Y should be tabulated, array[0..M-1]
                integrations starts at X[0], ends at X[M-1],  intermediate
                values at X[i] are returned too.
                SHOULD BE ORDERED BY ASCENDING OR BY DESCENDING!
    M       -   number of intermediate points + first point + last point:
                * M>2 means that you need both Y(X[M-1]) and M-2 values at
                  intermediate points
                * M=2 means that you want just to integrate from  X[0]  to
                  X[1] and don't interested in intermediate values.
                * M=1 means that you don't want to integrate :)
                  it is degenerate case, but it will be handled correctly.
                * M<1 means error
    Eps     -   tolerance (absolute/relative error on each  step  will  be
                less than Eps). When passing:
                * Eps>0, it means desired ABSOLUTE error
                * Eps<0, it means desired RELATIVE error.  Relative errors
                  are calculated with respect to maximum values of  Y seen
                  so far. Be careful to use this criterion  when  starting
                  from Y[] that are close to zero.
    H       -   initial  step  lenth,  it  will  be adjusted automatically
                after the first  step.  If  H=0,  step  will  be  selected
                automatically  (usualy  it  will  be  equal  to  0.001  of
                min(x[i]-x[j])).

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state between  subsequent
                calls of OdeSolverIteration. Used for reverse communication.
                This structure should be passed  to the OdeSolverIteration
                subroutine.

SEE ALSO
    AutoGKSmoothW, AutoGKSingular, AutoGKIteration, AutoGKResults.


  -- ALGLIB --
     Copyright 01.09.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::odesolverrkck(
    real_1d_array y,
    real_1d_array x,
    double eps,
    double h,
    odesolverstate& state,
    const xparams _params = alglib::xdefault);
void alglib::odesolverrkck(
    real_1d_array y,
    ae_int_t n,
    real_1d_array x,
    ae_int_t m,
    double eps,
    double h,
    odesolverstate& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`odesolversolve` function

/*************************************************************************
This function is used to launcn iterations of ODE solver

It accepts following parameters:
    diff    -   callback which calculates dy/dx for given y and x
    ptr     -   optional pointer which is passed to diff; can be NULL


  -- ALGLIB --
     Copyright 01.09.2009 by Bochkanov Sergey
*************************************************************************/
void odesolversolve(odesolverstate &state,
    void (*diff)(const real_1d_array &y, double x, real_1d_array &dy, void *ptr),
    void *ptr = NULL, const xparams _xparams = alglib::xdefault);

Examples: [1]

odesolver_d1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "diffequations.h"

using namespace alglib;
void ode_function_1_diff(const real_1d_array &y, double x, real_1d_array &dy, void *ptr) 
{
    // this callback calculates f(y[],x)=-y[0]
    dy[0] = -y[0];
}

int main(int argc, char **argv)
{
    real_1d_array y = "[1]";
    real_1d_array x = "[0, 1, 2, 3]";
    double eps = 0.00001;
    double h = 0;
    odesolverstate s;
    ae_int_t m;
    real_1d_array xtbl;
    real_2d_array ytbl;
    odesolverreport rep;
    odesolverrkck(y, x, eps, h, s);
    alglib::odesolversolve(s, ode_function_1_diff);
    odesolverresults(s, m, xtbl, ytbl, rep);
    printf("%d\n", int(m)); // EXPECTED: 4
    printf("%s\n", xtbl.tostring(2).c_str()); // EXPECTED: [0, 1, 2, 3]
    printf("%s\n", ytbl.tostring(2).c_str()); // EXPECTED: [[1], [0.367], [0.135], [0.050]]
    return 0;
}

`ortfac` subpackage

Functions

cmatrixlq
cmatrixlqunpackl
cmatrixlqunpackq
cmatrixqr
cmatrixqrunpackq
cmatrixqrunpackr
hmatrixtd
hmatrixtdunpackq
rmatrixbd
rmatrixbdmultiplybyp
rmatrixbdmultiplybyq
rmatrixbdunpackdiagonals
rmatrixbdunpackpt
rmatrixbdunpackq
rmatrixhessenberg
rmatrixhessenbergunpackh
rmatrixhessenbergunpackq
rmatrixlq
rmatrixlqunpackl
rmatrixlqunpackq
rmatrixqr
rmatrixqrunpackq
rmatrixqrunpackr
smatrixtd
smatrixtdunpackq

Examples

`cmatrixlq` function

/*************************************************************************
LQ decomposition of a rectangular complex matrix of size MxN

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A   -   matrix A whose indexes range within [0..M-1, 0..N-1]
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.

Output parameters:
    A   -   matrices Q and L in compact form
    Tau -   array of scalar factors which are used to form matrix Q. Array
            whose indexes range within [0.. Min(M,N)-1]

Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
MxM, L - lower triangular (or lower trapezoid) matrix of size MxN.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994
*************************************************************************/
void alglib::cmatrixlq(
    complex_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    complex_1d_array& tau,
    const xparams _params = alglib::xdefault);

`cmatrixlqunpackl` function

/*************************************************************************
Unpacking of matrix L from the LQ decomposition of a matrix A

Input parameters:
    A       -   matrices Q and L in compact form.
                Output of CMatrixLQ subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.

Output parameters:
    L       -   matrix L, array[0..M-1, 0..N-1].

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixlqunpackl(
    complex_2d_array a,
    ae_int_t m,
    ae_int_t n,
    complex_2d_array& l,
    const xparams _params = alglib::xdefault);

`cmatrixlqunpackq` function

/*************************************************************************
Partial unpacking of matrix Q from LQ decomposition of a complex matrix A.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A           -   matrices Q and R in compact form.
                    Output of CMatrixLQ subroutine .
    M           -   number of rows in matrix A. M>=0.
    N           -   number of columns in matrix A. N>=0.
    Tau         -   scalar factors which are used to form Q.
                    Output of CMatrixLQ subroutine .
    QRows       -   required number of rows in matrix Q. N>=QColumns>=0.

Output parameters:
    Q           -   first QRows rows of matrix Q.
                    Array whose index ranges within [0..QRows-1, 0..N-1].
                    If QRows=0, array isn't changed.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixlqunpackq(
    complex_2d_array a,
    ae_int_t m,
    ae_int_t n,
    complex_1d_array tau,
    ae_int_t qrows,
    complex_2d_array& q,
    const xparams _params = alglib::xdefault);

`cmatrixqr` function

/*************************************************************************
QR decomposition of a rectangular complex matrix of size MxN

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A   -   matrix A whose indexes range within [0..M-1, 0..N-1]
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.

Output parameters:
    A   -   matrices Q and R in compact form
    Tau -   array of scalar factors which are used to form matrix Q. Array
            whose indexes range within [0.. Min(M,N)-1]

Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
MxM, R - upper triangular (or upper trapezoid) matrix of size MxN.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994
*************************************************************************/
void alglib::cmatrixqr(
    complex_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    complex_1d_array& tau,
    const xparams _params = alglib::xdefault);

`cmatrixqrunpackq` function

/*************************************************************************
Partial unpacking of matrix Q from QR decomposition of a complex matrix A.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A           -   matrices Q and R in compact form.
                    Output of CMatrixQR subroutine .
    M           -   number of rows in matrix A. M>=0.
    N           -   number of columns in matrix A. N>=0.
    Tau         -   scalar factors which are used to form Q.
                    Output of CMatrixQR subroutine .
    QColumns    -   required number of columns in matrix Q. M>=QColumns>=0.

Output parameters:
    Q           -   first QColumns columns of matrix Q.
                    Array whose index ranges within [0..M-1, 0..QColumns-1].
                    If QColumns=0, array isn't changed.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixqrunpackq(
    complex_2d_array a,
    ae_int_t m,
    ae_int_t n,
    complex_1d_array tau,
    ae_int_t qcolumns,
    complex_2d_array& q,
    const xparams _params = alglib::xdefault);

`cmatrixqrunpackr` function

/*************************************************************************
Unpacking of matrix R from the QR decomposition of a matrix A

Input parameters:
    A       -   matrices Q and R in compact form.
                Output of CMatrixQR subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.

Output parameters:
    R       -   matrix R, array[0..M-1, 0..N-1].

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixqrunpackr(
    complex_2d_array a,
    ae_int_t m,
    ae_int_t n,
    complex_2d_array& r,
    const xparams _params = alglib::xdefault);

`hmatrixtd` function

/*************************************************************************
Reduction of a Hermitian matrix which is given  by  its  higher  or  lower
triangular part to a real  tridiagonal  matrix  using  unitary  similarity
transformation: Q'*A*Q = T.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix to be transformed
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format. If IsUpper = True, then matrix A is  given
                by its upper triangle, and the lower triangle is not  used
                and not modified by the algorithm, and vice versa
                if IsUpper = False.

Output parameters:
    A       -   matrices T and Q in  compact form (see lower)
    Tau     -   array of factors which are forming matrices H(i)
                array with elements [0..N-2].
    D       -   main diagonal of real symmetric matrix T.
                array with elements [0..N-1].
    E       -   secondary diagonal of real symmetric matrix T.
                array with elements [0..N-2].


  If IsUpper=True, the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-2) . . . H(2) H(0).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a complex scalar, and v is a complex vector with
  v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
  A(0:i-1,i+1), and tau in TAU(i).

  If IsUpper=False, the matrix Q is represented as a product of elementary
  reflectors

     Q = H(0) H(2) . . . H(n-2).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a complex scalar, and v is a complex vector with
  v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
  and tau in TAU(i).

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  d   e   v1  v2  v3 )              (  d                  )
    (      d   e   v2  v3 )              (  e   d              )
    (          d   e   v3 )              (  v0  e   d          )
    (              d   e  )              (  v0  v1  e   d      )
    (                  d  )              (  v0  v1  v2  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1992
*************************************************************************/
void alglib::hmatrixtd(
    complex_2d_array& a,
    ae_int_t n,
    bool isupper,
    complex_1d_array& tau,
    real_1d_array& d,
    real_1d_array& e,
    const xparams _params = alglib::xdefault);

`hmatrixtdunpackq` function

/*************************************************************************
Unpacking matrix Q which reduces a Hermitian matrix to a real  tridiagonal
form.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   the result of a HMatrixTD subroutine
    N       -   size of matrix A.
    IsUpper -   storage format (a parameter of HMatrixTD subroutine)
    Tau     -   the result of a HMatrixTD subroutine

Output parameters:
    Q       -   transformation matrix.
                array with elements [0..N-1, 0..N-1].

  -- ALGLIB --
     Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void alglib::hmatrixtdunpackq(
    complex_2d_array a,
    ae_int_t n,
    bool isupper,
    complex_1d_array tau,
    complex_2d_array& q,
    const xparams _params = alglib::xdefault);

`rmatrixbd` function

/*************************************************************************
Reduction of a rectangular matrix to  bidiagonal form

The algorithm reduces the rectangular matrix A to  bidiagonal form by
orthogonal transformations P and Q: A = Q*B*(P^T).

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   source matrix. array[0..M-1, 0..N-1]
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.

Output parameters:
    A       -   matrices Q, B, P in compact form (see below).
    TauQ    -   scalar factors which are used to form matrix Q.
    TauP    -   scalar factors which are used to form matrix P.

The main diagonal and one of the  secondary  diagonals  of  matrix  A  are
replaced with bidiagonal  matrix  B.  Other  elements  contain  elementary
reflections which form MxM matrix Q and NxN matrix P, respectively.

If M>=N, B is the upper  bidiagonal  MxN  matrix  and  is  stored  in  the
corresponding  elements  of  matrix  A.  Matrix  Q  is  represented  as  a
product   of   elementary   reflections   Q = H(0)*H(1)*...*H(n-1),  where
H(i) = 1-tau*v*v'. Here tau is a scalar which is stored  in  TauQ[i],  and
vector v has the following  structure:  v(0:i-1)=0, v(i)=1, v(i+1:m-1)  is
stored   in   elements   A(i+1:m-1,i).   Matrix   P  is  as  follows:  P =
G(0)*G(1)*...*G(n-2), where G(i) = 1 - tau*u*u'. Tau is stored in TauP[i],
u(0:i)=0, u(i+1)=1, u(i+2:n-1) is stored in elements A(i,i+2:n-1).

If M<N, B is the  lower  bidiagonal  MxN  matrix  and  is  stored  in  the
corresponding   elements  of  matrix  A.  Q = H(0)*H(1)*...*H(m-2),  where
H(i) = 1 - tau*v*v', tau is stored in TauQ, v(0:i)=0, v(i+1)=1, v(i+2:m-1)
is    stored    in   elements   A(i+2:m-1,i).    P = G(0)*G(1)*...*G(m-1),
G(i) = 1-tau*u*u', tau is stored in  TauP,  u(0:i-1)=0, u(i)=1, u(i+1:n-1)
is stored in A(i,i+1:n-1).

EXAMPLE:

m=6, n=5 (m > n):               m=5, n=6 (m < n):

(  d   e   u1  u1  u1 )         (  d   u1  u1  u1  u1  u1 )
(  v1  d   e   u2  u2 )         (  e   d   u2  u2  u2  u2 )
(  v1  v2  d   e   u3 )         (  v1  e   d   u3  u3  u3 )
(  v1  v2  v3  d   e  )         (  v1  v2  e   d   u4  u4 )
(  v1  v2  v3  v4  d  )         (  v1  v2  v3  e   d   u5 )
(  v1  v2  v3  v4  v5 )

Here vi and ui are vectors which form H(i) and G(i), and d and e -
are the diagonal and off-diagonal elements of matrix B.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994.
     Sergey Bochkanov, ALGLIB project, translation from FORTRAN to
     pseudocode, 2007-2010.
*************************************************************************/
void alglib::rmatrixbd(
    real_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    real_1d_array& tauq,
    real_1d_array& taup,
    const xparams _params = alglib::xdefault);

`rmatrixbdmultiplybyp` function

/*************************************************************************
Multiplication by matrix P which reduces matrix A to  bidiagonal form.

The algorithm allows pre- or post-multiply by P or P'.

Input parameters:
    QP          -   matrices Q and P in compact form.
                    Output of RMatrixBD subroutine.
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    TAUP        -   scalar factors which are used to form P.
                    Output of RMatrixBD subroutine.
    Z           -   multiplied matrix.
                    Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
    ZRows       -   number of rows in matrix Z. If FromTheRight=False,
                    ZRows=N, otherwise ZRows can be arbitrary.
    ZColumns    -   number of columns in matrix Z. If FromTheRight=True,
                    ZColumns=N, otherwise ZColumns can be arbitrary.
    FromTheRight -  pre- or post-multiply.
    DoTranspose -   multiply by P or P'.

Output parameters:
    Z - product of Z and P.
                Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
                If ZRows=0 or ZColumns=0, the array is not modified.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixbdmultiplybyp(
    real_2d_array qp,
    ae_int_t m,
    ae_int_t n,
    real_1d_array taup,
    real_2d_array& z,
    ae_int_t zrows,
    ae_int_t zcolumns,
    bool fromtheright,
    bool dotranspose,
    const xparams _params = alglib::xdefault);

`rmatrixbdmultiplybyq` function

/*************************************************************************
Multiplication by matrix Q which reduces matrix A to  bidiagonal form.

The algorithm allows pre- or post-multiply by Q or Q'.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    QP          -   matrices Q and P in compact form.
                    Output of ToBidiagonal subroutine.
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    TAUQ        -   scalar factors which are used to form Q.
                    Output of ToBidiagonal subroutine.
    Z           -   multiplied matrix.
                    array[0..ZRows-1,0..ZColumns-1]
    ZRows       -   number of rows in matrix Z. If FromTheRight=False,
                    ZRows=M, otherwise ZRows can be arbitrary.
    ZColumns    -   number of columns in matrix Z. If FromTheRight=True,
                    ZColumns=M, otherwise ZColumns can be arbitrary.
    FromTheRight -  pre- or post-multiply.
    DoTranspose -   multiply by Q or Q'.

Output parameters:
    Z           -   product of Z and Q.
                    Array[0..ZRows-1,0..ZColumns-1]
                    If ZRows=0 or ZColumns=0, the array is not modified.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixbdmultiplybyq(
    real_2d_array qp,
    ae_int_t m,
    ae_int_t n,
    real_1d_array tauq,
    real_2d_array& z,
    ae_int_t zrows,
    ae_int_t zcolumns,
    bool fromtheright,
    bool dotranspose,
    const xparams _params = alglib::xdefault);

`rmatrixbdunpackdiagonals` function

/*************************************************************************
Unpacking of the main and secondary diagonals of bidiagonal decomposition
of matrix A.

Input parameters:
    B   -   output of RMatrixBD subroutine.
    M   -   number of rows in matrix B.
    N   -   number of columns in matrix B.

Output parameters:
    IsUpper -   True, if the matrix is upper bidiagonal.
                otherwise IsUpper is False.
    D       -   the main diagonal.
                Array whose index ranges within [0..Min(M,N)-1].
    E       -   the secondary diagonal (upper or lower, depending on
                the value of IsUpper).
                Array index ranges within [0..Min(M,N)-1], the last
                element is not used.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixbdunpackdiagonals(
    real_2d_array b,
    ae_int_t m,
    ae_int_t n,
    bool& isupper,
    real_1d_array& d,
    real_1d_array& e,
    const xparams _params = alglib::xdefault);

`rmatrixbdunpackpt` function

/*************************************************************************
Unpacking matrix P which reduces matrix A to bidiagonal form.
The subroutine returns transposed matrix P.

Input parameters:
    QP      -   matrices Q and P in compact form.
                Output of ToBidiagonal subroutine.
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.
    TAUP    -   scalar factors which are used to form P.
                Output of ToBidiagonal subroutine.
    PTRows  -   required number of rows of matrix P^T. N >= PTRows >= 0.

Output parameters:
    PT      -   first PTRows columns of matrix P^T
                Array[0..PTRows-1, 0..N-1]
                If PTRows=0, the array is not modified.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixbdunpackpt(
    real_2d_array qp,
    ae_int_t m,
    ae_int_t n,
    real_1d_array taup,
    ae_int_t ptrows,
    real_2d_array& pt,
    const xparams _params = alglib::xdefault);

`rmatrixbdunpackq` function

/*************************************************************************
Unpacking matrix Q which reduces a matrix to bidiagonal form.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    QP          -   matrices Q and P in compact form.
                    Output of ToBidiagonal subroutine.
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    TAUQ        -   scalar factors which are used to form Q.
                    Output of ToBidiagonal subroutine.
    QColumns    -   required number of columns in matrix Q.
                    M>=QColumns>=0.

Output parameters:
    Q           -   first QColumns columns of matrix Q.
                    Array[0..M-1, 0..QColumns-1]
                    If QColumns=0, the array is not modified.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixbdunpackq(
    real_2d_array qp,
    ae_int_t m,
    ae_int_t n,
    real_1d_array tauq,
    ae_int_t qcolumns,
    real_2d_array& q,
    const xparams _params = alglib::xdefault);

`rmatrixhessenberg` function

/*************************************************************************
Reduction of a square matrix to  upper Hessenberg form: Q'*A*Q = H,
where Q is an orthogonal matrix, H - Hessenberg matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix A with elements [0..N-1, 0..N-1]
    N       -   size of matrix A.

Output parameters:
    A       -   matrices Q and P in  compact form (see below).
    Tau     -   array of scalar factors which are used to form matrix Q.
                Array whose index ranges within [0..N-2]

Matrix H is located on the main diagonal, on the lower secondary  diagonal
and above the main diagonal of matrix A. The elements which are used to
form matrix Q are situated in array Tau and below the lower secondary
diagonal of matrix A as follows:

Matrix Q is represented as a product of elementary reflections

Q = H(0)*H(2)*...*H(n-2),

where each H(i) is given by

H(i) = 1 - tau * v * (v^T)

where tau is a scalar stored in Tau[I]; v - is a real vector,
so that v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) stored in A(i+2:n-1,i).

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1992
*************************************************************************/
void alglib::rmatrixhessenberg(
    real_2d_array& a,
    ae_int_t n,
    real_1d_array& tau,
    const xparams _params = alglib::xdefault);

`rmatrixhessenbergunpackh` function

/*************************************************************************
Unpacking matrix H (the result of matrix A reduction to upper Hessenberg form)

Input parameters:
    A   -   output of RMatrixHessenberg subroutine.
    N   -   size of matrix A.

Output parameters:
    H   -   matrix H. Array whose indexes range within [0..N-1, 0..N-1].

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixhessenbergunpackh(
    real_2d_array a,
    ae_int_t n,
    real_2d_array& h,
    const xparams _params = alglib::xdefault);

`rmatrixhessenbergunpackq` function

/*************************************************************************
Unpacking matrix Q which reduces matrix A to upper Hessenberg form

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A   -   output of RMatrixHessenberg subroutine.
    N   -   size of matrix A.
    Tau -   scalar factors which are used to form Q.
            Output of RMatrixHessenberg subroutine.

Output parameters:
    Q   -   matrix Q.
            Array whose indexes range within [0..N-1, 0..N-1].

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixhessenbergunpackq(
    real_2d_array a,
    ae_int_t n,
    real_1d_array tau,
    real_2d_array& q,
    const xparams _params = alglib::xdefault);

`rmatrixlq` function

/*************************************************************************
LQ decomposition of a rectangular matrix of size MxN

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A   -   matrix A whose indexes range within [0..M-1, 0..N-1].
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.

Output parameters:
    A   -   matrices L and Q in compact form (see below)
    Tau -   array of scalar factors which are used to form
            matrix Q. Array whose index ranges within [0..Min(M,N)-1].

Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
MxM, L - lower triangular (or lower trapezoid) matrix of size M x N.

The elements of matrix L are located on and below  the  main  diagonal  of
matrix A. The elements which are located in Tau array and above  the  main
diagonal of matrix A are used to form matrix Q as follows:

Matrix Q is represented as a product of elementary reflections

Q = H(k-1)*H(k-2)*...*H(1)*H(0),

where k = min(m,n), and each H(i) is of the form

H(i) = 1 - tau * v * (v^T)

where tau is a scalar stored in Tau[I]; v - real vector, so that v(0:i-1)=0,
v(i) = 1, v(i+1:n-1) stored in A(i,i+1:n-1).

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixlq(
    real_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    real_1d_array& tau,
    const xparams _params = alglib::xdefault);

`rmatrixlqunpackl` function

/*************************************************************************
Unpacking of matrix L from the LQ decomposition of a matrix A

Input parameters:
    A       -   matrices Q and L in compact form.
                Output of RMatrixLQ subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.

Output parameters:
    L       -   matrix L, array[0..M-1, 0..N-1].

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixlqunpackl(
    real_2d_array a,
    ae_int_t m,
    ae_int_t n,
    real_2d_array& l,
    const xparams _params = alglib::xdefault);

`rmatrixlqunpackq` function

/*************************************************************************
Partial unpacking of matrix Q from the LQ decomposition of a matrix A

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrices L and Q in compact form.
                Output of RMatrixLQ subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.
    Tau     -   scalar factors which are used to form Q.
                Output of the RMatrixLQ subroutine.
    QRows   -   required number of rows in matrix Q. N>=QRows>=0.

Output parameters:
    Q       -   first QRows rows of matrix Q. Array whose indexes range
                within [0..QRows-1, 0..N-1]. If QRows=0, the array remains
                unchanged.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixlqunpackq(
    real_2d_array a,
    ae_int_t m,
    ae_int_t n,
    real_1d_array tau,
    ae_int_t qrows,
    real_2d_array& q,
    const xparams _params = alglib::xdefault);

`rmatrixqr` function

/*************************************************************************
QR decomposition of a rectangular matrix of size MxN

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A   -   matrix A whose indexes range within [0..M-1, 0..N-1].
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.

Output parameters:
    A   -   matrices Q and R in compact form (see below).
    Tau -   array of scalar factors which are used to form
            matrix Q. Array whose index ranges within [0.. Min(M-1,N-1)].

Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
MxM, R - upper triangular (or upper trapezoid) matrix of size M x N.

The elements of matrix R are located on and above the main diagonal of
matrix A. The elements which are located in Tau array and below the main
diagonal of matrix A are used to form matrix Q as follows:

Matrix Q is represented as a product of elementary reflections

Q = H(0)*H(2)*...*H(k-1),

where k = min(m,n), and each H(i) is in the form

H(i) = 1 - tau * v * (v^T)

where tau is a scalar stored in Tau[I]; v - real vector,
so that v(0:i-1) = 0, v(i) = 1, v(i+1:m-1) stored in A(i+1:m-1,i).

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixqr(
    real_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    real_1d_array& tau,
    const xparams _params = alglib::xdefault);

`rmatrixqrunpackq` function

/*************************************************************************
Partial unpacking of matrix Q from the QR decomposition of a matrix A

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrices Q and R in compact form.
                Output of RMatrixQR subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.
    Tau     -   scalar factors which are used to form Q.
                Output of the RMatrixQR subroutine.
    QColumns -  required number of columns of matrix Q. M>=QColumns>=0.

Output parameters:
    Q       -   first QColumns columns of matrix Q.
                Array whose indexes range within [0..M-1, 0..QColumns-1].
                If QColumns=0, the array remains unchanged.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixqrunpackq(
    real_2d_array a,
    ae_int_t m,
    ae_int_t n,
    real_1d_array tau,
    ae_int_t qcolumns,
    real_2d_array& q,
    const xparams _params = alglib::xdefault);

`rmatrixqrunpackr` function

/*************************************************************************
Unpacking of matrix R from the QR decomposition of a matrix A

Input parameters:
    A       -   matrices Q and R in compact form.
                Output of RMatrixQR subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.

Output parameters:
    R       -   matrix R, array[0..M-1, 0..N-1].

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixqrunpackr(
    real_2d_array a,
    ae_int_t m,
    ae_int_t n,
    real_2d_array& r,
    const xparams _params = alglib::xdefault);

`smatrixtd` function

/*************************************************************************
Reduction of a symmetric matrix which is given by its higher or lower
triangular part to a tridiagonal matrix using orthogonal similarity
transformation: Q'*A*Q=T.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix to be transformed
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format. If IsUpper = True, then matrix A is given
                by its upper triangle, and the lower triangle is not used
                and not modified by the algorithm, and vice versa
                if IsUpper = False.

Output parameters:
    A       -   matrices T and Q in  compact form (see lower)
    Tau     -   array of factors which are forming matrices H(i)
                array with elements [0..N-2].
    D       -   main diagonal of symmetric matrix T.
                array with elements [0..N-1].
    E       -   secondary diagonal of symmetric matrix T.
                array with elements [0..N-2].


  If IsUpper=True, the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-2) . . . H(2) H(0).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
  A(0:i-1,i+1), and tau in TAU(i).

  If IsUpper=False, the matrix Q is represented as a product of elementary
  reflectors

     Q = H(0) H(2) . . . H(n-2).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
  and tau in TAU(i).

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  d   e   v1  v2  v3 )              (  d                  )
    (      d   e   v2  v3 )              (  e   d              )
    (          d   e   v3 )              (  v0  e   d          )
    (              d   e  )              (  v0  v1  e   d      )
    (                  d  )              (  v0  v1  v2  e   d  )

  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i).

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1992
*************************************************************************/
void alglib::smatrixtd(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    real_1d_array& tau,
    real_1d_array& d,
    real_1d_array& e,
    const xparams _params = alglib::xdefault);

`smatrixtdunpackq` function

/*************************************************************************
Unpacking matrix Q which reduces symmetric matrix to a tridiagonal
form.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   the result of a SMatrixTD subroutine
    N       -   size of matrix A.
    IsUpper -   storage format (a parameter of SMatrixTD subroutine)
    Tau     -   the result of a SMatrixTD subroutine

Output parameters:
    Q       -   transformation matrix.
                array with elements [0..N-1, 0..N-1].

  -- ALGLIB --
     Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void alglib::smatrixtdunpackq(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    real_1d_array tau,
    real_2d_array& q,
    const xparams _params = alglib::xdefault);

`parametric` subpackage

Classes

pspline2interpolant
pspline3interpolant

Functions

parametricrdpfixed
pspline2arclength
pspline2build
pspline2buildperiodic
pspline2calc
pspline2diff
pspline2diff2
pspline2parametervalues
pspline2tangent
pspline3arclength
pspline3build
pspline3buildperiodic
pspline3calc
pspline3diff
pspline3diff2
pspline3parametervalues
pspline3tangent

Examples

parametric_rdp

Parametric Ramer-Douglas-Peucker approximation

`pspline2interpolant` class

/*************************************************************************
Parametric spline inteprolant: 2-dimensional curve.

You should not try to access its members directly - use PSpline2XXXXXXXX()
functions instead.
*************************************************************************/
class pspline2interpolant
{
};

`pspline3interpolant` class

/*************************************************************************
Parametric spline inteprolant: 3-dimensional curve.

You should not try to access its members directly - use PSpline3XXXXXXXX()
functions instead.
*************************************************************************/
class pspline3interpolant
{
};

`parametricrdpfixed` function

/*************************************************************************
This  subroutine fits piecewise linear curve to points with Ramer-Douglas-
Peucker algorithm. This  function  performs PARAMETRIC fit, i.e. it can be
used to fit curves like circles.

On  input  it  accepts dataset which describes parametric multidimensional
curve X(t), with X being vector, and t taking values in [0,N), where N  is
a number of points in dataset. As result, it returns reduced  dataset  X2,
which can be used to build  parametric  curve  X2(t),  which  approximates
X(t) with desired precision (or has specified number of sections).


INPUT PARAMETERS:
    X       -   array of multidimensional points:
                * at least N elements, leading N elements are used if more
                  than N elements were specified
                * order of points is IMPORTANT because  it  is  parametric
                  fit
                * each row of array is one point which has D coordinates
    N       -   number of elements in X
    D       -   number of dimensions (elements per row of X)
    StopM   -   stopping condition - desired number of sections:
                * at most M sections are generated by this function
                * less than M sections can be generated if we have N<M
                  (or some X are non-distinct).
                * zero StopM means that algorithm does not stop after
                  achieving some pre-specified section count
    StopEps -   stopping condition - desired precision:
                * algorithm stops after error in each section is at most Eps
                * zero Eps means that algorithm does not stop after
                  achieving some pre-specified precision

OUTPUT PARAMETERS:
    X2      -   array of corner points for piecewise approximation,
                has length NSections+1 or zero (for NSections=0).
    Idx2    -   array of indexes (parameter values):
                * has length NSections+1 or zero (for NSections=0).
                * each element of Idx2 corresponds to same-numbered
                  element of X2
                * each element of Idx2 is index of  corresponding  element
                  of X2 at original array X, i.e. I-th  row  of  X2  is
                  Idx2[I]-th row of X.
                * elements of Idx2 can be treated as parameter values
                  which should be used when building new parametric curve
                * Idx2[0]=0, Idx2[NSections]=N-1
    NSections-  number of sections found by algorithm, NSections<=M,
                NSections can be zero for degenerate datasets
                (N<=1 or all X[] are non-distinct).

NOTE: algorithm stops after:
      a) dividing curve into StopM sections
      b) achieving required precision StopEps
      c) dividing curve into N-1 sections
      If both StopM and StopEps are non-zero, algorithm is stopped by  the
      FIRST criterion which is satisfied. In case both StopM  and  StopEps
      are zero, algorithm stops because of (c).

  -- ALGLIB --
     Copyright 02.10.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::parametricrdpfixed(
    real_2d_array x,
    ae_int_t n,
    ae_int_t d,
    ae_int_t stopm,
    double stopeps,
    real_2d_array& x2,
    integer_1d_array& idx2,
    ae_int_t& nsections,
    const xparams _params = alglib::xdefault);

Examples: [1]

`pspline2arclength` function

/*************************************************************************
This function  calculates  arc length, i.e. length of  curve  between  t=a
and t=b.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    A,B -   parameter values corresponding to arc ends:
            * B>A will result in positive length returned
            * B<A will result in negative length returned

RESULT:
    length of arc starting at T=A and ending at T=B.


  -- ALGLIB PROJECT --
     Copyright 30.05.2010 by Bochkanov Sergey
*************************************************************************/
double alglib::pspline2arclength(
    pspline2interpolant p,
    double a,
    double b,
    const xparams _params = alglib::xdefault);

`pspline2build` function

/*************************************************************************
This function  builds  non-periodic 2-dimensional parametric spline  which
starts at (X[0],Y[0]) and ends at (X[N-1],Y[N-1]).

INPUT PARAMETERS:
    XY  -   points, array[0..N-1,0..1].
            XY[I,0:1] corresponds to the Ith point.
            Order of points is important!
    N   -   points count, N>=5 for Akima splines, N>=2 for other types  of
            splines.
    ST  -   spline type:
            * 0     Akima spline
            * 1     parabolically terminated Catmull-Rom spline (Tension=0)
            * 2     parabolically terminated cubic spline
    PT  -   parameterization type:
            * 0     uniform
            * 1     chord length
            * 2     centripetal

OUTPUT PARAMETERS:
    P   -   parametric spline interpolant


NOTES:
* this function  assumes  that  there all consequent points  are distinct.
  I.e. (x0,y0)<>(x1,y1),  (x1,y1)<>(x2,y2),  (x2,y2)<>(x3,y3)  and  so on.
  However, non-consequent points may coincide, i.e. we can  have  (x0,y0)=
  =(x2,y2).

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline2build(
    real_2d_array xy,
    ae_int_t n,
    ae_int_t st,
    ae_int_t pt,
    pspline2interpolant& p,
    const xparams _params = alglib::xdefault);

`pspline2buildperiodic` function

/*************************************************************************
This  function  builds  periodic  2-dimensional  parametric  spline  which
starts at (X[0],Y[0]), goes through all points to (X[N-1],Y[N-1]) and then
back to (X[0],Y[0]).

INPUT PARAMETERS:
    XY  -   points, array[0..N-1,0..1].
            XY[I,0:1] corresponds to the Ith point.
            XY[N-1,0:1] must be different from XY[0,0:1].
            Order of points is important!
    N   -   points count, N>=3 for other types of splines.
    ST  -   spline type:
            * 1     Catmull-Rom spline (Tension=0) with cyclic boundary conditions
            * 2     cubic spline with cyclic boundary conditions
    PT  -   parameterization type:
            * 0     uniform
            * 1     chord length
            * 2     centripetal

OUTPUT PARAMETERS:
    P   -   parametric spline interpolant


NOTES:
* this function  assumes  that there all consequent points  are  distinct.
  I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2),  (x2,y2)<>(x3,y3)  and  so  on.
  However, non-consequent points may coincide, i.e. we can  have  (x0,y0)=
  =(x2,y2).
* last point of sequence is NOT equal to the first  point.  You  shouldn't
  make curve "explicitly periodic" by making them equal.

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline2buildperiodic(
    real_2d_array xy,
    ae_int_t n,
    ae_int_t st,
    ae_int_t pt,
    pspline2interpolant& p,
    const xparams _params = alglib::xdefault);

`pspline2calc` function

/*************************************************************************
This function  calculates  the value of the parametric spline for a  given
value of parameter T

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-position
    Y   -   Y-position


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline2calc(
    pspline2interpolant p,
    double t,
    double& x,
    double& y,
    const xparams _params = alglib::xdefault);

`pspline2diff` function

/*************************************************************************
This function calculates derivative, i.e. it returns (dX/dT,dY/dT).

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-value
    DX  -   X-derivative
    Y   -   Y-value
    DY  -   Y-derivative


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline2diff(
    pspline2interpolant p,
    double t,
    double& x,
    double& dx,
    double& y,
    double& dy,
    const xparams _params = alglib::xdefault);

`pspline2diff2` function

/*************************************************************************
This function calculates first and second derivative with respect to T.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-value
    DX  -   derivative
    D2X -   second derivative
    Y   -   Y-value
    DY  -   derivative
    D2Y -   second derivative


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline2diff2(
    pspline2interpolant p,
    double t,
    double& x,
    double& dx,
    double& d2x,
    double& y,
    double& dy,
    double& d2y,
    const xparams _params = alglib::xdefault);

`pspline2parametervalues` function

/*************************************************************************
This function returns vector of parameter values correspoding to points.

I.e. for P created from (X[0],Y[0])...(X[N-1],Y[N-1]) and U=TValues(P)  we
have
    (X[0],Y[0]) = PSpline2Calc(P,U[0]),
    (X[1],Y[1]) = PSpline2Calc(P,U[1]),
    (X[2],Y[2]) = PSpline2Calc(P,U[2]),
    ...

INPUT PARAMETERS:
    P   -   parametric spline interpolant

OUTPUT PARAMETERS:
    N   -   array size
    T   -   array[0..N-1]


NOTES:
* for non-periodic splines U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]=1
* for periodic splines     U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]<1

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline2parametervalues(
    pspline2interpolant p,
    ae_int_t& n,
    real_1d_array& t,
    const xparams _params = alglib::xdefault);

`pspline2tangent` function

/*************************************************************************
This function  calculates  tangent vector for a given value of parameter T

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X    -   X-component of tangent vector (normalized)
    Y    -   Y-component of tangent vector (normalized)

NOTE:
    X^2+Y^2 is either 1 (for non-zero tangent vector) or 0.


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline2tangent(
    pspline2interpolant p,
    double t,
    double& x,
    double& y,
    const xparams _params = alglib::xdefault);

`pspline3arclength` function

/*************************************************************************
This function  calculates  arc length, i.e. length of  curve  between  t=a
and t=b.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    A,B -   parameter values corresponding to arc ends:
            * B>A will result in positive length returned
            * B<A will result in negative length returned

RESULT:
    length of arc starting at T=A and ending at T=B.


  -- ALGLIB PROJECT --
     Copyright 30.05.2010 by Bochkanov Sergey
*************************************************************************/
double alglib::pspline3arclength(
    pspline3interpolant p,
    double a,
    double b,
    const xparams _params = alglib::xdefault);

`pspline3build` function

/*************************************************************************
This function  builds  non-periodic 3-dimensional parametric spline  which
starts at (X[0],Y[0],Z[0]) and ends at (X[N-1],Y[N-1],Z[N-1]).

Same as PSpline2Build() function, but for 3D, so we  won't  duplicate  its
description here.

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline3build(
    real_2d_array xy,
    ae_int_t n,
    ae_int_t st,
    ae_int_t pt,
    pspline3interpolant& p,
    const xparams _params = alglib::xdefault);

`pspline3buildperiodic` function

/*************************************************************************
This  function  builds  periodic  3-dimensional  parametric  spline  which
starts at (X[0],Y[0],Z[0]), goes through all points to (X[N-1],Y[N-1],Z[N-1])
and then back to (X[0],Y[0],Z[0]).

Same as PSpline2Build() function, but for 3D, so we  won't  duplicate  its
description here.

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline3buildperiodic(
    real_2d_array xy,
    ae_int_t n,
    ae_int_t st,
    ae_int_t pt,
    pspline3interpolant& p,
    const xparams _params = alglib::xdefault);

`pspline3calc` function

/*************************************************************************
This function  calculates  the value of the parametric spline for a  given
value of parameter T.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-position
    Y   -   Y-position
    Z   -   Z-position


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline3calc(
    pspline3interpolant p,
    double t,
    double& x,
    double& y,
    double& z,
    const xparams _params = alglib::xdefault);

`pspline3diff` function

/*************************************************************************
This function calculates derivative, i.e. it returns (dX/dT,dY/dT,dZ/dT).

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-value
    DX  -   X-derivative
    Y   -   Y-value
    DY  -   Y-derivative
    Z   -   Z-value
    DZ  -   Z-derivative


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline3diff(
    pspline3interpolant p,
    double t,
    double& x,
    double& dx,
    double& y,
    double& dy,
    double& z,
    double& dz,
    const xparams _params = alglib::xdefault);

`pspline3diff2` function

/*************************************************************************
This function calculates first and second derivative with respect to T.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-value
    DX  -   derivative
    D2X -   second derivative
    Y   -   Y-value
    DY  -   derivative
    D2Y -   second derivative
    Z   -   Z-value
    DZ  -   derivative
    D2Z -   second derivative


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline3diff2(
    pspline3interpolant p,
    double t,
    double& x,
    double& dx,
    double& d2x,
    double& y,
    double& dy,
    double& d2y,
    double& z,
    double& dz,
    double& d2z,
    const xparams _params = alglib::xdefault);

`pspline3parametervalues` function

/*************************************************************************
This function returns vector of parameter values correspoding to points.

Same as PSpline2ParameterValues(), but for 3D.

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline3parametervalues(
    pspline3interpolant p,
    ae_int_t& n,
    real_1d_array& t,
    const xparams _params = alglib::xdefault);

`pspline3tangent` function

/*************************************************************************
This function  calculates  tangent vector for a given value of parameter T

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X    -   X-component of tangent vector (normalized)
    Y    -   Y-component of tangent vector (normalized)
    Z    -   Z-component of tangent vector (normalized)

NOTE:
    X^2+Y^2+Z^2 is either 1 (for non-zero tangent vector) or 0.


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::pspline3tangent(
    pspline3interpolant p,
    double t,
    double& x,
    double& y,
    double& z,
    const xparams _params = alglib::xdefault);

parametric_rdp example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use RDP algorithm to approximate parametric 2D curve given by
    // locations in t=0,1,2,3 (see below), which form piecewise linear
    // trajectory through D-dimensional space (2-dimensional in our example).
    // 
    //     |
    //     |
    //     -     *     *     X2................X3
    //     |                .
    //     |               .
    //     -     *     *  .  *     *     *     *
    //     |             .
    //     |            .
    //     -     *     X1    *     *     *     *
    //     |      .....
    //     |  ....
    //     X0----|-----|-----|-----|-----|-----|---
    //
    ae_int_t npoints = 4;
    ae_int_t ndimensions = 2;
    real_2d_array x = "[[0,0],[2,1],[3,3],[6,3]]";

    //
    // Approximation of parametric curve is performed by another parametric curve
    // with lesser amount of points. It allows to work with "compressed"
    // representation, which needs smaller amount of memory. Say, in our example
    // (we allow points with error smaller than 0.8) approximation will have
    // just two sequential sections connecting X0 with X2, and X2 with X3.
    // 
    //     |
    //     |
    //     -     *     *     X2................X3
    //     |               . 
    //     |             .  
    //     -     *     .     *     *     *     *
    //     |         .    
    //     |       .     
    //     -     .     X1    *     *     *     *
    //     |   .       
    //     | .    
    //     X0----|-----|-----|-----|-----|-----|---
    //
    //
    real_2d_array y;
    integer_1d_array idxy;
    ae_int_t nsections;
    ae_int_t limitcnt = 0;
    double limiteps = 0.8;
    parametricrdpfixed(x, npoints, ndimensions, limitcnt, limiteps, y, idxy, nsections);
    printf("%d\n", int(nsections)); // EXPECTED: 2
    printf("%s\n", idxy.tostring().c_str()); // EXPECTED: [0,2,3]
    return 0;
}

`pca` subpackage

Functions

pcabuildbasis
pcatruncatedsubspace
pcatruncatedsubspacesparse

Examples

`pcabuildbasis` function

/*************************************************************************
Principal components analysis

This function builds orthogonal basis  where  first  axis  corresponds  to
direction with maximum variance, second axis  maximizes  variance  in  the
subspace orthogonal to first axis and so on.

This function builds FULL basis, i.e. returns N vectors  corresponding  to
ALL directions, no matter how informative. If you need  just a  few  (say,
10 or 50) of the most important directions, you may find it faster to  use
one of the reduced versions:
* pcatruncatedsubspace() - for subspace iteration based method

It should be noted that, unlike LDA, PCA does not use class labels.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X           -   dataset, array[0..NPoints-1,0..NVars-1].
                    matrix contains ONLY INDEPENDENT VARIABLES.
    NPoints     -   dataset size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1

OUTPUT PARAMETERS:
    Info        -   return code:
                    * -4, if SVD subroutine haven't converged
                    * -1, if wrong parameters has been passed (NPoints<0,
                          NVars<1)
                    *  1, if task is solved
    S2          -   array[0..NVars-1]. variance values corresponding
                    to basis vectors.
    V           -   array[0..NVars-1,0..NVars-1]
                    matrix, whose columns store basis vectors.

  -- ALGLIB --
     Copyright 25.08.2008 by Bochkanov Sergey
*************************************************************************/
void alglib::pcabuildbasis(
    real_2d_array x,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t& info,
    real_1d_array& s2,
    real_2d_array& v,
    const xparams _params = alglib::xdefault);

`pcatruncatedsubspace` function

/*************************************************************************
Principal components analysis

This function performs truncated PCA, i.e. returns just a few most important
directions.

Internally it uses iterative eigensolver which is very efficient when only
a minor fraction of full basis is required. Thus, if you need full  basis,
it is better to use pcabuildbasis() function.

It should be noted that, unlike LDA, PCA does not use class labels.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X           -   dataset, array[0..NPoints-1,0..NVars-1].
                    matrix contains ONLY INDEPENDENT VARIABLES.
    NPoints     -   dataset size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1
    NNeeded     -   number of requested components, in [1,NVars] range;
                    this function is efficient only for NNeeded<<NVars.
    Eps         -   desired  precision  of  vectors  returned;  underlying
                    solver will stop iterations as soon as absolute  error
                    in corresponding singular values  reduces  to  roughly
                    eps*MAX(lambda[]), with lambda[] being array of  eigen
                    values.
                    Zero value means that  algorithm  performs  number  of
                    iterations  specified  by  maxits  parameter,  without
                    paying attention to precision.
    MaxIts      -   number of iterations performed by  subspace  iteration
                    method. Zero value means that no  limit  on  iteration
                    count is placed (eps-based stopping condition is used).


OUTPUT PARAMETERS:
    S2          -   array[NNeeded]. Variance values corresponding
                    to basis vectors.
    V           -   array[NVars,NNeeded]
                    matrix, whose columns store basis vectors.

NOTE: passing eps=0 and maxits=0 results in small eps  being  selected  as
stopping condition. Exact value of automatically selected eps is  version-
-dependent.

  -- ALGLIB --
     Copyright 10.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::pcatruncatedsubspace(
    real_2d_array x,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t nneeded,
    double eps,
    ae_int_t maxits,
    real_1d_array& s2,
    real_2d_array& v,
    const xparams _params = alglib::xdefault);

`pcatruncatedsubspacesparse` function

/*************************************************************************
Sparse truncated principal components analysis

This function performs sparse truncated PCA, i.e. returns just a few  most
important principal components for a sparse input X.

Internally it uses iterative eigensolver which is very efficient when only
a minor fraction of full basis is required.

It should be noted that, unlike LDA, PCA does not use class labels.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X           -   sparse dataset, sparse  npoints*nvars  matrix.  It  is
                    recommended to use CRS sparse storage format;  non-CRS
                    input will be internally converted to CRS.
                    Matrix contains ONLY INDEPENDENT VARIABLES,  and  must
                    be EXACTLY npoints*nvars.
    NPoints     -   dataset size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1
    NNeeded     -   number of requested components, in [1,NVars] range;
                    this function is efficient only for NNeeded<<NVars.
    Eps         -   desired  precision  of  vectors  returned;  underlying
                    solver will stop iterations as soon as absolute  error
                    in corresponding singular values  reduces  to  roughly
                    eps*MAX(lambda[]), with lambda[] being array of  eigen
                    values.
                    Zero value means that  algorithm  performs  number  of
                    iterations  specified  by  maxits  parameter,  without
                    paying attention to precision.
    MaxIts      -   number of iterations performed by  subspace  iteration
                    method. Zero value means that no  limit  on  iteration
                    count is placed (eps-based stopping condition is used).


OUTPUT PARAMETERS:
    S2          -   array[NNeeded]. Variance values corresponding
                    to basis vectors.
    V           -   array[NVars,NNeeded]
                    matrix, whose columns store basis vectors.

NOTE: passing eps=0 and maxits=0 results in small eps  being  selected  as
stopping condition. Exact value of automatically selected eps is  version-
-dependent.

  -- ALGLIB --
     Copyright 10.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::pcatruncatedsubspacesparse(
    sparsematrix x,
    ae_int_t npoints,
    ae_int_t nvars,
    ae_int_t nneeded,
    double eps,
    ae_int_t maxits,
    real_1d_array& s2,
    real_2d_array& v,
    const xparams _params = alglib::xdefault);

`poissondistr` subpackage

Functions

invpoissondistribution
poissoncdistribution
poissondistribution

Examples

`invpoissondistribution` function

/*************************************************************************
Inverse Poisson distribution

Finds the Poisson variable x such that the integral
from 0 to x of the Poisson density is equal to the
given probability y.

This is accomplished using the inverse gamma integral
function and the relation

   m = igami( k+1, y ).

ACCURACY:

See inverse incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::invpoissondistribution(
    ae_int_t k,
    double y,
    const xparams _params = alglib::xdefault);

`poissoncdistribution` function

/*************************************************************************
Complemented Poisson distribution

Returns the sum of the terms k+1 to infinity of the Poisson
distribution:

 inf.       j
  --   -m  m
  >   e    --
  --       j!
 j=k+1

The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the formula

y = pdtrc( k, m ) = igam( k+1, m ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::poissoncdistribution(
    ae_int_t k,
    double m,
    const xparams _params = alglib::xdefault);

`poissondistribution` function

/*************************************************************************
Poisson distribution

Returns the sum of the first k+1 terms of the Poisson
distribution:

  k         j
  --   -m  m
  >   e    --
  --       j!
 j=0

The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the relation

y = pdtr( k, m ) = igamc( k+1, m ).

The arguments must both be positive.
ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::poissondistribution(
    ae_int_t k,
    double m,
    const xparams _params = alglib::xdefault);

`polint` subpackage

Functions

polynomialbar2cheb
polynomialbar2pow
polynomialbuild
polynomialbuildcheb1
polynomialbuildcheb2
polynomialbuildeqdist
polynomialcalccheb1
polynomialcalccheb2
polynomialcalceqdist
polynomialcheb2bar
polynomialpow2bar

Examples

polint_d_calcdiff		Interpolation and differentiation using barycentric representation
polint_d_conv		Conversion between power basis and barycentric representation
polint_d_spec		Polynomial interpolation on special grids (equidistant, Chebyshev I/II)

`polynomialbar2cheb` function

/*************************************************************************
Conversion from barycentric representation to Chebyshev basis.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    P   -   polynomial in barycentric form
    A,B -   base interval for Chebyshev polynomials (see below)
            A<>B

OUTPUT PARAMETERS
    T   -   coefficients of Chebyshev representation;
            P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N-1 },
            where Ti - I-th Chebyshev polynomial.

NOTES:
    barycentric interpolant passed as P may be either polynomial  obtained
    from  polynomial  interpolation/ fitting or rational function which is
    NOT polynomial. We can't distinguish between these two cases, and this
    algorithm just tries to work assuming that P IS a polynomial.  If not,
    algorithm will return results, but they won't have any meaning.

  -- ALGLIB --
     Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialbar2cheb(
    barycentricinterpolant p,
    double a,
    double b,
    real_1d_array& t,
    const xparams _params = alglib::xdefault);

`polynomialbar2pow` function

/*************************************************************************
Conversion from barycentric representation to power basis.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    P   -   polynomial in barycentric form
    C   -   offset (see below); 0.0 is used as default value.
    S   -   scale (see below);  1.0 is used as default value. S<>0.

OUTPUT PARAMETERS
    A   -   coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
    N   -   number of coefficients (polynomial degree plus 1)

NOTES:
1.  this function accepts offset and scale, which can be  set  to  improve
    numerical properties of polynomial. For example, if P was obtained  as
    result of interpolation on [-1,+1],  you  can  set  C=0  and  S=1  and
    represent  P  as sum of 1, x, x^2, x^3 and so on. In most cases you it
    is exactly what you need.

    However, if your interpolation model was built on [999,1001], you will
    see significant growth of numerical errors when using {1, x, x^2, x^3}
    as basis. Representing P as sum of 1, (x-1000), (x-1000)^2, (x-1000)^3
    will be better option. Such representation can be  obtained  by  using
    1000.0 as offset C and 1.0 as scale S.

2.  power basis is ill-conditioned and tricks described above can't  solve
    this problem completely. This function  will  return  coefficients  in
    any  case,  but  for  N>8  they  will  become unreliable. However, N's
    less than 5 are pretty safe.

3.  barycentric interpolant passed as P may be either polynomial  obtained
    from  polynomial  interpolation/ fitting or rational function which is
    NOT polynomial. We can't distinguish between these two cases, and this
    algorithm just tries to work assuming that P IS a polynomial.  If not,
    algorithm will return results, but they won't have any meaning.

  -- ALGLIB --
     Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialbar2pow(
    barycentricinterpolant p,
    real_1d_array& a,
    const xparams _params = alglib::xdefault);
void alglib::polynomialbar2pow(
    barycentricinterpolant p,
    double c,
    double s,
    real_1d_array& a,
    const xparams _params = alglib::xdefault);

Examples: [1]

`polynomialbuild` function

/*************************************************************************
Lagrange intepolant: generation of the model on the general grid.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    X   -   abscissas, array[0..N-1]
    Y   -   function values, array[0..N-1]
    N   -   number of points, N>=1

OUTPUT PARAMETERS
    P   -   barycentric model which represents Lagrange interpolant
            (see ratint unit info and BarycentricCalc() description for
            more information).

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialbuild(
    real_1d_array x,
    real_1d_array y,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);
void alglib::polynomialbuild(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);

Examples: [1]

`polynomialbuildcheb1` function

/*************************************************************************
Lagrange intepolant on Chebyshev grid (first kind).
This function has O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    Y   -   function values at the nodes, array[0..N-1],
            Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n)))
    N   -   number of points, N>=1
            for N=1 a constant model is constructed.

OUTPUT PARAMETERS
    P   -   barycentric model which represents Lagrange interpolant
            (see ratint unit info and BarycentricCalc() description for
            more information).

  -- ALGLIB --
     Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialbuildcheb1(
    double a,
    double b,
    real_1d_array y,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);
void alglib::polynomialbuildcheb1(
    double a,
    double b,
    real_1d_array y,
    ae_int_t n,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);

Examples: [1]

`polynomialbuildcheb2` function

/*************************************************************************
Lagrange intepolant on Chebyshev grid (second kind).
This function has O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    Y   -   function values at the nodes, array[0..N-1],
            Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1)))
    N   -   number of points, N>=1
            for N=1 a constant model is constructed.

OUTPUT PARAMETERS
    P   -   barycentric model which represents Lagrange interpolant
            (see ratint unit info and BarycentricCalc() description for
            more information).

  -- ALGLIB --
     Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialbuildcheb2(
    double a,
    double b,
    real_1d_array y,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);
void alglib::polynomialbuildcheb2(
    double a,
    double b,
    real_1d_array y,
    ae_int_t n,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);

Examples: [1]

`polynomialbuildeqdist` function

/*************************************************************************
Lagrange intepolant: generation of the model on equidistant grid.
This function has O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    Y   -   function values at the nodes, array[0..N-1]
    N   -   number of points, N>=1
            for N=1 a constant model is constructed.

OUTPUT PARAMETERS
    P   -   barycentric model which represents Lagrange interpolant
            (see ratint unit info and BarycentricCalc() description for
            more information).

  -- ALGLIB --
     Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialbuildeqdist(
    double a,
    double b,
    real_1d_array y,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);
void alglib::polynomialbuildeqdist(
    double a,
    double b,
    real_1d_array y,
    ae_int_t n,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);

Examples: [1]

`polynomialcalccheb1` function

/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (first kind)
with O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    F   -   function values, array[0..N-1]
    N   -   number of points on Chebyshev grid (first kind),
            X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n))
            for N=1 a constant model is constructed.
    T   -   position where P(x) is calculated

RESULT
    value of the Lagrange interpolant at T

IMPORTANT
    this function provides fast interface which is not overflow-safe
    nor it is very precise.
    the best option is to use  PolIntBuildCheb1()/BarycentricCalc()
    subroutines unless you are pretty sure that your data will not result
    in overflow.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::polynomialcalccheb1(
    double a,
    double b,
    real_1d_array f,
    double t,
    const xparams _params = alglib::xdefault);
double alglib::polynomialcalccheb1(
    double a,
    double b,
    real_1d_array f,
    ae_int_t n,
    double t,
    const xparams _params = alglib::xdefault);

Examples: [1]

`polynomialcalccheb2` function

/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (second kind)
with O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    F   -   function values, array[0..N-1]
    N   -   number of points on Chebyshev grid (second kind),
            X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1))
            for N=1 a constant model is constructed.
    T   -   position where P(x) is calculated

RESULT
    value of the Lagrange interpolant at T

IMPORTANT
    this function provides fast interface which is not overflow-safe
    nor it is very precise.
    the best option is to use PolIntBuildCheb2()/BarycentricCalc()
    subroutines unless you are pretty sure that your data will not result
    in overflow.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::polynomialcalccheb2(
    double a,
    double b,
    real_1d_array f,
    double t,
    const xparams _params = alglib::xdefault);
double alglib::polynomialcalccheb2(
    double a,
    double b,
    real_1d_array f,
    ae_int_t n,
    double t,
    const xparams _params = alglib::xdefault);

Examples: [1]

`polynomialcalceqdist` function

/*************************************************************************
Fast equidistant polynomial interpolation function with O(N) complexity

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    F   -   function values, array[0..N-1]
    N   -   number of points on equidistant grid, N>=1
            for N=1 a constant model is constructed.
    T   -   position where P(x) is calculated

RESULT
    value of the Lagrange interpolant at T

IMPORTANT
    this function provides fast interface which is not overflow-safe
    nor it is very precise.
    the best option is to use  PolynomialBuildEqDist()/BarycentricCalc()
    subroutines unless you are pretty sure that your data will not result
    in overflow.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::polynomialcalceqdist(
    double a,
    double b,
    real_1d_array f,
    double t,
    const xparams _params = alglib::xdefault);
double alglib::polynomialcalceqdist(
    double a,
    double b,
    real_1d_array f,
    ae_int_t n,
    double t,
    const xparams _params = alglib::xdefault);

Examples: [1]

`polynomialcheb2bar` function

/*************************************************************************
Conversion from Chebyshev basis to barycentric representation.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    T   -   coefficients of Chebyshev representation;
            P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N },
            where Ti - I-th Chebyshev polynomial.
    N   -   number of coefficients:
            * if given, only leading N elements of T are used
            * if not given, automatically determined from size of T
    A,B -   base interval for Chebyshev polynomials (see above)
            A<B

OUTPUT PARAMETERS
    P   -   polynomial in barycentric form

  -- ALGLIB --
     Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialcheb2bar(
    real_1d_array t,
    double a,
    double b,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);
void alglib::polynomialcheb2bar(
    real_1d_array t,
    ae_int_t n,
    double a,
    double b,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);

`polynomialpow2bar` function

/*************************************************************************
Conversion from power basis to barycentric representation.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    A   -   coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
    N   -   number of coefficients (polynomial degree plus 1)
            * if given, only leading N elements of A are used
            * if not given, automatically determined from size of A
    C   -   offset (see below); 0.0 is used as default value.
    S   -   scale (see below);  1.0 is used as default value. S<>0.

OUTPUT PARAMETERS
    P   -   polynomial in barycentric form


NOTES:
1.  this function accepts offset and scale, which can be  set  to  improve
    numerical properties of polynomial. For example, if you interpolate on
    [-1,+1],  you  can  set C=0 and S=1 and convert from sum of 1, x, x^2,
    x^3 and so on. In most cases you it is exactly what you need.

    However, if your interpolation model was built on [999,1001], you will
    see significant growth of numerical errors when using {1, x, x^2, x^3}
    as  input  basis.  Converting  from  sum  of  1, (x-1000), (x-1000)^2,
    (x-1000)^3 will be better option (you have to specify 1000.0 as offset
    C and 1.0 as scale S).

2.  power basis is ill-conditioned and tricks described above can't  solve
    this problem completely. This function  will  return barycentric model
    in any case, but for N>8 accuracy well degrade. However, N's less than
    5 are pretty safe.

  -- ALGLIB --
     Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialpow2bar(
    real_1d_array a,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);
void alglib::polynomialpow2bar(
    real_1d_array a,
    ae_int_t n,
    double c,
    double s,
    barycentricinterpolant& p,
    const xparams _params = alglib::xdefault);

Examples: [1]

polint_d_calcdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Here we demonstrate polynomial interpolation and differentiation
    // of y=x^2-x sampled at [0,1,2]. Barycentric representation of polynomial is used.
    //
    real_1d_array x = "[0,1,2]";
    real_1d_array y = "[0,0,2]";
    double t = -1;
    double v;
    double dv;
    double d2v;
    barycentricinterpolant p;

    // barycentric model is created
    polynomialbuild(x, y, p);

    // barycentric interpolation is demonstrated
    v = barycentriccalc(p, t);
    printf("%.4f\n", double(v)); // EXPECTED: 2.0

    // barycentric differentation is demonstrated
    barycentricdiff1(p, t, v, dv);
    printf("%.4f\n", double(v)); // EXPECTED: 2.0
    printf("%.4f\n", double(dv)); // EXPECTED: -3.0

    // second derivatives with barycentric representation
    barycentricdiff1(p, t, v, dv);
    printf("%.4f\n", double(v)); // EXPECTED: 2.0
    printf("%.4f\n", double(dv)); // EXPECTED: -3.0
    barycentricdiff2(p, t, v, dv, d2v);
    printf("%.4f\n", double(v)); // EXPECTED: 2.0
    printf("%.4f\n", double(dv)); // EXPECTED: -3.0
    printf("%.4f\n", double(d2v)); // EXPECTED: 2.0
    return 0;
}

polint_d_conv example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Here we demonstrate conversion of y=x^2-x
    // between power basis and barycentric representation.
    //
    real_1d_array a = "[0,-1,+1]";
    double t = 2;
    real_1d_array a2;
    double v;
    barycentricinterpolant p;

    //
    // a=[0,-1,+1] is decomposition of y=x^2-x in the power basis:
    //
    //     y = 0 - 1*x + 1*x^2
    //
    // We convert it to the barycentric form.
    //
    polynomialpow2bar(a, p);

    // now we have barycentric interpolation; we can use it for interpolation
    v = barycentriccalc(p, t);
    printf("%.2f\n", double(v)); // EXPECTED: 2.0

    // we can also convert back from barycentric representation to power basis
    polynomialbar2pow(p, a2);
    printf("%s\n", a2.tostring(2).c_str()); // EXPECTED: [0,-1,+1]
    return 0;
}

polint_d_spec example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Temporaries:
    // * values of y=x^2-x sampled at three special grids:
    //   * equdistant grid spanning [0,2],     x[i] = 2*i/(N-1), i=0..N-1
    //   * Chebyshev-I grid spanning [-1,+1],  x[i] = 1 + Cos(PI*(2*i+1)/(2*n)), i=0..N-1
    //   * Chebyshev-II grid spanning [-1,+1], x[i] = 1 + Cos(PI*i/(n-1)), i=0..N-1
    // * barycentric interpolants for these three grids
    // * vectors to store coefficients of quadratic representation
    //
    real_1d_array y_eqdist = "[0,0,2]";
    real_1d_array y_cheb1 = "[-0.116025,0.000000,1.616025]";
    real_1d_array y_cheb2 = "[0,0,2]";
    barycentricinterpolant p_eqdist;
    barycentricinterpolant p_cheb1;
    barycentricinterpolant p_cheb2;
    real_1d_array a_eqdist;
    real_1d_array a_cheb1;
    real_1d_array a_cheb2;

    //
    // First, we demonstrate construction of barycentric interpolants on
    // special grids. We unpack power representation to ensure that
    // interpolant was built correctly.
    //
    // In all three cases we should get same quadratic function.
    //
    polynomialbuildeqdist(0.0, 2.0, y_eqdist, p_eqdist);
    polynomialbar2pow(p_eqdist, a_eqdist);
    printf("%s\n", a_eqdist.tostring(4).c_str()); // EXPECTED: [0,-1,+1]

    polynomialbuildcheb1(-1, +1, y_cheb1, p_cheb1);
    polynomialbar2pow(p_cheb1, a_cheb1);
    printf("%s\n", a_cheb1.tostring(4).c_str()); // EXPECTED: [0,-1,+1]

    polynomialbuildcheb2(-1, +1, y_cheb2, p_cheb2);
    polynomialbar2pow(p_cheb2, a_cheb2);
    printf("%s\n", a_cheb2.tostring(4).c_str()); // EXPECTED: [0,-1,+1]

    //
    // Now we demonstrate polynomial interpolation without construction 
    // of the barycentricinterpolant structure.
    //
    // We calculate interpolant value at x=-2.
    // In all three cases we should get same f=6
    //
    double t = -2;
    double v;
    v = polynomialcalceqdist(0.0, 2.0, y_eqdist, t);
    printf("%.4f\n", double(v)); // EXPECTED: 6.0

    v = polynomialcalccheb1(-1, +1, y_cheb1, t);
    printf("%.4f\n", double(v)); // EXPECTED: 6.0

    v = polynomialcalccheb2(-1, +1, y_cheb2, t);
    printf("%.4f\n", double(v)); // EXPECTED: 6.0
    return 0;
}

`polynomialsolver` subpackage

Classes

polynomialsolverreport

Functions

polynomialsolve

Examples

`polynomialsolverreport` class

/*************************************************************************

*************************************************************************/
class polynomialsolverreport
{
    double               maxerr;
};

`polynomialsolve` function

/*************************************************************************
Polynomial root finding.

This function returns all roots of the polynomial
    P(x) = a0 + a1*x + a2*x^2 + ... + an*x^n
Both real and complex roots are returned (see below).

INPUT PARAMETERS:
    A       -   array[N+1], polynomial coefficients:
                * A[0] is constant term
                * A[N] is a coefficient of X^N
    N       -   polynomial degree

OUTPUT PARAMETERS:
    X       -   array of complex roots:
                * for isolated real root, X[I] is strictly real: IMAGE(X[I])=0
                * complex roots are always returned in pairs - roots occupy
                  positions I and I+1, with:
                  * X[I+1]=Conj(X[I])
                  * IMAGE(X[I]) > 0
                  * IMAGE(X[I+1]) = -IMAGE(X[I]) < 0
                * multiple real roots may have non-zero imaginary part due
                  to roundoff errors. There is no reliable way to distinguish
                  real root of multiplicity 2 from two  complex  roots  in
                  the presence of roundoff errors.
    Rep     -   report, additional information, following fields are set:
                * Rep.MaxErr - max( |P(xi)| )  for  i=0..N-1.  This  field
                  allows to quickly estimate "quality" of the roots  being
                  returned.

NOTE:   this function uses companion matrix method to find roots. In  case
        internal EVD  solver  fails  do  find  eigenvalues,  exception  is
        generated.

NOTE:   roots are not "polished" and  no  matrix  balancing  is  performed
        for them.

  -- ALGLIB --
     Copyright 24.02.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::polynomialsolve(
    real_1d_array a,
    ae_int_t n,
    complex_1d_array& x,
    polynomialsolverreport& rep,
    const xparams _params = alglib::xdefault);

`psif` subpackage

Functions

psi

Examples

`psi` function

/*************************************************************************
Psi (digamma) function

             d      -
  psi(x)  =  -- ln | (x)
             dx

is the logarithmic derivative of the gamma function.
For integer x,
                  n-1
                   -
psi(n) = -EUL  +   >  1/k.
                   -
                  k=1

This formula is used for 0 < n <= 10.  If x is negative, it
is transformed to a positive argument by the reflection
formula  psi(1-x) = psi(x) + pi cot(pi x).
For general positive x, the argument is made greater than 10
using the recurrence  psi(x+1) = psi(x) + 1/x.
Then the following asymptotic expansion is applied:

                          inf.   B
                           -      2k
psi(x) = log(x) - 1/2x -   >   -------
                           -        2k
                          k=1   2k x

where the B2k are Bernoulli numbers.

ACCURACY:
   Relative error (except absolute when |psi| < 1):
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        30000       1.3e-15     1.4e-16
   IEEE      -30,0       40000       1.5e-15     2.2e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::psi(double x, const xparams _params = alglib::xdefault);

`ratint` subpackage

Classes

barycentricinterpolant

Functions

barycentricbuildfloaterhormann
barycentricbuildxyw
barycentriccalc
barycentricdiff1
barycentricdiff2
barycentriclintransx
barycentriclintransy
barycentricunpack

Examples

`barycentricinterpolant` class

/*************************************************************************
Barycentric interpolant.
*************************************************************************/
class barycentricinterpolant
{
};

`barycentricbuildfloaterhormann` function

/*************************************************************************
Rational interpolant without poles

The subroutine constructs the rational interpolating function without real
poles  (see  'Barycentric rational interpolation with no  poles  and  high
rates of approximation', Michael S. Floater. and  Kai  Hormann,  for  more
information on this subject).

Input parameters:
    X   -   interpolation nodes, array[0..N-1].
    Y   -   function values, array[0..N-1].
    N   -   number of nodes, N>0.
    D   -   order of the interpolation scheme, 0 <= D <= N-1.
            D<0 will cause an error.
            D>=N it will be replaced with D=N-1.
            if you don't know what D to choose, use small value about 3-5.

Output parameters:
    B   -   barycentric interpolant.

Note:
    this algorithm always succeeds and calculates the weights  with  close
    to machine precision.

  -- ALGLIB PROJECT --
     Copyright 17.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::barycentricbuildfloaterhormann(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t d,
    barycentricinterpolant& b,
    const xparams _params = alglib::xdefault);

`barycentricbuildxyw` function

/*************************************************************************
Rational interpolant from X/Y/W arrays

F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))

INPUT PARAMETERS:
    X   -   interpolation nodes, array[0..N-1]
    F   -   function values, array[0..N-1]
    W   -   barycentric weights, array[0..N-1]
    N   -   nodes count, N>0

OUTPUT PARAMETERS:
    B   -   barycentric interpolant built from (X, Y, W)

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::barycentricbuildxyw(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    ae_int_t n,
    barycentricinterpolant& b,
    const xparams _params = alglib::xdefault);

`barycentriccalc` function

/*************************************************************************
Rational interpolation using barycentric formula

F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))

Input parameters:
    B   -   barycentric interpolant built with one of model building
            subroutines.
    T   -   interpolation point

Result:
    barycentric interpolant F(t)

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
double alglib::barycentriccalc(
    barycentricinterpolant b,
    double t,
    const xparams _params = alglib::xdefault);

`barycentricdiff1` function

/*************************************************************************
Differentiation of barycentric interpolant: first derivative.

Algorithm used in this subroutine is very robust and should not fail until
provided with values too close to MaxRealNumber  (usually  MaxRealNumber/N
or greater will overflow).

INPUT PARAMETERS:
    B   -   barycentric interpolant built with one of model building
            subroutines.
    T   -   interpolation point

OUTPUT PARAMETERS:
    F   -   barycentric interpolant at T
    DF  -   first derivative

NOTE


  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::barycentricdiff1(
    barycentricinterpolant b,
    double t,
    double& f,
    double& df,
    const xparams _params = alglib::xdefault);

`barycentricdiff2` function

/*************************************************************************
Differentiation of barycentric interpolant: first/second derivatives.

INPUT PARAMETERS:
    B   -   barycentric interpolant built with one of model building
            subroutines.
    T   -   interpolation point

OUTPUT PARAMETERS:
    F   -   barycentric interpolant at T
    DF  -   first derivative
    D2F -   second derivative

NOTE: this algorithm may fail due to overflow/underflor if  used  on  data
whose values are close to MaxRealNumber or MinRealNumber.  Use more robust
BarycentricDiff1() subroutine in such cases.


  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::barycentricdiff2(
    barycentricinterpolant b,
    double t,
    double& f,
    double& df,
    double& d2f,
    const xparams _params = alglib::xdefault);

`barycentriclintransx` function

/*************************************************************************
This subroutine performs linear transformation of the argument.

INPUT PARAMETERS:
    B       -   rational interpolant in barycentric form
    CA, CB  -   transformation coefficients: x = CA*t + CB

OUTPUT PARAMETERS:
    B       -   transformed interpolant with X replaced by T

  -- ALGLIB PROJECT --
     Copyright 19.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::barycentriclintransx(
    barycentricinterpolant b,
    double ca,
    double cb,
    const xparams _params = alglib::xdefault);

`barycentriclintransy` function

/*************************************************************************
This  subroutine   performs   linear  transformation  of  the  barycentric
interpolant.

INPUT PARAMETERS:
    B       -   rational interpolant in barycentric form
    CA, CB  -   transformation coefficients: B2(x) = CA*B(x) + CB

OUTPUT PARAMETERS:
    B       -   transformed interpolant

  -- ALGLIB PROJECT --
     Copyright 19.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::barycentriclintransy(
    barycentricinterpolant b,
    double ca,
    double cb,
    const xparams _params = alglib::xdefault);

`barycentricunpack` function

/*************************************************************************
Extracts X/Y/W arrays from rational interpolant

INPUT PARAMETERS:
    B   -   barycentric interpolant

OUTPUT PARAMETERS:
    N   -   nodes count, N>0
    X   -   interpolation nodes, array[0..N-1]
    F   -   function values, array[0..N-1]
    W   -   barycentric weights, array[0..N-1]

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::barycentricunpack(
    barycentricinterpolant b,
    ae_int_t& n,
    real_1d_array& x,
    real_1d_array& y,
    real_1d_array& w,
    const xparams _params = alglib::xdefault);

`rbf` subpackage

Classes

rbfcalcbuffer
rbfmodel
rbfreport

Functions

rbfbuildmodel
rbfcalc
rbfcalc1
rbfcalc2
rbfcalc3
rbfcalcbuf
rbfcreate
rbfcreatecalcbuffer
rbfgetmodelversion
rbfgridcalc2
rbfgridcalc2v
rbfgridcalc2vsubset
rbfgridcalc3v
rbfgridcalc3vsubset
rbfserialize
rbfsetalgohierarchical
rbfsetalgomultilayer
rbfsetalgoqnn
rbfsetconstterm
rbfsetlinterm
rbfsetpoints
rbfsetpointsandscales
rbfsetv2bf
rbfsetv2its
rbfsetv2supportr
rbfsetzeroterm
rbftscalcbuf
rbfunpack
rbfunserialize

Examples

rbf_d_hrbf		Simple model built with HRBF algorithm
rbf_d_polterm		RBF models - working with polynomial term
rbf_d_serialize		Serialization/unserialization
rbf_d_vector		Working with vector functions

`rbfcalcbuffer` class

/*************************************************************************
Buffer object which is used to perform nearest neighbor  requests  in  the
multithreaded mode (multiple threads working with same KD-tree object).

This object should be created with KDTreeCreateBuffer().
*************************************************************************/
class rbfcalcbuffer
{
};

`rbfmodel` class

/*************************************************************************
RBF model.

Never try to directly work with fields of this object - always use  ALGLIB
functions to use this object.
*************************************************************************/
class rbfmodel
{
};

`rbfreport` class

/*************************************************************************
RBF solution report:
* TerminationType   -   termination type, positive values - success,
                        non-positive - failure.

Fields which are set by modern RBF solvers (hierarchical):
* RMSError          -   root-mean-square error; NAN for old solvers (ML, QNN)
* MaxError          -   maximum error; NAN for old solvers (ML, QNN)
*************************************************************************/
class rbfreport
{
    double               rmserror;
    double               maxerror;
    ae_int_t             arows;
    ae_int_t             acols;
    ae_int_t             annz;
    ae_int_t             iterationscount;
    ae_int_t             nmv;
    ae_int_t             terminationtype;
};

`rbfbuildmodel` function

/*************************************************************************
This   function  builds  RBF  model  and  returns  report  (contains  some
information which can be used for evaluation of the algorithm properties).

Call to this function modifies RBF model by calculating its centers/radii/
weights  and  saving  them  into  RBFModel  structure.  Initially RBFModel
contain zero coefficients, but after call to this function  we  will  have
coefficients which were calculated in order to fit our dataset.

After you called this function you can call RBFCalc(),  RBFGridCalc()  and
other model calculation functions.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    Rep     -   report:
                * Rep.TerminationType:
                  * -5 - non-distinct basis function centers were detected,
                         interpolation  aborted;  only  QNN  returns  this
                         error   code, other  algorithms  can  handle non-
                         distinct nodes.
                  * -4 - nonconvergence of the internal SVD solver
                  * -3   incorrect model construction algorithm was chosen:
                         QNN or RBF-ML, combined with one of the incompatible
                         features - NX=1 or NX>3; points with per-dimension
                         scales.
                  *  1 - successful termination

                Fields which are set only by modern RBF solvers (hierarchical
                or nonnegative; older solvers like QNN and ML initialize these
                fields by NANs):
                * rep.rmserror - root-mean-square error at nodes
                * rep.maxerror - maximum error at nodes

                Fields are used for debugging purposes:
                * Rep.IterationsCount - iterations count of the LSQR solver
                * Rep.NMV - number of matrix-vector products
                * Rep.ARows - rows count for the system matrix
                * Rep.ACols - columns count for the system matrix
                * Rep.ANNZ - number of significantly non-zero elements
                  (elements above some algorithm-determined threshold)

NOTE:  failure  to  build  model will leave current state of the structure
unchanged.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfbuildmodel(
    rbfmodel s,
    rbfreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`rbfcalc` function

/*************************************************************************
This function calculates values of the RBF model at the given point.

This is general function which can be used for arbitrary NX (dimension  of
the space of arguments) and NY (dimension of the function itself). However
when  you  have  NY=1  you  may  find more convenient to use rbfcalc2() or
rbfcalc3().

If you want to perform parallel model evaluation  from  multiple  threads,
use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 when model is not initialized.

INPUT PARAMETERS:
    S       -   RBF model
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is out-parameter and
                reallocated after call to this function. In case you  want
                to reuse previously allocated Y, you may use RBFCalcBuf(),
                which reallocates Y only when it is too small.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfcalc(
    rbfmodel s,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rbfcalc1` function

/*************************************************************************
This function calculates values of the RBF model in the given point.

IMPORTANT: this function works only with modern  (hierarchical)  RBFs.  It
           can not be used with legacy (version 1) RBFs because older  RBF
           code does not support 1-dimensional models.

This function should be used when we have NY=1 (scalar function) and  NX=1
(1-dimensional space). If you have 3-dimensional space, use rbfcalc3(). If
you  have  2-dimensional  space,  use  rbfcalc3().  If  you  have  general
situation (NX-dimensional space, NY-dimensional function)  you  should use
generic rbfcalc().

If you want to perform parallel model evaluation  from  multiple  threads,
use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 when:
* model is not initialized
* NX<>1
* NY<>1

INPUT PARAMETERS:
    S       -   RBF model
    X0      -   X-coordinate, finite number

RESULT:
    value of the model or 0.0 (as defined above)

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double alglib::rbfcalc1(
    rbfmodel s,
    double x0,
    const xparams _params = alglib::xdefault);

`rbfcalc2` function

/*************************************************************************
This function calculates values of the RBF model in the given point.

This function should be used when we have NY=1 (scalar function) and  NX=2
(2-dimensional space). If you have 3-dimensional space, use rbfcalc3(). If
you have general situation (NX-dimensional space, NY-dimensional function)
you should use generic rbfcalc().

If  you  want  to  calculate  function  values  many times, consider using
rbfgridcalc2v(), which is far more efficient than many subsequent calls to
rbfcalc2().

If you want to perform parallel model evaluation  from  multiple  threads,
use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 when:
* model is not initialized
* NX<>2
 *NY<>1

INPUT PARAMETERS:
    S       -   RBF model
    X0      -   first coordinate, finite number
    X1      -   second coordinate, finite number

RESULT:
    value of the model or 0.0 (as defined above)

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double alglib::rbfcalc2(
    rbfmodel s,
    double x0,
    double x1,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`rbfcalc3` function

/*************************************************************************
This function calculates value of the RBF model in the given point.

This function should be used when we have NY=1 (scalar function) and  NX=3
(3-dimensional space). If you have 2-dimensional space, use rbfcalc2(). If
you have general situation (NX-dimensional space, NY-dimensional function)
you should use generic rbfcalc().

If  you  want  to  calculate  function  values  many times, consider using
rbfgridcalc3v(), which is far more efficient than many subsequent calls to
rbfcalc3().

If you want to perform parallel model evaluation  from  multiple  threads,
use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 when:
* model is not initialized
* NX<>3
 *NY<>1

INPUT PARAMETERS:
    S       -   RBF model
    X0      -   first coordinate, finite number
    X1      -   second coordinate, finite number
    X2      -   third coordinate, finite number

RESULT:
    value of the model or 0.0 (as defined above)

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double alglib::rbfcalc3(
    rbfmodel s,
    double x0,
    double x1,
    double x2,
    const xparams _params = alglib::xdefault);

`rbfcalcbuf` function

/*************************************************************************
This function calculates values of the RBF model at the given point.

Same as rbfcalc(), but does not reallocate Y when in is large enough to
store function values.

If you want to perform parallel model evaluation  from  multiple  threads,
use rbftscalcbuf() with per-thread buffer object.

INPUT PARAMETERS:
    S       -   RBF model
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.
    Y       -   possibly preallocated array

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is not reallocated when it
                is larger than NY.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfcalcbuf(
    rbfmodel s,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`rbfcreate` function

/*************************************************************************
This function creates RBF  model  for  a  scalar (NY=1)  or  vector (NY>1)
function in a NX-dimensional space (NX>=1).

Newly created model is empty. It can be used for interpolation right after
creation, but it just returns zeros. You have to add points to the  model,
tune interpolation settings, and then  call  model  construction  function
rbfbuildmodel() which will update model according to your specification.

USAGE:
1. User creates model with rbfcreate()
2. User adds dataset with rbfsetpoints() (points do NOT have to  be  on  a
   regular grid) or rbfsetpointsandscales().
3. (OPTIONAL) User chooses polynomial term by calling:
   * rbflinterm() to set linear term
   * rbfconstterm() to set constant term
   * rbfzeroterm() to set zero term
   By default, linear term is used.
4. User tweaks algorithm properties with  rbfsetalgohierarchical()  method
   (or chooses one of the legacy algorithms - QNN  (rbfsetalgoqnn)  or  ML
   (rbfsetalgomultilayer)).
5. User calls rbfbuildmodel() function which rebuilds model  according  to
   the specification
6. User may call rbfcalc() to calculate model value at the specified point,
   rbfgridcalc() to  calculate   model  values at the points of the regular
   grid. User may extract model coefficients with rbfunpack() call.

IMPORTANT: we recommend you to use latest model construction  algorithm  -
           hierarchical RBFs, which is activated by rbfsetalgohierarchical()
           function. This algorithm is the fastest one, and  most  memory-
           efficient.
           However,  it  is  incompatible  with older versions  of  ALGLIB
           (pre-3.11). So, if you serialize hierarchical model,  you  will
           be unable to load it in pre-3.11 ALGLIB. Other model types (QNN
           and RBF-ML) are still backward-compatible.

INPUT PARAMETERS:
    NX      -   dimension of the space, NX>=1
    NY      -   function dimension, NY>=1

OUTPUT PARAMETERS:
    S       -   RBF model (initially equals to zero)

NOTE 1: memory requirements. RBF models require amount of memory  which is
        proportional  to the number of data points. Some additional memory
        is allocated during model construction, but most of this memory is
        freed after model coefficients  are  calculated.  Amount  of  this
        additional memory depends on model  construction  algorithm  being
        used.

NOTE 2: prior to ALGLIB version 3.11, RBF models supported  only  NX=2  or
        NX=3. Any  attempt  to  create  single-dimensional  or  more  than
        3-dimensional RBF model resulted in exception.

        ALGLIB 3.11 supports any NX>0, but models created with  NX!=2  and
        NX!=3 are incompatible with (a) older versions of ALGLIB, (b)  old
        model construction algorithms (QNN or RBF-ML).

        So, if you create a model with NX=2 or NX=3,  then,  depending  on
        specific  model construction algorithm being chosen, you will (QNN
        and RBF-ML) or will not (HierarchicalRBF) get backward compatibility
        with older versions of ALGLIB. You have a choice here.

        However, if you create a model with NX neither 2 nor 3,  you  have
        no backward compatibility from the start, and you  are  forced  to
        use hierarchical RBFs and ALGLIB 3.11 or later.

  -- ALGLIB --
     Copyright 13.12.2011, 20.06.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfcreate(
    ae_int_t nx,
    ae_int_t ny,
    rbfmodel& s,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3] [4]

`rbfcreatecalcbuffer` function

/*************************************************************************
This function creates buffer  structure  which  can  be  used  to  perform
parallel  RBF  model  evaluations  (with  one  RBF  model  instance  being
used from multiple threads, as long as  different  threads  use  different
instances of buffer).

This buffer object can be used with  rbftscalcbuf()  function  (here  "ts"
stands for "thread-safe", "buf" is a suffix which denotes  function  which
reuses previously allocated output space).

How to use it:
* create RBF model structure with rbfcreate()
* load data, tune parameters
* call rbfbuildmodel()
* call rbfcreatecalcbuffer(), once per thread working with RBF model  (you
  should call this function only AFTER call to rbfbuildmodel(), see  below
  for more information)
* call rbftscalcbuf() from different threads,  with  each  thread  working
  with its own copy of buffer object.

INPUT PARAMETERS
    S           -   RBF model

OUTPUT PARAMETERS
    Buf         -   external buffer.


IMPORTANT: buffer object should be used only with  RBF model object  which
           was used to initialize buffer. Any attempt to use buffer   with
           different object is dangerous - you may  get  memory  violation
           error because sizes of internal arrays do not fit to dimensions
           of RBF structure.

IMPORTANT: you  should  call  this function only for model which was built
           with rbfbuildmodel() function, after successful  invocation  of
           rbfbuildmodel().  Sizes   of   some   internal  structures  are
           determined only after model is built, so buffer object  created
           before model  construction  stage  will  be  useless  (and  any
           attempt to use it will result in exception).

  -- ALGLIB --
     Copyright 02.04.2016 by Sergey Bochkanov
*************************************************************************/
void alglib::rbfcreatecalcbuffer(
    rbfmodel s,
    rbfcalcbuffer& buf,
    const xparams _params = alglib::xdefault);

`rbfgetmodelversion` function

/*************************************************************************
This function returns model version.

INPUT PARAMETERS:
    S       -   RBF model

RESULT:
    * 1 - for models created by QNN and RBF-ML algorithms,
      compatible with ALGLIB 3.10 or earlier.
    * 2 - for models created by HierarchicalRBF, requires
      ALGLIB 3.11 or later

  -- ALGLIB --
     Copyright 06.07.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::rbfgetmodelversion(
    rbfmodel s,
    const xparams _params = alglib::xdefault);

`rbfgridcalc2` function

/*************************************************************************
This is legacy function for gridded calculation of RBF model.

It is superseded by rbfgridcalc2v() and  rbfgridcalc2vsubset()  functions.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfgridcalc2(
    rbfmodel s,
    real_1d_array x0,
    ae_int_t n0,
    real_1d_array x1,
    ae_int_t n1,
    real_2d_array& y,
    const xparams _params = alglib::xdefault);

`rbfgridcalc2v` function

/*************************************************************************
This function calculates values of the RBF  model  at  the  regular  grid,
which  has  N0*N1 points, with Point[I,J] = (X0[I], X1[J]).  Vector-valued
RBF models are supported.

This function returns 0.0 when:
* model is not initialized
* NX<>2

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel  processing  is  implemented only for modern (hierarchical)
      RBFs. Legacy version 1 RBFs (created  by  QNN  or  RBF-ML) are still
      processed serially.

INPUT PARAMETERS:
    S       -   RBF model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1], where NY is a  number of
                "output" vector values (this  function   supports  vector-
                valued RBF models). Y is out-variable and  is  reallocated
                by this function.
                Y[K+NY*(I0+I1*N0)]=F_k(X0[I0],X1[I1]), for:
                *  K=0...NY-1
                * I0=0...N0-1
                * I1=0...N1-1

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  rbfmodel
      structure in multiple threads calling  this function  for  different
      grids.

NOTE: if you need function values on some subset  of  regular  grid, which
      may be described as "several compact and  dense  islands",  you  may
      use rbfgridcalc2vsubset().

  -- ALGLIB --
     Copyright 27.01.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfgridcalc2v(
    rbfmodel s,
    real_1d_array x0,
    ae_int_t n0,
    real_1d_array x1,
    ae_int_t n1,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`rbfgridcalc2vsubset` function

/*************************************************************************
This function calculates values of the RBF model at some subset of regular
grid:
* grid has N0*N1 points, with Point[I,J] = (X0[I], X1[J])
* only values at some subset of this grid are required
Vector-valued RBF models are supported.

This function returns 0.0 when:
* model is not initialized
* NX<>2

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel  processing  is  implemented only for modern (hierarchical)
      RBFs. Legacy version 1 RBFs (created  by  QNN  or  RBF-ML) are still
      processed serially.

INPUT PARAMETERS:
    S       -   RBF model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension

    FlagY   -   array[N0*N1]:
                * Y[I0+I1*N0] corresponds to node (X0[I0],X1[I1])
                * it is a "bitmap" array which contains  False  for  nodes
                  which are NOT calculated, and True for nodes  which  are
                  required.

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1*N2], where NY is a  number
                of "output" vector values (this function  supports vector-
                valued RBF models):
                * Y[K+NY*(I0+I1*N0)]=F_k(X0[I0],X1[I1]),
                  for K=0...NY-1, I0=0...N0-1, I1=0...N1-1.
                * elements of Y[] which correspond  to  FlagY[]=True   are
                  loaded by model values (which may be  exactly  zero  for
                  some nodes).
                * elements of Y[] which correspond to FlagY[]=False MAY be
                  initialized by zeros OR may be calculated. This function
                  processes  grid  as  a  hierarchy  of  nested blocks and
                  micro-rows. If just one element of micro-row is required,
                  entire micro-row (up to 8 nodes in the current  version,
                  but no promises) is calculated.

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  rbfmodel
      structure in multiple threads calling  this function  for  different
      grids.

  -- ALGLIB --
     Copyright 04.03.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfgridcalc2vsubset(
    rbfmodel s,
    real_1d_array x0,
    ae_int_t n0,
    real_1d_array x1,
    ae_int_t n1,
    boolean_1d_array flagy,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`rbfgridcalc3v` function

/*************************************************************************
This function calculates values of the RBF  model  at  the  regular  grid,
which  has  N0*N1*N2  points,  with  Point[I,J,K] = (X0[I], X1[J], X2[K]).
Vector-valued RBF models are supported.

This function returns 0.0 when:
* model is not initialized
* NX<>3

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel  processing  is  implemented only for modern (hierarchical)
      RBFs. Legacy version 1 RBFs (created  by  QNN  or  RBF-ML) are still
      processed serially.

INPUT PARAMETERS:
    S       -   RBF model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension

    X2      -   array of grid nodes, third coordinates, array[N2]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N2      -   grid size (number of nodes) in the third dimension

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1*N2], where NY is a  number
                of "output" vector values (this function  supports vector-
                valued RBF models). Y is out-variable and  is  reallocated
                by this function.
                Y[K+NY*(I0+I1*N0+I2*N0*N1)]=F_k(X0[I0],X1[I1],X2[I2]), for:
                *  K=0...NY-1
                * I0=0...N0-1
                * I1=0...N1-1
                * I2=0...N2-1

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  rbfmodel
      structure in multiple threads calling  this function  for  different
      grids.

NOTE: if you need function values on some subset  of  regular  grid, which
      may be described as "several compact and  dense  islands",  you  may
      use rbfgridcalc3vsubset().

  -- ALGLIB --
     Copyright 04.03.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfgridcalc3v(
    rbfmodel s,
    real_1d_array x0,
    ae_int_t n0,
    real_1d_array x1,
    ae_int_t n1,
    real_1d_array x2,
    ae_int_t n2,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`rbfgridcalc3vsubset` function

/*************************************************************************
This function calculates values of the RBF model at some subset of regular
grid:
* grid has N0*N1*N2 points, with Point[I,J,K] = (X0[I], X1[J], X2[K])
* only values at some subset of this grid are required
Vector-valued RBF models are supported.

This function returns 0.0 when:
* model is not initialized
* NX<>3

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel  processing  is  implemented only for modern (hierarchical)
      RBFs. Legacy version 1 RBFs (created  by  QNN  or  RBF-ML) are still
      processed serially.

INPUT PARAMETERS:
    S       -   RBF model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension

    X2      -   array of grid nodes, third coordinates, array[N2]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N2      -   grid size (number of nodes) in the third dimension

    FlagY   -   array[N0*N1*N2]:
                * Y[I0+I1*N0+I2*N0*N1] corresponds to node (X0[I0],X1[I1],X2[I2])
                * it is a "bitmap" array which contains  False  for  nodes
                  which are NOT calculated, and True for nodes  which  are
                  required.

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1*N2], where NY is a  number
                of "output" vector values (this function  supports vector-
                valued RBF models):
                * Y[K+NY*(I0+I1*N0+I2*N0*N1)]=F_k(X0[I0],X1[I1],X2[I2]),
                  for K=0...NY-1, I0=0...N0-1, I1=0...N1-1, I2=0...N2-1.
                * elements of Y[] which correspond  to  FlagY[]=True   are
                  loaded by model values (which may be  exactly  zero  for
                  some nodes).
                * elements of Y[] which correspond to FlagY[]=False MAY be
                  initialized by zeros OR may be calculated. This function
                  processes  grid  as  a  hierarchy  of  nested blocks and
                  micro-rows. If just one element of micro-row is required,
                  entire micro-row (up to 8 nodes in the current  version,
                  but no promises) is calculated.

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  rbfmodel
      structure in multiple threads calling  this function  for  different
      grids.

  -- ALGLIB --
     Copyright 04.03.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfgridcalc3vsubset(
    rbfmodel s,
    real_1d_array x0,
    ae_int_t n0,
    real_1d_array x1,
    ae_int_t n1,
    real_1d_array x2,
    ae_int_t n2,
    boolean_1d_array flagy,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`rbfserialize` function

/*************************************************************************
This function serializes data structure to string.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string  in  case you want to include it into text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor you should change case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize  it  in  C++ version of ALGLIB and unserialize in C# one, 
  and vice versa.
*************************************************************************/
void rbfserialize(rbfmodel &obj, std::string &s_out);
void rbfserialize(rbfmodel &obj, std::ostream &s_out);

`rbfsetalgohierarchical` function

/*************************************************************************
This  function  sets  RBF interpolation algorithm. ALGLIB supports several
RBF algorithms with different properties.

This  algorithm is called Hierarchical RBF. It  similar  to  its  previous
incarnation, RBF-ML, i.e.  it  also  builds  a  sequence  of  models  with
decreasing radii. However, it uses more economical way of  building  upper
layers (ones with large radii), which results in faster model construction
and evaluation, as well as smaller memory footprint during construction.

This algorithm has following important features:
* ability to handle millions of points
* controllable smoothing via nonlinearity penalization
* support for NX-dimensional models with NX=1 or NX>3 (unlike QNN or RBF-ML)
* support for specification of per-dimensional  radii  via  scale  vector,
  which is set by means of rbfsetpointsandscales() function. This  feature
  is useful if you solve  spatio-temporal  interpolation  problems,  where
  different radii are required for spatial and temporal dimensions.

Running times are roughly proportional to:
* N*log(N)*NLayers - for model construction
* N*NLayers - for model evaluation
You may see that running time does not depend on search radius  or  points
density, just on number of layers in the hierarchy.

IMPORTANT: this model construction algorithm was introduced in ALGLIB 3.11
           and  produces  models  which  are  INCOMPATIBLE  with  previous
           versions of ALGLIB. You can  not  unserialize  models  produced
           with this function in ALGLIB 3.10 or earlier.

INPUT PARAMETERS:
    S       -   RBF model, initialized by rbfcreate() call
    RBase   -   RBase parameter, RBase>0
    NLayers -   NLayers parameter, NLayers>0, recommended value  to  start
                with - about 5.
    LambdaNS-   >=0, nonlinearity penalty coefficient, negative values are
                not allowed. This parameter adds controllable smoothing to
                the problem, which may reduce noise. Specification of non-
                zero lambda means that in addition to fitting error solver
                will  also  minimize   LambdaNS*|S''(x)|^2  (appropriately
                generalized to multiple dimensions.

                Specification of exactly zero value means that no  penalty
                is added  (we  do  not  even  evaluate  matrix  of  second
                derivatives which is necessary for smoothing).

                Calculation of nonlinearity penalty is costly - it results
                in  several-fold  increase  of  model  construction  time.
                Evaluation time remains the same.

                Optimal  lambda  is  problem-dependent and requires  trial
                and  error.  Good  value to  start  from  is  1e-5...1e-6,
                which corresponds to slightly noticeable smoothing  of the
                function.  Value  1e-2  usually  means  that  quite  heavy
                smoothing is applied.

TUNING ALGORITHM

In order to use this algorithm you have to choose three parameters:
* initial radius RBase
* number of layers in the model NLayers
* penalty coefficient LambdaNS

Initial radius is easy to choose - you can pick any number  several  times
larger  than  the  average  distance between points. Algorithm won't break
down if you choose radius which is too large (model construction time will
increase, but model will be built correctly).

Choose such number of layers that RLast=RBase/2^(NLayers-1)  (radius  used
by  the  last  layer)  will  be  smaller than the typical distance between
points.  In  case  model  error  is  too large, you can increase number of
layers.  Having  more  layers  will make model construction and evaluation
proportionally slower, but it will allow you to have model which precisely
fits your data. From the other side, if you want to  suppress  noise,  you
can DECREASE number of layers to make your model less flexible (or specify
non-zero LambdaNS).

TYPICAL ERRORS

1. Using too small number of layers - RBF models with large radius are not
   flexible enough to reproduce small variations in the  target  function.
   You  need  many  layers  with  different radii, from large to small, in
   order to have good model.

2. Using  initial  radius  which  is  too  small.  You will get model with
   "holes" in the areas which are too far away from interpolation centers.
   However, algorithm will work correctly (and quickly) in this case.

  -- ALGLIB --
     Copyright 20.06.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetalgohierarchical(
    rbfmodel s,
    double rbase,
    ae_int_t nlayers,
    double lambdans,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`rbfsetalgomultilayer` function

/*************************************************************************
DEPRECATED:since version 3.11 ALGLIB includes new RBF  model  construction
           algorithm, Hierarchical  RBF.  This  algorithm  is  faster  and
           requires less memory than QNN and RBF-ML. It is especially good
           for large-scale interpolation problems. So, we recommend you to
           consider Hierarchical RBF as default option.

==========================================================================

This  function  sets  RBF interpolation algorithm. ALGLIB supports several
RBF algorithms with different properties.

This  algorithm is called RBF-ML. It builds  multilayer  RBF  model,  i.e.
model with subsequently decreasing  radii,  which  allows  us  to  combine
smoothness (due to  large radii of  the first layers) with  exactness (due
to small radii of the last layers) and fast convergence.

Internally RBF-ML uses many different  means  of acceleration, from sparse
matrices  to  KD-trees,  which  results in algorithm whose working time is
roughly proportional to N*log(N)*Density*RBase^2*NLayers,  where  N  is  a
number of points, Density is an average density if points per unit of  the
interpolation space, RBase is an initial radius, NLayers is  a  number  of
layers.

RBF-ML is good for following kinds of interpolation problems:
1. "exact" problems (perfect fit) with well separated points
2. least squares problems with arbitrary distribution of points (algorithm
   gives  perfect  fit  where it is possible, and resorts to least squares
   fit in the hard areas).
3. noisy problems where  we  want  to  apply  some  controlled  amount  of
   smoothing.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    RBase   -   RBase parameter, RBase>0
    NLayers -   NLayers parameter, NLayers>0, recommended value  to  start
                with - about 5.
    LambdaV -   regularization value, can be useful when  solving  problem
                in the least squares sense.  Optimal  lambda  is  problem-
                dependent and require trial and error. In our  experience,
                good lambda can be as large as 0.1, and you can use  0.001
                as initial guess.
                Default  value  - 0.01, which is used when LambdaV is  not
                given.  You  can  specify  zero  value,  but  it  is   not
                recommended to do so.

TUNING ALGORITHM

In order to use this algorithm you have to choose three parameters:
* initial radius RBase
* number of layers in the model NLayers
* regularization coefficient LambdaV

Initial radius is easy to choose - you can pick any number  several  times
larger  than  the  average  distance between points. Algorithm won't break
down if you choose radius which is too large (model construction time will
increase, but model will be built correctly).

Choose such number of layers that RLast=RBase/2^(NLayers-1)  (radius  used
by  the  last  layer)  will  be  smaller than the typical distance between
points.  In  case  model  error  is  too large, you can increase number of
layers.  Having  more  layers  will make model construction and evaluation
proportionally slower, but it will allow you to have model which precisely
fits your data. From the other side, if you want to  suppress  noise,  you
can DECREASE number of layers to make your model less flexible.

Regularization coefficient LambdaV controls smoothness of  the  individual
models built for each layer. We recommend you to use default value in case
you don't want to tune this parameter,  because  having  non-zero  LambdaV
accelerates and stabilizes internal iterative algorithm. In case you  want
to suppress noise you can use  LambdaV  as  additional  parameter  (larger
value = more smoothness) to tune.

TYPICAL ERRORS

1. Using  initial  radius  which is too large. Memory requirements  of the
   RBF-ML are roughly proportional to N*Density*RBase^2 (where Density  is
   an average density of points per unit of the interpolation  space).  In
   the extreme case of the very large RBase we will need O(N^2)  units  of
   memory - and many layers in order to decrease radius to some reasonably
   small value.

2. Using too small number of layers - RBF models with large radius are not
   flexible enough to reproduce small variations in the  target  function.
   You  need  many  layers  with  different radii, from large to small, in
   order to have good model.

3. Using  initial  radius  which  is  too  small.  You will get model with
   "holes" in the areas which are too far away from interpolation centers.
   However, algorithm will work correctly (and quickly) in this case.

4. Using too many layers - you will get too large and too slow model. This
   model  will  perfectly  reproduce  your function, but maybe you will be
   able to achieve similar results with less layers (and less memory).

  -- ALGLIB --
     Copyright 02.03.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetalgomultilayer(
    rbfmodel s,
    double rbase,
    ae_int_t nlayers,
    const xparams _params = alglib::xdefault);
void alglib::rbfsetalgomultilayer(
    rbfmodel s,
    double rbase,
    ae_int_t nlayers,
    double lambdav,
    const xparams _params = alglib::xdefault);

`rbfsetalgoqnn` function

/*************************************************************************
DEPRECATED:since version 3.11 ALGLIB includes new RBF  model  construction
           algorithm, Hierarchical  RBF.  This  algorithm  is  faster  and
           requires less memory than QNN and RBF-ML. It is especially good
           for large-scale interpolation problems. So, we recommend you to
           consider Hierarchical RBF as default option.

==========================================================================

This  function  sets  RBF interpolation algorithm. ALGLIB supports several
RBF algorithms with different properties.

This algorithm is called RBF-QNN and  it  is  good  for  point  sets  with
following properties:
a) all points are distinct
b) all points are well separated.
c) points  distribution  is  approximately  uniform.  There is no "contour
   lines", clusters of points, or other small-scale structures.

Algorithm description:
1) interpolation centers are allocated to data points
2) interpolation radii are calculated as distances to the  nearest centers
   times Q coefficient (where Q is a value from [0.75,1.50]).
3) after  performing (2) radii are transformed in order to avoid situation
   when single outlier has very large radius and  influences  many  points
   across all dataset. Transformation has following form:
       new_r[i] = min(r[i],Z*median(r[]))
   where r[i] is I-th radius, median()  is a median  radius across  entire
   dataset, Z is user-specified value which controls amount  of  deviation
   from median radius.

When (a) is violated,  we  will  be unable to build RBF model. When (b) or
(c) are violated, model will be built, but interpolation quality  will  be
low. See http://www.alglib.net/interpolation/ for more information on this
subject.

This algorithm is used by default.

Additional Q parameter controls smoothness properties of the RBF basis:
* Q<0.75 will give perfectly conditioned basis,  but  terrible  smoothness
  properties (RBF interpolant will have sharp peaks around function values)
* Q around 1.0 gives good balance between smoothness and condition number
* Q>1.5 will lead to badly conditioned systems and slow convergence of the
  underlying linear solver (although smoothness will be very good)
* Q>2.0 will effectively make optimizer useless because it won't  converge
  within reasonable amount of iterations. It is possible to set such large
  Q, but it is advised not to do so.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    Q       -   Q parameter, Q>0, recommended value - 1.0
    Z       -   Z parameter, Z>0, recommended value - 5.0

NOTE: this   function  has   some   serialization-related  subtleties.  We
      recommend you to study serialization examples from ALGLIB  Reference
      Manual if you want to perform serialization of your models.


  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetalgoqnn(
    rbfmodel s,
    const xparams _params = alglib::xdefault);
void alglib::rbfsetalgoqnn(
    rbfmodel s,
    double q,
    double z,
    const xparams _params = alglib::xdefault);

`rbfsetconstterm` function

/*************************************************************************
This function sets constant term (model is a sum of radial basis functions
plus constant).  This  function  won't  have  effect  until  next  call to
RBFBuildModel().

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call

NOTE: this   function  has   some   serialization-related  subtleties.  We
      recommend you to study serialization examples from ALGLIB  Reference
      Manual if you want to perform serialization of your models.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetconstterm(
    rbfmodel s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rbfsetlinterm` function

/*************************************************************************
This function sets linear term (model is a sum of radial  basis  functions
plus linear polynomial). This function won't have effect until  next  call
to RBFBuildModel().

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call

NOTE: this   function  has   some   serialization-related  subtleties.  We
      recommend you to study serialization examples from ALGLIB  Reference
      Manual if you want to perform serialization of your models.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetlinterm(
    rbfmodel s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rbfsetpoints` function

/*************************************************************************
This function adds dataset.

This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.

IMPORTANT: ALGLIB version 3.11 and later allows you to specify  a  set  of
           per-dimension scales. Interpolation radii are multiplied by the
           scale vector. It may be useful if you have mixed spatio-temporal
           data (say, a set of 3D slices recorded at different times).
           You should call rbfsetpointsandscales() function  to  use  this
           feature.

INPUT PARAMETERS:
    S       -   RBF model, initialized by rbfcreate() call.
    XY      -   points, array[N,NX+NY]. One row corresponds to  one  point
                in the dataset. First NX elements  are  coordinates,  next
                NY elements are function values. Array may  be larger than
                specified, in  this  case  only leading [N,NX+NY] elements
                will be used.
    N       -   number of points in the dataset

After you've added dataset and (optionally) tuned algorithm  settings  you
should call rbfbuildmodel() in order to build a model for you.

NOTE: dataset added by this function is not saved during model serialization.
      MODEL ITSELF is serialized, but data used to build it are not.

      So, if you 1) add dataset to  empty  RBF  model,  2)  serialize  and
      unserialize it, then you will get an empty RBF model with no dataset
      being attached.

      From the other side, if you call rbfbuildmodel() between (1) and (2),
      then after (2) you will get your fully constructed RBF model  -  but
      again with no dataset attached, so subsequent calls to rbfbuildmodel()
      will produce empty model.


  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetpoints(
    rbfmodel s,
    real_2d_array xy,
    const xparams _params = alglib::xdefault);
void alglib::rbfsetpoints(
    rbfmodel s,
    real_2d_array xy,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`rbfsetpointsandscales` function

/*************************************************************************
This function adds dataset and a vector of per-dimension scales.

It may be useful if you have mixed spatio-temporal data - say, a set of 3D
slices recorded at different times. Such data typically require  different
RBF radii for spatial and temporal dimensions. ALGLIB solves this  problem
by specifying single RBF radius, which is (optionally) multiplied  by  the
scale vector.

This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.

IMPORTANT: only HierarchicalRBF algorithm can work with scaled points. So,
           using this function results in RBF models which can be used  in
           ALGLIB 3.11 or later. Previous versions of the library will  be
           unable  to unserialize models produced by HierarchicalRBF algo.

           Any attempt to use this function with RBF-ML or QNN  algorithms
           will result  in  -3  error  code   being   returned  (incorrect
           algorithm).

INPUT PARAMETERS:
    R       -   RBF model, initialized by rbfcreate() call.
    XY      -   points, array[N,NX+NY]. One row corresponds to  one  point
                in the dataset. First NX elements  are  coordinates,  next
                NY elements are function values. Array may  be larger than
                specified, in  this  case  only leading [N,NX+NY] elements
                will be used.
    N       -   number of points in the dataset
    S       -   array[NX], scale vector, S[i]>0.

After you've added dataset and (optionally) tuned algorithm  settings  you
should call rbfbuildmodel() in order to build a model for you.

NOTE: dataset added by this function is not saved during model serialization.
      MODEL ITSELF is serialized, but data used to build it are not.

      So, if you 1) add dataset to  empty  RBF  model,  2)  serialize  and
      unserialize it, then you will get an empty RBF model with no dataset
      being attached.

      From the other side, if you call rbfbuildmodel() between (1) and (2),
      then after (2) you will get your fully constructed RBF model  -  but
      again with no dataset attached, so subsequent calls to rbfbuildmodel()
      will produce empty model.


  -- ALGLIB --
     Copyright 20.06.2016 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetpointsandscales(
    rbfmodel r,
    real_2d_array xy,
    real_1d_array s,
    const xparams _params = alglib::xdefault);
void alglib::rbfsetpointsandscales(
    rbfmodel r,
    real_2d_array xy,
    ae_int_t n,
    real_1d_array s,
    const xparams _params = alglib::xdefault);

`rbfsetv2bf` function

/*************************************************************************
This function sets basis function type, which can be:
* 0 for classic Gaussian
* 1 for fast and compact bell-like basis function, which  becomes  exactly
  zero at distance equal to 3*R (default option).

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    BF      -   basis function type:
                * 0 - classic Gaussian
                * 1 - fast and compact one

  -- ALGLIB --
     Copyright 01.02.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetv2bf(
    rbfmodel s,
    ae_int_t bf,
    const xparams _params = alglib::xdefault);

`rbfsetv2its` function

/*************************************************************************
This function sets stopping criteria of the underlying linear  solver  for
hierarchical (version 2) RBF constructor.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    MaxIts  -   this criterion will stop algorithm after MaxIts iterations.
                Typically a few hundreds iterations is required,  with 400
                being a good default value to start experimentation.
                Zero value means that default value will be selected.

  -- ALGLIB --
     Copyright 01.02.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetv2its(
    rbfmodel s,
    ae_int_t maxits,
    const xparams _params = alglib::xdefault);

`rbfsetv2supportr` function

/*************************************************************************
This function sets support radius parameter  of  hierarchical  (version 2)
RBF constructor.

Hierarchical RBF model achieves great speed-up  by removing from the model
excessive (too dense) nodes. Say, if you have RBF radius equal to 1 meter,
and two nodes are just 1 millimeter apart, you  may  remove  one  of  them
without reducing model quality.

Support radius parameter is used to justify which points need removal, and
which do not. If two points are less than  SUPPORT_R*CUR_RADIUS  units  of
distance apart, one of them is removed from the model. The larger  support
radius  is, the faster model  construction  AND  evaluation are.  However,
too large values result in "bumpy" models.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    R       -   support radius coefficient, >=0.
                Recommended values are [0.1,0.4] range, with 0.1 being
                default value.

  -- ALGLIB --
     Copyright 01.02.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetv2supportr(
    rbfmodel s,
    double r,
    const xparams _params = alglib::xdefault);

`rbfsetzeroterm` function

/*************************************************************************
This  function  sets  zero  term (model is a sum of radial basis functions
without polynomial term). This function won't have effect until next  call
to RBFBuildModel().

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call

NOTE: this   function  has   some   serialization-related  subtleties.  We
      recommend you to study serialization examples from ALGLIB  Reference
      Manual if you want to perform serialization of your models.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfsetzeroterm(
    rbfmodel s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rbftscalcbuf` function

/*************************************************************************
This function calculates values of the RBF model at the given point, using
external  buffer  object  (internal  temporaries  of  RBF  model  are  not
modified).

This function allows to use same RBF model object  in  different  threads,
assuming  that  different   threads  use  different  instances  of  buffer
structure.

INPUT PARAMETERS:
    S       -   RBF model, may be shared between different threads
    Buf     -   buffer object created for this particular instance of  RBF
                model with rbfcreatecalcbuffer().
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.
    Y       -   possibly preallocated array

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is not reallocated when it
                is larger than NY.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbftscalcbuf(
    rbfmodel s,
    rbfcalcbuffer buf,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`rbfunpack` function

/*************************************************************************
This function "unpacks" RBF model by extracting its coefficients.

INPUT PARAMETERS:
    S       -   RBF model

OUTPUT PARAMETERS:
    NX      -   dimensionality of argument
    NY      -   dimensionality of the target function
    XWR     -   model information, array[NC,NX+NY+1].
                One row of the array corresponds to one basis function:
                * first NX columns  - coordinates of the center
                * next NY columns   - weights, one per dimension of the
                                      function being modelled
                For ModelVersion=1:
                * last column       - radius, same for all dimensions of
                                      the function being modelled
                For ModelVersion=2:
                * last NX columns   - radii, one per dimension
    NC      -   number of the centers
    V       -   polynomial  term , array[NY,NX+1]. One row per one
                dimension of the function being modelled. First NX
                elements are linear coefficients, V[NX] is equal to the
                constant part.
    ModelVersion-version of the RBF model:
                * 1 - for models created by QNN and RBF-ML algorithms,
                  compatible with ALGLIB 3.10 or earlier.
                * 2 - for models created by HierarchicalRBF, requires
                  ALGLIB 3.11 or later

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::rbfunpack(
    rbfmodel s,
    ae_int_t& nx,
    ae_int_t& ny,
    real_2d_array& xwr,
    ae_int_t& nc,
    real_2d_array& v,
    ae_int_t& modelversion,
    const xparams _params = alglib::xdefault);

Examples: [1]

`rbfunserialize` function

/*************************************************************************
This function unserializes data structure from string.
*************************************************************************/
void rbfunserialize(const std::string &s_in, rbfmodel &obj);
void rbfunserialize(const std::istream &s_in, rbfmodel &obj);

rbf_d_hrbf example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example illustrates basic concepts of the RBF models: creation, modification,
    // evaluation.
    // 
    // Suppose that we have set of 2-dimensional points with associated
    // scalar function values, and we want to build a RBF model using
    // our data.
    // 
    // NOTE: we can work with 3D models too :)
    // 
    // Typical sequence of steps is given below:
    // 1. we create RBF model object
    // 2. we attach our dataset to the RBF model and tune algorithm settings
    // 3. we rebuild RBF model using QNN algorithm on new data
    // 4. we use RBF model (evaluate, serialize, etc.)
    //
    double v;

    //
    // Step 1: RBF model creation.
    //
    // We have to specify dimensionality of the space (2 or 3) and
    // dimensionality of the function (scalar or vector).
    //
    // New model is empty - it can be evaluated,
    // but we just get zero value at any point.
    //
    rbfmodel model;
    rbfcreate(2, 1, model);

    v = rbfcalc2(model, 0.0, 0.0);
    printf("%.2f\n", double(v)); // EXPECTED: 0.000

    //
    // Step 2: we add dataset.
    //
    // XY contains two points - x0=(-1,0) and x1=(+1,0) -
    // and two function values f(x0)=2, f(x1)=3.
    //
    // We added points, but model was not rebuild yet.
    // If we call rbfcalc2(), we still will get 0.0 as result.
    //
    real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
    rbfsetpoints(model, xy);

    v = rbfcalc2(model, 0.0, 0.0);
    printf("%.2f\n", double(v)); // EXPECTED: 0.000

    //
    // Step 3: rebuild model
    //
    // After we've configured model, we should rebuild it -
    // it will change coefficients stored internally in the
    // rbfmodel structure.
    //
    // We use hierarchical RBF algorithm with following parameters:
    // * RBase - set to 1.0
    // * NLayers - three layers are used (although such simple problem
    //   does not need more than 1 layer)
    // * LambdaReg - is set to zero value, no smoothing is required
    //
    rbfreport rep;
    rbfsetalgohierarchical(model, 1.0, 3, 0.0);
    rbfbuildmodel(model, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1

    //
    // Step 4: model was built
    //
    // After call of rbfbuildmodel(), rbfcalc2() will return
    // value of the new model.
    //
    v = rbfcalc2(model, 0.0, 0.0);
    printf("%.2f\n", double(v)); // EXPECTED: 2.500
    return 0;
}

rbf_d_polterm example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example show how to work with polynomial term
    // 
    // Suppose that we have set of 2-dimensional points with associated
    // scalar function values, and we want to build a RBF model using
    // our data.
    //
    // We use hierarchical RBF algorithm with following parameters:
    // * RBase - set to 1.0
    // * NLayers - three layers are used (although such simple problem
    //   does not need more than 1 layer)
    // * LambdaReg - is set to zero value, no smoothing is required
    //
    double v;
    rbfmodel model;
    real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
    rbfreport rep;

    rbfcreate(2, 1, model);
    rbfsetpoints(model, xy);
    rbfsetalgohierarchical(model, 1.0, 3, 0.0);

    //
    // By default, RBF model uses linear term. It means that model
    // looks like
    //     f(x,y) = SUM(RBF[i]) + a*x + b*y + c
    // where RBF[i] is I-th radial basis function and a*x+by+c is a
    // linear term. Having linear terms in a model gives us:
    // (1) improved extrapolation properties
    // (2) linearity of the model when data can be perfectly fitted
    //     by the linear function
    // (3) linear asymptotic behavior
    //
    // Our simple dataset can be modelled by the linear function
    //     f(x,y) = 0.5*x + 2.5
    // and rbfbuildmodel() with default settings should preserve this
    // linearity.
    //
    ae_int_t nx;
    ae_int_t ny;
    ae_int_t nc;
    ae_int_t modelversion;
    real_2d_array xwr;
    real_2d_array c;
    rbfbuildmodel(model, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
    rbfunpack(model, nx, ny, xwr, nc, c, modelversion);
    printf("%s\n", c.tostring(2).c_str()); // EXPECTED: [[0.500,0.000,2.500]]

    // asymptotic behavior of our function is linear
    v = rbfcalc2(model, 1000.0, 0.0);
    printf("%.1f\n", double(v)); // EXPECTED: 502.50

    //
    // Instead of linear term we can use constant term. In this case
    // we will get model which has form
    //     f(x,y) = SUM(RBF[i]) + c
    // where RBF[i] is I-th radial basis function and c is a constant,
    // which is equal to the average function value on the dataset.
    //
    // Because we've already attached dataset to the model the only
    // thing we have to do is to call rbfsetconstterm() and then
    // rebuild model with rbfbuildmodel().
    //
    rbfsetconstterm(model);
    rbfbuildmodel(model, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
    rbfunpack(model, nx, ny, xwr, nc, c, modelversion);
    printf("%s\n", c.tostring(2).c_str()); // EXPECTED: [[0.000,0.000,2.500]]

    // asymptotic behavior of our function is constant
    v = rbfcalc2(model, 1000.0, 0.0);
    printf("%.2f\n", double(v)); // EXPECTED: 2.500

    //
    // Finally, we can use zero term. Just plain RBF without polynomial
    // part:
    //     f(x,y) = SUM(RBF[i])
    // where RBF[i] is I-th radial basis function.
    //
    rbfsetzeroterm(model);
    rbfbuildmodel(model, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
    rbfunpack(model, nx, ny, xwr, nc, c, modelversion);
    printf("%s\n", c.tostring(2).c_str()); // EXPECTED: [[0.000,0.000,0.000]]

    // asymptotic behavior of our function is just zero constant
    v = rbfcalc2(model, 1000.0, 0.0);
    printf("%.2f\n", double(v)); // EXPECTED: 0.000
    return 0;
}

rbf_d_serialize example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example show how to serialize and unserialize RBF model
    // 
    // Suppose that we have set of 2-dimensional points with associated
    // scalar function values, and we want to build a RBF model using
    // our data. Then we want to serialize it to string and to unserialize
    // from string, loading to another instance of RBF model.
    //
    // Here we assume that you already know how to create RBF models.
    //
    std::string s;
    double v;
    rbfmodel model0;
    rbfmodel model1;
    real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
    rbfreport rep;

    // model initialization
    rbfcreate(2, 1, model0);
    rbfsetpoints(model0, xy);
    rbfsetalgohierarchical(model0, 1.0, 3, 0.0);
    rbfbuildmodel(model0, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1

    //
    // Serialization - it looks easy,
    // but you should carefully read next section.
    //
    alglib::rbfserialize(model0, s);
    alglib::rbfunserialize(s, model1);

    // both models return same value
    v = rbfcalc2(model0, 0.0, 0.0);
    printf("%.2f\n", double(v)); // EXPECTED: 2.500
    v = rbfcalc2(model1, 0.0, 0.0);
    printf("%.2f\n", double(v)); // EXPECTED: 2.500

    //
    // Previous section shows that model state is saved/restored during
    // serialization. However, some properties are NOT serialized.
    //
    // Serialization saves/restores RBF model, but it does NOT saves/restores
    // settings which were used to build current model. In particular, dataset
    // which was used to build model, is not preserved.
    //
    // What does it mean in for us?
    //
    // Do you remember this sequence: rbfcreate-rbfsetpoints-rbfbuildmodel?
    // First step creates model, second step adds dataset and tunes model
    // settings, third step builds model using current dataset and model
    // construction settings.
    //
    // If you call rbfbuildmodel() without calling rbfsetpoints() first, you
    // will get empty (zero) RBF model. In our example, model0 contains
    // dataset which was added by rbfsetpoints() call. However, model1 does
    // NOT contain dataset - because dataset is NOT serialized.
    //
    // This, if we call rbfbuildmodel(model0,rep), we will get same model,
    // which returns 2.5 at (x,y)=(0,0). However, after same call model1 will
    // return zero - because it contains RBF model (coefficients), but does NOT
    // contain dataset which was used to build this model.
    //
    // Basically, it means that:
    // * serialization of the RBF model preserves anything related to the model
    //   EVALUATION
    // * but it does NOT creates perfect copy of the original object.
    //
    rbfbuildmodel(model0, rep);
    v = rbfcalc2(model0, 0.0, 0.0);
    printf("%.2f\n", double(v)); // EXPECTED: 2.500

    rbfbuildmodel(model1, rep);
    v = rbfcalc2(model1, 0.0, 0.0);
    printf("%.2f\n", double(v)); // EXPECTED: 0.000
    return 0;
}

rbf_d_vector example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Suppose that we have set of 2-dimensional points with associated VECTOR
    // function values, and we want to build a RBF model using our data.
    // 
    // Typical sequence of steps is given below:
    // 1. we create RBF model object
    // 2. we attach our dataset to the RBF model and tune algorithm settings
    // 3. we rebuild RBF model using new data
    // 4. we use RBF model (evaluate, serialize, etc.)
    //
    real_1d_array x;
    real_1d_array y;

    //
    // Step 1: RBF model creation.
    //
    // We have to specify dimensionality of the space (equal to 2) and
    // dimensionality of the function (2-dimensional vector function).
    //
    // New model is empty - it can be evaluated,
    // but we just get zero value at any point.
    //
    rbfmodel model;
    rbfcreate(2, 2, model);

    x = "[+1,+1]";
    rbfcalc(model, x, y);
    printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [0.000,0.000]

    //
    // Step 2: we add dataset.
    //
    // XY arrays containt four points:
    // * (x0,y0) = (+1,+1), f(x0,y0)=(0,-1)
    // * (x1,y1) = (+1,-1), f(x1,y1)=(-1,0)
    // * (x2,y2) = (-1,-1), f(x2,y2)=(0,+1)
    // * (x3,y3) = (-1,+1), f(x3,y3)=(+1,0)
    //
    real_2d_array xy = "[[+1,+1,0,-1],[+1,-1,-1,0],[-1,-1,0,+1],[-1,+1,+1,0]]";
    rbfsetpoints(model, xy);

    // We added points, but model was not rebuild yet.
    // If we call rbfcalc(), we still will get 0.0 as result.
    rbfcalc(model, x, y);
    printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [0.000,0.000]

    //
    // Step 3: rebuild model
    //
    // We use hierarchical RBF algorithm with following parameters:
    // * RBase - set to 1.0
    // * NLayers - three layers are used (although such simple problem
    //   does not need more than 1 layer)
    // * LambdaReg - is set to zero value, no smoothing is required
    //
    // After we've configured model, we should rebuild it -
    // it will change coefficients stored internally in the
    // rbfmodel structure.
    //
    rbfreport rep;
    rbfsetalgohierarchical(model, 1.0, 3, 0.0);
    rbfbuildmodel(model, rep);
    printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1

    //
    // Step 4: model was built
    //
    // After call of rbfbuildmodel(), rbfcalc() will return
    // value of the new model.
    //
    rbfcalc(model, x, y);
    printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [0.000,-1.000]
    return 0;
}

`rcond` subpackage

Functions

cmatrixlurcond1
cmatrixlurcondinf
cmatrixrcond1
cmatrixrcondinf
cmatrixtrrcond1
cmatrixtrrcondinf
hpdmatrixcholeskyrcond
hpdmatrixrcond
rmatrixlurcond1
rmatrixlurcondinf
rmatrixrcond1
rmatrixrcondinf
rmatrixtrrcond1
rmatrixtrrcondinf
spdmatrixcholeskyrcond
spdmatrixrcond

Examples

`cmatrixlurcond1` function

/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    LUA         -   LU decomposition of a matrix in compact form. Output of
                    the CMatrixLU subroutine.
    N           -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::cmatrixlurcond1(
    complex_2d_array lua,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`cmatrixlurcondinf` function

/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition
(infinity norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    LUA     -   LU decomposition of a matrix in compact form. Output of
                the CMatrixLU subroutine.
    N       -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::cmatrixlurcondinf(
    complex_2d_array lua,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`cmatrixrcond1` function

/*************************************************************************
Estimate of a matrix condition number (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::cmatrixrcond1(
    complex_2d_array a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`cmatrixrcondinf` function

/*************************************************************************
Estimate of a matrix condition number (infinity-norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::cmatrixrcondinf(
    complex_2d_array a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`cmatrixtrrcond1` function

/*************************************************************************
Triangular matrix: estimate of a condition number (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A       -   matrix. Array[0..N-1, 0..N-1].
    N       -   size of A.
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   True, if the matrix has a unit diagonal.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::cmatrixtrrcond1(
    complex_2d_array a,
    ae_int_t n,
    bool isupper,
    bool isunit,
    const xparams _params = alglib::xdefault);

`cmatrixtrrcondinf` function

/*************************************************************************
Triangular matrix: estimate of a matrix condition number (infinity-norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   True, if the matrix has a unit diagonal.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::cmatrixtrrcondinf(
    complex_2d_array a,
    ae_int_t n,
    bool isupper,
    bool isunit,
    const xparams _params = alglib::xdefault);

`hpdmatrixcholeskyrcond` function

/*************************************************************************
Condition number estimate of a Hermitian positive definite matrix given by
Cholesky decomposition.

The algorithm calculates a lower bound of the condition number. In this
case, the algorithm does not return a lower bound of the condition number,
but an inverse number (to avoid an overflow in case of a singular matrix).

It should be noted that 1-norm and inf-norm condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.

Input parameters:
    CD  - Cholesky decomposition of matrix A,
          output of SMatrixCholesky subroutine.
    N   - size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::hpdmatrixcholeskyrcond(
    complex_2d_array a,
    ae_int_t n,
    bool isupper,
    const xparams _params = alglib::xdefault);

`hpdmatrixrcond` function

/*************************************************************************
Condition number estimate of a Hermitian positive definite matrix.

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

It should be noted that 1-norm and inf-norm of condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.

Input parameters:
    A       -   Hermitian positive definite matrix which is given by its
                upper or lower triangle depending on the value of
                IsUpper. Array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format.

Result:
    1/LowerBound(cond(A)), if matrix A is positive definite,
   -1, if matrix A is not positive definite, and its condition number
    could not be found by this algorithm.

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::hpdmatrixrcond(
    complex_2d_array a,
    ae_int_t n,
    bool isupper,
    const xparams _params = alglib::xdefault);

`rmatrixlurcond1` function

/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    LUA         -   LU decomposition of a matrix in compact form. Output of
                    the RMatrixLU subroutine.
    N           -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::rmatrixlurcond1(
    real_2d_array lua,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`rmatrixlurcondinf` function

/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition
(infinity norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    LUA     -   LU decomposition of a matrix in compact form. Output of
                the RMatrixLU subroutine.
    N       -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::rmatrixlurcondinf(
    real_2d_array lua,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`rmatrixrcond1` function

/*************************************************************************
Estimate of a matrix condition number (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::rmatrixrcond1(
    real_2d_array a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`rmatrixrcondinf` function

/*************************************************************************
Estimate of a matrix condition number (infinity-norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::rmatrixrcondinf(
    real_2d_array a,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`rmatrixtrrcond1` function

/*************************************************************************
Triangular matrix: estimate of a condition number (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A       -   matrix. Array[0..N-1, 0..N-1].
    N       -   size of A.
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   True, if the matrix has a unit diagonal.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::rmatrixtrrcond1(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    bool isunit,
    const xparams _params = alglib::xdefault);

`rmatrixtrrcondinf` function

/*************************************************************************
Triangular matrix: estimate of a matrix condition number (infinity-norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   True, if the matrix has a unit diagonal.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::rmatrixtrrcondinf(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    bool isunit,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskyrcond` function

/*************************************************************************
Condition number estimate of a symmetric positive definite matrix given by
Cholesky decomposition.

The algorithm calculates a lower bound of the condition number. In this
case, the algorithm does not return a lower bound of the condition number,
but an inverse number (to avoid an overflow in case of a singular matrix).

It should be noted that 1-norm and inf-norm condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.

Input parameters:
    CD  - Cholesky decomposition of matrix A,
          output of SMatrixCholesky subroutine.
    N   - size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::spdmatrixcholeskyrcond(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    const xparams _params = alglib::xdefault);

`spdmatrixrcond` function

/*************************************************************************
Condition number estimate of a symmetric positive definite matrix.

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

It should be noted that 1-norm and inf-norm of condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.

Input parameters:
    A       -   symmetric positive definite matrix which is given by its
                upper or lower triangle depending on the value of
                IsUpper. Array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format.

Result:
    1/LowerBound(cond(A)), if matrix A is positive definite,
   -1, if matrix A is not positive definite, and its condition number
    could not be found by this algorithm.

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double alglib::spdmatrixrcond(
    real_2d_array a,
    ae_int_t n,
    bool isupper,
    const xparams _params = alglib::xdefault);

`schur` subpackage

Functions

rmatrixschur

Examples

`rmatrixschur` function

/*************************************************************************
Subroutine performing the Schur decomposition of a general matrix by using
the QR algorithm with multiple shifts.

COMMERCIAL EDITION OF ALGLIB:

  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives approximately constant  (with  respect  to  number  of
  ! worker threads) acceleration factor which depends on CPU  being  used,
  ! problem  size  and  "baseline"  ALGLIB  edition  which  is  used   for
  ! comparison.
  !
  ! Multithreaded acceleration is NOT supported for this function.
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

The source matrix A is represented as S'*A*S = T, where S is an orthogonal
matrix (Schur vectors), T - upper quasi-triangular matrix (with blocks of
sizes 1x1 and 2x2 on the main diagonal).

Input parameters:
    A   -   matrix to be decomposed.
            Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of A, N>=0.


Output parameters:
    A   -   contains matrix T.
            Array whose indexes range within [0..N-1, 0..N-1].
    S   -   contains Schur vectors.
            Array whose indexes range within [0..N-1, 0..N-1].

Note 1:
    The block structure of matrix T can be easily recognized: since all
    the elements below the blocks are zeros, the elements a[i+1,i] which
    are equal to 0 show the block border.

Note 2:
    The algorithm performance depends on the value of the internal parameter
    NS of the InternalSchurDecomposition subroutine which defines the number
    of shifts in the QR algorithm (similarly to the block width in block-matrix
    algorithms in linear algebra). If you require maximum performance on
    your machine, it is recommended to adjust this parameter manually.

Result:
    True,
        if the algorithm has converged and parameters A and S contain the result.
    False,
        if the algorithm has not converged.

Algorithm implemented on the basis of the DHSEQR subroutine (LAPACK 3.0 library).
*************************************************************************/
bool alglib::rmatrixschur(
    real_2d_array& a,
    ae_int_t n,
    real_2d_array& s,
    const xparams _params = alglib::xdefault);

`sparse` subpackage

Classes

sparsebuffers
sparsematrix

Functions

sparseadd
sparseconvertto
sparseconverttocrs
sparseconverttohash
sparseconverttosks
sparsecopy
sparsecopybuf
sparsecopytobuf
sparsecopytocrs
sparsecopytocrsbuf
sparsecopytohash
sparsecopytohashbuf
sparsecopytosks
sparsecopytosksbuf
sparsecreate
sparsecreatebuf
sparsecreatecrs
sparsecreatecrsbuf
sparsecreatesks
sparsecreatesksband
sparsecreatesksbandbuf
sparsecreatesksbuf
sparseenumerate
sparsefree
sparseget
sparsegetcompressedrow
sparsegetdiagonal
sparsegetlowercount
sparsegetmatrixtype
sparsegetncols
sparsegetnrows
sparsegetrow
sparsegetuppercount
sparseiscrs
sparseishash
sparseissks
sparsemm
sparsemm2
sparsemtm
sparsemtv
sparsemv
sparsemv2
sparseresizematrix
sparserewriteexisting
sparseset
sparsesmm
sparsesmv
sparseswap
sparsetransposecrs
sparsetransposesks
sparsetrmv
sparsetrsv
sparsevsmv

Examples

sparse_d_1		Basic operations with sparse matrices
sparse_d_crs		Advanced topic: creation in the CRS format.

`sparsebuffers` class

/*************************************************************************
Temporary buffers for sparse matrix operations.

You should pass an instance of this structure to factorization  functions.
It allows to reuse memory during repeated sparse  factorizations.  You  do
not have to call some initialization function - simply passing an instance
to factorization function is enough.
*************************************************************************/
class sparsebuffers
{
};

`sparsematrix` class

/*************************************************************************
Sparse matrix structure.

You should use ALGLIB functions to work with sparse matrix. Never  try  to
access its fields directly!

NOTES ON THE SPARSE STORAGE FORMATS

Sparse matrices can be stored using several formats:
* Hash-Table representation
* Compressed Row Storage (CRS)
* Skyline matrix storage (SKS)

Each of the formats has benefits and drawbacks:
* Hash-table is good for dynamic operations (insertion of new elements),
  but does not support linear algebra operations
* CRS is good for operations like matrix-vector or matrix-matrix products,
  but its initialization is less convenient - you have to tell row   sizes
  at the initialization, and you have to fill  matrix  only  row  by  row,
  from left to right.
* SKS is a special format which is used to store triangular  factors  from
  Cholesky factorization. It does not support  dynamic  modification,  and
  support for linear algebra operations is very limited.

Tables below outline information about these two formats:

    OPERATIONS WITH MATRIX      HASH        CRS         SKS
    creation                    +           +           +
    SparseGet                   +           +           +
    SparseRewriteExisting       +           +           +
    SparseSet                   +           +           +
    SparseAdd                   +
    SparseGetRow                            +           +
    SparseGetCompressedRow                  +           +
    sparse-dense linear algebra             +           +
*************************************************************************/
class sparsematrix
{
};

`sparseadd` function

/*************************************************************************
This function adds value to S[i,j] - element of the sparse matrix. Matrix
must be in a Hash-Table mode.

In case S[i,j] already exists in the table, V i added to  its  value.  In
case  S[i,j]  is  non-existent,  it  is  inserted  in  the  table.  Table
automatically grows when necessary.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    Exception will be thrown for CRS matrix.
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N
    V           -   value to add, must be finite number

OUTPUT PARAMETERS
    S           -   modified matrix

NOTE 1:  when  S[i,j]  is exactly zero after modification, it is  deleted
from the table.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparseadd(
    sparsematrix s,
    ae_int_t i,
    ae_int_t j,
    double v,
    const xparams _params = alglib::xdefault);

Examples: [1]

`sparseconvertto` function

/*************************************************************************
This  function  performs  in-place  conversion  to  desired sparse storage
format.

INPUT PARAMETERS
    S0      -   sparse matrix in any format.
    Fmt     -   desired storage format  of  the  output,  as  returned  by
                SparseGetMatrixType() function:
                * 0 for hash-based storage
                * 1 for CRS
                * 2 for SKS

OUTPUT PARAMETERS
    S0          -   sparse matrix in requested format.

NOTE: in-place conversion wastes a lot of memory which is  used  to  store
      temporaries.  If  you  perform  a  lot  of  repeated conversions, we
      recommend to use out-of-place buffered  conversion  functions,  like
      SparseCopyToBuf(), which can reuse already allocated memory.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparseconvertto(
    sparsematrix s0,
    ae_int_t fmt,
    const xparams _params = alglib::xdefault);

`sparseconverttocrs` function

/*************************************************************************
This function converts matrix to CRS format.

Some  algorithms  (linear  algebra ones, for example) require matrices in
CRS format. This function allows to perform in-place conversion.

INPUT PARAMETERS
    S           -   sparse M*N matrix in any format

OUTPUT PARAMETERS
    S           -   matrix in CRS format

NOTE: this   function  has  no  effect  when  called with matrix which is
      already in CRS mode.

NOTE: this function allocates temporary memory to store a   copy  of  the
      matrix. If you perform a lot of repeated conversions, we  recommend
      you  to  use  SparseCopyToCRSBuf()  function,   which   can   reuse
      previously allocated memory.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparseconverttocrs(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`sparseconverttohash` function

/*************************************************************************
This function performs in-place conversion to Hash table storage.

INPUT PARAMETERS
    S           -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S           -   sparse matrix in Hash table format.

NOTE: this  function  has   no  effect  when  called with matrix which  is
      already in Hash table mode.

NOTE: in-place conversion involves allocation of temporary arrays. If  you
      perform a lot of repeated in- place  conversions,  it  may  lead  to
      memory fragmentation. Consider using out-of-place SparseCopyToHashBuf()
      function in this case.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::sparseconverttohash(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparseconverttosks` function

/*************************************************************************
This function performs in-place conversion to SKS format.

INPUT PARAMETERS
    S           -   sparse matrix in any format.

OUTPUT PARAMETERS
    S           -   sparse matrix in SKS format.

NOTE: this  function  has   no  effect  when  called with matrix which  is
      already in SKS mode.

NOTE: in-place conversion involves allocation of temporary arrays. If  you
      perform a lot of repeated in- place  conversions,  it  may  lead  to
      memory fragmentation. Consider using out-of-place SparseCopyToSKSBuf()
      function in this case.

  -- ALGLIB PROJECT --
     Copyright 15.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparseconverttosks(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsecopy` function

/*************************************************************************
This function copies S0 to S1.
This function completely deallocates memory owned by S1 before creating a
copy of S0. If you want to reuse memory, use SparseCopyBuf.

NOTE:  this  function  does  not verify its arguments, it just copies all
fields of the structure.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecopy(
    sparsematrix s0,
    sparsematrix& s1,
    const xparams _params = alglib::xdefault);

`sparsecopybuf` function

/*************************************************************************
This function copies S0 to S1.
Memory already allocated in S1 is reused as much as possible.

NOTE:  this  function  does  not verify its arguments, it just copies all
fields of the structure.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecopybuf(
    sparsematrix s0,
    sparsematrix s1,
    const xparams _params = alglib::xdefault);

`sparsecopytobuf` function

/*************************************************************************
This  function  performs out-of-place conversion to desired sparse storage
format. S0 is copied to S1 and converted on-the-fly. Memory  allocated  in
S1 is reused to maximum extent possible.

INPUT PARAMETERS
    S0      -   sparse matrix in any format.
    Fmt     -   desired storage format  of  the  output,  as  returned  by
                SparseGetMatrixType() function:
                * 0 for hash-based storage
                * 1 for CRS
                * 2 for SKS

OUTPUT PARAMETERS
    S1          -   sparse matrix in requested format.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecopytobuf(
    sparsematrix s0,
    ae_int_t fmt,
    sparsematrix s1,
    const xparams _params = alglib::xdefault);

`sparsecopytocrs` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to  CRS format.  S0 is
copied to S1 and converted on-the-fly.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in CRS format.

NOTE: if S0 is stored as CRS, it is just copied without conversion.

NOTE: this function de-allocates memory occupied by S1 before starting CRS
      conversion. If you perform a lot of repeated CRS conversions, it may
      lead to memory fragmentation. In this case we recommend you  to  use
      SparseCopyToCRSBuf() function which re-uses memory in S1 as much  as
      possible.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecopytocrs(
    sparsematrix s0,
    sparsematrix& s1,
    const xparams _params = alglib::xdefault);

`sparsecopytocrsbuf` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to  CRS format.  S0 is
copied to S1 and converted on-the-fly. Memory allocated in S1 is reused to
maximum extent possible.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.
    S1          -   matrix which may contain some pre-allocated memory, or
                    can be just uninitialized structure.

OUTPUT PARAMETERS
    S1          -   sparse matrix in CRS format.

NOTE: if S0 is stored as CRS, it is just copied without conversion.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecopytocrsbuf(
    sparsematrix s0,
    sparsematrix s1,
    const xparams _params = alglib::xdefault);

`sparsecopytohash` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to  Hash table storage
format. S0 is copied to S1 and converted on-the-fly.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in Hash table format.

NOTE: if S0 is stored as Hash-table, it is just copied without conversion.

NOTE: this function de-allocates memory  occupied  by  S1 before  starting
      conversion. If you perform a  lot  of  repeated  conversions, it may
      lead to memory fragmentation. In this case we recommend you  to  use
      SparseCopyToHashBuf() function which re-uses memory in S1 as much as
      possible.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecopytohash(
    sparsematrix s0,
    sparsematrix& s1,
    const xparams _params = alglib::xdefault);

`sparsecopytohashbuf` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to  Hash table storage
format. S0 is copied to S1 and converted on-the-fly. Memory  allocated  in
S1 is reused to maximum extent possible.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in Hash table format.

NOTE: if S0 is stored as Hash-table, it is just copied without conversion.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecopytohashbuf(
    sparsematrix s0,
    sparsematrix s1,
    const xparams _params = alglib::xdefault);

`sparsecopytosks` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to SKS storage format.
S0 is copied to S1 and converted on-the-fly.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in SKS format.

NOTE: if S0 is stored as SKS, it is just copied without conversion.

NOTE: this function de-allocates memory  occupied  by  S1 before  starting
      conversion. If you perform a  lot  of  repeated  conversions, it may
      lead to memory fragmentation. In this case we recommend you  to  use
      SparseCopyToSKSBuf() function which re-uses memory in S1 as much  as
      possible.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecopytosks(
    sparsematrix s0,
    sparsematrix& s1,
    const xparams _params = alglib::xdefault);

`sparsecopytosksbuf` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to SKS format.  S0  is
copied to S1 and converted on-the-fly. Memory  allocated  in S1 is  reused
to maximum extent possible.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in SKS format.

NOTE: if S0 is stored as SKS, it is just copied without conversion.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecopytosksbuf(
    sparsematrix s0,
    sparsematrix s1,
    const xparams _params = alglib::xdefault);

`sparsecreate` function

/*************************************************************************
This function creates sparse matrix in a Hash-Table format.

This function creates Hast-Table matrix, which can be  converted  to  CRS
format after its initialization is over. Typical  usage  scenario  for  a
sparse matrix is:
1. creation in a Hash-Table format
2. insertion of the matrix elements
3. conversion to the CRS representation
4. matrix is passed to some linear algebra algorithm

Some  information  about  different matrix formats can be found below, in
the "NOTES" section.

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    K           -   K>=0, expected number of non-zero elements in a matrix.
                    K can be inexact approximation, can be less than actual
                    number  of  elements  (table will grow when needed) or
                    even zero).
                    It is important to understand that although hash-table
                    may grow automatically, it is better to  provide  good
                    estimate of data size.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    All elements of the matrix are zero.

NOTE 1

Hash-tables use memory inefficiently, and they have to keep  some  amount
of the "spare memory" in order to have good performance. Hash  table  for
matrix with K non-zero elements will  need  C*K*(8+2*sizeof(int))  bytes,
where C is a small constant, about 1.5-2 in magnitude.

CRS storage, from the other side, is  more  memory-efficient,  and  needs
just K*(8+sizeof(int))+M*sizeof(int) bytes, where M is a number  of  rows
in a matrix.

When you convert from the Hash-Table to CRS  representation, all unneeded
memory will be freed.

NOTE 2

Comments of SparseMatrix structure outline  information  about  different
sparse storage formats. We recommend you to read them before starting  to
use ALGLIB sparse matrices.

NOTE 3

This function completely  overwrites S with new sparse matrix. Previously
allocated storage is NOT reused. If you  want  to reuse already allocated
memory, call SparseCreateBuf function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecreate(
    ae_int_t m,
    ae_int_t n,
    sparsematrix& s,
    const xparams _params = alglib::xdefault);
void alglib::sparsecreate(
    ae_int_t m,
    ae_int_t n,
    ae_int_t k,
    sparsematrix& s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`sparsecreatebuf` function

/*************************************************************************
This version of SparseCreate function creates sparse matrix in Hash-Table
format, reusing previously allocated storage as much  as  possible.  Read
comments for SparseCreate() for more information.

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    K           -   K>=0, expected number of non-zero elements in a matrix.
                    K can be inexact approximation, can be less than actual
                    number  of  elements  (table will grow when needed) or
                    even zero).
                    It is important to understand that although hash-table
                    may grow automatically, it is better to  provide  good
                    estimate of data size.
    S           -   SparseMatrix structure which MAY contain some  already
                    allocated storage.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    All elements of the matrix are zero.
                    Previously allocated storage is reused, if  its  size
                    is compatible with expected number of non-zeros K.

  -- ALGLIB PROJECT --
     Copyright 14.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecreatebuf(
    ae_int_t m,
    ae_int_t n,
    sparsematrix s,
    const xparams _params = alglib::xdefault);
void alglib::sparsecreatebuf(
    ae_int_t m,
    ae_int_t n,
    ae_int_t k,
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsecreatecrs` function

/*************************************************************************
This function creates sparse matrix in a CRS format (expert function for
situations when you are running out of memory).

This function creates CRS matrix. Typical usage scenario for a CRS matrix
is:
1. creation (you have to tell number of non-zero elements at each row  at
   this moment)
2. insertion of the matrix elements (row by row, from left to right)
3. matrix is passed to some linear algebra algorithm

This function is a memory-efficient alternative to SparseCreate(), but it
is more complex because it requires you to know in advance how large your
matrix is. Some  information about  different matrix formats can be found
in comments on SparseMatrix structure.  We recommend  you  to  read  them
before starting to use ALGLIB sparse matrices..

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    NER         -   number of elements at each row, array[M], NER[I]>=0

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in CRS representation.
                    You have to fill ALL non-zero elements by calling
                    SparseSet() BEFORE you try to use this matrix.

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateCRSBuf function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecreatecrs(
    ae_int_t m,
    ae_int_t n,
    integer_1d_array ner,
    sparsematrix& s,
    const xparams _params = alglib::xdefault);

Examples: [1]

`sparsecreatecrsbuf` function

/*************************************************************************
This function creates sparse matrix in a CRS format (expert function  for
situations when you are running out  of  memory).  This  version  of  CRS
matrix creation function may reuse memory already allocated in S.

This function creates CRS matrix. Typical usage scenario for a CRS matrix
is:
1. creation (you have to tell number of non-zero elements at each row  at
   this moment)
2. insertion of the matrix elements (row by row, from left to right)
3. matrix is passed to some linear algebra algorithm

This function is a memory-efficient alternative to SparseCreate(), but it
is more complex because it requires you to know in advance how large your
matrix is. Some  information about  different matrix formats can be found
in comments on SparseMatrix structure.  We recommend  you  to  read  them
before starting to use ALGLIB sparse matrices..

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    NER         -   number of elements at each row, array[M], NER[I]>=0
    S           -   sparse matrix structure with possibly preallocated
                    memory.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in CRS representation.
                    You have to fill ALL non-zero elements by calling
                    SparseSet() BEFORE you try to use this matrix.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecreatecrsbuf(
    ae_int_t m,
    ae_int_t n,
    integer_1d_array ner,
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsecreatesks` function

/*************************************************************************
This function creates sparse matrix in  a  SKS  format  (skyline  storage
format). In most cases you do not need this function - CRS format  better
suits most use cases.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    D           -   "bottom" bandwidths, array[M], D[I]>=0.
                    I-th element stores number of non-zeros at I-th  row,
                    below the diagonal (diagonal itself is not  included)
    U           -   "top" bandwidths, array[N], U[I]>=0.
                    I-th element stores number of non-zeros  at I-th row,
                    above the diagonal (diagonal itself  is not included)

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to change their values.

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateSKSBuf function.

  -- ALGLIB PROJECT --
     Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecreatesks(
    ae_int_t m,
    ae_int_t n,
    integer_1d_array d,
    integer_1d_array u,
    sparsematrix& s,
    const xparams _params = alglib::xdefault);

`sparsecreatesksband` function

/*************************************************************************
This function creates sparse matrix in  a  SKS  format  (skyline  storage
format). Unlike more general  sparsecreatesks(),  this  function  creates
sparse matrix with constant bandwidth.

You may want to use this function instead of sparsecreatesks() when  your
matrix has  constant  or  nearly-constant  bandwidth,  and  you  want  to
simplify source code.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    BW          -   matrix bandwidth, BW>=0

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to  change  their values.

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call sparsecreatesksbandbuf function.

  -- ALGLIB PROJECT --
     Copyright 25.12.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecreatesksband(
    ae_int_t m,
    ae_int_t n,
    ae_int_t bw,
    sparsematrix& s,
    const xparams _params = alglib::xdefault);

`sparsecreatesksbandbuf` function

/*************************************************************************
This is "buffered" version  of  sparsecreatesksband() which reuses memory
previously allocated in S (of course, memory is reallocated if needed).

You may want to use this function instead  of  sparsecreatesksbuf()  when
your matrix has  constant or nearly-constant  bandwidth,  and you want to
simplify source code.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    BW          -   bandwidth, BW>=0

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to change their values.

  -- ALGLIB PROJECT --
     Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecreatesksbandbuf(
    ae_int_t m,
    ae_int_t n,
    ae_int_t bw,
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsecreatesksbuf` function

/*************************************************************************
This is "buffered"  version  of  SparseCreateSKS()  which  reuses  memory
previously allocated in S (of course, memory is reallocated if needed).

This function creates sparse matrix in  a  SKS  format  (skyline  storage
format). In most cases you do not need this function - CRS format  better
suits most use cases.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    D           -   "bottom" bandwidths, array[M], 0<=D[I]<=I.
                    I-th element stores number of non-zeros at I-th row,
                    below the diagonal (diagonal itself is not included)
    U           -   "top" bandwidths, array[N], 0<=U[I]<=I.
                    I-th element stores number of non-zeros at I-th row,
                    above the diagonal (diagonal itself is not included)

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to change their values.

  -- ALGLIB PROJECT --
     Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsecreatesksbuf(
    ae_int_t m,
    ae_int_t n,
    integer_1d_array d,
    integer_1d_array u,
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparseenumerate` function

/*************************************************************************
This  function  is  used  to enumerate all elements of the sparse matrix.
Before  first  call  user  initializes  T0 and T1 counters by zero. These
counters are used to remember current position in a  matrix;  after  each
call they are updated by the function.

Subsequent calls to this function return non-zero elements of the  sparse
matrix, one by one. If you enumerate CRS matrix, matrix is traversed from
left to right, from top to bottom. In case you enumerate matrix stored as
Hash table, elements are returned in random order.

EXAMPLE
    > T0=0
    > T1=0
    > while SparseEnumerate(S,T0,T1,I,J,V) do
    >     ....do something with I,J,V

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table or CRS representation.
    T0          -   internal counter
    T1          -   internal counter

OUTPUT PARAMETERS
    T0          -   new value of the internal counter
    T1          -   new value of the internal counter
    I           -   row index of non-zero element, 0<=I<M.
    J           -   column index of non-zero element, 0<=J<N
    V           -   value of the T-th element

RESULT
    True in case of success (next non-zero element was retrieved)
    False in case all non-zero elements were enumerated

NOTE: you may call SparseRewriteExisting() during enumeration, but it  is
      THE  ONLY  matrix  modification  function  you  can  call!!!  Other
      matrix modification functions should not be called during enumeration!

  -- ALGLIB PROJECT --
     Copyright 14.03.2012 by Bochkanov Sergey
*************************************************************************/
bool alglib::sparseenumerate(
    sparsematrix s,
    ae_int_t& t0,
    ae_int_t& t1,
    ae_int_t& i,
    ae_int_t& j,
    double& v,
    const xparams _params = alglib::xdefault);

`sparsefree` function

/*************************************************************************
The function frees all memory occupied by  sparse  matrix.  Sparse  matrix
structure becomes unusable after this call.

OUTPUT PARAMETERS
    S   -   sparse matrix to delete

  -- ALGLIB PROJECT --
     Copyright 24.07.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsefree(
    sparsematrix& s,
    const xparams _params = alglib::xdefault);

`sparseget` function

/*************************************************************************
This function returns S[i,j] - element of the sparse matrix.  Matrix  can
be in any mode (Hash-Table, CRS, SKS), but this function is less efficient
for CRS matrices. Hash-Table and SKS matrices can find  element  in  O(1)
time, while  CRS  matrices need O(log(RS)) time, where RS is an number of
non-zero elements in a row.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    Exception will be thrown for CRS matrix.
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N

RESULT
    value of S[I,J] or zero (in case no element with such index is found)

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
double alglib::sparseget(
    sparsematrix s,
    ae_int_t i,
    ae_int_t j,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`sparsegetcompressedrow` function

/*************************************************************************
This function returns I-th row of the sparse matrix IN COMPRESSED FORMAT -
only non-zero elements are returned (with their indexes). Matrix  must  be
stored in CRS or SKS format.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format
    I           -   row index, 0<=I<M
    ColIdx      -   output buffer for column indexes, can be preallocated.
                    In case buffer size is too small to store I-th row, it
                    is automatically reallocated.
    Vals        -   output buffer for values, can be preallocated. In case
                    buffer size is too small to  store  I-th  row,  it  is
                    automatically reallocated.

OUTPUT PARAMETERS:
    ColIdx      -   column   indexes   of  non-zero  elements,  sorted  by
                    ascending. Symbolically non-zero elements are  counted
                    (i.e. if you allocated place for element, but  it  has
                    zero numerical value - it is counted).
    Vals        -   values. Vals[K] stores value of  matrix  element  with
                    indexes (I,ColIdx[K]). Symbolically non-zero  elements
                    are counted (i.e. if you allocated place for  element,
                    but it has zero numerical value - it is counted).
    NZCnt       -   number of symbolically non-zero elements per row.

NOTE: when  incorrect  I  (outside  of  [0,M-1]) or  matrix (non  CRS/SKS)
      is passed, this function throws exception.

NOTE: this function may allocate additional, unnecessary place for  ColIdx
      and Vals arrays. It is dictated by  performance  reasons  -  on  SKS
      matrices it is faster  to  allocate  space  at  the  beginning  with
      some "extra"-space, than performing two passes over matrix  -  first
      time to calculate exact space required for data, second  time  -  to
      store data itself.

  -- ALGLIB PROJECT --
     Copyright 10.12.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsegetcompressedrow(
    sparsematrix s,
    ae_int_t i,
    integer_1d_array& colidx,
    real_1d_array& vals,
    ae_int_t& nzcnt,
    const xparams _params = alglib::xdefault);

`sparsegetdiagonal` function

/*************************************************************************
This function returns I-th diagonal element of the sparse matrix.

Matrix can be in any mode (Hash-Table or CRS storage), but this  function
is most efficient for CRS matrices - it requires less than 50 CPU  cycles
to extract diagonal element. For Hash-Table matrices we still  have  O(1)
query time, but function is many times slower.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    Exception will be thrown for CRS matrix.
    I           -   index of the element to modify, 0<=I<min(M,N)

RESULT
    value of S[I,I] or zero (in case no element with such index is found)

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
double alglib::sparsegetdiagonal(
    sparsematrix s,
    ae_int_t i,
    const xparams _params = alglib::xdefault);

`sparsegetlowercount` function

/*************************************************************************
The function returns number of strictly lower triangular non-zero elements
in  the  matrix.  It  counts  SYMBOLICALLY non-zero elements, i.e. entries
in the sparse matrix data structure. If some element  has  zero  numerical
value, it is still counted.

This function has different cost for different types of matrices:
* for hash-based matrices it involves complete pass over entire hash-table
  with O(NNZ) cost, where NNZ is number of non-zero elements
* for CRS and SKS matrix types cost of counting is O(N) (N - matrix size).

RESULT: number of non-zero elements strictly below main diagonal

  -- ALGLIB PROJECT --
     Copyright 12.02.2014 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::sparsegetlowercount(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsegetmatrixtype` function

/*************************************************************************
This function returns type of the matrix storage format.

INPUT PARAMETERS:
    S           -   sparse matrix.

RESULT:
    sparse storage format used by matrix:
        0   -   Hash-table
        1   -   CRS (compressed row storage)
        2   -   SKS (skyline)

NOTE: future  versions  of  ALGLIB  may  include additional sparse storage
      formats.


  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::sparsegetmatrixtype(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsegetncols` function

/*************************************************************************
The function returns number of columns of a sparse matrix.

RESULT: number of columns of a sparse matrix.

  -- ALGLIB PROJECT --
     Copyright 23.08.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::sparsegetncols(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsegetnrows` function

/*************************************************************************
The function returns number of rows of a sparse matrix.

RESULT: number of rows of a sparse matrix.

  -- ALGLIB PROJECT --
     Copyright 23.08.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::sparsegetnrows(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsegetrow` function

/*************************************************************************
This function returns I-th row of the sparse matrix. Matrix must be stored
in CRS or SKS format.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format
    I           -   row index, 0<=I<M
    IRow        -   output buffer, can be  preallocated.  In  case  buffer
                    size  is  too  small  to  store  I-th   row,   it   is
                    automatically reallocated.

OUTPUT PARAMETERS:
    IRow        -   array[M], I-th row.

NOTE: this function has O(N) running time, where N is a  column  count. It
      allocates and fills N-element  array,  even  although  most  of  its
      elemets are zero.

NOTE: If you have O(non-zeros-per-row) time and memory  requirements,  use
      SparseGetCompressedRow() function. It  returns  data  in  compressed
      format.

NOTE: when  incorrect  I  (outside  of  [0,M-1]) or  matrix (non  CRS/SKS)
      is passed, this function throws exception.

  -- ALGLIB PROJECT --
     Copyright 10.12.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsegetrow(
    sparsematrix s,
    ae_int_t i,
    real_1d_array& irow,
    const xparams _params = alglib::xdefault);

`sparsegetuppercount` function

/*************************************************************************
The function returns number of strictly upper triangular non-zero elements
in  the  matrix.  It  counts  SYMBOLICALLY non-zero elements, i.e. entries
in the sparse matrix data structure. If some element  has  zero  numerical
value, it is still counted.

This function has different cost for different types of matrices:
* for hash-based matrices it involves complete pass over entire hash-table
  with O(NNZ) cost, where NNZ is number of non-zero elements
* for CRS and SKS matrix types cost of counting is O(N) (N - matrix size).

RESULT: number of non-zero elements strictly above main diagonal

  -- ALGLIB PROJECT --
     Copyright 12.02.2014 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::sparsegetuppercount(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparseiscrs` function

/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using CRS representation.

INPUT PARAMETERS:
    S   -   sparse matrix.

RESULT:
    True if matrix type is CRS
    False if matrix type is not CRS

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool alglib::sparseiscrs(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparseishash` function

/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using Hash table representation.

INPUT PARAMETERS:
    S   -   sparse matrix.

RESULT:
    True if matrix type is Hash table
    False if matrix type is not Hash table

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool alglib::sparseishash(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparseissks` function

/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using SKS representation.

INPUT PARAMETERS:
    S   -   sparse matrix.

RESULT:
    True if matrix type is SKS
    False if matrix type is not SKS

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool alglib::sparseissks(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsemm` function

/*************************************************************************
This function calculates matrix-matrix product  S*A.  Matrix  S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    A           -   array[N][K], input dense matrix. For  performance reasons
                    we make only quick checks - we check that array size
                    is at least N, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B           -   array[M][K], S*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsemm(
    sparsematrix s,
    real_2d_array a,
    ae_int_t k,
    real_2d_array& b,
    const xparams _params = alglib::xdefault);

`sparsemm2` function

/*************************************************************************
This function simultaneously calculates two matrix-matrix products:
    S*A and S^T*A.
S  must  be  square (non-rectangular) matrix stored in CRS or  SKS  format
(exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse N*N matrix in CRS or SKS format.
    A           -   array[N][K], input dense matrix. For performance reasons
                    we make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B0          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.
    B1          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B0          -   array[N][K], S*A
    B1          -   array[N][K], S^T*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsemm2(
    sparsematrix s,
    real_2d_array a,
    ae_int_t k,
    real_2d_array& b0,
    real_2d_array& b1,
    const xparams _params = alglib::xdefault);

`sparsemtm` function

/*************************************************************************
This function calculates matrix-matrix product  S^T*A. Matrix S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    A           -   array[M][K], input dense matrix. For performance reasons
                    we make only quick checks - we check that array size  is
                    at least M, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B           -   array[N][K], S^T*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsemtm(
    sparsematrix s,
    real_2d_array a,
    ae_int_t k,
    real_2d_array& b,
    const xparams _params = alglib::xdefault);

`sparsemtv` function

/*************************************************************************
This function calculates matrix-vector product  S^T*x. Matrix S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    X           -   array[M], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least M, but we do not check for NAN's or INF's.
    Y           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y           -   array[N], S^T*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsemtv(
    sparsematrix s,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`sparsemv` function

/*************************************************************************
This function calculates matrix-vector product  S*x.  Matrix  S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    Y           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y           -   array[M], S*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsemv(
    sparsematrix s,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`sparsemv2` function

/*************************************************************************
This function simultaneously calculates two matrix-vector products:
    S*x and S^T*x.
S must be square (non-rectangular) matrix stored in  CRS  or  SKS  format
(exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse N*N matrix in CRS or SKS format.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    Y0          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.
    Y1          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y0          -   array[N], S*x
    Y1          -   array[N], S^T*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsemv2(
    sparsematrix s,
    real_1d_array x,
    real_1d_array& y0,
    real_1d_array& y1,
    const xparams _params = alglib::xdefault);

`sparseresizematrix` function

/*************************************************************************
This procedure resizes Hash-Table matrix. It can be called when you  have
deleted too many elements from the matrix, and you want to  free unneeded
memory.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparseresizematrix(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparserewriteexisting` function

/*************************************************************************
This function rewrites existing (non-zero) element. It  returns  True   if
element  exists  or  False,  when  it  is  called for non-existing  (zero)
element.

This function works with any kind of the matrix.

The purpose of this function is to provide convenient thread-safe  way  to
modify  sparse  matrix.  Such  modification  (already  existing element is
rewritten) is guaranteed to be thread-safe without any synchronization, as
long as different threads modify different elements.

INPUT PARAMETERS
    S           -   sparse M*N matrix in any kind of representation
                    (Hash, SKS, CRS).
    I           -   row index of non-zero element to modify, 0<=I<M
    J           -   column index of non-zero element to modify, 0<=J<N
    V           -   value to rewrite, must be finite number

OUTPUT PARAMETERS
    S           -   modified matrix
RESULT
    True in case when element exists
    False in case when element doesn't exist or it is zero

  -- ALGLIB PROJECT --
     Copyright 14.03.2012 by Bochkanov Sergey
*************************************************************************/
bool alglib::sparserewriteexisting(
    sparsematrix s,
    ae_int_t i,
    ae_int_t j,
    double v,
    const xparams _params = alglib::xdefault);

`sparseset` function

/*************************************************************************
This function modifies S[i,j] - element of the sparse matrix.

For Hash-based storage format:
* this function can be called at any moment - during matrix initialization
  or later
* new value can be zero or non-zero.  In case new value of S[i,j] is zero,
  this element is deleted from the table.
* this  function  has  no  effect when called with zero V for non-existent
  element.

For CRS-bases storage format:
* this function can be called ONLY DURING MATRIX INITIALIZATION
* zero values are stored in the matrix similarly to non-zero ones
* elements must be initialized in correct order -  from top row to bottom,
  within row - from left to right.

For SKS storage:
* this function can be called at any moment - during matrix initialization
  or later
* zero values are stored in the matrix similarly to non-zero ones
* this function CAN NOT be called for non-existent (outside  of  the  band
  specified during SKS matrix creation) elements. Say, if you created  SKS
  matrix  with  bandwidth=2  and  tried to call sparseset(s,0,10,VAL),  an
  exception will be generated.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table, SKS or CRS format.
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N
    V           -   value to set, must be finite number, can be zero

OUTPUT PARAMETERS
    S           -   modified matrix

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparseset(
    sparsematrix s,
    ae_int_t i,
    ae_int_t j,
    double v,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`sparsesmm` function

/*************************************************************************
This function calculates matrix-matrix product  S*A, when S  is  symmetric
matrix. Matrix S must be stored in CRS or SKS format  (exception  will  be
thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*M matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is given:
                    * if upper triangle is given,  only   S[i,j] for j>=i
                      are used, and lower triangle is ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for j<=i
                      are used, and upper triangle is ignored.
    A           -   array[N][K], input dense matrix. For performance reasons
                    we make only quick checks - we check that array size is
                    at least N, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B           -   array[M][K], S*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsesmm(
    sparsematrix s,
    bool isupper,
    real_2d_array a,
    ae_int_t k,
    real_2d_array& b,
    const xparams _params = alglib::xdefault);

`sparsesmv` function

/*************************************************************************
This function calculates matrix-vector product  S*x, when S is  symmetric
matrix. Matrix S  must be stored in CRS or SKS format  (exception will be
thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*M matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is given:
                    * if upper triangle is given,  only   S[i,j] for j>=i
                      are used, and lower triangle is ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for j<=i
                      are used, and upper triangle is ignored.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    Y           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y           -   array[M], S*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsesmv(
    sparsematrix s,
    bool isupper,
    real_1d_array x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`sparseswap` function

/*************************************************************************
This function efficiently swaps contents of S0 and S1.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparseswap(
    sparsematrix s0,
    sparsematrix s1,
    const xparams _params = alglib::xdefault);

`sparsetransposecrs` function

/*************************************************************************
This function performs transpose of CRS matrix.

INPUT PARAMETERS
    S       -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S           -   sparse matrix, transposed.

NOTE: internal  temporary  copy  is  allocated   for   the   purposes   of
      transposition. It is deallocated after transposition.

  -- ALGLIB PROJECT --
     Copyright 30.01.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsetransposecrs(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsetransposesks` function

/*************************************************************************
This function performs efficient in-place  transpose  of  SKS  matrix.  No
additional memory is allocated during transposition.

This function supports only skyline storage format (SKS).

INPUT PARAMETERS
    S       -   sparse matrix in SKS format.

OUTPUT PARAMETERS
    S           -   sparse matrix, transposed.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsetransposesks(
    sparsematrix s,
    const xparams _params = alglib::xdefault);

`sparsetrmv` function

/*************************************************************************
This function calculates matrix-vector product op(S)*x, when x is  vector,
S is symmetric triangular matrix, op(S) is transposition or no  operation.
Matrix S must be stored in CRS or SKS format  (exception  will  be  thrown
otherwise).

INPUT PARAMETERS
    S           -   sparse square matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is used:
                    * if upper triangle is given,  only   S[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    IsUnit      -   unit or non-unit diagonal:
                    * if True, diagonal elements of triangular matrix  are
                      considered equal to 1.0. Actual elements  stored  in
                      S are not referenced at all.
                    * if False, diagonal stored in S is used
    OpType      -   operation type:
                    * if 0, S*x is calculated
                    * if 1, (S^T)*x is calculated (transposition)
    X           -   array[N] which stores input  vector.  For  performance
                    reasons we make only quick  checks  -  we  check  that
                    array  size  is  at  least  N, but we do not check for
                    NAN's or INF's.
    Y           -   possibly  preallocated  input   buffer.  Automatically
                    resized if its size is too small.

OUTPUT PARAMETERS
    Y           -   array[N], op(S)*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 20.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsetrmv(
    sparsematrix s,
    bool isupper,
    bool isunit,
    ae_int_t optype,
    real_1d_array& x,
    real_1d_array& y,
    const xparams _params = alglib::xdefault);

`sparsetrsv` function

/*************************************************************************
This function solves linear system op(S)*y=x  where  x  is  vector,  S  is
symmetric  triangular  matrix,  op(S)  is  transposition  or no operation.
Matrix S must be stored in CRS or SKS format  (exception  will  be  thrown
otherwise).

INPUT PARAMETERS
    S           -   sparse square matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is used:
                    * if upper triangle is given,  only   S[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    IsUnit      -   unit or non-unit diagonal:
                    * if True, diagonal elements of triangular matrix  are
                      considered equal to 1.0. Actual elements  stored  in
                      S are not referenced at all.
                    * if False, diagonal stored in S is used. It  is  your
                      responsibility  to  make  sure  that   diagonal   is
                      non-zero.
    OpType      -   operation type:
                    * if 0, S*x is calculated
                    * if 1, (S^T)*x is calculated (transposition)
    X           -   array[N] which stores input  vector.  For  performance
                    reasons we make only quick  checks  -  we  check  that
                    array  size  is  at  least  N, but we do not check for
                    NAN's or INF's.

OUTPUT PARAMETERS
    X           -   array[N], inv(op(S))*x

NOTE: this function throws exception when called for  non-CRS/SKS  matrix.
      You must convert your matrix  with  SparseConvertToCRS/SKS()  before
      using this function.

NOTE: no assertion or tests are done during algorithm  operation.   It  is
      your responsibility to provide invertible matrix to algorithm.

  -- ALGLIB PROJECT --
     Copyright 20.01.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::sparsetrsv(
    sparsematrix s,
    bool isupper,
    bool isunit,
    ae_int_t optype,
    real_1d_array& x,
    const xparams _params = alglib::xdefault);

`sparsevsmv` function

/*************************************************************************
This function calculates vector-matrix-vector product x'*S*x, where  S is
symmetric matrix. Matrix S must be stored in CRS or SKS format (exception
will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*M matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is given:
                    * if upper triangle is given,  only   S[i,j] for j>=i
                      are used, and lower triangle is ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for j<=i
                      are used, and upper triangle is ignored.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.

RESULT
    x'*S*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 27.01.2014 by Bochkanov Sergey
*************************************************************************/
double alglib::sparsevsmv(
    sparsematrix s,
    bool isupper,
    real_1d_array x,
    const xparams _params = alglib::xdefault);

sparse_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates creation/initialization of the sparse matrix
    // and matrix-vector multiplication.
    //
    // First, we have to create matrix and initialize it. Matrix is initially created
    // in the Hash-Table format, which allows convenient initialization. We can modify
    // Hash-Table matrix with sparseset() and sparseadd() functions.
    //
    // NOTE: Unlike CRS format, Hash-Table representation allows you to initialize
    // elements in the arbitrary order. You may see that we initialize a[0][0] first,
    // then move to the second row, and then move back to the first row.
    //
    sparsematrix s;
    sparsecreate(2, 2, s);
    sparseset(s, 0, 0, 2.0);
    sparseset(s, 1, 1, 1.0);
    sparseset(s, 0, 1, 1.0);

    sparseadd(s, 1, 1, 4.0);

    //
    // Now S is equal to
    //   [ 2 1 ]
    //   [   5 ]
    // Lets check it by reading matrix contents with sparseget().
    // You may see that with sparseget() you may read both non-zero
    // and zero elements.
    //
    double v;
    v = sparseget(s, 0, 0);
    printf("%.2f\n", double(v)); // EXPECTED: 2.0000
    v = sparseget(s, 0, 1);
    printf("%.2f\n", double(v)); // EXPECTED: 1.0000
    v = sparseget(s, 1, 0);
    printf("%.2f\n", double(v)); // EXPECTED: 0.0000
    v = sparseget(s, 1, 1);
    printf("%.2f\n", double(v)); // EXPECTED: 5.0000

    //
    // After successful creation we can use our matrix for linear operations.
    //
    // However, there is one more thing we MUST do before using S in linear
    // operations: we have to convert it from HashTable representation (used for
    // initialization and dynamic operations) to CRS format with sparseconverttocrs()
    // call. If you omit this call, ALGLIB will generate exception on the first
    // attempt to use S in linear operations. 
    //
    sparseconverttocrs(s);

    //
    // Now S is in the CRS format and we are ready to do linear operations.
    // Lets calculate A*x for some x.
    //
    real_1d_array x = "[1,-1]";
    real_1d_array y = "[]";
    sparsemv(s, x, y);
    printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [1.000,-5.000]
    return 0;
}

sparse_d_crs example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // This example demonstrates creation/initialization of the sparse matrix in the
    // CRS format.
    //
    // Hash-Table format used by default is very convenient (it allows easy
    // insertion of elements, automatic memory reallocation), but has
    // significant memory and performance overhead. Insertion of one element 
    // costs hundreds of CPU cycles, and memory consumption is several times
    // higher than that of CRS.
    //
    // When you work with really large matrices and when you can tell in 
    // advance how many elements EXACTLY you need, it can be beneficial to 
    // create matrix in the CRS format from the very beginning.
    //
    // If you want to create matrix in the CRS format, you should:
    // * use sparsecreatecrs() function
    // * know row sizes in advance (number of non-zero entries in the each row)
    // * initialize matrix with sparseset() - another function, sparseadd(), is not allowed
    // * initialize elements from left to right, from top to bottom, each
    //   element is initialized only once.
    //
    sparsematrix s;
    integer_1d_array row_sizes = "[2,2,2,1]";
    sparsecreatecrs(4, 4, row_sizes, s);
    sparseset(s, 0, 0, 2.0);
    sparseset(s, 0, 1, 1.0);
    sparseset(s, 1, 1, 4.0);
    sparseset(s, 1, 2, 2.0);
    sparseset(s, 2, 2, 3.0);
    sparseset(s, 2, 3, 1.0);
    sparseset(s, 3, 3, 9.0);

    //
    // Now S is equal to
    //   [ 2 1     ]
    //   [   4 2   ]
    //   [     3 1 ]
    //   [       9 ]
    //
    // We should point that we have initialized S elements from left to right,
    // from top to bottom. CRS representation does NOT allow you to do so in
    // the different order. Try to change order of the sparseset() calls above,
    // and you will see that your program generates exception.
    //
    // We can check it by reading matrix contents with sparseget().
    // However, you should remember that sparseget() is inefficient on
    // CRS matrices (it may have to pass through all elements of the row 
    // until it finds element you need).
    //
    double v;
    v = sparseget(s, 0, 0);
    printf("%.2f\n", double(v)); // EXPECTED: 2.0000
    v = sparseget(s, 2, 3);
    printf("%.2f\n", double(v)); // EXPECTED: 1.0000

    // you may see that you can read zero elements (which are not stored) with sparseget()
    v = sparseget(s, 3, 2);
    printf("%.2f\n", double(v)); // EXPECTED: 0.0000

    //
    // After successful creation we can use our matrix for linear operations.
    // Lets calculate A*x for some x.
    //
    real_1d_array x = "[1,-1,1,-1]";
    real_1d_array y = "[]";
    sparsemv(s, x, y);
    printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [1.000,-2.000,2.000,-9]
    return 0;
}

`spdgevd` subpackage

Functions

smatrixgevd
smatrixgevdreduce

Examples

`smatrixgevd` function

/*************************************************************************
Algorithm for solving the following generalized symmetric positive-definite
eigenproblem:
    A*x = lambda*B*x (1) or
    A*B*x = lambda*x (2) or
    B*A*x = lambda*x (3).
where A is a symmetric matrix, B - symmetric positive-definite matrix.
The problem is solved by reducing it to an ordinary  symmetric  eigenvalue
problem.

Input parameters:
    A           -   symmetric matrix which is given by its upper or lower
                    triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
    N           -   size of matrices A and B.
    IsUpperA    -   storage format of matrix A.
    B           -   symmetric positive-definite matrix which is given by
                    its upper or lower triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
    IsUpperB    -   storage format of matrix B.
    ZNeeded     -   if ZNeeded is equal to:
                     * 0, the eigenvectors are not returned;
                     * 1, the eigenvectors are returned.
    ProblemType -   if ProblemType is equal to:
                     * 1, the following problem is solved: A*x = lambda*B*x;
                     * 2, the following problem is solved: A*B*x = lambda*x;
                     * 3, the following problem is solved: B*A*x = lambda*x.

Output parameters:
    D           -   eigenvalues in ascending order.
                    Array whose index ranges within [0..N-1].
    Z           -   if ZNeeded is equal to:
                     * 0, Z hasn't changed;
                     * 1, Z contains eigenvectors.
                    Array whose indexes range within [0..N-1, 0..N-1].
                    The eigenvectors are stored in matrix columns. It should
                    be noted that the eigenvectors in such problems do not
                    form an orthogonal system.

Result:
    True, if the problem was solved successfully.
    False, if the error occurred during the Cholesky decomposition of matrix
    B (the matrix isn't positive-definite) or during the work of the iterative
    algorithm for solving the symmetric eigenproblem.

See also the GeneralizedSymmetricDefiniteEVDReduce subroutine.

  -- ALGLIB --
     Copyright 1.28.2006 by Bochkanov Sergey
*************************************************************************/
bool alglib::smatrixgevd(
    real_2d_array a,
    ae_int_t n,
    bool isuppera,
    real_2d_array b,
    bool isupperb,
    ae_int_t zneeded,
    ae_int_t problemtype,
    real_1d_array& d,
    real_2d_array& z,
    const xparams _params = alglib::xdefault);

`smatrixgevdreduce` function

/*************************************************************************
Algorithm for reduction of the following generalized symmetric positive-
definite eigenvalue problem:
    A*x = lambda*B*x (1) or
    A*B*x = lambda*x (2) or
    B*A*x = lambda*x (3)
to the symmetric eigenvalues problem C*y = lambda*y (eigenvalues of this and
the given problems are the same, and the eigenvectors of the given problem
could be obtained by multiplying the obtained eigenvectors by the
transformation matrix x = R*y).

Here A is a symmetric matrix, B - symmetric positive-definite matrix.

Input parameters:
    A           -   symmetric matrix which is given by its upper or lower
                    triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
    N           -   size of matrices A and B.
    IsUpperA    -   storage format of matrix A.
    B           -   symmetric positive-definite matrix which is given by
                    its upper or lower triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
    IsUpperB    -   storage format of matrix B.
    ProblemType -   if ProblemType is equal to:
                     * 1, the following problem is solved: A*x = lambda*B*x;
                     * 2, the following problem is solved: A*B*x = lambda*x;
                     * 3, the following problem is solved: B*A*x = lambda*x.

Output parameters:
    A           -   symmetric matrix which is given by its upper or lower
                    triangle depending on IsUpperA. Contains matrix C.
                    Array whose indexes range within [0..N-1, 0..N-1].
    R           -   upper triangular or low triangular transformation matrix
                    which is used to obtain the eigenvectors of a given problem
                    as the product of eigenvectors of C (from the right) and
                    matrix R (from the left). If the matrix is upper
                    triangular, the elements below the main diagonal
                    are equal to 0 (and vice versa). Thus, we can perform
                    the multiplication without taking into account the
                    internal structure (which is an easier though less
                    effective way).
                    Array whose indexes range within [0..N-1, 0..N-1].
    IsUpperR    -   type of matrix R (upper or lower triangular).

Result:
    True, if the problem was reduced successfully.
    False, if the error occurred during the Cholesky decomposition of
        matrix B (the matrix is not positive-definite).

  -- ALGLIB --
     Copyright 1.28.2006 by Bochkanov Sergey
*************************************************************************/
bool alglib::smatrixgevdreduce(
    real_2d_array& a,
    ae_int_t n,
    bool isuppera,
    real_2d_array b,
    bool isupperb,
    ae_int_t problemtype,
    real_2d_array& r,
    bool& isupperr,
    const xparams _params = alglib::xdefault);

`spline1d` subpackage

Classes

spline1dfitreport
spline1dinterpolant

Functions

spline1dbuildakima
spline1dbuildcatmullrom
spline1dbuildcubic
spline1dbuildhermite
spline1dbuildlinear
spline1dbuildmonotone
spline1dcalc
spline1dconvcubic
spline1dconvdiff2cubic
spline1dconvdiffcubic
spline1ddiff
spline1dfitpenalized
spline1dfitpenalizedw
spline1dgriddiff2cubic
spline1dgriddiffcubic
spline1dintegrate
spline1dlintransx
spline1dlintransy
spline1dunpack

Examples

spline1d_d_convdiff		Resampling using cubic splines
spline1d_d_cubic		Cubic spline interpolation
spline1d_d_griddiff		Differentiation on the grid using cubic splines
spline1d_d_linear		Piecewise linear spline interpolation
spline1d_d_monotone		Monotone interpolation

`spline1dfitreport` class

/*************************************************************************
Spline fitting report:
    RMSError        RMS error
    AvgError        average error
    AvgRelError     average relative error (for non-zero Y[I])
    MaxError        maximum error

Fields  below are  filled  by   obsolete    functions   (Spline1DFitCubic,
Spline1DFitHermite). Modern fitting functions do NOT fill these fields:
    TaskRCond       reciprocal of task's condition number
*************************************************************************/
class spline1dfitreport
{
    double               taskrcond;
    double               rmserror;
    double               avgerror;
    double               avgrelerror;
    double               maxerror;
};

`spline1dinterpolant` class

/*************************************************************************
1-dimensional spline interpolant
*************************************************************************/
class spline1dinterpolant
{
};

`spline1dbuildakima` function

/*************************************************************************
This subroutine builds Akima spline interpolant

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1]
    Y           -   function values, array[0..N-1]
    N           -   points count (optional):
                    * N>=2
                    * if given, only first N points are used to build spline
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))

OUTPUT PARAMETERS:
    C           -   spline interpolant


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

  -- ALGLIB PROJECT --
     Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dbuildakima(
    real_1d_array x,
    real_1d_array y,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);
void alglib::spline1dbuildakima(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);

`spline1dbuildcatmullrom` function

/*************************************************************************
This subroutine builds Catmull-Rom spline interpolant.

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1].
    Y           -   function values, array[0..N-1].

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points are used to build spline
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundType   -   boundary condition type:
                    * -1 for periodic boundary condition
                    *  0 for parabolically terminated spline (default)
    Tension     -   tension parameter:
                    * tension=0   corresponds to classic Catmull-Rom spline (default)
                    * 0<tension<1 corresponds to more general form - cardinal spline

OUTPUT PARAMETERS:
    C           -   spline interpolant


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dbuildcatmullrom(
    real_1d_array x,
    real_1d_array y,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);
void alglib::spline1dbuildcatmullrom(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t boundtype,
    double tension,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);

`spline1dbuildcubic` function

/*************************************************************************
This subroutine builds cubic spline interpolant.

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1].
    Y           -   function values, array[0..N-1].

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points are used to build spline
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)

OUTPUT PARAMETERS:
    C           -   spline interpolant

ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dbuildcubic(
    real_1d_array x,
    real_1d_array y,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);
void alglib::spline1dbuildcubic(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t boundltype,
    double boundl,
    ae_int_t boundrtype,
    double boundr,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);

`spline1dbuildhermite` function

/*************************************************************************
This subroutine builds Hermite spline interpolant.

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1]
    Y           -   function values, array[0..N-1]
    D           -   derivatives, array[0..N-1]
    N           -   points count (optional):
                    * N>=2
                    * if given, only first N points are used to build spline
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))

OUTPUT PARAMETERS:
    C           -   spline interpolant.


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dbuildhermite(
    real_1d_array x,
    real_1d_array y,
    real_1d_array d,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);
void alglib::spline1dbuildhermite(
    real_1d_array x,
    real_1d_array y,
    real_1d_array d,
    ae_int_t n,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);

`spline1dbuildlinear` function

/*************************************************************************
This subroutine builds linear spline interpolant

INPUT PARAMETERS:
    X   -   spline nodes, array[0..N-1]
    Y   -   function values, array[0..N-1]
    N   -   points count (optional):
            * N>=2
            * if given, only first N points are used to build spline
            * if not given, automatically detected from X/Y sizes
              (len(X) must be equal to len(Y))

OUTPUT PARAMETERS:
    C   -   spline interpolant


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

  -- ALGLIB PROJECT --
     Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dbuildlinear(
    real_1d_array x,
    real_1d_array y,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);
void alglib::spline1dbuildlinear(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`spline1dbuildmonotone` function

/*************************************************************************
This function builds monotone cubic Hermite interpolant. This interpolant
is monotonic in [x(0),x(n-1)] and is constant outside of this interval.

In  case  y[]  form  non-monotonic  sequence,  interpolant  is  piecewise
monotonic.  Say, for x=(0,1,2,3,4)  and  y=(0,1,2,1,0)  interpolant  will
monotonically grow at [0..2] and monotonically decrease at [2..4].

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1]. Subroutine automatically
                    sorts points, so caller may pass unsorted array.
    Y           -   function values, array[0..N-1]
    N           -   the number of points(N>=2).

OUTPUT PARAMETERS:
    C           -   spline interpolant.

 -- ALGLIB PROJECT --
     Copyright 21.06.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dbuildmonotone(
    real_1d_array x,
    real_1d_array y,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);
void alglib::spline1dbuildmonotone(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    spline1dinterpolant& c,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline1dcalc` function

/*************************************************************************
This subroutine calculates the value of the spline at the given point X.

INPUT PARAMETERS:
    C   -   spline interpolant
    X   -   point

Result:
    S(x)

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::spline1dcalc(
    spline1dinterpolant c,
    double x,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`spline1dconvcubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[]  and new nodes  x2[],  it calculates and returns table of
function values y2[] (calculated at x2[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DDiff() calls, but it can be several times faster  when
called for ordered X[] and X2[].

INPUT PARAMETERS:
    X           -   old spline nodes
    Y           -   function values
    X2           -  new spline nodes

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points from X/Y are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)
    N2          -   new points count:
                    * N2>=2
                    * if given, only first N2 points from X2 are used
                    * if not given, automatically detected from X2 size

OUTPUT PARAMETERS:
    F2          -   function values at X2[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller  may pass unsorted array.
Function  values  are correctly reordered on  return, so F2[I]  is  always
equal to S(X2[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dconvcubic(
    real_1d_array x,
    real_1d_array y,
    real_1d_array x2,
    real_1d_array& y2,
    const xparams _params = alglib::xdefault);
void alglib::spline1dconvcubic(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t boundltype,
    double boundl,
    ae_int_t boundrtype,
    double boundr,
    real_1d_array x2,
    ae_int_t n2,
    real_1d_array& y2,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline1dconvdiff2cubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[]  and new nodes  x2[],  it calculates and returns table of
function  values  y2[],  first  and  second  derivatives  d2[]  and  dd2[]
(calculated at x2[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DDiff() calls, but it can be several times faster  when
called for ordered X[] and X2[].

INPUT PARAMETERS:
    X           -   old spline nodes
    Y           -   function values
    X2           -  new spline nodes

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points from X/Y are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)
    N2          -   new points count:
                    * N2>=2
                    * if given, only first N2 points from X2 are used
                    * if not given, automatically detected from X2 size

OUTPUT PARAMETERS:
    F2          -   function values at X2[]
    D2          -   first derivatives at X2[]
    DD2         -   second derivatives at X2[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller  may pass unsorted array.
Function  values  are correctly reordered on  return, so F2[I]  is  always
equal to S(X2[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dconvdiff2cubic(
    real_1d_array x,
    real_1d_array y,
    real_1d_array x2,
    real_1d_array& y2,
    real_1d_array& d2,
    real_1d_array& dd2,
    const xparams _params = alglib::xdefault);
void alglib::spline1dconvdiff2cubic(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t boundltype,
    double boundl,
    ae_int_t boundrtype,
    double boundr,
    real_1d_array x2,
    ae_int_t n2,
    real_1d_array& y2,
    real_1d_array& d2,
    real_1d_array& dd2,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline1dconvdiffcubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[]  and new nodes  x2[],  it calculates and returns table of
function values y2[] and derivatives d2[] (calculated at x2[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DDiff() calls, but it can be several times faster  when
called for ordered X[] and X2[].

INPUT PARAMETERS:
    X           -   old spline nodes
    Y           -   function values
    X2           -  new spline nodes

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points from X/Y are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)
    N2          -   new points count:
                    * N2>=2
                    * if given, only first N2 points from X2 are used
                    * if not given, automatically detected from X2 size

OUTPUT PARAMETERS:
    F2          -   function values at X2[]
    D2          -   first derivatives at X2[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller  may pass unsorted array.
Function  values  are correctly reordered on  return, so F2[I]  is  always
equal to S(X2[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dconvdiffcubic(
    real_1d_array x,
    real_1d_array y,
    real_1d_array x2,
    real_1d_array& y2,
    real_1d_array& d2,
    const xparams _params = alglib::xdefault);
void alglib::spline1dconvdiffcubic(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t boundltype,
    double boundl,
    ae_int_t boundrtype,
    double boundr,
    real_1d_array x2,
    ae_int_t n2,
    real_1d_array& y2,
    real_1d_array& d2,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline1ddiff` function

/*************************************************************************
This subroutine differentiates the spline.

INPUT PARAMETERS:
    C   -   spline interpolant.
    X   -   point

Result:
    S   -   S(x)
    DS  -   S'(x)
    D2S -   S''(x)

  -- ALGLIB PROJECT --
     Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1ddiff(
    spline1dinterpolant c,
    double x,
    double& s,
    double& ds,
    double& d2s,
    const xparams _params = alglib::xdefault);

`spline1dfitpenalized` function

/*************************************************************************
Fitting by penalized cubic spline.

Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is  used  to  build
basis functions. Basis functions are cubic splines with  natural  boundary
conditions. Problem is regularized by  adding non-linearity penalty to the
usual least squares penalty function:

    S(x) = arg min { LS + P }, where
    LS   = SUM { w[i]^2*(y[i] - S(x[i]))^2 } - least squares penalty
    P    = C*10^rho*integral{ S''(x)^2*dx } - non-linearity penalty
    rho  - tunable constant given by user
    C    - automatically determined scale parameter,
           makes penalty invariant with respect to scaling of X, Y, W.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    N   -   number of points (optional):
            * N>0
            * if given, only first N elements of X/Y are processed
            * if not given, automatically determined from X/Y sizes
    M   -   number of basis functions ( = number_of_nodes), M>=4.
    Rho -   regularization  constant  passed   by   user.   It   penalizes
            nonlinearity in the regression spline. It  is  logarithmically
            scaled,  i.e.  actual  value  of  regularization  constant  is
            calculated as 10^Rho. It is automatically scaled so that:
            * Rho=2.0 corresponds to moderate amount of nonlinearity
            * generally, it should be somewhere in the [-8.0,+8.0]
            If you do not want to penalize nonlineary,
            pass small Rho. Values as low as -15 should work.

OUTPUT PARAMETERS:
    Info-   same format as in LSFitLinearWC() subroutine.
            * Info>0    task is solved
            * Info<=0   an error occured:
                        -4 means inconvergence of internal SVD or
                           Cholesky decomposition; problem may be
                           too ill-conditioned (very rare)
    S   -   spline interpolant.
    Rep -   Following fields are set:
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

NOTE 1: additional nodes are added to the spline outside  of  the  fitting
interval to force linearity when x<min(x,xc) or x>max(x,xc).  It  is  done
for consistency - we penalize non-linearity  at [min(x,xc),max(x,xc)],  so
it is natural to force linearity outside of this interval.

NOTE 2: function automatically sorts points,  so  caller may pass unsorted
array.

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dfitpenalized(
    real_1d_array x,
    real_1d_array y,
    ae_int_t m,
    double rho,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::spline1dfitpenalized(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t m,
    double rho,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline1dfitpenalizedw` function

/*************************************************************************
Weighted fitting by penalized cubic spline.

Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is  used  to  build
basis functions. Basis functions are cubic splines with  natural  boundary
conditions. Problem is regularized by  adding non-linearity penalty to the
usual least squares penalty function:

    S(x) = arg min { LS + P }, where
    LS   = SUM { w[i]^2*(y[i] - S(x[i]))^2 } - least squares penalty
    P    = C*10^rho*integral{ S''(x)^2*dx } - non-linearity penalty
    rho  - tunable constant given by user
    C    - automatically determined scale parameter,
           makes penalty invariant with respect to scaling of X, Y, W.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    W   -   weights, array[0..N-1]
            Each summand in square  sum  of  approximation deviations from
            given  values  is  multiplied  by  the square of corresponding
            weight. Fill it by 1's if you don't  want  to  solve  weighted
            problem.
    N   -   number of points (optional):
            * N>0
            * if given, only first N elements of X/Y/W are processed
            * if not given, automatically determined from X/Y/W sizes
    M   -   number of basis functions ( = number_of_nodes), M>=4.
    Rho -   regularization  constant  passed   by   user.   It   penalizes
            nonlinearity in the regression spline. It  is  logarithmically
            scaled,  i.e.  actual  value  of  regularization  constant  is
            calculated as 10^Rho. It is automatically scaled so that:
            * Rho=2.0 corresponds to moderate amount of nonlinearity
            * generally, it should be somewhere in the [-8.0,+8.0]
            If you do not want to penalize nonlineary,
            pass small Rho. Values as low as -15 should work.

OUTPUT PARAMETERS:
    Info-   same format as in LSFitLinearWC() subroutine.
            * Info>0    task is solved
            * Info<=0   an error occured:
                        -4 means inconvergence of internal SVD or
                           Cholesky decomposition; problem may be
                           too ill-conditioned (very rare)
    S   -   spline interpolant.
    Rep -   Following fields are set:
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

NOTE 1: additional nodes are added to the spline outside  of  the  fitting
interval to force linearity when x<min(x,xc) or x>max(x,xc).  It  is  done
for consistency - we penalize non-linearity  at [min(x,xc),max(x,xc)],  so
it is natural to force linearity outside of this interval.

NOTE 2: function automatically sorts points,  so  caller may pass unsorted
array.

  -- ALGLIB PROJECT --
     Copyright 19.10.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dfitpenalizedw(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    ae_int_t m,
    double rho,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);
void alglib::spline1dfitpenalizedw(
    real_1d_array x,
    real_1d_array y,
    real_1d_array w,
    ae_int_t n,
    ae_int_t m,
    double rho,
    ae_int_t& info,
    spline1dinterpolant& s,
    spline1dfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline1dgriddiff2cubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at  nodes  x[],  it  calculates  and  returns  tables  of first and second
function derivatives d1[] and d2[] (calculated at the same nodes x[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DDiff() calls, but it can be several times faster  when
called for ordered X[] and X2[].

INPUT PARAMETERS:
    X           -   spline nodes
    Y           -   function values

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)

OUTPUT PARAMETERS:
    D1          -   S' values at X[]
    D2          -   S'' values at X[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so  D[I]  is  always
equal to S'(X[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dgriddiff2cubic(
    real_1d_array x,
    real_1d_array y,
    real_1d_array& d1,
    real_1d_array& d2,
    const xparams _params = alglib::xdefault);
void alglib::spline1dgriddiff2cubic(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t boundltype,
    double boundl,
    ae_int_t boundrtype,
    double boundr,
    real_1d_array& d1,
    real_1d_array& d2,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline1dgriddiffcubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at nodes x[], it calculates and returns table of function derivatives  d[]
(calculated at the same nodes x[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DDiff() calls, but it can be several times faster  when
called for ordered X[] and X2[].

INPUT PARAMETERS:
    X           -   spline nodes
    Y           -   function values

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)

OUTPUT PARAMETERS:
    D           -   derivative values at X[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so  D[I]  is  always
equal to S'(X[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dgriddiffcubic(
    real_1d_array x,
    real_1d_array y,
    real_1d_array& d,
    const xparams _params = alglib::xdefault);
void alglib::spline1dgriddiffcubic(
    real_1d_array x,
    real_1d_array y,
    ae_int_t n,
    ae_int_t boundltype,
    double boundl,
    ae_int_t boundrtype,
    double boundr,
    real_1d_array& d,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline1dintegrate` function

/*************************************************************************
This subroutine integrates the spline.

INPUT PARAMETERS:
    C   -   spline interpolant.
    X   -   right bound of the integration interval [a, x],
            here 'a' denotes min(x[])
Result:
    integral(S(t)dt,a,x)

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::spline1dintegrate(
    spline1dinterpolant c,
    double x,
    const xparams _params = alglib::xdefault);

`spline1dlintransx` function

/*************************************************************************
This subroutine performs linear transformation of the spline argument.

INPUT PARAMETERS:
    C   -   spline interpolant.
    A, B-   transformation coefficients: x = A*t + B
Result:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dlintransx(
    spline1dinterpolant c,
    double a,
    double b,
    const xparams _params = alglib::xdefault);

`spline1dlintransy` function

/*************************************************************************
This subroutine performs linear transformation of the spline.

INPUT PARAMETERS:
    C   -   spline interpolant.
    A, B-   transformation coefficients: S2(x) = A*S(x) + B
Result:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dlintransy(
    spline1dinterpolant c,
    double a,
    double b,
    const xparams _params = alglib::xdefault);

`spline1dunpack` function

/*************************************************************************
This subroutine unpacks the spline into the coefficients table.

INPUT PARAMETERS:
    C   -   spline interpolant.
    X   -   point

OUTPUT PARAMETERS:
    Tbl -   coefficients table, unpacked format, array[0..N-2, 0..5].
            For I = 0...N-2:
                Tbl[I,0] = X[i]
                Tbl[I,1] = X[i+1]
                Tbl[I,2] = C0
                Tbl[I,3] = C1
                Tbl[I,4] = C2
                Tbl[I,5] = C3
            On [x[i], x[i+1]] spline is equals to:
                S(x) = C0 + C1*t + C2*t^2 + C3*t^3
                t = x-x[i]

NOTE:
    You  can rebuild spline with  Spline1DBuildHermite()  function,  which
    accepts as inputs function values and derivatives at nodes, which  are
    easy to calculate when you have coefficients.

  -- ALGLIB PROJECT --
     Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline1dunpack(
    spline1dinterpolant c,
    ae_int_t& n,
    real_2d_array& tbl,
    const xparams _params = alglib::xdefault);

spline1d_d_convdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use cubic spline to do resampling, i.e. having
    // values of f(x)=x^2 sampled at 5 equidistant nodes on [-1,+1]
    // we calculate values/derivatives of cubic spline on 
    // another grid (equidistant with 9 nodes on [-1,+1])
    // WITHOUT CONSTRUCTION OF SPLINE OBJECT.
    //
    // There are efficient functions spline1dconvcubic(),
    // spline1dconvdiffcubic() and spline1dconvdiff2cubic() 
    // for such calculations.
    //
    // We use default boundary conditions ("parabolically terminated
    // spline") because cubic spline built with such boundary conditions 
    // will exactly reproduce any quadratic f(x).
    //
    // Actually, we could use natural conditions, but we feel that 
    // spline which exactly reproduces f() will show us more 
    // understandable results.
    //
    real_1d_array x_old = "[-1.0,-0.5,0.0,+0.5,+1.0]";
    real_1d_array y_old = "[+1.0,0.25,0.0,0.25,+1.0]";
    real_1d_array x_new = "[-1.00,-0.75,-0.50,-0.25,0.00,+0.25,+0.50,+0.75,+1.00]";
    real_1d_array y_new;
    real_1d_array d1_new;
    real_1d_array d2_new;

    //
    // First, conversion without differentiation.
    //
    //
    spline1dconvcubic(x_old, y_old, x_new, y_new);
    printf("%s\n", y_new.tostring(3).c_str()); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]

    //
    // Then, conversion with differentiation (first derivatives only)
    //
    //
    spline1dconvdiffcubic(x_old, y_old, x_new, y_new, d1_new);
    printf("%s\n", y_new.tostring(3).c_str()); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]
    printf("%s\n", d1_new.tostring(3).c_str()); // EXPECTED: [-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0]

    //
    // Finally, conversion with first and second derivatives
    //
    //
    spline1dconvdiff2cubic(x_old, y_old, x_new, y_new, d1_new, d2_new);
    printf("%s\n", y_new.tostring(3).c_str()); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]
    printf("%s\n", d1_new.tostring(3).c_str()); // EXPECTED: [-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0]
    printf("%s\n", d2_new.tostring(3).c_str()); // EXPECTED: [2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0]
    return 0;
}

spline1d_d_cubic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use cubic spline to interpolate f(x)=x^2 sampled 
    // at 5 equidistant nodes on [-1,+1].
    //
    // First, we use default boundary conditions ("parabolically terminated
    // spline") because cubic spline built with such boundary conditions 
    // will exactly reproduce any quadratic f(x).
    //
    // Then we try to use natural boundary conditions
    //     d2S(-1)/dx^2 = 0.0
    //     d2S(+1)/dx^2 = 0.0
    // and see that such spline interpolated f(x) with small error.
    //
    real_1d_array x = "[-1.0,-0.5,0.0,+0.5,+1.0]";
    real_1d_array y = "[+1.0,0.25,0.0,0.25,+1.0]";
    double t = 0.25;
    double v;
    spline1dinterpolant s;
    ae_int_t natural_bound_type = 2;
    //
    // Test exact boundary conditions: build S(x), calculare S(0.25)
    // (almost same as original function)
    //
    spline1dbuildcubic(x, y, s);
    v = spline1dcalc(s, t);
    printf("%.4f\n", double(v)); // EXPECTED: 0.0625

    //
    // Test natural boundary conditions: build S(x), calculare S(0.25)
    // (small interpolation error)
    //
    spline1dbuildcubic(x, y, 5, natural_bound_type, 0.0, natural_bound_type, 0.0, s);
    v = spline1dcalc(s, t);
    printf("%.3f\n", double(v)); // EXPECTED: 0.0580
    return 0;
}

spline1d_d_griddiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use cubic spline to do grid differentiation, i.e. having
    // values of f(x)=x^2 sampled at 5 equidistant nodes on [-1,+1]
    // we calculate derivatives of cubic spline at nodes WITHOUT
    // CONSTRUCTION OF SPLINE OBJECT.
    //
    // There are efficient functions spline1dgriddiffcubic() and
    // spline1dgriddiff2cubic() for such calculations.
    //
    // We use default boundary conditions ("parabolically terminated
    // spline") because cubic spline built with such boundary conditions 
    // will exactly reproduce any quadratic f(x).
    //
    // Actually, we could use natural conditions, but we feel that 
    // spline which exactly reproduces f() will show us more 
    // understandable results.
    //
    real_1d_array x = "[-1.0,-0.5,0.0,+0.5,+1.0]";
    real_1d_array y = "[+1.0,0.25,0.0,0.25,+1.0]";
    real_1d_array d1;
    real_1d_array d2;

    //
    // We calculate first derivatives: they must be equal to 2*x
    //
    spline1dgriddiffcubic(x, y, d1);
    printf("%s\n", d1.tostring(3).c_str()); // EXPECTED: [-2.0, -1.0, 0.0, +1.0, +2.0]

    //
    // Now test griddiff2, which returns first AND second derivatives.
    // First derivative is 2*x, second is equal to 2.0
    //
    spline1dgriddiff2cubic(x, y, d1, d2);
    printf("%s\n", d1.tostring(3).c_str()); // EXPECTED: [-2.0, -1.0, 0.0, +1.0, +2.0]
    printf("%s\n", d2.tostring(3).c_str()); // EXPECTED: [ 2.0,  2.0, 2.0,  2.0,  2.0]
    return 0;
}

spline1d_d_linear example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use piecewise linear spline to interpolate f(x)=x^2 sampled 
    // at 5 equidistant nodes on [-1,+1].
    //
    real_1d_array x = "[-1.0,-0.5,0.0,+0.5,+1.0]";
    real_1d_array y = "[+1.0,0.25,0.0,0.25,+1.0]";
    double t = 0.25;
    double v;
    spline1dinterpolant s;

    // build spline
    spline1dbuildlinear(x, y, s);

    // calculate S(0.25) - it is quite different from 0.25^2=0.0625
    v = spline1dcalc(s, t);
    printf("%.4f\n", double(v)); // EXPECTED: 0.125
    return 0;
}

spline1d_d_monotone example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Spline built witn spline1dbuildcubic() can be non-monotone even when
    // Y-values form monotone sequence. Say, for x=[0,1,2] and y=[0,1,1]
    // cubic spline will monotonically grow until x=1.5 and then start
    // decreasing.
    //
    // That's why ALGLIB provides special spline construction function
    // which builds spline which preserves monotonicity of the original
    // dataset.
    //
    // NOTE: in case original dataset is non-monotonic, ALGLIB splits it
    // into monotone subsequences and builds piecewise monotonic spline.
    //
    real_1d_array x = "[0,1,2]";
    real_1d_array y = "[0,1,1]";
    spline1dinterpolant s;

    // build spline
    spline1dbuildmonotone(x, y, s);

    // calculate S at x = [-0.5, 0.0, 0.5, 1.0, 1.5, 2.0]
    // you may see that spline is really monotonic
    double v;
    v = spline1dcalc(s, -0.5);
    printf("%.4f\n", double(v)); // EXPECTED: 0.0000
    v = spline1dcalc(s, 0.0);
    printf("%.4f\n", double(v)); // EXPECTED: 0.0000
    v = spline1dcalc(s, +0.5);
    printf("%.4f\n", double(v)); // EXPECTED: 0.5000
    v = spline1dcalc(s, 1.0);
    printf("%.4f\n", double(v)); // EXPECTED: 1.0000
    v = spline1dcalc(s, 1.5);
    printf("%.4f\n", double(v)); // EXPECTED: 1.0000
    v = spline1dcalc(s, 2.0);
    printf("%.4f\n", double(v)); // EXPECTED: 1.0000
    return 0;
}

`spline2d` subpackage

Classes

spline2dbuilder
spline2dfitreport
spline2dinterpolant

Functions

spline2dbuildbicubic
spline2dbuildbicubicv
spline2dbuildbilinear
spline2dbuildbilinearv
spline2dbuildercreate
spline2dbuildersetalgoblocklls
spline2dbuildersetalgofastddm
spline2dbuildersetalgonaivells
spline2dbuildersetarea
spline2dbuildersetareaauto
spline2dbuildersetconstterm
spline2dbuildersetgrid
spline2dbuildersetlinterm
spline2dbuildersetpoints
spline2dbuildersetuserterm
spline2dbuildersetzeroterm
spline2dcalc
spline2dcalcv
spline2dcalcvbuf
spline2dcalcvi
spline2dcopy
spline2ddiff
spline2ddiffvi
spline2dfit
spline2dlintransf
spline2dlintransxy
spline2dresamplebicubic
spline2dresamplebilinear
spline2dserialize
spline2dunpack
spline2dunpackv
spline2dunserialize

Examples

spline2d_bicubic		Bilinear spline interpolation
spline2d_bilinear		Bilinear spline interpolation
spline2d_copytrans		Copy and transform
spline2d_fit_blocklls		Fitting bicubic spline to irregular data
spline2d_unpack		Unpacking bilinear spline
spline2d_vector		Copy and transform

`spline2dbuilder` class

/*************************************************************************
Nonlinear least squares solver used to fit 2D splines to data
*************************************************************************/
class spline2dbuilder
{
};

`spline2dfitreport` class

/*************************************************************************
Spline 2D fitting report:
    rmserror        RMS error
    avgerror        average error
    maxerror        maximum error
    r2              coefficient of determination,  R-squared, 1-RSS/TSS
*************************************************************************/
class spline2dfitreport
{
    double               rmserror;
    double               avgerror;
    double               maxerror;
    double               r2;
};

`spline2dinterpolant` class

/*************************************************************************
2-dimensional spline inteprolant
*************************************************************************/
class spline2dinterpolant
{
};

`spline2dbuildbicubic` function

/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0

We recommend you to switch  to  Spline2DBuildBicubicV(),  which  is  more
flexible and accepts its arguments in more convenient order.

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildbicubic(
    real_1d_array x,
    real_1d_array y,
    real_2d_array f,
    ae_int_t m,
    ae_int_t n,
    spline2dinterpolant& c,
    const xparams _params = alglib::xdefault);

`spline2dbuildbicubicv` function

/*************************************************************************
This subroutine builds bicubic vector-valued spline.

Input parameters:
    X   -   spline abscissas, array[0..N-1]
    Y   -   spline ordinates, array[0..M-1]
    F   -   function values, array[0..M*N*D-1]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
    M,N -   grid size, M>=2, N>=2
    D   -   vector dimension, D>=1

Output parameters:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildbicubicv(
    real_1d_array x,
    ae_int_t n,
    real_1d_array y,
    ae_int_t m,
    real_1d_array f,
    ae_int_t d,
    spline2dinterpolant& c,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline2dbuildbilinear` function

/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0

We recommend you to switch  to  Spline2DBuildBilinearV(),  which  is  more
flexible and accepts its arguments in more convenient order.

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildbilinear(
    real_1d_array x,
    real_1d_array y,
    real_2d_array f,
    ae_int_t m,
    ae_int_t n,
    spline2dinterpolant& c,
    const xparams _params = alglib::xdefault);

`spline2dbuildbilinearv` function

/*************************************************************************
This subroutine builds bilinear vector-valued spline.

Input parameters:
    X   -   spline abscissas, array[0..N-1]
    Y   -   spline ordinates, array[0..M-1]
    F   -   function values, array[0..M*N*D-1]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
    M,N -   grid size, M>=2, N>=2
    D   -   vector dimension, D>=1

Output parameters:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildbilinearv(
    real_1d_array x,
    ae_int_t n,
    real_1d_array y,
    ae_int_t m,
    real_1d_array f,
    ae_int_t d,
    spline2dinterpolant& c,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`spline2dbuildercreate` function

/*************************************************************************
This subroutine creates least squares solver used to  fit  2D  splines  to
irregularly sampled (scattered) data.

Solver object is used to perform spline fits as follows:
* solver object is created with spline2dbuildercreate() function
* dataset is added with spline2dbuildersetpoints() function
* fit area is chosen:
  * spline2dbuildersetarea()     - for user-defined area
  * spline2dbuildersetareaauto() - for automatically chosen area
* number of grid nodes is chosen with spline2dbuildersetgrid()
* prior term is chosen with one of the following functions:
  * spline2dbuildersetlinterm()   to set linear prior
  * spline2dbuildersetconstterm() to set constant prior
  * spline2dbuildersetzeroterm()  to set zero prior
  * spline2dbuildersetuserterm()  to set user-defined constant prior
* solver algorithm is chosen with either:
  * spline2dbuildersetalgoblocklls() - BlockLLS algorithm, medium-scale problems
  * spline2dbuildersetalgofastddm()  - FastDDM algorithm, large-scale problems
* finally, fitting itself is performed with spline2dfit() function.

Most of the steps above can be omitted,  solver  is  configured with  good
defaults. The minimum is to call:
* spline2dbuildercreate() to create solver object
* spline2dbuildersetpoints() to specify dataset
* spline2dbuildersetgrid() to tell how many nodes you need
* spline2dfit() to perform fit

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    D   -   positive number, number of Y-components: D=1 for simple scalar
            fit, D>1 for vector-valued spline fitting.

OUTPUT PARAMETERS:
    S   -   solver object

  -- ALGLIB PROJECT --
     Copyright 29.01.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildercreate(
    ae_int_t d,
    spline2dbuilder& state,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline2dbuildersetalgoblocklls` function

/*************************************************************************
This  function  allows  you to choose least squares solver used to perform
fitting. This function sets solver algorithm to "BlockLLS", which performs
least squares fitting  with  fast  sparse  direct  solver,  with  optional
nonsmoothness penalty being applied.

Nonlinearity penalty has the following form:

                          [                                            ]
    P() ~ Lambda* integral[ (d2S/dx2)^2 + 2*(d2S/dxdy)^2 + (d2S/dy2)^2 ]dxdy
                          [                                            ]

here integral is calculated over entire grid, and "~" means "proportional"
because integral is normalized after calcilation. Extremely  large  values
of Lambda result in linear fit being performed.

NOTE: this algorithm is the most robust and controllable one,  but  it  is
      limited by 512x512 grids and (say) up to 1.000.000 points.  However,
      ALGLIB has one more  spline  solver:  FastDDM  algorithm,  which  is
      intended for really large-scale problems (in 10M-100M range). FastDDM
      algorithm also has better parallelism properties.

More information on BlockLLS solver:
* memory requirements: ~[32*K^3+256*NPoints]  bytes  for  KxK  grid   with
  NPoints-sized dataset
* serial running time: O(K^4+NPoints)
* parallelism potential: limited. You may get some sublinear gain when
  working with large grids (K's in 256..512 range)

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    LambdaNS-   non-negative value:
                * positive value means that some smoothing is applied
                * zero value means  that  no  smoothing  is  applied,  and
                  corresponding entries of design matrix  are  numerically
                  zero and dropped from consideration.

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetalgoblocklls(
    spline2dbuilder state,
    double lambdans,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetalgofastddm` function

/*************************************************************************
This  function  allows  you to choose least squares solver used to perform
fitting. This function sets solver algorithm to "FastDDM", which  performs
fast parallel fitting by splitting problem into smaller chunks and merging
results together.

This solver is optimized for large-scale problems, starting  from  256x256
grids, and up to 10000x10000 grids. Of course, it will  work  for  smaller
grids too.

More detailed description of the algorithm is given below:
* algorithm generates hierarchy  of  nested  grids,  ranging  from  ~16x16
  (topmost "layer" of the model) to ~KX*KY one (final layer). Upper layers
  model global behavior of the function, lower layers are  used  to  model
  fine details. Moving from layer to layer doubles grid density.
* fitting  is  started  from  topmost  layer, subsequent layers are fitted
  using residuals from previous ones.
* user may choose to skip generation of upper layers and generate  only  a
  few bottom ones, which  will  result  in  much  better  performance  and
  parallelization efficiency, at the cost of algorithm inability to "patch"
  large holes in the dataset.
* every layer is regularized using progressively increasing regularization
  coefficient; thus, increasing  LambdaV  penalizes  fine  details  first,
  leaving lower frequencies almost intact for a while.
* after fitting is done, all layers are merged together into  one  bicubic
  spline

IMPORTANT: regularization coefficient used by  this  solver  is  different
           from the one used by  BlockLLS.  Latter  utilizes  nonlinearity
           penalty,  which  is  global  in  nature  (large  regularization
           results in global linear trend being  extracted);  this  solver
           uses another, localized form of penalty, which is suitable  for
           parallel processing.

Notes on memory and performance:
* memory requirements: most memory is consumed  during  modeling   of  the
  higher layers; ~[512*NPoints] bytes is required for a  model  with  full
  hierarchy of grids being generated. However, if you skip a  few  topmost
  layers, you will get nearly constant (wrt. points count and  grid  size)
  memory consumption.
* serial running time: O(K*K)+O(NPoints) for a KxK grid
* parallelism potential: good. You may get  nearly  linear  speed-up  when
  performing fitting with just a few layers. Adding more layers results in
  model becoming more global, which somewhat  reduces  efficiency  of  the
  parallel code.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    NLayers -   number of layers in the model:
                * NLayers>=1 means that up  to  chosen  number  of  bottom
                  layers is fitted
                * NLayers=0 means that maximum number of layers is  chosen
                  (according to current grid size)
                * NLayers<=-1 means that up to |NLayers| topmost layers is
                  skipped
                Recommendations:
                * good "default" value is 2 layers
                * you may need  more  layers,  if  your  dataset  is  very
                  irregular and you want to "patch"  large  holes.  For  a
                  grid step H (equal to AreaWidth/GridSize) you may expect
                  that last layer reproduces variations at distance H (and
                  can patch holes that wide); that higher  layers  operate
                  at distances 2*H, 4*H, 8*H and so on.
                * good value for "bullletproof" mode is  NLayers=0,  which
                  results in complete hierarchy of layers being generated.
    LambdaV -   regularization coefficient, chosen in such a way  that  it
                penalizes bottom layers (fine details) first.
                LambdaV>=0, zero value means that no penalty is applied.

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetalgofastddm(
    spline2dbuilder state,
    ae_int_t nlayers,
    double lambdav,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetalgonaivells` function

/*************************************************************************
This  function  allows  you to choose least squares solver used to perform
fitting. This function sets solver algorithm to "NaiveLLS".

IMPORTANT: NaiveLLS is NOT intended to be used in  real  life  code!  This
           algorithm solves problem by generated dense (K^2)x(K^2+NPoints)
           matrix and solves  linear  least  squares  problem  with  dense
           solver.

           It is here just  to  test  BlockLLS  against  reference  solver
           (and maybe for someone trying to compare well optimized  solver
           against straightforward approach to the LLS problem).

More information on naive LLS solver:
* memory requirements: ~[8*K^4+256*NPoints] bytes for KxK grid.
* serial running time: O(K^6+NPoints) for KxK grid
* when compared with BlockLLS,  NaiveLLS  has ~K  larger memory demand and
  ~K^2  larger running time.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    LambdaNS-   nonsmoothness penalty

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetalgonaivells(
    spline2dbuilder state,
    double lambdans,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetarea` function

/*************************************************************************
This  function  sets  area  where  2D  spline  interpolant  is   built  to
user-defined one: [XA,XB]*[YA,YB]

INPUT PARAMETERS:
    S       -   spline 2D builder object
    XA,XB   -   spatial extent in the first (X) dimension, XA<XB
    YA,YB   -   spatial extent in the second (Y) dimension, YA<YB

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetarea(
    spline2dbuilder state,
    double xa,
    double xb,
    double ya,
    double yb,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetareaauto` function

/*************************************************************************
This function sets area where 2D spline interpolant is built. "Auto" means
that area extent is determined automatically from dataset extent.

INPUT PARAMETERS:
    S       -   spline 2D builder object

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetareaauto(
    spline2dbuilder state,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetconstterm` function

/*************************************************************************
This function sets constant prior term (model is a sum of  bicubic  spline
and global prior, which can be linear, constant, user-defined  constant or
zero).

Constant prior term is determined by least squares fitting.

INPUT PARAMETERS:
    S       -   spline builder

  -- ALGLIB --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetconstterm(
    spline2dbuilder state,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetgrid` function

/*************************************************************************
This  function  sets  nodes  count  for  2D spline interpolant. Fitting is
performed on area defined with one of the "setarea"  functions;  this  one
sets number of nodes placed upon the fitting area.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    KX      -   nodes count for the first (X) dimension; fitting  interval
                [XA,XB] is separated into KX-1 subintervals, with KX nodes
                created at the boundaries.
    KY      -   nodes count for the first (Y) dimension; fitting  interval
                [YA,YB] is separated into KY-1 subintervals, with KY nodes
                created at the boundaries.

NOTE: at  least  4  nodes  is  created in each dimension, so KX and KY are
      silently increased if needed.

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetgrid(
    spline2dbuilder state,
    ae_int_t kx,
    ae_int_t ky,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetlinterm` function

/*************************************************************************
This function sets linear prior term (model is a sum of bicubic spline and
global  prior,  which  can  be  linear, constant, user-defined constant or
zero).

Linear prior term is determined by least squares fitting.

INPUT PARAMETERS:
    S       -   spline builder

  -- ALGLIB --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetlinterm(
    spline2dbuilder state,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetpoints` function

/*************************************************************************
This function adds dataset to the builder object.

This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    XY      -   points, array[N,2+D]. One  row  corresponds to  one  point
                in the dataset. First 2  elements  are  coordinates,  next
                D  elements are function values. Array may  be larger than
                specified, in  this  case  only leading [N,NX+NY] elements
                will be used.
    N       -   number of points in the dataset

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetpoints(
    spline2dbuilder state,
    real_2d_array xy,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetuserterm` function

/*************************************************************************
This function sets constant prior term (model is a sum of  bicubic  spline
and global prior, which can be linear, constant, user-defined  constant or
zero).

Constant prior term is determined by least squares fitting.

INPUT PARAMETERS:
    S       -   spline builder
    V       -   value for user-defined prior

  -- ALGLIB --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetuserterm(
    spline2dbuilder state,
    double v,
    const xparams _params = alglib::xdefault);

`spline2dbuildersetzeroterm` function

/*************************************************************************
This function sets zero prior term (model is a sum of bicubic  spline  and
global  prior,  which  can  be  linear, constant, user-defined constant or
zero).

INPUT PARAMETERS:
    S       -   spline builder

  -- ALGLIB --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dbuildersetzeroterm(
    spline2dbuilder state,
    const xparams _params = alglib::xdefault);

`spline2dcalc` function

/*************************************************************************
This subroutine calculates the value of the bilinear or bicubic spline  at
the given point X.

Input parameters:
    C   -   2D spline object.
            Built by spline2dbuildbilinearv or spline2dbuildbicubicv.
    X, Y-   point

Result:
    S(x,y)

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
double alglib::spline2dcalc(
    spline2dinterpolant c,
    double x,
    double y,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`spline2dcalcv` function

/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y).

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y-   point

OUTPUT PARAMETERS:
    F   -   array[D] which stores function values.  F is out-parameter and
            it  is  reallocated  after  call to this function. In case you
            want  to    reuse  previously  allocated  F,   you   may   use
            Spline2DCalcVBuf(),  which  reallocates  F only when it is too
            small.

  -- ALGLIB PROJECT --
     Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dcalcv(
    spline2dinterpolant c,
    double x,
    double y,
    real_1d_array& f,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline2dcalcvbuf` function

/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y).

If you need just some specific component of vector-valued spline, you  can
use spline2dcalcvi() function.

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y-   point
    F   -   output buffer, possibly preallocated array. In case array size
            is large enough to store result, it is not reallocated.  Array
            which is too short will be reallocated

OUTPUT PARAMETERS:
    F   -   array[D] (or larger) which stores function values

  -- ALGLIB PROJECT --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dcalcvbuf(
    spline2dinterpolant c,
    double x,
    double y,
    real_1d_array& f,
    const xparams _params = alglib::xdefault);

`spline2dcalcvi` function

/*************************************************************************
This subroutine calculates specific component of vector-valued bilinear or
bicubic spline at the given point (X,Y).

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y-   point
    I   -   component index, in [0,D). An exception is generated for out
            of range values.

RESULT:
    value of I-th component

  -- ALGLIB PROJECT --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
double alglib::spline2dcalcvi(
    spline2dinterpolant c,
    double x,
    double y,
    ae_int_t i,
    const xparams _params = alglib::xdefault);

`spline2dcopy` function

/*************************************************************************
This subroutine makes the copy of the spline model.

Input parameters:
    C   -   spline interpolant

Output parameters:
    CC  -   spline copy

  -- ALGLIB PROJECT --
     Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dcopy(
    spline2dinterpolant c,
    spline2dinterpolant& cc,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline2ddiff` function

/*************************************************************************
This subroutine calculates the value of the bilinear or bicubic spline  at
the given point X and its derivatives.

Input parameters:
    C   -   spline interpolant.
    X, Y-   point

Output parameters:
    F   -   S(x,y)
    FX  -   dS(x,y)/dX
    FY  -   dS(x,y)/dY
    FXY -   d2S(x,y)/dXdY

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2ddiff(
    spline2dinterpolant c,
    double x,
    double y,
    double& f,
    double& fx,
    double& fy,
    double& fxy,
    const xparams _params = alglib::xdefault);

`spline2ddiffvi` function

/*************************************************************************
This subroutine calculates value of  specific  component  of  bilinear  or
bicubic vector-valued spline and its derivatives.

Input parameters:
    C   -   spline interpolant.
    X, Y-   point
    I   -   component index, in [0,D)

Output parameters:
    F   -   S(x,y)
    FX  -   dS(x,y)/dX
    FY  -   dS(x,y)/dY
    FXY -   d2S(x,y)/dXdY

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2ddiffvi(
    spline2dinterpolant c,
    double x,
    double y,
    ae_int_t i,
    double& f,
    double& fx,
    double& fy,
    double& fxy,
    const xparams _params = alglib::xdefault);

`spline2dfit` function

/*************************************************************************
This function fits bicubic spline to current dataset, using current  area/
grid and current LLS solver.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    State   -   spline 2D builder object

OUTPUT PARAMETERS:
    S       -   2D spline, fit result
    Rep     -   fitting report, which provides some additional info  about
                errors, R2 coefficient and so on.

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dfit(
    spline2dbuilder state,
    spline2dinterpolant& s,
    spline2dfitreport& rep,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline2dlintransf` function

/*************************************************************************
This subroutine performs linear transformation of the spline.

Input parameters:
    C   -   spline interpolant.
    A, B-   transformation coefficients: S2(x,y) = A*S(x,y) + B

Output parameters:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dlintransf(
    spline2dinterpolant c,
    double a,
    double b,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline2dlintransxy` function

/*************************************************************************
This subroutine performs linear transformation of the spline argument.

Input parameters:
    C       -   spline interpolant
    AX, BX  -   transformation coefficients: x = A*t + B
    AY, BY  -   transformation coefficients: y = A*u + B
Result:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dlintransxy(
    spline2dinterpolant c,
    double ax,
    double bx,
    double ay,
    double by,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline2dresamplebicubic` function

/*************************************************************************
Bicubic spline resampling

Input parameters:
    A           -   function values at the old grid,
                    array[0..OldHeight-1, 0..OldWidth-1]
    OldHeight   -   old grid height, OldHeight>1
    OldWidth    -   old grid width, OldWidth>1
    NewHeight   -   new grid height, NewHeight>1
    NewWidth    -   new grid width, NewWidth>1

Output parameters:
    B           -   function values at the new grid,
                    array[0..NewHeight-1, 0..NewWidth-1]

  -- ALGLIB routine --
     15 May, 2007
     Copyright by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dresamplebicubic(
    real_2d_array a,
    ae_int_t oldheight,
    ae_int_t oldwidth,
    real_2d_array& b,
    ae_int_t newheight,
    ae_int_t newwidth,
    const xparams _params = alglib::xdefault);

`spline2dresamplebilinear` function

/*************************************************************************
Bilinear spline resampling

Input parameters:
    A           -   function values at the old grid,
                    array[0..OldHeight-1, 0..OldWidth-1]
    OldHeight   -   old grid height, OldHeight>1
    OldWidth    -   old grid width, OldWidth>1
    NewHeight   -   new grid height, NewHeight>1
    NewWidth    -   new grid width, NewWidth>1

Output parameters:
    B           -   function values at the new grid,
                    array[0..NewHeight-1, 0..NewWidth-1]

  -- ALGLIB routine --
     09.07.2007
     Copyright by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dresamplebilinear(
    real_2d_array a,
    ae_int_t oldheight,
    ae_int_t oldwidth,
    real_2d_array& b,
    ae_int_t newheight,
    ae_int_t newwidth,
    const xparams _params = alglib::xdefault);

`spline2dserialize` function

/*************************************************************************
This function serializes data structure to string.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string  in  case you want to include it into text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor you should change case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize  it  in  C++ version of ALGLIB and unserialize in C# one, 
  and vice versa.
*************************************************************************/
void spline2dserialize(spline2dinterpolant &obj, std::string &s_out);
void spline2dserialize(spline2dinterpolant &obj, std::ostream &s_out);

`spline2dunpack` function

/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0

We recommend you to switch  to  Spline2DUnpackV(),  which is more flexible
and accepts its arguments in more convenient order.

  -- ALGLIB PROJECT --
     Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dunpack(
    spline2dinterpolant c,
    ae_int_t& m,
    ae_int_t& n,
    real_2d_array& tbl,
    const xparams _params = alglib::xdefault);

`spline2dunpackv` function

/*************************************************************************
This subroutine unpacks two-dimensional spline into the coefficients table

Input parameters:
    C   -   spline interpolant.

Result:
    M, N-   grid size (x-axis and y-axis)
    D   -   number of components
    Tbl -   coefficients table, unpacked format,
            D - components: [0..(N-1)*(M-1)*D-1, 0..19].
            For T=0..D-1 (component index), I = 0...N-2 (x index),
            J=0..M-2 (y index):
                K :=  T + I*D + J*D*(N-1)

                K-th row stores decomposition for T-th component of the
                vector-valued function

                Tbl[K,0] = X[i]
                Tbl[K,1] = X[i+1]
                Tbl[K,2] = Y[j]
                Tbl[K,3] = Y[j+1]
                Tbl[K,4] = C00
                Tbl[K,5] = C01
                Tbl[K,6] = C02
                Tbl[K,7] = C03
                Tbl[K,8] = C10
                Tbl[K,9] = C11
                ...
                Tbl[K,19] = C33
            On each grid square spline is equals to:
                S(x) = SUM(c[i,j]*(t^i)*(u^j), i=0..3, j=0..3)
                t = x-x[j]
                u = y-y[i]

  -- ALGLIB PROJECT --
     Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline2dunpackv(
    spline2dinterpolant c,
    ae_int_t& m,
    ae_int_t& n,
    ae_int_t& d,
    real_2d_array& tbl,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline2dunserialize` function

/*************************************************************************
This function unserializes data structure from string.
*************************************************************************/
void spline2dunserialize(const std::string &s_in, spline2dinterpolant &obj);
void spline2dunserialize(const std::istream &s_in, spline2dinterpolant &obj);

spline2d_bicubic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use bilinear spline to interpolate f(x,y)=x^2+2*y^2 sampled 
    // at (x,y) from [0.0, 0.5, 1.0] X [0.0, 1.0].
    //
    real_1d_array x = "[0.0, 0.5, 1.0]";
    real_1d_array y = "[0.0, 1.0]";
    real_1d_array f = "[0.00,0.25,1.00,2.00,2.25,3.00]";
    double vx = 0.25;
    double vy = 0.50;
    double v;
    double dx;
    double dy;
    double dxy;
    spline2dinterpolant s;

    // build spline
    spline2dbuildbicubicv(x, 3, y, 2, f, 1, s);

    // calculate S(0.25,0.50)
    v = spline2dcalc(s, vx, vy);
    printf("%.4f\n", double(v)); // EXPECTED: 1.0625

    // calculate derivatives
    spline2ddiff(s, vx, vy, v, dx, dy, dxy);
    printf("%.4f\n", double(v)); // EXPECTED: 1.0625
    printf("%.4f\n", double(dx)); // EXPECTED: 0.5000
    printf("%.4f\n", double(dy)); // EXPECTED: 2.0000
    return 0;
}

spline2d_bilinear example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use bilinear spline to interpolate f(x,y)=x^2+2*y^2 sampled 
    // at (x,y) from [0.0, 0.5, 1.0] X [0.0, 1.0].
    //
    real_1d_array x = "[0.0, 0.5, 1.0]";
    real_1d_array y = "[0.0, 1.0]";
    real_1d_array f = "[0.00,0.25,1.00,2.00,2.25,3.00]";
    double vx = 0.25;
    double vy = 0.50;
    double v;
    spline2dinterpolant s;

    // build spline
    spline2dbuildbilinearv(x, 3, y, 2, f, 1, s);

    // calculate S(0.25,0.50)
    v = spline2dcalc(s, vx, vy);
    printf("%.4f\n", double(v)); // EXPECTED: 1.1250
    return 0;
}

spline2d_copytrans example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We build bilinear spline for f(x,y)=x+2*y for (x,y) in [0,1].
    // Then we apply several transformations to this spline.
    //
    real_1d_array x = "[0.0, 1.0]";
    real_1d_array y = "[0.0, 1.0]";
    real_1d_array f = "[0.00,1.00,2.00,3.00]";
    spline2dinterpolant s;
    spline2dinterpolant snew;
    double v;
    spline2dbuildbilinearv(x, 2, y, 2, f, 1, s);

    // copy spline, apply transformation x:=2*xnew, y:=4*ynew
    // evaluate at (xnew,ynew) = (0.25,0.25) - should be same as (x,y)=(0.5,1.0)
    spline2dcopy(s, snew);
    spline2dlintransxy(snew, 2.0, 0.0, 4.0, 0.0);
    v = spline2dcalc(snew, 0.25, 0.25);
    printf("%.4f\n", double(v)); // EXPECTED: 2.500

    // copy spline, apply transformation SNew:=2*S+3
    spline2dcopy(s, snew);
    spline2dlintransf(snew, 2.0, 3.0);
    v = spline2dcalc(snew, 0.5, 1.0);
    printf("%.4f\n", double(v)); // EXPECTED: 8.000

    //
    // Same example, but for vector spline (f0,f1) = {x+2*y, 2*x+y}
    //
    real_1d_array f2 = "[0.00,0.00, 1.00,2.00, 2.00,1.00, 3.00,3.00]";
    real_1d_array vr;
    spline2dbuildbilinearv(x, 2, y, 2, f2, 2, s);

    // copy spline, apply transformation x:=2*xnew, y:=4*ynew
    spline2dcopy(s, snew);
    spline2dlintransxy(snew, 2.0, 0.0, 4.0, 0.0);
    spline2dcalcv(snew, 0.25, 0.25, vr);
    printf("%s\n", vr.tostring(4).c_str()); // EXPECTED: [2.500,2.000]

    // copy spline, apply transformation SNew:=2*S+3
    spline2dcopy(s, snew);
    spline2dlintransf(snew, 2.0, 3.0);
    spline2dcalcv(snew, 0.5, 1.0, vr);
    printf("%s\n", vr.tostring(4).c_str()); // EXPECTED: [8.000,7.000]
    return 0;
}

spline2d_fit_blocklls example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use bicubic spline to reproduce f(x,y)=1/(1+x^2+2*y^2) sampled
    // at irregular points (x,y) from [-1,+1]*[-1,+1]
    //
    // We have 5 such points, located approximately at corners of the area
    // and its center -  but not exactly at the grid. Thus, we have to FIT
    // the spline, i.e. to solve least squares problem
    //
    real_2d_array xy = "[[-0.987,-0.902,0.359],[0.948,-0.992,0.347],[-1.000,1.000,0.333],[1.000,0.973,0.339],[0.017,0.180,0.968]]";

    //
    // First step is to create spline2dbuilder object and set its properties:
    // * d=1 means that we create vector-valued spline with 1 component
    // * we specify dataset xy
    // * we rely on automatic selection of interpolation area
    // * we tell builder that we want to use 5x5 grid for an underlying spline
    // * we choose least squares solver named BlockLLS and configure it by
    //   telling that we want to apply zero nonlinearity penalty.
    //
    // NOTE: ALGLIB has two solvers which fit bicubic splines to irregular data,
    //       one of them is BlockLLS and another one is FastDDM. Former is
    //       intended for moderately sized grids (up to 512x512 nodes, although
    //       it may take up to few minutes); it is the most easy to use and
    //       control spline fitting function in the library. Latter, FastDDM,
    //       is intended for efficient solution of large-scale problems
    //       (up to 100.000.000 nodes). Both solvers can be parallelized, but
    //       FastDDM is much more efficient. See comments for more information.
    //
    spline2dbuilder builder;
    ae_int_t d = 1;
    spline2dbuildercreate(d, builder);
    spline2dbuildersetpoints(builder, xy, 5);
    spline2dbuildersetgrid(builder, 5, 5);
    spline2dbuildersetalgoblocklls(builder, 0.000);

    //
    // Now we are ready to fit and evaluate our results
    //
    spline2dinterpolant s;
    spline2dfitreport rep;
    spline2dfit(builder, s, rep);

    // evaluate results - function value at the grid is reproduced exactly
    double v;
    v = spline2dcalc(s, -1, 1);
    printf("%.2f\n", double(v)); // EXPECTED: 0.333000

    // check maximum error - it must be nearly zero
    printf("%.2f\n", double(rep.maxerror)); // EXPECTED: 0.000
    return 0;
}

spline2d_unpack example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We build bilinear spline for f(x,y)=x+2*y+3*xy for (x,y) in [0,1].
    // Then we demonstrate how to unpack it.
    //
    real_1d_array x = "[0.0, 1.0]";
    real_1d_array y = "[0.0, 1.0]";
    real_1d_array f = "[0.00,1.00,2.00,6.00]";
    real_2d_array c;
    ae_int_t m;
    ae_int_t n;
    ae_int_t d;
    spline2dinterpolant s;

    // build spline
    spline2dbuildbilinearv(x, 2, y, 2, f, 1, s);

    // unpack and test
    spline2dunpackv(s, m, n, d, c);
    printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [[0, 1, 0, 1, 0,2,0,0, 1,3,0,0, 0,0,0,0, 0,0,0,0 ]]
    return 0;
}

spline2d_vector example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We build bilinear vector-valued spline (f0,f1) = {x+2*y, 2*x+y}
    // Spline is built using function values at 2x2 grid: (x,y)=[0,1]*[0,1]
    // Then we perform evaluation at (x,y)=(0.1,0.3)
    //
    real_1d_array x = "[0.0, 1.0]";
    real_1d_array y = "[0.0, 1.0]";
    real_1d_array f = "[0.00,0.00, 1.00,2.00, 2.00,1.00, 3.00,3.00]";
    spline2dinterpolant s;
    real_1d_array vr;
    spline2dbuildbilinearv(x, 2, y, 2, f, 2, s);
    spline2dcalcv(s, 0.1, 0.3, vr);
    printf("%s\n", vr.tostring(4).c_str()); // EXPECTED: [0.700,0.500]
    return 0;
}

`spline3d` subpackage

Classes

spline3dinterpolant

Functions

spline3dbuildtrilinearv
spline3dcalc
spline3dcalcv
spline3dcalcvbuf
spline3dlintransf
spline3dlintransxyz
spline3dresampletrilinear
spline3dunpackv

Examples

spline3d_trilinear		Trilinear spline interpolation
spline3d_vector		Vector-valued trilinear spline interpolation

`spline3dinterpolant` class

/*************************************************************************
3-dimensional spline inteprolant
*************************************************************************/
class spline3dinterpolant
{
};

`spline3dbuildtrilinearv` function

/*************************************************************************
This subroutine builds trilinear vector-valued spline.

INPUT PARAMETERS:
    X   -   spline abscissas,  array[0..N-1]
    Y   -   spline ordinates,  array[0..M-1]
    Z   -   spline applicates, array[0..L-1]
    F   -   function values, array[0..M*N*L*D-1]:
            * first D elements store D values at (X[0],Y[0],Z[0])
            * next D elements store D values at (X[1],Y[0],Z[0])
            * next D elements store D values at (X[2],Y[0],Z[0])
            * ...
            * next D elements store D values at (X[0],Y[1],Z[0])
            * next D elements store D values at (X[1],Y[1],Z[0])
            * next D elements store D values at (X[2],Y[1],Z[0])
            * ...
            * next D elements store D values at (X[0],Y[0],Z[1])
            * next D elements store D values at (X[1],Y[0],Z[1])
            * next D elements store D values at (X[2],Y[0],Z[1])
            * ...
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(N*(M*K+J)+I)...D*(N*(M*K+J)+I)+D-1].
    M,N,
    L   -   grid size, M>=2, N>=2, L>=2
    D   -   vector dimension, D>=1

OUTPUT PARAMETERS:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline3dbuildtrilinearv(
    real_1d_array x,
    ae_int_t n,
    real_1d_array y,
    ae_int_t m,
    real_1d_array z,
    ae_int_t l,
    real_1d_array f,
    ae_int_t d,
    spline3dinterpolant& c,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`spline3dcalc` function

/*************************************************************************
This subroutine calculates the value of the trilinear or tricubic spline at
the given point (X,Y,Z).

INPUT PARAMETERS:
    C   -   coefficients table.
            Built by BuildBilinearSpline or BuildBicubicSpline.
    X, Y,
    Z   -   point

Result:
    S(x,y,z)

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
double alglib::spline3dcalc(
    spline3dinterpolant c,
    double x,
    double y,
    double z,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline3dcalcv` function

/*************************************************************************
This subroutine calculates trilinear or tricubic vector-valued spline at the
given point (X,Y,Z).

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y,
    Z   -   point

OUTPUT PARAMETERS:
    F   -   array[D] which stores function values.  F is out-parameter and
            it  is  reallocated  after  call to this function. In case you
            want  to    reuse  previously  allocated  F,   you   may   use
            Spline2DCalcVBuf(),  which  reallocates  F only when it is too
            small.

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline3dcalcv(
    spline3dinterpolant c,
    double x,
    double y,
    double z,
    real_1d_array& f,
    const xparams _params = alglib::xdefault);

Examples: [1]

`spline3dcalcvbuf` function

/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y,Z).

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y,
    Z   -   point
    F   -   output buffer, possibly preallocated array. In case array size
            is large enough to store result, it is not reallocated.  Array
            which is too short will be reallocated

OUTPUT PARAMETERS:
    F   -   array[D] (or larger) which stores function values

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline3dcalcvbuf(
    spline3dinterpolant c,
    double x,
    double y,
    double z,
    real_1d_array& f,
    const xparams _params = alglib::xdefault);

`spline3dlintransf` function

/*************************************************************************
This subroutine performs linear transformation of the spline.

INPUT PARAMETERS:
    C   -   spline interpolant.
    A, B-   transformation coefficients: S2(x,y) = A*S(x,y,z) + B

OUTPUT PARAMETERS:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline3dlintransf(
    spline3dinterpolant c,
    double a,
    double b,
    const xparams _params = alglib::xdefault);

`spline3dlintransxyz` function

/*************************************************************************
This subroutine performs linear transformation of the spline argument.

INPUT PARAMETERS:
    C       -   spline interpolant
    AX, BX  -   transformation coefficients: x = A*u + B
    AY, BY  -   transformation coefficients: y = A*v + B
    AZ, BZ  -   transformation coefficients: z = A*w + B

OUTPUT PARAMETERS:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline3dlintransxyz(
    spline3dinterpolant c,
    double ax,
    double bx,
    double ay,
    double by,
    double az,
    double bz,
    const xparams _params = alglib::xdefault);

`spline3dresampletrilinear` function

/*************************************************************************
Trilinear spline resampling

INPUT PARAMETERS:
    A           -   array[0..OldXCount*OldYCount*OldZCount-1], function
                    values at the old grid, :
                        A[0]        x=0,y=0,z=0
                        A[1]        x=1,y=0,z=0
                        A[..]       ...
                        A[..]       x=oldxcount-1,y=0,z=0
                        A[..]       x=0,y=1,z=0
                        A[..]       ...
                        ...
    OldZCount   -   old Z-count, OldZCount>1
    OldYCount   -   old Y-count, OldYCount>1
    OldXCount   -   old X-count, OldXCount>1
    NewZCount   -   new Z-count, NewZCount>1
    NewYCount   -   new Y-count, NewYCount>1
    NewXCount   -   new X-count, NewXCount>1

OUTPUT PARAMETERS:
    B           -   array[0..NewXCount*NewYCount*NewZCount-1], function
                    values at the new grid:
                        B[0]        x=0,y=0,z=0
                        B[1]        x=1,y=0,z=0
                        B[..]       ...
                        B[..]       x=newxcount-1,y=0,z=0
                        B[..]       x=0,y=1,z=0
                        B[..]       ...
                        ...

  -- ALGLIB routine --
     26.04.2012
     Copyright by Bochkanov Sergey
*************************************************************************/
void alglib::spline3dresampletrilinear(
    real_1d_array a,
    ae_int_t oldzcount,
    ae_int_t oldycount,
    ae_int_t oldxcount,
    ae_int_t newzcount,
    ae_int_t newycount,
    ae_int_t newxcount,
    real_1d_array& b,
    const xparams _params = alglib::xdefault);

`spline3dunpackv` function

/*************************************************************************
This subroutine unpacks tri-dimensional spline into the coefficients table

INPUT PARAMETERS:
    C   -   spline interpolant.

Result:
    N   -   grid size (X)
    M   -   grid size (Y)
    L   -   grid size (Z)
    D   -   number of components
    SType-  spline type. Currently, only one spline type is supported:
            trilinear spline, as indicated by SType=1.
    Tbl -   spline coefficients: [0..(N-1)*(M-1)*(L-1)*D-1, 0..13].
            For T=0..D-1 (component index), I = 0...N-2 (x index),
            J=0..M-2 (y index), K=0..L-2 (z index):
                Q := T + I*D + J*D*(N-1) + K*D*(N-1)*(M-1),

                Q-th row stores decomposition for T-th component of the
                vector-valued function

                Tbl[Q,0] = X[i]
                Tbl[Q,1] = X[i+1]
                Tbl[Q,2] = Y[j]
                Tbl[Q,3] = Y[j+1]
                Tbl[Q,4] = Z[k]
                Tbl[Q,5] = Z[k+1]

                Tbl[Q,6] = C000
                Tbl[Q,7] = C100
                Tbl[Q,8] = C010
                Tbl[Q,9] = C110
                Tbl[Q,10]= C001
                Tbl[Q,11]= C101
                Tbl[Q,12]= C011
                Tbl[Q,13]= C111
            On each grid square spline is equals to:
                S(x) = SUM(c[i,j,k]*(x^i)*(y^j)*(z^k), i=0..1, j=0..1, k=0..1)
                t = x-x[j]
                u = y-y[i]
                v = z-z[k]

            NOTE: format of Tbl is given for SType=1. Future versions of
                  ALGLIB can use different formats for different values of
                  SType.

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void alglib::spline3dunpackv(
    spline3dinterpolant c,
    ae_int_t& n,
    ae_int_t& m,
    ae_int_t& l,
    ae_int_t& d,
    ae_int_t& stype,
    real_2d_array& tbl,
    const xparams _params = alglib::xdefault);

spline3d_trilinear example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use trilinear spline to interpolate f(x,y,z)=x+xy+z sampled 
    // at (x,y,z) from [0.0, 1.0] X [0.0, 1.0] X [0.0, 1.0].
    //
    // We store x, y and z-values at local arrays with same names.
    // Function values are stored in the array F as follows:
    //     f[0]     (x,y,z) = (0,0,0)
    //     f[1]     (x,y,z) = (1,0,0)
    //     f[2]     (x,y,z) = (0,1,0)
    //     f[3]     (x,y,z) = (1,1,0)
    //     f[4]     (x,y,z) = (0,0,1)
    //     f[5]     (x,y,z) = (1,0,1)
    //     f[6]     (x,y,z) = (0,1,1)
    //     f[7]     (x,y,z) = (1,1,1)
    //
    real_1d_array x = "[0.0, 1.0]";
    real_1d_array y = "[0.0, 1.0]";
    real_1d_array z = "[0.0, 1.0]";
    real_1d_array f = "[0,1,0,2,1,2,1,3]";
    double vx = 0.50;
    double vy = 0.50;
    double vz = 0.50;
    double v;
    spline3dinterpolant s;

    // build spline
    spline3dbuildtrilinearv(x, 2, y, 2, z, 2, f, 1, s);

    // calculate S(0.5,0.5,0.5)
    v = spline3dcalc(s, vx, vy, vz);
    printf("%.4f\n", double(v)); // EXPECTED: 1.2500
    return 0;
}

spline3d_vector example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // We use trilinear vector-valued spline to interpolate {f0,f1}={x+xy+z,x+xy+yz+z}
    // sampled at (x,y,z) from [0.0, 1.0] X [0.0, 1.0] X [0.0, 1.0].
    //
    // We store x, y and z-values at local arrays with same names.
    // Function values are stored in the array F as follows:
    //     f[0]     f0, (x,y,z) = (0,0,0)
    //     f[1]     f1, (x,y,z) = (0,0,0)
    //     f[2]     f0, (x,y,z) = (1,0,0)
    //     f[3]     f1, (x,y,z) = (1,0,0)
    //     f[4]     f0, (x,y,z) = (0,1,0)
    //     f[5]     f1, (x,y,z) = (0,1,0)
    //     f[6]     f0, (x,y,z) = (1,1,0)
    //     f[7]     f1, (x,y,z) = (1,1,0)
    //     f[8]     f0, (x,y,z) = (0,0,1)
    //     f[9]     f1, (x,y,z) = (0,0,1)
    //     f[10]    f0, (x,y,z) = (1,0,1)
    //     f[11]    f1, (x,y,z) = (1,0,1)
    //     f[12]    f0, (x,y,z) = (0,1,1)
    //     f[13]    f1, (x,y,z) = (0,1,1)
    //     f[14]    f0, (x,y,z) = (1,1,1)
    //     f[15]    f1, (x,y,z) = (1,1,1)
    //
    real_1d_array x = "[0.0, 1.0]";
    real_1d_array y = "[0.0, 1.0]";
    real_1d_array z = "[0.0, 1.0]";
    real_1d_array f = "[0,0, 1,1, 0,0, 2,2, 1,1, 2,2, 1,2, 3,4]";
    double vx = 0.50;
    double vy = 0.50;
    double vz = 0.50;
    spline3dinterpolant s;

    // build spline
    spline3dbuildtrilinearv(x, 2, y, 2, z, 2, f, 2, s);

    // calculate S(0.5,0.5,0.5) - we have vector of values instead of single value
    real_1d_array v;
    spline3dcalcv(s, vx, vy, vz, v);
    printf("%s\n", v.tostring(4).c_str()); // EXPECTED: [1.2500,1.5000]
    return 0;
}

`ssa` subpackage

Classes

ssamodel

Functions

ssaaddsequence
ssaanalyzelast
ssaanalyzelastwindow
ssaanalyzesequence
ssaappendpointandupdate
ssaappendsequenceandupdate
ssacleardata
ssacreate
ssaforecastavglast
ssaforecastavgsequence
ssaforecastlast
ssaforecastsequence
ssagetbasis
ssagetlrr
ssasetalgoprecomputed
ssasetalgotopkdirect
ssasetalgotopkrealtime
ssasetmemorylimit
ssasetpoweruplength
ssasetseed
ssasetwindow

Examples

ssa_d_basic		Simple SSA analysis demo
ssa_d_forecast		Simple SSA forecasting demo
ssa_d_realtime		Real-time SSA algorithm with fast incremental updates

`ssamodel` class

/*************************************************************************
This object stores state of the SSA model.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class ssamodel
{
};

`ssaaddsequence` function

/*************************************************************************
This function adds data sequence to SSA  model.  Only   single-dimensional
sequences are supported.

What is a sequences? Following definitions/requirements apply:
* a sequence  is  an  array of  values  measured  in  subsequent,  equally
  separated time moments (ticks).
* you may have many sequences  in your  dataset;  say,  one  sequence  may
  correspond to one trading session.
* sequence length should be larger  than current  window  length  (shorter
  sequences will be ignored during analysis).
* analysis is performed within a  sequence; different  sequences  are  NOT
  stacked together to produce one large contiguous stream of data.
* analysis is performed for all  sequences at once, i.e. same set of basis
  vectors is computed for all sequences

INCREMENTAL ANALYSIS

This function is non intended for  incremental updates of previously found
SSA basis. Calling it invalidates  all previous analysis results (basis is
reset and will be recalculated from zero during next analysis).

If  you  want  to  perform   incremental/real-time  SSA,  consider   using
following functions:
* ssaappendpointandupdate() for appending one point
* ssaappendsequenceandupdate() for appending new sequence

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()
    X               -   array[N], data, can be larger (additional values
                        are ignored)
    N               -   data length, can be automatically determined from
                        the array length. N>=0.

OUTPUT PARAMETERS:
    S               -   SSA model, updated

NOTE: you can clear dataset with ssacleardata()

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaaddsequence(
    ssamodel s,
    real_1d_array x,
    const xparams _params = alglib::xdefault);
void alglib::ssaaddsequence(
    ssamodel s,
    real_1d_array x,
    ae_int_t n,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`ssaanalyzelast` function

/*************************************************************************
This function:
* builds SSA basis using internally stored (entire) dataset
* returns reconstruction for the last NTicks of the last sequence

If you want to analyze some other sequence, use ssaanalyzesequence().

Reconstruction phase involves  generation  of  NTicks-WindowWidth  sliding
windows, their decomposition using empirical orthogonal functions found by
SSA, followed by averaging of each data point across  several  overlapping
windows. Thus, every point in the output trend is reconstructed  using  up
to WindowWidth overlapping  windows  (WindowWidth windows exactly  in  the
inner points, just one window at the extremal points).

IMPORTANT: due to averaging this function returns  different  results  for
           different values of NTicks. It is expected and not a bug.

           For example:
           * Trend[NTicks-1] is always same because it is not averaged  in
             any case (same applies to Trend[0]).
           * Trend[NTicks-2] has different values  for  NTicks=WindowWidth
             and NTicks=WindowWidth+1 because former  case  means that  no
             averaging is performed, and latter  case means that averaging
             using two sliding windows  is  performed.  Larger  values  of
             NTicks produce same results as NTicks=WindowWidth+1.
           * ...and so on...

PERFORMANCE: this  function has O((NTicks-WindowWidth)*WindowWidth*NBasis)
             running time. If you work  in  time-constrained  setting  and
             have to analyze just a few last ticks, choosing NTicks  equal
             to WindowWidth+SmoothingLen, with SmoothingLen=1...WindowWidth
             will result in good compromise between noise cancellation and
             analysis speed.

INPUT PARAMETERS:
    S               -   SSA model
    NTicks          -   number of ticks to analyze, Nticks>=1.
                        * special case of NTicks<=WindowWidth  is  handled
                          by analyzing last window and  returning   NTicks
                          last ticks.
                        * special case NTicks>LastSequenceLen  is  handled
                          by prepending result with NTicks-LastSequenceLen
                          zeros.

OUTPUT PARAMETERS:
    Trend           -   array[NTicks], reconstructed trend line
    Noise           -   array[NTicks], the rest of the signal;
                        it holds that ActualData = Trend+Noise.


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.

In  any  case,  only  basis  is  reused. Reconstruction is performed  from
scratch every time you call this function.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* last sequence is shorter than the window length (analysis  can  be done,
  but we can not perform reconstruction on the last sequence)

Calling this function in degenerate cases returns following result:
* in any case, NTicks ticks is returned
* trend is assumed to be zero
* noise is initialized by the last sequence; if last sequence  is  shorter
  than the window size, it is moved to  the  end  of  the  array, and  the
  beginning of the noise array is filled by zeros

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaanalyzelast(
    ssamodel s,
    ae_int_t nticks,
    real_1d_array& trend,
    real_1d_array& noise,
    const xparams _params = alglib::xdefault);

`ssaanalyzelastwindow` function

/*************************************************************************
This  function  executes  SSA  on  internally  stored  dataset and returns
analysis  for  the  last  window  of  the  last sequence. Such analysis is
an lightweight alternative for full scale reconstruction (see below).

Typical use case for this function is  real-time  setting,  when  you  are
interested in quick-and-dirty (very quick and very  dirty)  processing  of
just a few last ticks of the trend.

IMPORTANT: full  scale  SSA  involves  analysis  of  the  ENTIRE  dataset,
           with reconstruction being done for  all  positions  of  sliding
           window with subsequent hankelization  (diagonal  averaging)  of
           the resulting matrix.

           Such analysis requires O((DataLen-Window)*Window*NBasis)  FLOPs
           and can be quite costly. However, it has  nice  noise-canceling
           effects due to averaging.

           This function performs REDUCED analysis of the last window.  It
           is much faster - just O(Window*NBasis),  but  its  results  are
           DIFFERENT from that of ssaanalyzelast(). In  particular,  first
           few points of the trend are much more prone to noise.

INPUT PARAMETERS:
    S               -   SSA model

OUTPUT PARAMETERS:
    Trend           -   array[WindowSize], reconstructed trend line
    Noise           -   array[WindowSize], the rest of the signal;
                        it holds that ActualData = Trend+Noise.
    NTicks          -   current WindowSize


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.

In  any  case,  only  basis  is  reused. Reconstruction is performed  from
scratch every time you call this function.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* last sequence is shorter than the window length (analysis can  be  done,
  but we can not perform reconstruction on the last sequence)

Calling this function in degenerate cases returns following result:
* in any case, WindowWidth ticks is returned
* trend is assumed to be zero
* noise is initialized by the last sequence; if last sequence  is  shorter
  than the window size, it is moved to  the  end  of  the  array, and  the
  beginning of the noise array is filled by zeros

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaanalyzelastwindow(
    ssamodel s,
    real_1d_array& trend,
    real_1d_array& noise,
    ae_int_t& nticks,
    const xparams _params = alglib::xdefault);

`ssaanalyzesequence` function

/*************************************************************************
This function:
* builds SSA basis using internally stored (entire) dataset
* returns reconstruction for the sequence being passed to this function

If  you  want  to  analyze  last  sequence  stored  in   the   model,  use
ssaanalyzelast().

Reconstruction phase involves  generation  of  NTicks-WindowWidth  sliding
windows, their decomposition using empirical orthogonal functions found by
SSA, followed by averaging of each data point across  several  overlapping
windows. Thus, every point in the output trend is reconstructed  using  up
to WindowWidth overlapping  windows  (WindowWidth windows exactly  in  the
inner points, just one window at the extremal points).

PERFORMANCE: this  function has O((NTicks-WindowWidth)*WindowWidth*NBasis)
             running time. If you work  in  time-constrained  setting  and
             have to analyze just a few last ticks, choosing NTicks  equal
             to WindowWidth+SmoothingLen, with SmoothingLen=1...WindowWidth
             will result in good compromise between noise cancellation and
             analysis speed.

INPUT PARAMETERS:
    S               -   SSA model
    Data            -   array[NTicks], can be larger (only NTicks  leading
                        elements will be used)
    NTicks          -   number of ticks to analyze, Nticks>=1.
                        * special case of NTicks<WindowWidth  is   handled
                          by returning zeros as trend, and signal as noise

OUTPUT PARAMETERS:
    Trend           -   array[NTicks], reconstructed trend line
    Noise           -   array[NTicks], the rest of the signal;
                        it holds that ActualData = Trend+Noise.


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.

In  any  case,  only  basis  is  reused. Reconstruction is performed  from
scratch every time you call this function.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* sequence being passed is shorter than the window length

Calling this function in degenerate cases returns following result:
* in any case, NTicks ticks is returned
* trend is assumed to be zero
* noise is initialized by the sequence.

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaanalyzesequence(
    ssamodel s,
    real_1d_array data,
    real_1d_array& trend,
    real_1d_array& noise,
    const xparams _params = alglib::xdefault);
void alglib::ssaanalyzesequence(
    ssamodel s,
    real_1d_array data,
    ae_int_t nticks,
    real_1d_array& trend,
    real_1d_array& noise,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`ssaappendpointandupdate` function

/*************************************************************************
This function appends single point to last data sequence stored in the SSA
model and tries to update model in the  incremental  manner  (if  possible
with current algorithm).

If you want to add more than one point at once:
* if you want to add M points to the same sequence, perform M-1 calls with
  UpdateIts parameter set to 0.0, and last call with non-zero UpdateIts.
* if you want to add new sequence, use ssaappendsequenceandupdate()

Running time of this function does NOT depend on  dataset  size,  only  on
window width and number of singular vectors. Depending on algorithm  being
used, incremental update has complexity:
* for top-K real time   - O(UpdateIts*K*Width^2), with fractional UpdateIts
* for top-K direct      - O(Width^3) for any non-zero UpdateIts
* for precomputed basis - O(1), no update is performed

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()
    X               -   new point
    UpdateIts       -   >=0,  floating  point (!)  value,  desired  update
                        frequency:
                        * zero value means that point is  stored,  but  no
                          update is performed
                        * integer part of the value means  that  specified
                          number of iterations is always performed
                        * fractional part of  the  value  means  that  one
                          iteration is performed with this probability.

                        Recommended value: 0<UpdateIts<=1.  Values  larger
                        than 1 are VERY seldom  needed.  If  your  dataset
                        changes slowly, you can set it  to  0.1  and  skip
                        90% of updates.

                        In any case, no information is lost even with zero
                        value of UpdateIts! It will be  incorporated  into
                        model, sooner or later.

OUTPUT PARAMETERS:
    S               -   SSA model, updated

NOTE: this function uses internal  RNG  to  handle  fractional  values  of
      UpdateIts. By default it  is  initialized  with  fixed  seed  during
      initial calculation of basis. Thus subsequent calls to this function
      will result in the same sequence of pseudorandom decisions.

      However, if  you  have  several  SSA  models  which  are  calculated
      simultaneously, and if you want to reduce computational  bottlenecks
      by performing random updates at random moments, then fixed  seed  is
      not an option - all updates will fire at same moments.

      You may change it with ssasetseed() function.

NOTE: this function throws an exception if called for empty dataset (there
      is no "last" sequence to modify).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaappendpointandupdate(
    ssamodel s,
    double x,
    double updateits,
    const xparams _params = alglib::xdefault);

`ssaappendsequenceandupdate` function

/*************************************************************************
This function appends new sequence to dataset stored in the SSA  model and
tries to update model in the incremental manner (if possible  with current
algorithm).

Notes:
* if you want to add M sequences at once, perform M-1 calls with UpdateIts
  parameter set to 0.0, and last call with non-zero UpdateIts.
* if you want to add just one point, use ssaappendpointandupdate()

Running time of this function does NOT depend on  dataset  size,  only  on
sequence length, window width and number of singular vectors. Depending on
algorithm being used, incremental update has complexity:
* for top-K real time   - O(UpdateIts*K*Width^2+(NTicks-Width)*Width^2)
* for top-K direct      - O(Width^3+(NTicks-Width)*Width^2)
* for precomputed basis - O(1), no update is performed

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()
    X               -   new sequence, array[NTicks] or larget
    NTicks          -   >=1, number of ticks in the sequence
    UpdateIts       -   >=0,  floating  point (!)  value,  desired  update
                        frequency:
                        * zero value means that point is  stored,  but  no
                          update is performed
                        * integer part of the value means  that  specified
                          number of iterations is always performed
                        * fractional part of  the  value  means  that  one
                          iteration is performed with this probability.

                        Recommended value: 0<UpdateIts<=1.  Values  larger
                        than 1 are VERY seldom  needed.  If  your  dataset
                        changes slowly, you can set it  to  0.1  and  skip
                        90% of updates.

                        In any case, no information is lost even with zero
                        value of UpdateIts! It will be  incorporated  into
                        model, sooner or later.

OUTPUT PARAMETERS:
    S               -   SSA model, updated

NOTE: this function uses internal  RNG  to  handle  fractional  values  of
      UpdateIts. By default it  is  initialized  with  fixed  seed  during
      initial calculation of basis. Thus subsequent calls to this function
      will result in the same sequence of pseudorandom decisions.

      However, if  you  have  several  SSA  models  which  are  calculated
      simultaneously, and if you want to reduce computational  bottlenecks
      by performing random updates at random moments, then fixed  seed  is
      not an option - all updates will fire at same moments.

      You may change it with ssasetseed() function.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaappendsequenceandupdate(
    ssamodel s,
    real_1d_array x,
    double updateits,
    const xparams _params = alglib::xdefault);
void alglib::ssaappendsequenceandupdate(
    ssamodel s,
    real_1d_array x,
    ae_int_t nticks,
    double updateits,
    const xparams _params = alglib::xdefault);

`ssacleardata` function

/*************************************************************************
This function clears all data stored in the  model  and  invalidates  all
basis components found so far.

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()

OUTPUT PARAMETERS:
    S               -   SSA model, updated

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssacleardata(
    ssamodel s,
    const xparams _params = alglib::xdefault);

`ssacreate` function

/*************************************************************************
This function creates SSA model object.  Right after creation model is  in
"dummy" mode - you can add data,  but   analyzing/prediction  will  return
just zeros (it assumes that basis is empty).

HOW TO USE SSA MODEL:

1. create model with ssacreate()
2. add data with one/many ssaaddsequence() calls
3. choose SSA algorithm with one of ssasetalgo...() functions:
   * ssasetalgotopkdirect() for direct one-run analysis
   * ssasetalgotopkrealtime() for algorithm optimized for many  subsequent
     runs with warm-start capabilities
   * ssasetalgoprecomputed() for user-supplied basis
4. set window width with ssasetwindow()
5. perform one of the analysis-related activities:
   a) call ssagetbasis() to get basis
   b) call ssaanalyzelast() ssaanalyzesequence() or ssaanalyzelastwindow()
      to perform analysis (trend/noise separation)
   c) call  one  of   the   forecasting   functions  (ssaforecastlast() or
      ssaforecastsequence()) to perform prediction; alternatively, you can
      extract linear recurrence coefficients with ssagetlrr().
   SSA analysis will be performed during first  call  to  analysis-related
   function. SSA model is smart enough to track all changes in the dataset
   and  model  settings,  to  cache  previously  computed  basis  and   to
   re-evaluate basis only when necessary.

Additionally, if your setting involves constant stream  of  incoming data,
you can perform quick update already calculated  model  with  one  of  the
incremental   append-and-update   functions:  ssaappendpointandupdate() or
ssaappendsequenceandupdate().

NOTE: steps (2), (3), (4) can be performed in arbitrary order.

INPUT PARAMETERS:
    none

OUTPUT PARAMETERS:
    S               -   structure which stores model state

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssacreate(
    ssamodel& s,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

`ssaforecastavglast` function

/*************************************************************************
This function builds SSA basis and performs forecasting  for  a  specified
number of ticks, returning value of trend.

Forecast is performed as follows:
* SSA  trend  extraction  is  applied to last  M  sliding windows  of  the
  internally stored dataset
* for each of M sliding windows, M predictions are built
* average value of M predictions is returned

This function has following running time:
* O(NBasis*WindowWidth*M) for trend extraction phase (always performed)
* O(WindowWidth*NTicks*M) for forecast phase

NOTE: noise reduction is ALWAYS applied by this algorithm; if you want  to
      apply recurrence relation  to  raw  unprocessed  data,  use  another
      function - ssaforecastsequence() which allows to  turn  on  and  off
      noise reduction phase.

NOTE: combination of several predictions results in lesser sensitivity  to
      noise, but it may produce undesirable discontinuities  between  last
      point of the trend and first point of the prediction. The reason  is
      that  last  point  of  the  trend is usually corrupted by noise, but
      average  value of  several  predictions  is less sensitive to noise,
      thus discontinuity appears. It is not a bug.

INPUT PARAMETERS:
    S               -   SSA model
    M               -   number  of  sliding  windows  to combine, M>=1. If
                        your dataset has less than M sliding windows, this
                        parameter will be silently reduced.
    NTicks          -   number of ticks to forecast, NTicks>=1

OUTPUT PARAMETERS:
    Trend           -   array[NTicks], predicted trend line


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* last sequence is shorter than the WindowWidth   (analysis  can  be done,
  but we can not perform forecasting on the last sequence)
* window lentgh is 1 (impossible to use for forecasting)
* SSA analysis algorithm is  configured  to  extract  basis  whose size is
  equal to window length (impossible to use for  forecasting;  only  basis
  whose size is less than window length can be used).

Calling this function in degenerate cases returns following result:
* NTicks  copies  of  the  last  value is returned for non-empty task with
  large enough dataset, but with overcomplete  basis  (window  width=1  or
  basis size is equal to window width)
* zero trend with length=NTicks is returned for empty task

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is ever constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaforecastavglast(
    ssamodel s,
    ae_int_t m,
    ae_int_t nticks,
    real_1d_array& trend,
    const xparams _params = alglib::xdefault);

`ssaforecastavgsequence` function

/*************************************************************************
This function builds SSA  basis  and  performs  forecasting  for  a  user-
specified sequence, returning value of trend.

Forecasting is done in two stages:
* first,  we  extract  trend  from M last sliding windows of the sequence.
  This stage is optional, you can  turn  it  off  if  you  pass data which
  are already processed with SSA. Of course, you  can  turn  it  off  even
  for raw data, but it is not recommended  -  noise  suppression  is  very
  important for correct prediction.
* then, we apply LRR independently for M sliding windows
* average of M predictions is returned

This function has following running time:
* O(NBasis*WindowWidth*M) for trend extraction phase
* O(WindowWidth*NTicks*M) for forecast phase

NOTE: combination of several predictions results in lesser sensitivity  to
      noise, but it may produce undesirable discontinuities  between  last
      point of the trend and first point of the prediction. The reason  is
      that  last  point  of  the  trend is usually corrupted by noise, but
      average  value of  several  predictions  is less sensitive to noise,
      thus discontinuity appears. It is not a bug.

INPUT PARAMETERS:
    S               -   SSA model
    Data            -   array[NTicks], data to forecast
    DataLen         -   number of ticks in the data, DataLen>=1
    M               -   number  of  sliding  windows  to combine, M>=1. If
                        your dataset has less than M sliding windows, this
                        parameter will be silently reduced.
    ForecastLen     -   number of ticks to predict, ForecastLen>=1
    ApplySmoothing  -   whether to apply smoothing trend extraction or not.
                        if you do not know what to specify, pass true.

OUTPUT PARAMETERS:
    Trend           -   array[ForecastLen], forecasted trend


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* data sequence is shorter than the WindowWidth   (analysis  can  be done,
  but we can not perform forecasting on the last sequence)
* window lentgh is 1 (impossible to use for forecasting)
* SSA analysis algorithm is  configured  to  extract  basis  whose size is
  equal to window length (impossible to use for  forecasting;  only  basis
  whose size is less than window length can be used).

Calling this function in degenerate cases returns following result:
* ForecastLen copies of the last value is returned for non-empty task with
  large enough dataset, but with overcomplete  basis  (window  width=1  or
  basis size is equal to window width)
* zero trend with length=ForecastLen is returned for empty task

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is ever constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaforecastavgsequence(
    ssamodel s,
    real_1d_array data,
    ae_int_t m,
    ae_int_t forecastlen,
    real_1d_array& trend,
    const xparams _params = alglib::xdefault);
void alglib::ssaforecastavgsequence(
    ssamodel s,
    real_1d_array data,
    ae_int_t datalen,
    ae_int_t m,
    ae_int_t forecastlen,
    bool applysmoothing,
    real_1d_array& trend,
    const xparams _params = alglib::xdefault);

`ssaforecastlast` function

/*************************************************************************
This function builds SSA basis and performs forecasting  for  a  specified
number of ticks, returning value of trend.

Forecast is performed as follows:
* SSA  trend  extraction  is  applied  to last WindowWidth elements of the
  internally stored dataset; this step is basically a noise reduction.
* linear recurrence relation is applied to extracted trend

This function has following running time:
* O(NBasis*WindowWidth) for trend extraction phase (always performed)
* O(WindowWidth*NTicks) for forecast phase

NOTE: noise reduction is ALWAYS applied by this algorithm; if you want  to
      apply recurrence relation  to  raw  unprocessed  data,  use  another
      function - ssaforecastsequence() which allows to  turn  on  and  off
      noise reduction phase.

NOTE: this algorithm performs prediction using only one - last  -  sliding
      window.  Predictions  produced   by   such   approach   are   smooth
      continuations of the reconstructed  trend  line,  but  they  can  be
      easily corrupted by noise. If you need  noise-resistant  prediction,
      use ssaforecastavglast() function, which averages predictions  built
      using several sliding windows.

INPUT PARAMETERS:
    S               -   SSA model
    NTicks          -   number of ticks to forecast, NTicks>=1

OUTPUT PARAMETERS:
    Trend           -   array[NTicks], predicted trend line


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* last sequence is shorter than the WindowWidth   (analysis  can  be done,
  but we can not perform forecasting on the last sequence)
* window lentgh is 1 (impossible to use for forecasting)
* SSA analysis algorithm is  configured  to  extract  basis  whose size is
  equal to window length (impossible to use for  forecasting;  only  basis
  whose size is less than window length can be used).

Calling this function in degenerate cases returns following result:
* NTicks  copies  of  the  last  value is returned for non-empty task with
  large enough dataset, but with overcomplete  basis  (window  width=1  or
  basis size is equal to window width)
* zero trend with length=NTicks is returned for empty task

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is ever constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaforecastlast(
    ssamodel s,
    ae_int_t nticks,
    real_1d_array& trend,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`ssaforecastsequence` function

/*************************************************************************
This function builds SSA  basis  and  performs  forecasting  for  a  user-
specified sequence, returning value of trend.

Forecasting is done in two stages:
* first,  we  extract  trend  from the WindowWidth  last  elements of  the
  sequence. This stage is optional, you  can  turn  it  off  if  you  pass
  data which are already processed with SSA. Of course, you  can  turn  it
  off even for raw data, but it is not recommended - noise suppression  is
  very important for correct prediction.
* then, we apply LRR for last  WindowWidth-1  elements  of  the  extracted
  trend.

This function has following running time:
* O(NBasis*WindowWidth) for trend extraction phase
* O(WindowWidth*NTicks) for forecast phase

NOTE: this algorithm performs prediction using only one - last  -  sliding
      window.  Predictions  produced   by   such   approach   are   smooth
      continuations of the reconstructed  trend  line,  but  they  can  be
      easily corrupted by noise. If you need  noise-resistant  prediction,
      use ssaforecastavgsequence() function,  which  averages  predictions
      built using several sliding windows.

INPUT PARAMETERS:
    S               -   SSA model
    Data            -   array[NTicks], data to forecast
    DataLen         -   number of ticks in the data, DataLen>=1
    ForecastLen     -   number of ticks to predict, ForecastLen>=1
    ApplySmoothing  -   whether to apply smoothing trend extraction or not;
                        if you do not know what to specify, pass True.

OUTPUT PARAMETERS:
    Trend           -   array[ForecastLen], forecasted trend


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* data sequence is shorter than the WindowWidth   (analysis  can  be done,
  but we can not perform forecasting on the last sequence)
* window lentgh is 1 (impossible to use for forecasting)
* SSA analysis algorithm is  configured  to  extract  basis  whose size is
  equal to window length (impossible to use for  forecasting;  only  basis
  whose size is less than window length can be used).

Calling this function in degenerate cases returns following result:
* ForecastLen copies of the last value is returned for non-empty task with
  large enough dataset, but with overcomplete  basis  (window  width=1  or
  basis size is equal to window width)
* zero trend with length=ForecastLen is returned for empty task

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is ever constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssaforecastsequence(
    ssamodel s,
    real_1d_array data,
    ae_int_t forecastlen,
    real_1d_array& trend,
    const xparams _params = alglib::xdefault);
void alglib::ssaforecastsequence(
    ssamodel s,
    real_1d_array data,
    ae_int_t datalen,
    ae_int_t forecastlen,
    bool applysmoothing,
    real_1d_array& trend,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`ssagetbasis` function

/*************************************************************************
This function executes SSA on internally stored dataset and returns  basis
found by current method.

INPUT PARAMETERS:
    S               -   SSA model

OUTPUT PARAMETERS:
    A               -   array[WindowWidth,NBasis],   basis;  vectors  are
                        stored in matrix columns, by descreasing variance
    SV              -   array[NBasis]:
                        * zeros - for model initialized with SSASetAlgoPrecomputed()
                        * singular values - for other algorithms
    WindowWidth     -   current window
    NBasis          -   basis size


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Calling  this  function  in  degenerate  cases  (no  data  or all data are
shorter than window size; no algorithm is specified)  returns  basis  with
just one zero vector.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssagetbasis(
    ssamodel s,
    real_2d_array& a,
    real_1d_array& sv,
    ae_int_t& windowwidth,
    ae_int_t& nbasis,
    const xparams _params = alglib::xdefault);

`ssagetlrr` function

/*************************************************************************
This function returns linear recurrence relation (LRR) coefficients  found
by current SSA algorithm.

INPUT PARAMETERS:
    S               -   SSA model

OUTPUT PARAMETERS:
    A               -   array[WindowWidth-1]. Coefficients  of  the
                        linear recurrence of the form:
                        X[W-1] = X[W-2]*A[W-2] + X[W-3]*A[W-3] + ... + X[0]*A[0].
                        Empty array for WindowWidth=1.
    WindowWidth     -   current window width


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Calling  this  function  in  degenerate  cases  (no  data  or all data are
shorter than window size; no algorithm is specified) returns zeros.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssagetlrr(
    ssamodel s,
    real_1d_array& a,
    ae_int_t& windowwidth,
    const xparams _params = alglib::xdefault);

`ssasetalgoprecomputed` function

/*************************************************************************
This  function sets SSA algorithm to "precomputed vectors" algorithm.

This  algorithm  uses  precomputed  set  of  orthonormal  (orthogonal  AND
normalized) basis vectors supplied by user. Thus, basis calculation  phase
is not performed -  we  already  have  our  basis  -  and  only  analysis/
forecasting phase requires actual calculations.

This algorithm may handle "append" requests which add just  one/few  ticks
to the end of the last sequence in O(1) time.

NOTE: this algorithm accepts both basis and window  width,  because  these
      two parameters are naturally aligned.  Calling  this  function  sets
      window width; if you call ssasetwindow() with  other  window  width,
      then during analysis stage algorithm will detect conflict and  reset
      to zero basis.

INPUT PARAMETERS:
    S               -   SSA model
    A               -   array[WindowWidth,NBasis], orthonormalized  basis;
                        this function does NOT control  orthogonality  and
                        does NOT perform any kind of  renormalization.  It
                        is your responsibility to provide it with  correct
                        basis.
    WindowWidth     -   window width, >=1
    NBasis          -   number of basis vectors, 1<=NBasis<=WindowWidth

OUTPUT PARAMETERS:
    S               -   updated model

NOTE: calling this function invalidates basis in all cases.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssasetalgoprecomputed(
    ssamodel s,
    real_2d_array a,
    const xparams _params = alglib::xdefault);
void alglib::ssasetalgoprecomputed(
    ssamodel s,
    real_2d_array a,
    ae_int_t windowwidth,
    ae_int_t nbasis,
    const xparams _params = alglib::xdefault);

`ssasetalgotopkdirect` function

/*************************************************************************
This  function sets SSA algorithm to "direct top-K" algorithm.

"Direct top-K" algorithm performs full  SVD  of  the  N*WINDOW  trajectory
matrix (hence its name - direct solver  is  used),  then  extracts  top  K
components. Overall running time is O(N*WINDOW^2), where N is a number  of
ticks in the dataset, WINDOW is window width.

This algorithm may handle "append" requests which add just  one/few  ticks
to the end of the last sequence in O(WINDOW^3) time,  which  is  ~N/WINDOW
times faster than re-computing everything from scratch.

INPUT PARAMETERS:
    S               -   SSA model
    TopK            -   number of components to analyze; TopK>=1.

OUTPUT PARAMETERS:
    S               -   updated model


NOTE: TopK>WindowWidth is silently decreased to WindowWidth during analysis
      phase

NOTE: calling this function invalidates basis, except  for  the  situation
      when this algorithm was already set with same parameters.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssasetalgotopkdirect(
    ssamodel s,
    ae_int_t topk,
    const xparams _params = alglib::xdefault);

Examples: [1] [2]

`ssasetalgotopkrealtime` function

/*************************************************************************
This function sets SSA algorithm to "top-K real time algorithm". This algo
extracts K components with largest singular values.

It  is  real-time  version  of  top-K  algorithm  which  is  optimized for
incremental processing and  fast  start-up. Internally  it  uses  subspace
eigensolver for truncated SVD. It results  in  ability  to  perform  quick
updates of the basis when only a few points/sequences is added to dataset.

Performance profile of the algorithm is given below:
* O(K*WindowWidth^2) running time for incremental update  of  the  dataset
  with one of the "append-and-update" functions (ssaappendpointandupdate()
  or ssaappendsequenceandupdate()).
* O(N*WindowWidth^2) running time for initial basis evaluation (N=size  of
  dataset)
* ability  to  split  costly  initialization  across  several  incremental
  updates of the basis (so called "Power-Up" functionality,  activated  by
  ssasetpoweruplength() function)

INPUT PARAMETERS:
    S               -   SSA model
    TopK            -   number of components to analyze; TopK>=1.

OUTPUT PARAMETERS:
    S               -   updated model

NOTE: this  algorithm  is  optimized  for  large-scale  tasks  with  large
      datasets. On toy problems with just  5-10 points it can return basis
      which is slightly different from that returned by  direct  algorithm
      (ssasetalgotopkdirect() function). However, the  difference  becomes
      negligible as dataset grows.

NOTE: TopK>WindowWidth is silently decreased to WindowWidth during analysis
      phase

NOTE: calling this function invalidates basis, except  for  the  situation
      when this algorithm was already set with same parameters.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssasetalgotopkrealtime(
    ssamodel s,
    ae_int_t topk,
    const xparams _params = alglib::xdefault);

Examples: [1]

`ssasetmemorylimit` function

/*************************************************************************
This function sets memory limit of SSA analysis.

Straightforward SSA with sequence length T and window width W needs O(T*W)
memory. It is possible to reduce memory consumption by splitting task into
smaller chunks.

Thus function allows you to specify approximate memory limit (measured  in
double precision numbers used for buffers). Actual memory consumption will
be comparable to the number specified by you.

Default memory limit is 50.000.000 (400Mbytes) in current version.

INPUT PARAMETERS:
    S       -   SSA model
    MemLimit-   memory limit, >=0. Zero value means no limit.

  -- ALGLIB --
     Copyright 20.12.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssasetmemorylimit(
    ssamodel s,
    ae_int_t memlimit,
    const xparams _params = alglib::xdefault);

`ssasetpoweruplength` function

/*************************************************************************
This function sets length of power-up cycle for real-time algorithm.

By default, this algorithm performs costly O(N*WindowWidth^2)  init  phase
followed by full run of truncated  EVD.  However,  if  you  are  ready  to
live with a bit lower-quality basis during first few iterations,  you  can
split this O(N*WindowWidth^2) initialization  between  several  subsequent
append-and-update rounds. It results in better latency of the algorithm.

This function invalidates basis/solver, next analysis call will result  in
full recalculation of everything.

INPUT PARAMETERS:
    S       -   SSA model
    PWLen   -   length of the power-up stage:
                * 0 means that no power-up is requested
                * 1 is the same as 0
                * >1 means that delayed power-up is performed

  -- ALGLIB --
     Copyright 03.11.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssasetpoweruplength(
    ssamodel s,
    ae_int_t pwlen,
    const xparams _params = alglib::xdefault);

Examples: [1]

`ssasetseed` function

/*************************************************************************
This  function  sets  seed  which  is used to initialize internal RNG when
we make pseudorandom decisions on model updates.

By default, deterministic seed is used - which results in same sequence of
pseudorandom decisions every time you run SSA model. If you  specify  non-
deterministic seed value, then SSA  model  may  return  slightly different
results after each run.

This function can be useful when you have several SSA models updated  with
sseappendpointandupdate() called with 0<UpdateIts<1 (fractional value) and
due to performance limitations want them to perform updates  at  different
moments.

INPUT PARAMETERS:
    S       -   SSA model
    Seed    -   seed:
                * positive values = use deterministic seed for each run of
                  algorithms which depend on random initialization
                * zero or negative values = use non-deterministic seed

  -- ALGLIB --
     Copyright 03.11.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssasetseed(
    ssamodel s,
    ae_int_t seed,
    const xparams _params = alglib::xdefault);

`ssasetwindow` function

/*************************************************************************
This function sets window width for SSA model. You should call  it  before
analysis phase. Default window width is 1 (not for real use).

Special notes:
* this function call can be performed at any moment before  first call  to
  analysis-related functions
* changing window width invalidates internally stored basis; if you change
  window width AFTER you call analysis-related  function,  next  analysis
  phase will require re-calculation of  the  basis  according  to  current
  algorithm.
* calling this function with exactly  same window width as current one has
  no effect
* if you specify window width larger  than any data sequence stored in the
  model, analysis will return zero basis.

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()
    WindowWidth     -   >=1, new window width

OUTPUT PARAMETERS:
    S               -   SSA model, updated

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void alglib::ssasetwindow(
    ssamodel s,
    ae_int_t windowwidth,
    const xparams _params = alglib::xdefault);

Examples: [1] [2] [3]

ssa_d_basic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Here we demonstrate SSA trend/noise separation for some toy problem:
    // small monotonically growing series X are analyzed with 3-tick window
    // and "top-K" version of SSA, which selects K largest singular vectors
    // for analysis, with K=1.
    //
    ssamodel s;
    real_1d_array x = "[0,0.5,1,1,1.5,2]";

    //
    // First, we create SSA model, set its properties and add dataset.
    //
    // We use window with width=3 and configure model to use direct SSA
    // algorithm - one which runs exact O(N*W^2) analysis - to extract
    // one top singular vector. Well, it is toy problem :)
    //
    // NOTE: SSA model may store and analyze more than one sequence
    //       (say, different sequences may correspond to data collected
    //       from different devices)
    //
    ssacreate(s);
    ssasetwindow(s, 3);
    ssaaddsequence(s, x);
    ssasetalgotopkdirect(s, 1);

    //
    // Now we begin analysis. Internally SSA model stores everything it needs:
    // data, settings, solvers and so on. Right after first call to analysis-
    // related function it will analyze dataset, build basis and perform analysis.
    //
    // Subsequent calls to analysis functions will reuse previously computed
    // basis, unless you invalidate it by changing model settings (or dataset).
    //
    real_1d_array trend;
    real_1d_array noise;
    ssaanalyzesequence(s, x, trend, noise);
    printf("%s\n", trend.tostring(2).c_str()); // EXPECTED: [0.3815,0.5582,0.7810,1.0794,1.5041,2.0105]
    return 0;
}

ssa_d_forecast example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Here we demonstrate SSA forecasting on some toy problem with clearly
    // visible linear trend and small amount of noise.
    //
    ssamodel s;
    real_1d_array x = "[0.05,0.96,2.04,3.11,3.97,5.03,5.98,7.02,8.02]";

    //
    // First, we create SSA model, set its properties and add dataset.
    //
    // We use window with width=3 and configure model to use direct SSA
    // algorithm - one which runs exact O(N*W^2) analysis - to extract
    // two top singular vectors. Well, it is toy problem :)
    //
    // NOTE: SSA model may store and analyze more than one sequence
    //       (say, different sequences may correspond to data collected
    //       from different devices)
    //
    ssacreate(s);
    ssasetwindow(s, 3);
    ssaaddsequence(s, x);
    ssasetalgotopkdirect(s, 2);

    //
    // Now we begin analysis. Internally SSA model stores everything it needs:
    // data, settings, solvers and so on. Right after first call to analysis-
    // related function it will analyze dataset, build basis and perform analysis.
    //
    // Subsequent calls to analysis functions will reuse previously computed
    // basis, unless you invalidate it by changing model settings (or dataset).
    //
    // In this example we show how to use ssaforecastlast() function, which
    // predicts changed in the last sequence of the dataset. If you want to
    // perform prediction for some other sequence, use ssaforecastsequence().
    //
    real_1d_array trend;
    ssaforecastlast(s, 3, trend);

    //
    // Well, we expected it to be [9,10,11]. There exists some difference,
    // which can be explained by the artificial noise in the dataset.
    //
    printf("%s\n", trend.tostring(2).c_str()); // EXPECTED: [9.0005,9.9322,10.8051]
    return 0;
}

ssa_d_realtime example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;


int main(int argc, char **argv)
{
    //
    // Suppose that you have a constant stream of incoming data, and you want
    // to regularly perform singular spectral analysis of this stream.
    //
    // One full run of direct algorithm costs O(N*Width^2) operations, so
    // the more points you have, the more it costs to rebuild basis from
    // scratch.
    // 
    // Luckily we have incremental SSA algorithm which can perform quick
    // updates of already computed basis in O(K*Width^2) ops, where K
    // is a number of singular vectors extracted. Usually it is orders of
    // magnitude faster than full update of the basis.
    //
    // In this example we start from some initial dataset x0. Then we
    // start appending elements one by one to the end of the last sequence.
    //
    // NOTE: direct algorithm also supports incremental updates, but
    //       with O(Width^3) cost. Typically K<<Width, so specialized
    //       incremental algorithm is still faster.
    //
    ssamodel s1;
    real_2d_array a1;
    real_1d_array sv1;
    ae_int_t w;
    ae_int_t k;
    real_1d_array x0 = "[0.009,0.976,1.999,2.984,3.977,5.002]";
    ssacreate(s1);
    ssasetwindow(s1, 3);
    ssaaddsequence(s1, x0);

    // set algorithm to the real-time version of top-K, K=2
    ssasetalgotopkrealtime(s1, 2);

    // one more interesting feature of the incremental algorithm is "power-up" cycle.
    // even with incremental algorithm initial basis calculation costs O(N*Width^2) ops.
    // if such startup cost is too high for your real-time app, then you may divide
    // initial basis calculation across several model updates. It results in better
    // latency at the price of somewhat lesser precision during first few updates.
    ssasetpoweruplength(s1, 3);

    // now, after we prepared everything, start to add incoming points one by one;
    // in the real life, of course, we will perform some work between subsequent update
    // (analyze something, predict, and so on).
    //
    // After each append we perform one iteration of the real-time solver. Usually
    // one iteration is more than enough to update basis. If you have REALLY tight
    // performance constraints, you may specify fractional amount of iterations,
    // which means that iteration is performed with required probability.
    double updateits = 1.0;
    ssaappendpointandupdate(s1, 5.951, updateits);
    ssagetbasis(s1, a1, sv1, w, k);

    ssaappendpointandupdate(s1, 7.074, updateits);
    ssagetbasis(s1, a1, sv1, w, k);

    ssaappendpointandupdate(s1, 7.925, updateits);
    ssagetbasis(s1, a1, sv1, w, k);

    ssaappendpointandupdate(s1, 8.992, updateits);
    ssagetbasis(s1, a1, sv1, w, k);

    ssaappendpointandupdate(s1, 9.942, updateits);
    ssagetbasis(s1, a1, sv1, w, k);

    ssaappendpointandupdate(s1, 11.051, updateits);
    ssagetbasis(s1, a1, sv1, w, k);

    ssaappendpointandupdate(s1, 11.965, updateits);
    ssagetbasis(s1, a1, sv1, w, k);

    ssaappendpointandupdate(s1, 13.047, updateits);
    ssagetbasis(s1, a1, sv1, w, k);

    ssaappendpointandupdate(s1, 13.970, updateits);
    ssagetbasis(s1, a1, sv1, w, k);

    // Ok, we have our basis in a1[] and singular values at sv1[].
    // But is it good enough? Let's print it.
    printf("%s\n", a1.tostring(3).c_str()); // EXPECTED: [[0.510607,0.753611],[0.575201,0.058445],[0.639081,-0.654717]]

    // Ok, two vectors with 3 components each.
    // But how to understand that is it really good basis?
    // Let's compare it with direct SSA algorithm on the entire sequence.
    ssamodel s2;
    real_2d_array a2;
    real_1d_array sv2;
    real_1d_array x2 = "[0.009,0.976,1.999,2.984,3.977,5.002,5.951,7.074,7.925,8.992,9.942,11.051,11.965,13.047,13.970]";
    ssacreate(s2);
    ssasetwindow(s2, 3);
    ssaaddsequence(s2, x2);
    ssasetalgotopkdirect(s2, 2);
    ssagetbasis(s2, a2, sv2, w, k);

    // it is exactly the same as one calculated with incremental approach!
    printf("%s\n", a2.tostring(3).c_str()); // EXPECTED: [[0.510607,0.753611],[0.575201,0.058445],[0.639081,-0.654717]]
    return 0;
}

`stest` subpackage

Functions

onesamplesigntest

Examples

`onesamplesigntest` function

/*************************************************************************
Sign test

This test checks three hypotheses about the median of  the  given  sample.
The following tests are performed:
    * two-tailed test (null hypothesis - the median is equal to the  given
      value)
    * left-tailed test (null hypothesis - the median is  greater  than  or
      equal to the given value)
    * right-tailed test (null hypothesis - the  median  is  less  than  or
      equal to the given value)

Requirements:
    * the scale of measurement should be ordinal, interval or ratio  (i.e.
      the test could not be applied to nominal variables).

The test is non-parametric and doesn't require distribution X to be normal

Input parameters:
    X       -   sample. Array whose index goes from 0 to N-1.
    N       -   size of the sample.
    Median  -   assumed median value.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

While   calculating   p-values   high-precision   binomial    distribution
approximation is used, so significance levels have about 15 exact digits.

  -- ALGLIB --
     Copyright 08.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::onesamplesigntest(
    real_1d_array x,
    ae_int_t n,
    double median,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`studenttdistr` subpackage

Functions

invstudenttdistribution
studenttdistribution

Examples

`invstudenttdistribution` function

/*************************************************************************
Functional inverse of Student's t distribution

Given probability p, finds the argument t such that stdtr(k,t)
is equal to p.

ACCURACY:

Tested at random 1 <= k <= 100.  The "domain" refers to p:
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE    .001,.999     25000       5.7e-15     8.0e-16
   IEEE    10^-6,.001    25000       2.0e-12     2.9e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::invstudenttdistribution(
    ae_int_t k,
    double p,
    const xparams _params = alglib::xdefault);

`studenttdistribution` function

/*************************************************************************
Student's t distribution

Computes the integral from minus infinity to t of the Student
t distribution with integer k > 0 degrees of freedom:

                                     t
                                     -
                                    | |
             -                      |         2   -(k+1)/2
            | ( (k+1)/2 )           |  (     x   )
      ----------------------        |  ( 1 + --- )        dx
                    -               |  (      k  )
      sqrt( k pi ) | ( k/2 )        |
                                  | |
                                   -
                                  -inf.

Relation to incomplete beta integral:

       1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
where
       z = k/(k + t**2).

For t < -2, this is the method of computation.  For higher t,
a direct method is derived from integration by parts.
Since the function is symmetric about t=0, the area under the
right tail of the density is found by calling the function
with -t instead of t.

ACCURACY:

Tested at random 1 <= k <= 25.  The "domain" refers to t.
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -100,-2      50000       5.9e-15     1.4e-15
   IEEE     -2,100      500000       2.7e-15     4.9e-17

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double alglib::studenttdistribution(
    ae_int_t k,
    double t,
    const xparams _params = alglib::xdefault);

`studentttests` subpackage

Functions

studentttest1
studentttest2
unequalvariancettest

Examples

`studentttest1` function

/*************************************************************************
One-sample t-test

This test checks three hypotheses about the mean of the given sample.  The
following tests are performed:
    * two-tailed test (null hypothesis - the mean is equal  to  the  given
      value)
    * left-tailed test (null hypothesis - the  mean  is  greater  than  or
      equal to the given value)
    * right-tailed test (null hypothesis - the mean is less than or  equal
      to the given value).

The test is based on the assumption that  a  given  sample  has  a  normal
distribution and  an  unknown  dispersion.  If  the  distribution  sharply
differs from normal, the test will work incorrectly.

INPUT PARAMETERS:
    X       -   sample. Array whose index goes from 0 to N-1.
    N       -   size of sample, N>=0
    Mean    -   assumed value of the mean.

OUTPUT PARAMETERS:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

NOTE: this function correctly handles degenerate cases:
      * when N=0, all p-values are set to 1.0
      * when variance of X[] is exactly zero, p-values are set
        to 1.0 or 0.0, depending on difference between sample mean and
        value of mean being tested.


  -- ALGLIB --
     Copyright 08.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::studentttest1(
    real_1d_array x,
    ae_int_t n,
    double mean,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`studentttest2` function

/*************************************************************************
Two-sample pooled test

This test checks three hypotheses about the mean of the given samples. The
following tests are performed:
    * two-tailed test (null hypothesis - the means are equal)
    * left-tailed test (null hypothesis - the mean of the first sample  is
      greater than or equal to the mean of the second sample)
    * right-tailed test (null hypothesis - the mean of the first sample is
      less than or equal to the mean of the second sample).

Test is based on the following assumptions:
    * given samples have normal distributions
    * dispersions are equal
    * samples are independent.

Input parameters:
    X       -   sample 1. Array whose index goes from 0 to N-1.
    N       -   size of sample.
    Y       -   sample 2. Array whose index goes from 0 to M-1.
    M       -   size of sample.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

NOTE: this function correctly handles degenerate cases:
      * when N=0 or M=0, all p-values are set to 1.0
      * when both samples has exactly zero variance, p-values are set
        to 1.0 or 0.0, depending on difference between means.

  -- ALGLIB --
     Copyright 18.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::studentttest2(
    real_1d_array x,
    ae_int_t n,
    real_1d_array y,
    ae_int_t m,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`unequalvariancettest` function

/*************************************************************************
Two-sample unpooled test

This test checks three hypotheses about the mean of the given samples. The
following tests are performed:
    * two-tailed test (null hypothesis - the means are equal)
    * left-tailed test (null hypothesis - the mean of the first sample  is
      greater than or equal to the mean of the second sample)
    * right-tailed test (null hypothesis - the mean of the first sample is
      less than or equal to the mean of the second sample).

Test is based on the following assumptions:
    * given samples have normal distributions
    * samples are independent.
Equality of variances is NOT required.

Input parameters:
    X - sample 1. Array whose index goes from 0 to N-1.
    N - size of the sample.
    Y - sample 2. Array whose index goes from 0 to M-1.
    M - size of the sample.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

NOTE: this function correctly handles degenerate cases:
      * when N=0 or M=0, all p-values are set to 1.0
      * when both samples has zero variance, p-values are set
        to 1.0 or 0.0, depending on difference between means.
      * when only one sample has zero variance, test reduces to 1-sample
        version.

  -- ALGLIB --
     Copyright 18.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::unequalvariancettest(
    real_1d_array x,
    ae_int_t n,
    real_1d_array y,
    ae_int_t m,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`svd` subpackage

Functions

rmatrixsvd

Examples

`rmatrixsvd` function

/*************************************************************************
Singular value decomposition of a rectangular matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

The algorithm calculates the singular value decomposition of a matrix of
size MxN: A = U * S * V^T

The algorithm finds the singular values and, optionally, matrices U and V^T.
The algorithm can find both first min(M,N) columns of matrix U and rows of
matrix V^T (singular vectors), and matrices U and V^T wholly (of sizes MxM
and NxN respectively).

Take into account that the subroutine does not return matrix V but V^T.

Input parameters:
    A           -   matrix to be decomposed.
                    Array whose indexes range within [0..M-1, 0..N-1].
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    UNeeded     -   0, 1 or 2. See the description of the parameter U.
    VTNeeded    -   0, 1 or 2. See the description of the parameter VT.
    AdditionalMemory -
                    If the parameter:
                     * equals 0, the algorithm doesn't use additional
                       memory (lower requirements, lower performance).
                     * equals 1, the algorithm uses additional
                       memory of size min(M,N)*min(M,N) of real numbers.
                       It often speeds up the algorithm.
                     * equals 2, the algorithm uses additional
                       memory of size M*min(M,N) of real numbers.
                       It allows to get a maximum performance.
                    The recommended value of the parameter is 2.

Output parameters:
    W           -   contains singular values in descending order.
    U           -   if UNeeded=0, U isn't changed, the left singular vectors
                    are not calculated.
                    if Uneeded=1, U contains left singular vectors (first
                    min(M,N) columns of matrix U). Array whose indexes range
                    within [0..M-1, 0..Min(M,N)-1].
                    if UNeeded=2, U contains matrix U wholly. Array whose
                    indexes range within [0..M-1, 0..M-1].
    VT          -   if VTNeeded=0, VT isn't changed, the right singular vectors
                    are not calculated.
                    if VTNeeded=1, VT contains right singular vectors (first
                    min(M,N) rows of matrix V^T). Array whose indexes range
                    within [0..min(M,N)-1, 0..N-1].
                    if VTNeeded=2, VT contains matrix V^T wholly. Array whose
                    indexes range within [0..N-1, 0..N-1].

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
bool alglib::rmatrixsvd(
    real_2d_array a,
    ae_int_t m,
    ae_int_t n,
    ae_int_t uneeded,
    ae_int_t vtneeded,
    ae_int_t additionalmemory,
    real_1d_array& w,
    real_2d_array& u,
    real_2d_array& vt,
    const xparams _params = alglib::xdefault);

`trfac` subpackage

Functions

cmatrixlu
hpdmatrixcholesky
rmatrixlu
sparsecholeskyskyline
spdmatrixcholesky
spdmatrixcholeskyupdateadd1
spdmatrixcholeskyupdateadd1buf
spdmatrixcholeskyupdatefix
spdmatrixcholeskyupdatefixbuf

Examples

`cmatrixlu` function

/*************************************************************************
LU decomposition of a general complex matrix with row pivoting

A is represented as A = P*L*U, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=min(M,N)-1,
  Pi - permutation matrix for I and Pivots[I]

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    A       -   array[0..M-1, 0..N-1].
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.


OUTPUT PARAMETERS:
    A       -   matrices L and U in compact form:
                * L is stored under main diagonal
                * U is stored on and above main diagonal
    Pivots  -   permutation matrix in compact form.
                array[0..Min(M-1,N-1)].

  -- ALGLIB routine --
     10.01.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::cmatrixlu(
    complex_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    integer_1d_array& pivots,
    const xparams _params = alglib::xdefault);

`hpdmatrixcholesky` function

/*************************************************************************
Cache-oblivious Cholesky decomposition

The algorithm computes Cholesky decomposition  of  a  Hermitian  positive-
definite matrix. The result of an algorithm is a representation  of  A  as
A=U'*U  or A=L*L' (here X' denotes conj(X^T)).

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    A       -   upper or lower triangle of a factorized matrix.
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   if IsUpper=True, then A contains an upper triangle of
                a symmetric matrix, otherwise A contains a lower one.

OUTPUT PARAMETERS:
    A       -   the result of factorization. If IsUpper=True, then
                the upper triangle contains matrix U, so that A = U'*U,
                and the elements below the main diagonal are not modified.
                Similarly, if IsUpper = False.

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False. Contents of A is not determined
    in such case.

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
bool alglib::hpdmatrixcholesky(
    complex_2d_array& a,
    ae_int_t n,
    bool isupper,
    const xparams _params = alglib::xdefault);

`rmatrixlu` function

/*************************************************************************
LU decomposition of a general real matrix with row pivoting

A is represented as A = P*L*U, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=min(M,N)-1,
  Pi - permutation matrix for I and Pivots[I]

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    A       -   array[0..M-1, 0..N-1].
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.


OUTPUT PARAMETERS:
    A       -   matrices L and U in compact form:
                * L is stored under main diagonal
                * U is stored on and above main diagonal
    Pivots  -   permutation matrix in compact form.
                array[0..Min(M-1,N-1)].

  -- ALGLIB routine --
     10.01.2010
     Bochkanov Sergey
*************************************************************************/
void alglib::rmatrixlu(
    real_2d_array& a,
    ae_int_t m,
    ae_int_t n,
    integer_1d_array& pivots,
    const xparams _params = alglib::xdefault);

`sparsecholeskyskyline` function

/*************************************************************************
Sparse Cholesky decomposition for skyline matrixm using in-place algorithm
without allocating additional storage.

The algorithm computes Cholesky decomposition  of  a  symmetric  positive-
definite sparse matrix. The result of an algorithm is a representation  of
A as A=U^T*U or A=L*L^T

This  function  is  a  more  efficient alternative to general, but  slower
SparseCholeskyX(), because it does not  create  temporary  copies  of  the
target. It performs factorization in-place, which gives  best  performance
on low-profile matrices. Its drawback, however, is that it can not perform
profile-reducing permutation of input matrix.

INPUT PARAMETERS:
    A       -   sparse matrix in skyline storage (SKS) format.
    N       -   size of matrix A (can be smaller than actual size of A)
    IsUpper -   if IsUpper=True, then factorization is performed on  upper
                triangle. Another triangle is ignored (it may contant some
                data, but it is not changed).


OUTPUT PARAMETERS:
    A       -   the result of factorization, stored in SKS. If IsUpper=True,
                then the upper  triangle  contains  matrix  U,  such  that
                A = U^T*U. Lower triangle is not changed.
                Similarly, if IsUpper = False. In this case L is returned,
                and we have A = L*(L^T).
                Note that THIS function does not  perform  permutation  of
                rows to reduce bandwidth.

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False. Contents of A is not determined
    in such case.

NOTE: for  performance  reasons  this  function  does NOT check that input
      matrix  includes  only  finite  values. It is your responsibility to
      make sure that there are no infinite or NAN values in the matrix.

  -- ALGLIB routine --
     16.01.2014
     Bochkanov Sergey
*************************************************************************/
bool alglib::sparsecholeskyskyline(
    sparsematrix a,
    ae_int_t n,
    bool isupper,
    const xparams _params = alglib::xdefault);

`spdmatrixcholesky` function

/*************************************************************************
Cache-oblivious Cholesky decomposition

The algorithm computes Cholesky decomposition  of  a  symmetric  positive-
definite matrix. The result of an algorithm is a representation  of  A  as
A=U^T*U  or A=L*L^T

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    A       -   upper or lower triangle of a factorized matrix.
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   if IsUpper=True, then A contains an upper triangle of
                a symmetric matrix, otherwise A contains a lower one.

OUTPUT PARAMETERS:
    A       -   the result of factorization. If IsUpper=True, then
                the upper triangle contains matrix U, so that A = U^T*U,
                and the elements below the main diagonal are not modified.
                Similarly, if IsUpper = False.

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False. Contents of A is not determined
    in such case.

  -- ALGLIB routine --
     15.12.2009
     Bochkanov Sergey
*************************************************************************/
bool alglib::spdmatrixcholesky(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskyupdateadd1` function

/*************************************************************************
Update of Cholesky decomposition: rank-1 update to original A.  "Buffered"
version which uses preallocated buffer which is saved  between  subsequent
function calls.

This function uses internally allocated buffer which is not saved  between
subsequent  calls.  So,  if  you  perform  a lot  of  subsequent  updates,
we  recommend   you   to   use   "buffered"   version   of  this function:
SPDMatrixCholeskyUpdateAdd1Buf().

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    U       -   array[N], rank-1 update to A: A_mod = A + u*u'
                Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

NOTE: this function always succeeds, so it does not return completion code

NOTE: this function checks sizes of input arrays, but it does  NOT  checks
      for presence of infinities or NAN's.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void alglib::spdmatrixcholeskyupdateadd1(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    real_1d_array u,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskyupdateadd1buf` function

/*************************************************************************
Update of Cholesky decomposition: rank-1 update to original A.  "Buffered"
version which uses preallocated buffer which is saved  between  subsequent
function calls.

See comments for SPDMatrixCholeskyUpdateAdd1() for more information.

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    U       -   array[N], rank-1 update to A: A_mod = A + u*u'
                Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void alglib::spdmatrixcholeskyupdateadd1buf(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    real_1d_array u,
    real_1d_array& bufr,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskyupdatefix` function

/*************************************************************************
Update of Cholesky decomposition: "fixing" some variables.

This function uses internally allocated buffer which is not saved  between
subsequent  calls.  So,  if  you  perform  a lot  of  subsequent  updates,
we  recommend   you   to   use   "buffered"   version   of  this function:
SPDMatrixCholeskyUpdateFixBuf().

"FIXING" EXPLAINED:

    Suppose we have N*N positive definite matrix A. "Fixing" some variable
    means filling corresponding row/column of  A  by  zeros,  and  setting
    diagonal element to 1.

    For example, if we fix 2nd variable in 4*4 matrix A, it becomes Af:

        ( A00  A01  A02  A03 )      ( Af00  0   Af02 Af03 )
        ( A10  A11  A12  A13 )      (  0    1    0    0   )
        ( A20  A21  A22  A23 )  =>  ( Af20  0   Af22 Af23 )
        ( A30  A31  A32  A33 )      ( Af30  0   Af32 Af33 )

    If we have Cholesky decomposition of A, it must be recalculated  after
    variables were  fixed.  However,  it  is  possible  to  use  efficient
    algorithm, which needs O(K*N^2)  time  to  "fix"  K  variables,  given
    Cholesky decomposition of original, "unfixed" A.

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    Fix     -   array[N], I-th element is True if I-th  variable  must  be
                fixed. Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

NOTE: this function always succeeds, so it does not return completion code

NOTE: this function checks sizes of input arrays, but it does  NOT  checks
      for presence of infinities or NAN's.

NOTE: this  function  is  efficient  only  for  moderate amount of updated
      variables - say, 0.1*N or 0.3*N. For larger amount of  variables  it
      will  still  work,  but  you  may  get   better   performance   with
      straightforward Cholesky.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void alglib::spdmatrixcholeskyupdatefix(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    boolean_1d_array fix,
    const xparams _params = alglib::xdefault);

`spdmatrixcholeskyupdatefixbuf` function

/*************************************************************************
Update of Cholesky  decomposition:  "fixing"  some  variables.  "Buffered"
version which uses preallocated buffer which is saved  between  subsequent
function calls.

See comments for SPDMatrixCholeskyUpdateFix() for more information.

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    Fix     -   array[N], I-th element is True if I-th  variable  must  be
                fixed. Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void alglib::spdmatrixcholeskyupdatefixbuf(
    real_2d_array& a,
    ae_int_t n,
    bool isupper,
    boolean_1d_array fix,
    real_1d_array& bufr,
    const xparams _params = alglib::xdefault);

`trigintegrals` subpackage

Functions

hyperbolicsinecosineintegrals
sinecosineintegrals

Examples

`hyperbolicsinecosineintegrals` function

/*************************************************************************
Hyperbolic sine and cosine integrals

Approximates the integrals

                           x
                           -
                          | |   cosh t - 1
  Chi(x) = eul + ln x +   |    -----------  dt,
                        | |          t
                         -
                         0

              x
              -
             | |  sinh t
  Shi(x) =   |    ------  dt
           | |       t
            -
            0

where eul = 0.57721566490153286061 is Euler's constant.
The integrals are evaluated by power series for x < 8
and by Chebyshev expansions for x between 8 and 88.
For large x, both functions approach exp(x)/2x.
Arguments greater than 88 in magnitude return MAXNUM.


ACCURACY:

Test interval 0 to 88.
                     Relative error:
arithmetic   function  # trials      peak         rms
   IEEE         Shi      30000       6.9e-16     1.6e-16
       Absolute error, except relative when |Chi| > 1:
   IEEE         Chi      30000       8.4e-16     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void alglib::hyperbolicsinecosineintegrals(
    double x,
    double& shi,
    double& chi,
    const xparams _params = alglib::xdefault);

`sinecosineintegrals` function

/*************************************************************************
Sine and cosine integrals

Evaluates the integrals

                         x
                         -
                        |  cos t - 1
  Ci(x) = eul + ln x +  |  --------- dt,
                        |      t
                       -
                        0
            x
            -
           |  sin t
  Si(x) =  |  ----- dt
           |    t
          -
           0

where eul = 0.57721566490153286061 is Euler's constant.
The integrals are approximated by rational functions.
For x > 8 auxiliary functions f(x) and g(x) are employed
such that

Ci(x) = f(x) sin(x) - g(x) cos(x)
Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)


ACCURACY:
   Test interval = [0,50].
Absolute error, except relative when > 1:
arithmetic   function   # trials      peak         rms
   IEEE        Si        30000       4.4e-16     7.3e-17
   IEEE        Ci        30000       6.9e-16     5.1e-17

Cephes Math Library Release 2.1:  January, 1989
Copyright 1984, 1987, 1989 by Stephen L. Moshier
*************************************************************************/
void alglib::sinecosineintegrals(
    double x,
    double& si,
    double& ci,
    const xparams _params = alglib::xdefault);

`variancetests` subpackage

Functions

ftest
onesamplevariancetest

Examples

`ftest` function

/*************************************************************************
Two-sample F-test

This test checks three hypotheses about dispersions of the given  samples.
The following tests are performed:
    * two-tailed test (null hypothesis - the dispersions are equal)
    * left-tailed test (null hypothesis  -  the  dispersion  of  the first
      sample is greater than or equal to  the  dispersion  of  the  second
      sample).
    * right-tailed test (null hypothesis - the  dispersion  of  the  first
      sample is less than or equal to the dispersion of the second sample)

The test is based on the following assumptions:
    * the given samples have normal distributions
    * the samples are independent.

Input parameters:
    X   -   sample 1. Array whose index goes from 0 to N-1.
    N   -   sample size.
    Y   -   sample 2. Array whose index goes from 0 to M-1.
    M   -   sample size.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

  -- ALGLIB --
     Copyright 19.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::ftest(
    real_1d_array x,
    ae_int_t n,
    real_1d_array y,
    ae_int_t m,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`onesamplevariancetest` function

/*************************************************************************
One-sample chi-square test

This test checks three hypotheses about the dispersion of the given sample
The following tests are performed:
    * two-tailed test (null hypothesis - the dispersion equals  the  given
      number)
    * left-tailed test (null hypothesis - the dispersion is  greater  than
      or equal to the given number)
    * right-tailed test (null hypothesis  -  dispersion is  less  than  or
      equal to the given number).

Test is based on the following assumptions:
    * the given sample has a normal distribution.

Input parameters:
    X           -   sample 1. Array whose index goes from 0 to N-1.
    N           -   size of the sample.
    Variance    -   dispersion value to compare with.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

  -- ALGLIB --
     Copyright 19.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::onesamplevariancetest(
    real_1d_array x,
    ae_int_t n,
    double variance,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`wsr` subpackage

Functions

wilcoxonsignedranktest

Examples

`wilcoxonsignedranktest` function

/*************************************************************************
Wilcoxon signed-rank test

This test checks three hypotheses about the median  of  the  given sample.
The following tests are performed:
    * two-tailed test (null hypothesis - the median is equal to the  given
      value)
    * left-tailed test (null hypothesis - the median is  greater  than  or
      equal to the given value)
    * right-tailed test (null hypothesis  -  the  median  is  less than or
      equal to the given value)

Requirements:
    * the scale of measurement should be ordinal, interval or  ratio (i.e.
      the test could not be applied to nominal variables).
    * the distribution should be continuous and symmetric relative to  its
      median.
    * number of distinct values in the X array should be greater than 4

The test is non-parametric and doesn't require distribution X to be normal

Input parameters:
    X       -   sample. Array whose index goes from 0 to N-1.
    N       -   size of the sample.
    Median  -   assumed median value.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

To calculate p-values, special approximation is used. This method lets  us
calculate p-values with two decimal places in interval [0.0001, 1].

"Two decimal places" does not sound very impressive, but in  practice  the
relative error of less than 1% is enough to make a decision.

There is no approximation outside the [0.0001, 1] interval. Therefore,  if
the significance level outlies this interval, the test returns 0.0001.

  -- ALGLIB --
     Copyright 08.09.2006 by Bochkanov Sergey
*************************************************************************/
void alglib::wilcoxonsignedranktest(
    real_1d_array x,
    ae_int_t n,
    double e,
    double& bothtails,
    double& lefttail,
    double& righttail,
    const xparams _params = alglib::xdefault);

`xdebug` subpackage

Classes

xdebugrecord1

Functions

xdebugb1appendcopy
xdebugb1count
xdebugb1not
xdebugb1outeven
xdebugb2count
xdebugb2not
xdebugb2outsin
xdebugb2transpose
xdebugc1appendcopy
xdebugc1neg
xdebugc1outeven
xdebugc1sum
xdebugc2neg
xdebugc2outsincos
xdebugc2sum
xdebugc2transpose
xdebugi1appendcopy
xdebugi1neg
xdebugi1outeven
xdebugi1sum
xdebugi2neg
xdebugi2outsin
xdebugi2sum
xdebugi2transpose
xdebuginitrecord1
xdebugmaskedbiasedproductsum
xdebugr1appendcopy
xdebugr1neg
xdebugr1outeven
xdebugr1sum
xdebugr2neg
xdebugr2outsin
xdebugr2sum
xdebugr2transpose

Examples

`xdebugrecord1` class

/*************************************************************************

*************************************************************************/
class xdebugrecord1
{
    ae_int_t             i;
    alglib::complex      c;
    real_1d_array        a;
};

`xdebugb1appendcopy` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Appends copy of array to itself.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugb1appendcopy(
    boolean_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugb1count` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Counts number of True values in the boolean 1D array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::xdebugb1count(
    boolean_1d_array a,
    const xparams _params = alglib::xdefault);

`xdebugb1not` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by NOT(a[i]).
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugb1not(
    boolean_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugb1outeven` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate N-element array with even-numbered elements set to True.
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugb1outeven(
    ae_int_t n,
    boolean_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugb2count` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Counts number of True values in the boolean 2D array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::xdebugb2count(
    boolean_2d_array a,
    const xparams _params = alglib::xdefault);

`xdebugb2not` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by NOT(a[i]).
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugb2not(
    boolean_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugb2outsin` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate MxN matrix with elements set to "Sin(3*I+5*J)>0"
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugb2outsin(
    ae_int_t m,
    ae_int_t n,
    boolean_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugb2transpose` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Transposes array.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugb2transpose(
    boolean_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugc1appendcopy` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Appends copy of array to itself.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugc1appendcopy(
    complex_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugc1neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -A[I]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugc1neg(
    complex_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugc1outeven` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate N-element array with even-numbered A[K] set to (x,y) = (K*0.25, K*0.125)
and odd-numbered ones are set to 0.

Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugc1outeven(
    ae_int_t n,
    complex_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugc1sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
alglib::complex alglib::xdebugc1sum(
    complex_1d_array a,
    const xparams _params = alglib::xdefault);

`xdebugc2neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -a[i,j]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugc2neg(
    complex_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugc2outsincos` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate MxN matrix with elements set to "Sin(3*I+5*J),Cos(3*I+5*J)"
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugc2outsincos(
    ae_int_t m,
    ae_int_t n,
    complex_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugc2sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
alglib::complex alglib::xdebugc2sum(
    complex_2d_array a,
    const xparams _params = alglib::xdefault);

`xdebugc2transpose` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Transposes array.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugc2transpose(
    complex_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugi1appendcopy` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Appends copy of array to itself.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugi1appendcopy(
    integer_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugi1neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -A[I]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugi1neg(
    integer_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugi1outeven` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate N-element array with even-numbered A[I] set to I, and odd-numbered
ones set to 0.

Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugi1outeven(
    ae_int_t n,
    integer_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugi1sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::xdebugi1sum(
    integer_1d_array a,
    const xparams _params = alglib::xdefault);

`xdebugi2neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -a[i,j]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugi2neg(
    integer_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugi2outsin` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate MxN matrix with elements set to "Sign(Sin(3*I+5*J))"
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugi2outsin(
    ae_int_t m,
    ae_int_t n,
    integer_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugi2sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
ae_int_t alglib::xdebugi2sum(
    integer_2d_array a,
    const xparams _params = alglib::xdefault);

`xdebugi2transpose` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Transposes array.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugi2transpose(
    integer_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebuginitrecord1` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Creates and returns XDebugRecord1 structure:
* integer and complex fields of Rec1 are set to 1 and 1+i correspondingly
* array field of Rec1 is set to [2,3]

  -- ALGLIB --
     Copyright 27.05.2014 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebuginitrecord1(
    xdebugrecord1& rec1,
    const xparams _params = alglib::xdefault);

`xdebugmaskedbiasedproductsum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of a[i,j]*(1+b[i,j]) such that c[i,j] is True

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
double alglib::xdebugmaskedbiasedproductsum(
    ae_int_t m,
    ae_int_t n,
    real_2d_array a,
    real_2d_array b,
    boolean_2d_array c,
    const xparams _params = alglib::xdefault);

`xdebugr1appendcopy` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Appends copy of array to itself.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugr1appendcopy(
    real_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugr1neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -A[I]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugr1neg(
    real_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugr1outeven` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate N-element array with even-numbered A[I] set to I*0.25,
and odd-numbered ones are set to 0.

Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugr1outeven(
    ae_int_t n,
    real_1d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugr1sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
double alglib::xdebugr1sum(
    real_1d_array a,
    const xparams _params = alglib::xdefault);

`xdebugr2neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -a[i,j]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugr2neg(
    real_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugr2outsin` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate MxN matrix with elements set to "Sin(3*I+5*J)"
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugr2outsin(
    ae_int_t m,
    ae_int_t n,
    real_2d_array& a,
    const xparams _params = alglib::xdefault);

`xdebugr2sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
double alglib::xdebugr2sum(
    real_2d_array a,
    const xparams _params = alglib::xdefault);

`xdebugr2transpose` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Transposes array.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void alglib::xdebugr2transpose(
    real_2d_array& a,
    const xparams _params = alglib::xdefault);

1 Introduction

1.1 What is ALGLIB

1.2 ALGLIB license

1.3 Documentation license

1.4 Reference Manual and User Guide

1.5 Acknowledgements

2 ALGLIB structure

2.1 Packages

2.2 Subpackages

2.3 Open Source and Commercial versions

3 Compatibility

3.1 CPU

3.2 OS

3.3 Compiler

3.4 Optimization settings

4 Compiling ALGLIB

4.1 Adding to your project

4.2 Configuring for your compiler

4.3 Improving performance (CPU-specific and OS-specific optimizations)

4.4 Examples (free and commercial editions)

4.4.1 Introduction

4.4.2 Compiling under Windows

4.4.3 Compiling under Linux

5 Using ALGLIB

5.1 Thread-safety

5.2 Global definitions

5.3 Datatypes

5.4 Constants

5.5 Functions

5.6 Working with vectors and matrices

5.7 Using functions: 'expert' and 'friendly' interfaces

5.8 Handling errors

5.9 Working with Level 1 BLAS functions

5.10 Reading data from CSV files

6 Working with commercial version

6.1 Benefits of commercial version

6.2 Working with SIMD support (Intel/AMD users)

6.3 Using multithreading

6.3.1 SMT (CMT/hyper-threading) issues

6.4 Linking with Intel MKL

6.4.1 Using lightweight Intel MKL supplied by ALGLIB Project

6.4.2 Using your own installation of Intel MKL

7 Advanced topics

7.1 Exception-free mode

7.2 Partial compilation

7.3 Testing ALGLIB

8 ALGLIB packages and subpackages

8.1 AlglibMisc package

8.2 DataAnalysis package

8.3 DiffEquations package

8.4 FastTransforms package

8.5 Integration package

8.6 Interpolation package

8.7 LinAlg package

8.8 Optimization package

8.9 Solvers package

8.10 SpecialFunctions package

8.11 Statistics package

ablas subpackage

Functions

Examples

cmatrixcopy function

cmatrixgemm function

cmatrixherk function

cmatrixlefttrsm function

cmatrixmv function

cmatrixrank1 function

cmatrixrighttrsm function

cmatrixsyrk function

cmatrixtranspose function

rmatrixcopy function

rmatrixenforcesymmetricity function

rmatrixgemm function

rmatrixgemv function

rmatrixger function

rmatrixlefttrsm function

rmatrixmv function

rmatrixrank1 function

rmatrixrighttrsm function

rmatrixsymv function

8.1 `AlglibMisc` package

8.2 `DataAnalysis` package

8.3 `DiffEquations` package

8.4 `FastTransforms` package

8.5 `Integration` package

8.6 `Interpolation` package

8.7 `LinAlg` package

8.8 `Optimization` package

8.9 `Solvers` package

8.10 `SpecialFunctions` package

8.11 `Statistics` package

`ablas` subpackage

`cmatrixcopy` function

`cmatrixgemm` function

`cmatrixherk` function

`cmatrixlefttrsm` function

`cmatrixmv` function

`cmatrixrank1` function

`cmatrixrighttrsm` function

`cmatrixsyrk` function

`cmatrixtranspose` function

`rmatrixcopy` function

`rmatrixenforcesymmetricity` function

`rmatrixgemm` function

`rmatrixgemv` function

`rmatrixger` function

`rmatrixlefttrsm` function

`rmatrixmv` function

`rmatrixrank1` function

`rmatrixrighttrsm` function

`rmatrixsymv` function

`rmatrixsyrk` function

`rmatrixsyvmv` function

`rmatrixtranspose` function

`rmatrixtrsv` function

`airyf` subpackage

`airy` function

`autogk` subpackage

`autogkreport` class

`autogkstate` class

`autogkintegrate` function

`autogkresults` function

`autogksingular` function

`autogksmooth` function

`autogksmoothw` function

`basestat` subpackage

`cov2` function

`covm` function

`covm2` function

`pearsoncorr2` function

`pearsoncorrelation` function

`pearsoncorrm` function

`pearsoncorrm2` function

`rankdata` function

`rankdatacentered` function

`sampleadev` function

`samplekurtosis` function

`samplemean` function

`samplemedian` function

`samplemoments` function

`samplepercentile` function

`sampleskewness` function

`samplevariance` function

`spearmancorr2` function

`spearmancorrm` function

`spearmancorrm2` function

`spearmanrankcorrelation` function

`bdss` subpackage

`dsoptimalsplit2` function

`dsoptimalsplit2fast` function

`bdsvd` subpackage

`rmatrixbdsvd` function

`bessel` subpackage

`besseli0` function

`besseli1` function

`besselj0` function

`besselj1` function

`besseljn` function

`besselk0` function

`besselk1` function