GNU Scientific Library

Reference Manual

Edition 1.0, for GSL Version 1.0

1 November 2001

Mark Galassi
Jim Davies
James Theiler
Brian Gough
Gerard Jungman
Michael Booth
Fabrice Rossi


@raggedbottom

@dircategory Scientific software * gsl-ref: (gsl-ref). GNU Scientific Library -- Reference

Los Alamos National Laboratory

Department of Computer Science, Georgia Institute of Technology

Astrophysics and Radiation Measurements Group, Los Alamos National Laboratory

Network Theory Limited

Theoretical Fluid Dynamics Group, Los Alamos National Laboratory

Department of Physics and Astronomy, The Johns Hopkins University

University of Paris-Dauphine Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001 The GSL Team.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

Introduction

The GNU Scientific Library (GSL) is a collection of routines for numerical computing. The routines have been written from scratch in C, and are meant to present a modern Applications Programming Interface (API) for C programmers, while allowing wrappers to be written for very high level languages. The source code is distributed under the GNU General Public License.

Routines available in GSL

The library covers a wide range of topics in numerical computing. Routines are available for the following areas,
Complex Numbers Roots of Polynomials
Special Functions Vectors and Matrices
Permutations Sorting
BLAS Support Linear Algebra
Eigensystems Fast Fourier Transforms
Quadrature Random Numbers
Quasi-Random Sequences Random Distributions
Statistics Histograms
N-Tuples Monte Carlo Integration
Simulated Annealing Differential Equations
Interpolation Numerical Differentiation
Chebyshev Approximations Series Acceleration
Discrete Hankel Transforms Root-Finding
Minimization Least-Squares Fitting
Physical Constants IEEE Floating-Point

The use of these routines is described in this manual. Each chapter provides detailed definitions of the functions, followed by example programs and references to the articles on which the algorithms are based.

GSL is Free Software

The subroutines in the GNU Scientific Library are "free software"; this means that everyone is free to use them, and to redistribute them in other free programs. The library is not in the public domain; it is copyrighted and there are conditions on its distribution. These conditions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of the software that they might get from you.

Specifically, we want to make sure that you have the right to give away copies of any programs related to the GNU Scientific Library, that you receive their source code or else can get it if you want it, that you can change these programs or use pieces of them in new free programs, and that you know you can do these things. The library should not be redistributed in proprietary programs.

To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of any related code, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights.

Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU Scientific Library. If these programs are modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.

The precise conditions for the distribution of software related to the GNU Scientific Library are found in the GNU General Public License (see section GNU General Public License). Further information about this license is available from the GNU Project webpage Frequently Asked Questions about the GNU GPL,

Obtaining GSL

The source code for the library can be obtained in different ways, by copying it from a friend, purchasing it on CDROM or downloading it from the internet. A list of public ftp servers which carry the source code can be found on the development website,

The preferred platform for the library is a GNU system, which allows it to take advantage of additional features. The library is portable and compiles on most Unix platforms. It is also available for Microsoft Windows. Precompiled versions of the library can be purchased from commercial redistributors listed on the website.

Announcements of new releases, updates and other relevant events are made on the gsl-announce mailing list. To subscribe to this low-volume list, send an email of the following form,

To: gsl-announce-request@sources.redhat.com 
Subject: subscribe

You will receive a response asking to you to reply in order to confirm your subscription.

An Example Program

The following short program demonstrates the use of the library by computing the value of the Bessel function J_0(x) for x=5,

#include <stdio.h>
#include <gsl/gsl_sf_bessel.h>

int
main (void)
{
  double x = 5.0;

  double y = gsl_sf_bessel_J0 (x);

  printf("J0(%g) = %.18e\n", x, y);

  return 0;
}

The output is shown below, and should be correct to double-precision accuracy,

J0(5) = -1.775967713143382920e-01

The steps needed to compile programs which use the library are described in the next chapter.

No Warranty

The software described in this manual has no warranty, it is provided "as is". It is your responsibility to validate the behavior of the routines and their accuracy using the source code provided. Consult the GNU General Public license for further details (see section GNU General Public License).

Further Information

Additional information, including online copies of this manual, links to related projects, and mailing list archives are available from the development website mentioned above. The developers of the library can be reached via the project's public mailing list,

This mailing list can be used to report bugs or to ask questions not covered by this manual.

Using the library

This chapter describes how to compile programs that use GSL, and introduces its conventions.

ANSI C Compliance

The library is written in ANSI C and is intended to conform to the ANSI C standard. It should be portable to any system with a working ANSI C compiler.

The library does not rely on any non-ANSI extensions in the interface it exports to the user. Programs you write using GSL can be ANSI compliant. Extensions which can be used in a way compatible with pure ANSI C are supported, however, via conditional compilation. This allows the library to take advantage of compiler extensions on those platforms which support them.

When an ANSI C feature is known to be broken on a particular system the library will exclude any related functions at compile-time. This should make it impossible to link a program that would use these functions and give incorrect results.

To avoid namespace conflicts all exported function names and variables have the prefix gsl_, while exported macros have the prefix GSL_.

Compiling and Linking

The library header files are installed in their own `gsl' directory. You should write any preprocessor include statements with a `gsl/' directory prefix thus,

#include <gsl/gsl_math.h>

If the directory is not installed on the standard search path of your compiler you will also need to provide its location to the preprocessor as a command line flag. The default location of the `gsl' directory is `/usr/local/include/gsl'.

The library is installed as a single file, `libgsl.a'. A shared version of the library is also installed on systems that support shared libraries. The default location of these files is `/usr/local/lib'. To link against the library you need to specify both the main library and a supporting CBLAS library, which provides standard basic linear algebra subroutines. A suitable CBLAS implementation is provided in the library `libgslcblas.a' if your system does not provide one. The following example shows how to link an application with the library,

gcc app.o -lgsl -lgslcblas -lm

The following command line shows how you would link the same application with an alternative blas library called `libcblas',

gcc app.o -lgsl -lcblas -lm

For the best performance an optimized platform-specific CBLAS library should be used for -lcblas. The library must conform to the CBLAS standard. The ATLAS package provides a portable high-performance BLAS library with a CBLAS interface. It is free software and should be installed for any work requiring fast vector and matrix operations. The following command line will link with the ATLAS library and its CBLAS interface,

gcc app.o -lgsl -lcblas -latlas -lm

For more information see section BLAS Support.

The program gsl-config provides information on the local version of the library. For example, the following command shows that the library has been installed under the directory `/usr/local',

bash$ gsl-config --prefix
/usr/local

Further information is available using the command gsl-config --help.

Shared Libraries

To run a program linked with the shared version of the library it may be necessary to define the shell variable LD_LIBRARY_PATH to include the directory where the library is installed. For example,

LD_LIBRARY_PATH=/usr/local/lib:$LD_LIBRARY_PATH ./app

To compile a statically linked version of the program instead, use the -static flag in gcc,

gcc -static app.o -lgsl -lgslcblas -lm

Autoconf macros

For applications using autoconf the standard macro AC_CHECK_LIB can be used to link with the library automatically from a configure script. The library itself depends on the presence of a CBLAS and math library as well, so these must also be located before linking with the main libgsl file. The following commands should be placed in the `configure.in' file to perform these tests,

AC_CHECK_LIB(m,main)
AC_CHECK_LIB(gslcblas,main)
AC_CHECK_LIB(gsl,main)

Assuming the libraries are found the output during the configure stage looks like this,

checking for main in -lm... yes
checking for main in -lgslcblas... yes
checking for main in -lgsl... yes

If the library is found then the tests will define the macros HAVE_LIBGSL, HAVE_LIBGSLCBLAS, HAVE_LIBM and add the options -lgsl -lgslcblas -lm to the variable LIBS.

The tests above will find any version of the library. They are suitable for general use, where the versions of the functions are not important. An alternative macro is available in the file `gsl.m4' to test for a specific version of the library. To use this macro simply add the following line to your `configure.in' file instead of the tests above:

AM_PATH_GSL(GSL_VERSION,
           [action-if-found],
           [action-if-not-found])

The argument GSL_VERSION should be the two or three digit MAJOR.MINOR or MAJOR.MINOR.MICRO version number of the release you require. A suitable choice for action-if-not-found is,

AC_MSG_ERROR(could not find required version of GSL)

Then you can add the variables GSL_LIBS and GSL_CFLAGS to your Makefile.am files to obtain the correct compiler flags. GSL_LIBS is equal to the output of the gsl-config --libs command and GSL_CFLAGS is equal to gsl-config --cflags command. For example,

libgsdv_la_LDFLAGS =    \
        $(GTK_LIBDIR) \
        $(GTK_LIBS) -lgsdvgsl $(GSL_LIBS) -lgslcblas

Note that the macro AM_PATH_GSL needs to use the C compiler so it should appear in the `configure.in' file before the macro AC_LANG_CPLUSPLUS for programs that use C++.

Inline functions

The inline keyword is not part of ANSI C and the library does not export any inline function definitions by default. However, the library provides optional inline versions of performance-critical functions by conditional compilation. The inline versions of these functions can be included by defining the macro HAVE_INLINE when compiling an application.

gcc -c -DHAVE_INLINE app.c

If you use autoconf this macro can be defined automatically. The following test should be placed in your `configure.in' file,

AC_C_INLINE

if test "$ac_cv_c_inline" != no ; then
  AC_DEFINE(HAVE_INLINE,1)
  AC_SUBST(HAVE_INLINE)
fi

and the macro will then be defined in the compilation flags or by including the file `config.h' before any library headers. If you do not define the macro HAVE_INLINE then the slower non-inlined versions of the functions will be used instead.

Note that the actual usage of the inline keyword is extern inline, which eliminates unnecessary function definitions in GCC. If the form extern inline causes problems with other compilers a stricter autoconf test can be used, see section Autoconf Macros.

Long double

The extended numerical type long double is part of the ANSI C standard and should be available in every modern compiler. However, the precision of long double is platform dependent, and this should be considered when using it. The IEEE standard only specifies the minimum precision of extended precision numbers, while the precision of double is the same on all platforms.

In some system libraries the stdio.h formatted input/output functions printf and scanf are not implemented correctly for long double. Undefined or incorrect results are avoided by testing these functions during the configure stage of library compilation and eliminating certain GSL functions which depend on them if necessary. The corresponding line in the configure output looks like this,

checking whether printf works with long double... no

Consequently when long double formatted input/output does not work on a given system it should be impossible to link a program which uses GSL functions dependent on this.

If it is necessary to work on a system which does not support formatted long double input/output then the options are to use binary formats or to convert long double results into double for reading and writing.

Portability functions

To help in writing portable applications GSL provides some implementations of functions that are found in other libraries, such as the BSD math library. You can write your application to use the native versions of these functions, and substitute the GSL versions via a preprocessor macro if they are unavailable on another platform. The substitution can be made automatically if you use autoconf. For example, to test whether the BSD function hypot is available you can include the following line in the configure file `configure.in' for your application,

AC_CHECK_FUNCS(hypot)

and place the following macro definitions in the file `config.h.in',

/* Substitute gsl_hypot for missing system hypot */

#ifndef HAVE_HYPOT
#define hypot gsl_hypot
#endif

The application source files can then use the include command #include <config.h> to substitute gsl_hypot for each occurrence of hypot when hypot is not available.

In most circumstances the best strategy is to use the native versions of these functions when available, and fall back to GSL versions otherwise, since this allows your application to take advantage of any platform-specific optimizations in the system library. This is the strategy used within GSL itself.

Alternative optimized functions

The main implementation of some functions in the library will not be optimal on all architectures. For example, there are several ways to compute a Gaussian random variate and their relative speeds are platform-dependent. In cases like this the library provides alternate implementations of these functions with the same interface. If you write your application using calls to the standard implementation you can select an alternative version later via a preprocessor definition. It is also possible to introduce your own optimized functions this way while retaining portability. The following lines demonstrate the use of a platform-dependent choice of methods for sampling from the Gaussian distribution,

#ifdef SPARC
#define gsl_ran_gaussian gsl_ran_gaussian_ratio_method
#endif
#ifdef INTEL
#define gsl_ran_gaussian my_gaussian
#endif

These lines would be placed in the configuration header file `config.h' of the application, which should then be included by all the source files. Note that the alternative implementations will not produce bit-for-bit identical results, and in the case of random number distributions will produce an entirely different stream of random variates.

Support for different numeric types

Many functions in the library are defined for different numeric types. This feature is implemented by varying the name of the function with a type-related modifier -- a primitive form of C++ templates. The modifier is inserted into the function name after the initial module prefix. The following table shows the function names defined for all the numeric types of an imaginary module gsl_foo with function fn,

gsl_foo_fn               double        
gsl_foo_long_double_fn   long double   
gsl_foo_float_fn         float         
gsl_foo_long_fn          long          
gsl_foo_ulong_fn         unsigned long 
gsl_foo_int_fn           int           
gsl_foo_uint_fn          unsigned int  
gsl_foo_short_fn         short         
gsl_foo_ushort_fn        unsigned short
gsl_foo_char_fn          char          
gsl_foo_uchar_fn         unsigned char 

The normal numeric precision double is considered the default and does not require a suffix. For example, the function gsl_stats_mean computes the mean of double precision numbers, while the function gsl_stats_int_mean computes the mean of integers.

A corresponding scheme is used for library defined types, such as gsl_vector and gsl_matrix. In this case the modifier is appended to the type name. For example, if a module defines a new type-dependent struct or typedef gsl_foo it is modified for other types in the following way,

gsl_foo                  double        
gsl_foo_long_double      long double   
gsl_foo_float            float         
gsl_foo_long             long          
gsl_foo_ulong            unsigned long 
gsl_foo_int              int           
gsl_foo_uint             unsigned int  
gsl_foo_short            short         
gsl_foo_ushort           unsigned short
gsl_foo_char             char          
gsl_foo_uchar            unsigned char 

When a module contains type-dependent definitions the library provides individual header files for each type. The filenames are modified as shown in the below. For convenience the default header includes the definitions for all the types. To include only the double precision header, or any other specific type, file use its individual filename.

#include <gsl/gsl_foo.h>               All types
#include <gsl/gsl_foo_double.h>        double        
#include <gsl/gsl_foo_long_double.h>   long double   
#include <gsl/gsl_foo_float.h>         float         
#include <gsl/gsl_foo_long.h>          long          
#include <gsl/gsl_foo_ulong.h>         unsigned long 
#include <gsl/gsl_foo_int.h>           int           
#include <gsl/gsl_foo_uint.h>          unsigned int  
#include <gsl/gsl_foo_short.h>         short         
#include <gsl/gsl_foo_ushort.h>        unsigned short
#include <gsl/gsl_foo_char.h>          char          
#include <gsl/gsl_foo_uchar.h>         unsigned char 

Compatibility with C++

The library header files automatically define functions to have extern "C" linkage when included in C++ programs.

Aliasing of arrays

The library assumes that arrays, vectors and matrices passed as modifiable arguments are not aliased and do not overlap with each other. This removes the need for the library to handle overlapping memory regions as a special case, and allows additional optimizations to be used. If overlapping memory regions are passed as modifiable arguments then the results of such functions will be undefined. If the arguments will not be modified (for example, if a function prototype declares them as const arguments) then overlapping or aliased memory regions can be safely used.

Code Reuse

Where possible the routines in the library have been written to avoid dependencies between modules and files. This should make it possible to extract individual functions for use in your own applications, without needing to have the whole library installed. You may need to define certain macros such as GSL_ERROR and remove some #include statements in order to compile the files as standalone units. Reuse of the library code in this way is encouraged, subject to the terms of the GNU General Public License.

Error Handling

This chapter describes the way that GSL functions report and handle errors. By examining the status information returned by every function you can determine whether it succeeded or failed, and if it failed you can find out what the precise cause of failure was. You can also define your own error handling functions to modify the default behavior of the library.

The functions described in this section are declared in the header file `gsl_errno.h'.

Error Reporting

The library follows the thread-safe error reporting conventions of the POSIX Threads library. Functions return a non-zero error code to indicate an error and 0 to indicate success.

int status = gsl_function(...)

if (status) { /* an error occurred */
  .....       
  /* status value specifies the type of error */
}

The routines report an error whenever they cannot perform the task requested of them. For example, a root-finding function would return a non-zero error code if could not converge to the requested accuracy, or exceeded a limit on the number of iterations. Situations like this are a normal occurrence when using any mathematical library and you should check the return status of the functions that you call.

Whenever a routine reports an error the return value specifies the type of error. The return value is analogous to the value of the variable errno in the C library. However, the C library's errno is a global variable, which is not thread-safe (There can be only one instance of a global variable per program. Different threads of execution may overwrite errno simultaneously). Returning the error number directly avoids this problem. The caller can examine the return code and decide what action to take, including ignoring the error if it is not considered serious.

The error code numbers are defined in the file `gsl_errno.h'. They all have the prefix GSL_ and expand to non-zero constant integer values. Many of the error codes use the same base name as a corresponding error code in C library. Here are some of the most common error codes,

Macro: int GSL_EDOM
Domain error; used by mathematical functions when an argument value does not fall into the domain over which the function is defined (like EDOM in the C library)

Macro: int GSL_ERANGE
Range error; used by mathematical functions when the result value is not representable because of overflow or underflow (like ERANGE in the C library)

Macro: int GSL_ENOMEM
No memory available. The system cannot allocate more virtual memory because its capacity is full (like ENOMEM in the C library). This error is reported when a GSL routine encounters problems when trying to allocate memory with malloc.

Macro: int GSL_EINVAL
Invalid argument. This is used to indicate various kinds of problems with passing the wrong argument to a library function (like EINVAL in the C library).
Here is an example of some code which checks the return value of a function where an error might be reported,
int status = gsl_fft_complex_radix2_forward (data, n);

if (status) {
  if (status == GSL_EINVAL) {
     fprintf (stderr, "invalid argument, n=%d\n", n);
  } else {
     fprintf (stderr, "failed, gsl_errno=%d\n", 
                      status);
  }
  exit (-1);
}

The function gsl_fft_complex_radix2 only accepts integer lengths which are a power of two. If the variable n is not a power of two then the call to the library function will return GSL_EINVAL, indicating that the length argument is invalid. The else clause catches any other possible errors.

The error codes can be converted into an error message using the function gsl_strerror.

Function: const char * gsl_strerror (const int gsl_errno)
This function returns a pointer to a string describing the error code gsl_errno. For example,
printf("error: %s\n", gsl_strerror (status));

would print an error message like error: output range error for a status value of GSL_ERANGE.

Error Handlers

In addition to reporting errors the library also provides an optional error handler. The error handler is called by library functions when they report an error, just before they return to the caller. The purpose of the handler is to provide a function where a breakpoint can be set that will catch library errors when running under the debugger. It is not intended for use in production programs, which should handle any errors using the error return codes described above.

The default behavior of the error handler is to print a short message and call abort() whenever an error is reported by the library. If this default is not turned off then any program using the library will stop with a core-dump whenever a library routine reports an error. This is intended as a fail-safe default for programs which do not check the return status of library routines (we don't encourage you to write programs this way). If you turn off the default error handler it is your responsibility to check the return values of the GSL routines. You can customize the error behavior by providing a new error handler. For example, an alternative error handler could log all errors to a file, ignore certain error conditions (such as underflows), or start the debugger and attach it to the current process when an error occurs.

All GSL error handlers have the type gsl_error_handler_t, which is defined in `gsl_errno.h',

Data Type: gsl_error_handler_t

This is the type of GSL error handler functions. An error handler will be passed four arguments which specify the reason for the error (a string), the name of the source file in which it occurred (also a string), the line number in that file (an integer) and the error number (an integer). The source file and line number are set at compile time using the __FILE__ and __LINE__ directives in the preprocessor. An error handler function returns type void. Error handler functions should be defined like this,

void handler (const char * reason, 
              const char * file, 
              int line, 
              int gsl_errno)

To request the use of your own error handler you need to call the function gsl_set_error_handler which is also declared in `gsl_errno.h',

Function: gsl_error_handler_t gsl_set_error_handler (gsl_error_handler_t new_handler)

This functions sets a new error handler, new_handler, for the GSL library routines. The previous handler is returned (so that you can restore it later). Note that the pointer to a user defined error handler function is stored in a static variable, so there can only be one error handler per program. This function should be not be used in multi-threaded programs except to set up a program-wide error handler from a master thread. The following example shows how to set and restore a new error handler,

/* save original handler, install new handler */
old_handler = gsl_set_error_handler (&my_handler); 

/* code uses new handler */
.....     

/* restore original handler */
gsl_set_error_handler (old_handler); 

To use the default behavior (abort on error) set the error handler to NULL,

old_handler = gsl_set_error_handler (NULL); 

Function: gsl_error_handler_t gsl_set_error_handler_off ()
This function turns off the error handler by defining an error handler which does nothing. This will cause the program to continue after any error, so the return values from any library routines must be checked. This is the recommended behavior for production programs. The previous handler is returned (so that you can restore it later).

The error behavior can be changed for specific applications by recompiling the library with a customized definition of the GSL_ERROR macro in the file `gsl_errno.h'.

Using GSL error reporting in your own functions

If you are writing numerical functions in a program which also uses GSL code you may find it convenient to adopt the same error reporting conventions as in the library.

To report an error you need to call the function gsl_error with a string describing the error and then return an appropriate error code from gsl_errno.h, or a special value, such as NaN. For convenience the file `gsl_errno.h' defines two macros which carry out these steps:

Macro: GSL_ERROR (reason, gsl_errno)

This macro reports an error using the GSL conventions and returns a status value of gsl_errno. It expands to the following code fragment,

gsl_error (reason, __FILE__, __LINE__, gsl_errno);
return gsl_errno;

The macro definition in `gsl_errno.h' actually wraps the code in a do { ... } while (0) block to prevent possible parsing problems.

Here is an example of how the macro could be used to report that a routine did not achieve a requested tolerance. To report the error the routine needs to return the error code GSL_ETOL.

if (residual > tolerance) 
  {
    GSL_ERROR("residual exceeds tolerance", GSL_ETOL);
  }

Macro: GSL_ERROR_VAL (reason, gsl_errno, value)

This macro is the same as GSL_ERROR but returns a user-defined status value of value instead of an error code. It can be used for mathematical functions that return a floating point value.

Here is an example where a function needs to return a NaN because of a mathematical singularity,

if (x == 0) 
  {
    GSL_ERROR_VAL("argument lies on singularity", 
                  GSL_ERANGE, GSL_NAN);
  }

Mathematical Functions

This chapter describes basic mathematical functions. Some of these functions are present in system libraries, but the alternative versions given here can be used as a substitute when the system functions are not available.

The functions and macros described in this chapter are defined in the header file `gsl_math.h'.

Mathematical Constants

The library ensures that the standard BSD mathematical constants are defined. For reference here is a list of the constants.

M_E
The base of exponentials, e
M_LOG2E
The base-2 logarithm of e, \log_2 (e)
M_LOG10E
The base-10 logarithm of e, @c{$\log_{10}(e)$} \log_10 (e)
M_SQRT2
The square root of two, \sqrt 2
M_SQRT1_2
The square root of one-half, @c{$\sqrt{1/2}$} \sqrt{1/2}
M_SQRT3
The square root of three, \sqrt 3
M_PI
The constant pi, \pi
M_PI_2
Pi divided by two, \pi/2
M_PI_4
Pi divided by four, \pi/4
M_SQRTPI
The square root of pi, \sqrt\pi
M_2_SQRTPI
Two divided by the square root of pi, 2/\sqrt\pi
M_1_PI
The reciprocal of pi, 1/\pi
M_2_PI
Twice the reciprocal of pi, 2/\pi
M_LN10
The natural logarithm of ten, \ln(10)
M_LN2
The natural logarithm of two, \ln(2)
M_LNPI
The natural logarithm of pi, \ln(\pi)
M_EULER
Euler's constant, \gamma

Infinities and Not-a-number

Macro: GSL_POSINF
This macro contains the IEEE representation of positive infinity, +\infty. It is computed from the expression +1.0/0.0.

Macro: GSL_NEGINF
This macro contains the IEEE representation of negative infinity, -\infty. It is computed from the expression -1.0/0.0.

Macro: GSL_NAN
This macro contains the IEEE representation of the Not-a-Number symbol, NaN. It is computed from the ratio 0.0/0.0.

Function: int gsl_isnan (const double x)
This function returns 1 if x is not-a-number.

Function: int gsl_isinf (const double x)
This function returns +1 if x is positive infinity, -1 if x is negative infinity and 0 otherwise.

Function: int gsl_finite (const double x)
This function returns 1 if x is a real number, and 0 if it is infinite or not-a-number.

Elementary Functions

The following routines provide portable implementations of functions found in the BSD math library. When native versions are not available the functions described here can be used instead. The substitution can be made automatically if you use autoconf to compile your application (see section Portability functions).

Function: double gsl_log1p (const double x)
This function computes the value of \log(1+x) in a way that is accurate for small x. It provides an alternative to the BSD math function log1p(x).

Function: double gsl_expm1 (const double x)
This function computes the value of \exp(x)-1 in a way that is accurate for small x. It provides an alternative to the BSD math function expm1(x).

Function: double gsl_hypot (const double x, const double y)
This function computes the value of \sqrt{x^2 + y^2} in a way that avoids overflow. It provides an alternative to the BSD math function hypot(x,y).

Function: double gsl_acosh (const double x)
This function computes the value of \arccosh(x). It provides an alternative to the standard math function acosh(x).

Function: double gsl_asinh (const double x)
This function computes the value of \arcsinh(x). It provides an alternative to the standard math function asinh(x).

Function: double gsl_atanh (const double x)
This function computes the value of \arctanh(x). It provides an alternative to the standard math function atanh(x).

Small integer powers

A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL provides a simple functions to fill this gap. For reasons of efficiency, these functions do not check for overflow or underflow conditions.

Function: double gsl_pow_int (double x, int n)
This routine computes the power x^n for integer n. The power is computed using the minimum number of multiplications. For example, x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications. A version of this function which also computes the numerical error in the result is available as gsl_sf_pow_int_e.

Function: double gsl_pow_2 (const double x)
Function: double gsl_pow_3 (const double x)
Function: double gsl_pow_4 (const double x)
Function: double gsl_pow_5 (const double x)
Function: double gsl_pow_6 (const double x)
Function: double gsl_pow_7 (const double x)
Function: double gsl_pow_8 (const double x)
Function: double gsl_pow_9 (const double x)
These functions can be used to compute small integer powers x^2, x^3, etc. efficiently. The functions will be inlined when possible so that use of these functions should be as efficient as explicitly writing the corresponding product expression.
#include <gsl/gsl_math.h>
double y = gsl_pow_4 (3.141)  /* compute 3.141**4 */

Testing the Sign of Numbers

Macro: GSL_SIGN (x)
This macro returns the sign of x. It is defined as ((x) >= 0 ? 1 : -1). Note that with this definition the sign of zero is positive (regardless of its IEEE sign bit).

Testing for Odd and Even Numbers

Macro: GSL_IS_ODD (n)
This macro evaluates to 1 if n is odd and 0 if n is even. The argument n must be of integer type.

Macro: GSL_IS_EVEN (n)
This macro is the opposite of GSL_IS_ODD(n). It evaluates to 1 if n is even and 0 if n is odd. The argument n must be of integer type.

Maximum and Minimum functions

Macro: GSL_MAX (a, b)
This macro returns the maximum of a and b. It is defined as ((a) > (b) ? (a):(b)).

Macro: GSL_MIN (a, b)
This macro returns the minimum of a and b. It is defined as ((a) < (b) ? (a):(b)).

Function: extern inline double GSL_MAX_DBL (double a, double b)
This function returns the maximum of the double precision numbers a and b using an inline function. The use of a function allows for type checking of the arguments as an extra safety feature. On platforms where inline functions are not available the macro GSL_MAX will be automatically substituted.

Function: extern inline double GSL_MIN_DBL (double a, double b)
This function returns the minimum of the double precision numbers a and b using an inline function. The use of a function allows for type checking of the arguments as an extra safety feature. On platforms where inline functions are not available the macro GSL_MIN will be automatically substituted.

Function: extern inline int GSL_MAX_INT (int a, int b)
Function: extern inline int GSL_MIN_INT (int a, int b)
These functions return the maximum or minimum of the integers a and b using an inline function. On platforms where inline functions are not available the macros GSL_MAX or GSL_MIN will be automatically substituted.

Function: extern inline long double GSL_MAX_LDBL (long double a, long double b)
Function: extern inline long double GSL_MIN_LDBL (long double a, long double b)
These functions return the maximum or minimum of the long doubles a and b using an inline function. On platforms where inline functions are not available the macros GSL_MAX or GSL_MIN will be automatically substituted.

Complex Numbers

The functions described in this chapter provide support for complex numbers. The algorithms take care to avoid unnecessary intermediate underflows and overflows, allowing the functions to evaluated over the as much of the complex plane as possible.

For multiple-valued functions the branch cuts have been chosen to follow the conventions of Abramowitz and Stegun in the Handbook of Mathematical Functions. The functions return principal values which are the same as those in GNU Calc, which in turn are the same as those in Common Lisp, The Language (Second Edition) (n.b. The second edition uses different definitions from the first edition) and the HP-28/48 series of calculators.

The complex types are defined in the header file `gsl_complex.h', while the corresponding complex functions and arithmetic operations are defined in `gsl_complex_math.h'.

Complex numbers

Complex numbers are represented using the type gsl_complex. The internal representation of this type may vary across platforms and should not be accessed directly. The functions and macros described below allow complex numbers to be manipulated in a portable way.

For reference, the default form of the gsl_complex type is given by the following struct,

typedef struct
{
  double dat[2];
} gsl_complex;

The real and imaginary part are stored in contiguous elements of a two element array. This eliminates any padding between the real and imaginary parts, dat[0] and dat[1], allowing the struct to be mapped correctly onto packed complex arrays.

Function: gsl_complex gsl_complex_rect (double x, double y)
This function uses the rectangular cartesian components (x,y) to return the complex number z = x + i y.

Function: gsl_complex gsl_complex_polar (double r, double theta)
This function returns the complex number z = r \exp(i \theta) = r (\cos(\theta) + i \sin(\theta)) from the polar representation (r,theta).

Macro: GSL_REAL (z)
Macro: GSL_IMAG (z)
These macros return the real and imaginary parts of the complex number z.

Macro: GSL_SET_COMPLEX (zp, x, y)
This macro uses the cartesian components (x,y) to set the real and imaginary parts of the complex number pointed to by zp. For example,
GSL_SET_COMPLEX(&z, 3, 4)

sets z to be 3 + 4i.

Macro: GSL_SET_REAL (zp,x)
Macro: GSL_SET_IMAG (zp,y)
These macros allow the real and imaginary parts of the complex number pointed to by zp to be set independently.

Properties of complex numbers

Function: double gsl_complex_arg (gsl_complex z)
This function returns the argument of the complex number z, \arg(z), where @c{$-\pi < \arg(z) \leq \pi$} -\pi < \arg(z) <= \pi.

Function: double gsl_complex_abs (gsl_complex z)
This function returns the magnitude of the complex number z, |z|.

Function: double gsl_complex_abs2 (gsl_complex z)
This function returns the squared magnitude of the complex number z, |z|^2.

Function: double gsl_complex_logabs (gsl_complex z)
This function returns the natural logarithm of the magnitude of the complex number z, \log|z|. It allows an accurate evaluation of \log|z| when |z| is close to one. The direct evaluation of log(gsl_complex_abs(z)) would lead to a loss of precision in this case.

Complex arithmetic operators

Function: gsl_complex gsl_complex_add (gsl_complex a, gsl_complex b)
This function returns the sum of the complex numbers a and b, z=a+b.

Function: gsl_complex gsl_complex_sub (gsl_complex a, gsl_complex b)
This function returns the difference of the complex numbers a and b, z=a-b.

Function: gsl_complex gsl_complex_mul (gsl_complex a, gsl_complex b)
This function returns the product of the complex numbers a and b, z=ab.

Function: gsl_complex gsl_complex_div (gsl_complex a, gsl_complex b)
This function returns the quotient of the complex numbers a and b, z=a/b.

Function: gsl_complex gsl_complex_add_real (gsl_complex a, double x)
This function returns the sum of the complex number a and the real number x, z=a+x.

Function: gsl_complex gsl_complex_sub_real (gsl_complex a, double x)
This function returns the difference of the complex number a and the real number x, z=a-x.

Function: gsl_complex gsl_complex_mul_real (gsl_complex a, double x)
This function returns the product of the complex number a and the real number x, z=ax.

Function: gsl_complex gsl_complex_div_real (gsl_complex a, double x)
This function returns the quotient of the complex number a and the real number x, z=a/x.

Function: gsl_complex gsl_complex_add_imag (gsl_complex a, double y)
This function returns the sum of the complex number a and the imaginary number iy, z=a+iy.

Function: gsl_complex gsl_complex_sub_imag (gsl_complex a, double y)
This function returns the difference of the complex number a and the imaginary number iy, z=a-iy.

Function: gsl_complex gsl_complex_mul_imag (gsl_complex a, double y)
This function returns the product of the complex number a and the imaginary number iy, z=a*(iy).

Function: gsl_complex gsl_complex_div_imag (gsl_complex a, double y)
This function returns the quotient of the complex number a and the imaginary number iy, z=a/(iy).

Function: gsl_complex gsl_complex_conjugate (gsl_complex z)
This function returns the complex conjugate of the complex number z, z^* = x - i y.

Function: gsl_complex gsl_complex_inverse (gsl_complex z)
This function returns the inverse, or reciprocal, of the complex number z, 1/z = (x - i y)/(x^2 + y^2).

Function: gsl_complex gsl_complex_negative (gsl_complex z)
This function returns the negative of the complex number z, -z = (-x) + i(-y).

Elementary Complex Functions

Function: gsl_complex gsl_complex_sqrt (gsl_complex z)
This function returns the square root of the complex number z, \sqrt z. The branch cut is the negative real axis. The result always lies in the right half of the complex plane.

Function: gsl_complex gsl_complex_sqrt_real (double x)
This function returns the complex square root of the real number x, where x may be negative.

Function: gsl_complex gsl_complex_pow (gsl_complex z, gsl_complex a)
The function returns the complex number z raised to the complex power a, z^a. This is computed as \exp(\log(z)*a) using complex logarithms and complex exponentials.

Function: gsl_complex gsl_complex_pow_real (gsl_complex z, double x)
This function returns the complex number z raised to the real power x, z^x.

Function: gsl_complex gsl_complex_exp (gsl_complex z)
This function returns the complex exponential of the complex number z, \exp(z).

Function: gsl_complex gsl_complex_log (gsl_complex z)
This function returns the complex natural logarithm (base e) of the complex number z, \log(z). The branch cut is the negative real axis.

Function: gsl_complex gsl_complex_log10 (gsl_complex z)
This function returns the complex base-10 logarithm of the complex number z, @c{$\log_{10}(z)$} \log_10 (z).

Function: gsl_complex gsl_complex_log_b (gsl_complex z, gsl_complex b)
This function returns the complex base-b logarithm of the complex number z, \log_b(z). This quantity is computed as the ratio \log(z)/\log(b).

Complex Trigonometric Functions

Function: gsl_complex gsl_complex_sin (gsl_complex z)
This function returns the complex sine of the complex number z, \sin(z) = (\exp(iz) - \exp(-iz))/(2i).

Function: gsl_complex gsl_complex_cos (gsl_complex z)
This function returns the complex cosine of the complex number z, \cos(z) = (\exp(iz) + \exp(-iz))/2.

Function: gsl_complex gsl_complex_tan (gsl_complex z)
This function returns the complex tangent of the complex number z, \tan(z) = \sin(z)/\cos(z).

Function: gsl_complex gsl_complex_sec (gsl_complex z)
This function returns the complex secant of the complex number z, \sec(z) = 1/\cos(z).

Function: gsl_complex gsl_complex_csc (gsl_complex z)
This function returns the complex cosecant of the complex number z, \csc(z) = 1/\sin(z).

Function: gsl_complex gsl_complex_cot (gsl_complex z)
This function returns the complex cotangent of the complex number z, \cot(z) = 1/\tan(z).

Inverse Complex Trigonometric Functions

Function: gsl_complex gsl_complex_arcsin (gsl_complex z)
This function returns the complex arcsine of the complex number z, \arcsin(z). The branch cuts are on the real axis, less than -1 and greater than 1.

Function: gsl_complex gsl_complex_arcsin_real (double z)
This function returns the complex arcsine of the real number z, \arcsin(z). For z between -1 and 1, the function returns a real value in the range (-\pi,\pi]. For z less than -1 the result has a real part of -\pi/2 and a positive imaginary part. For z greater than 1 the result has a real part of \pi/2 and a negative imaginary part.

Function: gsl_complex gsl_complex_arccos (gsl_complex z)
This function returns the complex arccosine of the complex number z, \arccos(z). The branch cuts are on the real axis, less than -1 and greater than 1.

Function: gsl_complex gsl_complex_arccos_real (double z)
This function returns the complex arccosine of the real number z, \arccos(z). For z between -1 and 1, the function returns a real value in the range [0,\pi]. For z less than -1 the result has a real part of \pi/2 and a negative imaginary part. For z greater than 1 the result is purely imaginary and positive.

Function: gsl_complex gsl_complex_arctan (gsl_complex z)
This function returns the complex arctangent of the complex number z, \arctan(z). The branch cuts are on the imaginary axis, below -i and above i.

Function: gsl_complex gsl_complex_arcsec (gsl_complex z)
This function returns the complex arcsecant of the complex number z, \arcsec(z) = \arccos(1/z).

Function: gsl_complex gsl_complex_arcsec_real (double z)
This function returns the complex arcsecant of the real number z, \arcsec(z) = \arccos(1/z).

Function: gsl_complex gsl_complex_arccsc (gsl_complex z)
This function returns the complex arccosecant of the complex number z, \arccsc(z) = \arcsin(1/z).

Function: gsl_complex gsl_complex_arccsc_real (double z)
This function returns the complex arccosecant of the real number z, \arccsc(z) = \arcsin(1/z).

Function: gsl_complex gsl_complex_arccot (gsl_complex z)
This function returns the complex arccotangent of the complex number z, \arccot(z) = \arctan(1/z).

Complex Hyperbolic Functions

Function: gsl_complex gsl_complex_sinh (gsl_complex z)
This function returns the complex hyperbolic sine of the complex number z, \sinh(z) = (\exp(z) - \exp(-z))/2.

Function: gsl_complex gsl_complex_cosh (gsl_complex z)
This function returns the complex hyperbolic cosine of the complex number z, \cosh(z) = (\exp(z) + \exp(-z))/2.

Function: gsl_complex gsl_complex_tanh (gsl_complex z)
This function returns the complex hyperbolic tangent of the complex number z, \tanh(z) = \sinh(z)/\cosh(z).

Function: gsl_complex gsl_complex_sech (gsl_complex z)
This function returns the complex hyperbolic secant of the complex number z, \sech(z) = 1/\cosh(z).

Function: gsl_complex gsl_complex_csch (gsl_complex z)
This function returns the complex hyperbolic cosecant of the complex number z, \csch(z) = 1/\sinh(z).

Function: gsl_complex gsl_complex_coth (gsl_complex z)
This function returns the complex hyperbolic cotangent of the complex number z, \coth(z) = 1/\tanh(z).

Inverse Complex Hyperbolic Functions

Function: gsl_complex gsl_complex_arcsinh (gsl_complex z)
This function returns the complex hyperbolic arcsine of the complex number z, \arcsinh(z). The branch cuts are on the imaginary axis, below -i and above i.

Function: gsl_complex gsl_complex_arccosh (gsl_complex z)
This function returns the complex hyperbolic arccosine of the complex number z, \arccosh(z). The branch cut is on the real axis, less than 1.

Function: gsl_complex gsl_complex_arccosh_real (double z)
This function returns the complex hyperbolic arccosine of the real number z, \arccosh(z).

Function: gsl_complex gsl_complex_arctanh (gsl_complex z)
This function returns the complex hyperbolic arctangent of the complex number z, \arctanh(z). The branch cuts are on the real axis, less than -1 and greater than 1.

Function: gsl_complex gsl_complex_arctanh_real (double z)
This function returns the complex hyperbolic arctangent of the real number z, \arctanh(z).

Function: gsl_complex gsl_complex_arcsech (gsl_complex z)
This function returns the complex hyperbolic arcsecant of the complex number z, \arcsech(z) = \arccosh(1/z).

Function: gsl_complex gsl_complex_arccsch (gsl_complex z)
This function returns the complex hyperbolic arccosecant of the complex number z, \arccsch(z) = \arcsin(1/z).

Function: gsl_complex gsl_complex_arccoth (gsl_complex z)
This function returns the complex hyperbolic arccotangent of the complex number z, \arccoth(z) = \arctanh(1/z).

References and Further Reading

The implementations of the elementary and trigonometric functions are based on the following papers,

The general formulas and details of branch cuts can be found in the following books,

Roots of Polynomials

This chapter describes functions for evaluating and solving polynomials. There are routines for finding real and complex roots of quadratic and cubic equations using analytic methods. An iterative polynomial solver is also available for finding the roots of general polynomials with real coefficients (of any order). The functions are declared in the header file gsl_poly.h.

Polynomial evaluation

Function: double gsl_poly_eval (const double c[], const int len, const double x)
This function evaluates the polynomial c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^{len-1} using Horner's method for stability. The function is inlined when possible.

Quadratic equations

Function: int gsl_poly_solve_quadratic (double a, double b, double c, double *x0, double *x1)
This function finds the real roots of the quadratic equation,

The number of real roots (either zero or two) is returned, and their locations are stored in x0 and x1. If no real roots are found then x0 and x1 are not modified. When two real roots are found they are stored in x0 and x1 in ascending order. The case of coincident roots is not considered special. For example (x-1)^2=0 will have two roots, which happen to have exactly equal values.

The number of roots found depends on the sign of the discriminant b^2 - 4 a c. This will be subject to rounding and cancellation errors when computed in double precision, and will also be subject to errors if the coefficients of the polynomial are inexact. These errors may cause a discrete change in the number of roots. However, for polynomials with small integer coefficients the discriminant can always be computed exactly.

Function: int gsl_poly_complex_solve_quadratic (double a, double b, double c, gsl_complex *z0, gsl_complex *z1)

This function finds the complex roots of the quadratic equation,

The number of complex roots is returned (always two) and the locations of the roots are stored in z0 and z1. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components.

Cubic equations

Function: int gsl_poly_solve_cubic (double a, double b, double c, double *x0, double *x1, double *x2)

This function finds the real roots of the cubic equation,

with a leading coefficient of unity. The number of real roots (either one or three) is returned, and their locations are stored in x0, x1 and x2. If one real root is found then only x0 is modified. When three real roots are found they are stored in x0, x1 and x2 in ascending order. The case of coincident roots is not considered special. For example, the equation (x-1)^3=0 will have three roots with exactly equal values.

Function: int gsl_poly_complex_solve_cubic (double a, double b, double c, gsl_complex *z0, gsl_complex *z1, gsl_complex *z2)

This function finds the complex roots of the cubic equation,

The number of complex roots is returned (always three) and the locations of the roots are stored in z0, z1 and z2. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components.

General polynomial equations

The roots of polynomial equations cannot be found analytically beyond the special cases of the quadratic, cubic and quartic equation. The algorithm described in this section uses an iterative method to find the approximate locations of roots of higher order polynomials.

Function: gsl_poly_complex_workspace * gsl_poly_complex_workspace_alloc (size_t n)
This function allocates space for a gsl_poly_complex_workspace struct and a workspace suitable for solving a polynomial with n coefficients using the routine gsl_poly_complex_solve.

The function returns a pointer to the newly allocated gsl_poly_complex_workspace if no errors were detected, and a null pointer in the case of error.

Function: void gsl_poly_complex_workspace_free (gsl_poly_complex_workspace * w)
This function frees all the memory associated with the workspace w.

Function: int gsl_poly_complex_solve (const double * a, size_t n, gsl_poly_complex_workspace * w, gsl_complex_packed_ptr z)
This function computes the roots of the general polynomial P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1} using balanced-QR reduction of the companion matrix. The parameter n specifies the length of the coefficient array. The coefficient of the highest order term must be non-zero. The function requires a workspace w of the appropriate size. The n-1 roots are returned in the packed complex array z of length 2(n-1), alternating real and imaginary parts.

The function returns GSL_SUCCESS if all the roots are found and GSL_EFAILED if the QR reduction does not converge.

Examples

To demonstrate the use of the general polynomial solver we will take the polynomial P(x) = x^5 - 1 which has the following roots,

The following program will find these roots.

#include <stdio.h>
#include <gsl/gsl_poly.h>

int
main (void)
{
  int i;
  /* coefficient of P(x) =  -1 + x^5  */
  double a[6] = { -1, 0, 0, 0, 0, 1 };  
  double z[10];

  gsl_poly_complex_workspace * w 
      = gsl_poly_complex_workspace_alloc (6);
  
  gsl_poly_complex_solve (a, 6, w, z);

  gsl_poly_complex_workspace_free (w);

  for (i = 0; i < 5; i++)
    {
      printf("z%d = %+.18f %+.18f\n", 
             i, z[2*i], z[2*i+1]);
    }

  return 0;
}

The output of the program is,

bash$ ./a.out 
z0 = -0.809016994374947451 +0.587785252292473137
z1 = -0.809016994374947451 -0.587785252292473137
z2 = +0.309016994374947451 +0.951056516295153642
z3 = +0.309016994374947451 -0.951056516295153642
z4 = +1.000000000000000000 +0.000000000000000000

which agrees with the analytic result, z_n = \exp(2 \pi n i/5).

References and Further Reading

The balanced-QR method and its error analysis is described in the following papers.

Special Functions

This chapter describes the GSL special function library. The library includes routines for calculating the values of Airy functions, Bessel functions, Clausen functions, Coulomb wave functions, Coupling coefficients, the Dawson function, Debye functions, Dilogarithms, Elliptic integrals, Jacobi elliptic functions, Error functions, Exponential integrals, Fermi-Dirac functions, Gamma functions, Gegenbauer functions, Hypergeometric functions, Laguerre functions, Legendre functions and Spherical Harmonics, the Psi (Digamma) Function, Synchrotron functions, Transport functions, Trigonometric functions and Zeta functions. Each routine also computes an estimate of the numerical error in the calculated value of the function.

The functions are declared in individual header files, such as `gsl_sf_airy.h', `gsl_sf_bessel.h', etc. The complete set of header files can be included using the file `gsl_sf.h'.

Usage

The special functions are available in two calling conventions, a natural form which returns the numerical value of the function and an error-handling form which returns an error code. The two types of function provide alternative ways of accessing the same underlying code.

The natural form returns only the value of the function and can be used directly in mathematical expressions.. For example, the following function call will compute the value of the Bessel function J_0(x),

double y = gsl_sf_bessel_J0 (x);

There is no way to access an error code or to estimate the error using this method. To allow access to this information the alternative error-handling form stores the value and error in a modifiable argument,

gsl_sf_result result;
int status = gsl_sf_bessel_J0_e (x, &result);

The error-handling functions have the suffix _e. The returned status value indicates error conditions such as overflow, underflow or loss of precision. If there are no errors the error-handling functions return GSL_SUCCESS.

The gsl_sf_result struct

The error handling form of the special functions always calculate an error estimate along with the value of the result. Therefore, structures are provided for amalgamating a value and error estimate. These structures are declared in the header file `gsl_sf_result.h'.

The gsl_sf_result struct contains value and error fields.

typedef struct
{
  double val;
  double err;
} gsl_sf_result;

The field val contains the value and the field err contains an estimate of the absolute error in the value.

In some cases, an overflow or underflow can be detected and handled by a function. In this case, it may be possible to return a scaling exponent as well as an error/value pair in order to save the result from exceeding the dynamic range of the built-in types. The gsl_sf_result_e10 struct contains value and error fields as well as an exponent field such that the actual result is obtained as result * 10^(e10).

typedef struct
{
  double val;
  double err;
  int    e10;
} gsl_sf_result_e10;

Modes

The goal of the library is to achieve double precision accuracy wherever possible. However the cost of evaluating some special functions to double precision can be significant, particularly where very high order terms are required. In these cases a mode argument allows the accuracy of the function to be reduced in order to improve performance. The following precision levels are available for the mode argument,

GSL_PREC_DOUBLE
Double-precision, a relative accuracy of approximately @c{$2 \times 10^{-16}$} 2 * 10^-16.
GSL_PREC_SINGLE
Single-precision, a relative accuracy of approximately @c{$1 \times 10^{-7}$} 10^-7.
GSL_PREC_APPROX
Approximate values, a relative accuracy of approximately @c{$5 \times 10^{-4}$} 5 * 10^-4.

The approximate mode provides the fastest evaluation at the lowest accuracy.

Airy Functions and Derivatives

The Airy functions Ai(x) and Bi(x) are defined by the integral representations,

For further information see Abramowitz & Stegun, Section 10.4. The Airy functions are defined in the header file `gsl_sf_airy.h'.

Airy Functions

Function: double gsl_sf_airy_Ai (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Ai_e (double x, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the Airy function Ai(x) with an accuracy specified by mode.

Function: double gsl_sf_airy_Bi (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Bi_e (double x, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the Airy function Bi(x) with an accuracy specified by mode.

Function: double gsl_sf_airy_Ai_scaled (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Ai_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
These routines compute a scaled version of the Airy function S_A(x) Ai(x). For x>0 the scaling factor S_A(x) is @c{$\exp(+(2/3) x^{3/2})$} \exp(+(2/3) x^(3/2)), and is 1 for x<0.

Function: double gsl_sf_airy_Bi_scaled (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Bi_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
These routines compute a scaled version of the Airy function S_B(x) Bi(x). For x>0 the scaling factor S_B(x) is @c{$\exp(-(2/3) x^{3/2})$} exp(-(2/3) x^(3/2)), and is 1 for x<0.

Derivatives of Airy Functions

Function: double gsl_sf_airy_Ai_deriv (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Ai_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the Airy function derivative Ai'(x) with an accuracy specified by mode.

Function: double gsl_sf_airy_Bi_deriv (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Bi_deriv_e (double x, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the Airy function derivative Bi'(x) with an accuracy specified by mode.

Function: double gsl_sf_airy_Ai_deriv_scaled (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Ai_deriv_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the derivative of the scaled Airy function S_A(x) Ai(x).

Function: double gsl_sf_airy_Bi_deriv_scaled (double x, gsl_mode_t mode)
Function: int gsl_sf_airy_Bi_deriv_scaled_e (double x, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the derivative of the scaled Airy function S_B(x) Bi(x).

Zeros of Airy Functions

Function: double gsl_sf_airy_zero_Ai (unsigned int s)
Function: int gsl_sf_airy_zero_Ai_e (unsigned int s, gsl_sf_result * result)
These routines compute the location of the s-th zero of the Airy function Ai(x).

Function: double gsl_sf_airy_zero_Bi (unsigned int s)
Function: int gsl_sf_airy_zero_Bi_e (unsigned int s, gsl_sf_result * result)
These routines compute the location of the s-th zero of the Airy function Bi(x).

Zeros of Derivatives of Airy Functions

Function: double gsl_sf_airy_zero_Ai_deriv (unsigned int s)
Function: int gsl_sf_airy_zero_Ai_deriv_e (unsigned int s, gsl_sf_result * result)
These routines compute the location of the s-th zero of the Airy function derivative Ai'(x).

Function: double gsl_sf_airy_zero_Bi_deriv (unsigned int s)
Function: int gsl_sf_airy_zero_Bi_deriv_e (unsigned int s, gsl_sf_result * result)
These routines compute the location of the s-th zero of the Airy function derivative Bi'(x).

Bessel Functions

The routines described in this section compute the Cylindrical Bessel functions J_n(x), Y_n(x), Modified cylindrical Bessel functions I_n(x), K_n(x), Spherical Bessel functions j_l(x), y_l(x), and Modified Spherical Bessel functions i_l(x), k_l(x). For more information see Abramowitz & Stegun, Chapters 9 and 10. The Bessel functions are defined in the header file `gsl_sf_bessel.h'.

Regular Cylindrical Bessel Functions

Function: double gsl_sf_bessel_J0 (double x)
Function: int gsl_sf_bessel_J0_e (double x, gsl_sf_result * result)
These routines compute the regular cylindrical Bessel function of zeroth order, J_0(x).

Function: double gsl_sf_bessel_J1 (double x)
Function: int gsl_sf_bessel_J1_e (double x, gsl_sf_result * result)
These routines compute the regular cylindrical Bessel function of first order, J_1(x).

Function: double gsl_sf_bessel_Jn (int n, double x)
Function: int gsl_sf_bessel_Jn_e (int n, double x, gsl_sf_result * result)
These routines compute the regular cylindrical Bessel function of order n, J_n(x).

Function: int gsl_sf_bessel_Jn_array (int nmin, int nmax, double x, double result_array[])
This routine computes the values of the regular cylindrical Bessel functions J_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Irregular Cylindrical Bessel Functions

Function: double gsl_sf_bessel_Y0 (double x)
Function: int gsl_sf_bessel_Y0_e (double x, gsl_sf_result * result)
These routines compute the irregular cylindrical Bessel function of zeroth order, Y_0(x), for x>0.

Function: double gsl_sf_bessel_Y1 (double x)
Function: int gsl_sf_bessel_Y1_e (double x, gsl_sf_result * result)
These routines compute the irregular cylindrical Bessel function of first order, Y_1(x), for x>0.

Function: double gsl_sf_bessel_Yn (int n,double x)
Function: int gsl_sf_bessel_Yn_e (int n,double x, gsl_sf_result * result)
These routines compute the irregular cylindrical Bessel function of order n, Y_n(x), for x>0.

Function: int gsl_sf_bessel_Yn_array (int nmin, int nmax, double x, double result_array[])
This routine computes the values of the irregular cylindrical Bessel functions Y_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The domain of the function is x>0. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Regular Modified Cylindrical Bessel Functions

Function: double gsl_sf_bessel_I0 (double x)
Function: int gsl_sf_bessel_I0_e (double x, gsl_sf_result * result)
These routines compute the regular modified cylindrical Bessel function of zeroth order, I_0(x).

Function: double gsl_sf_bessel_I1 (double x)
Function: int gsl_sf_bessel_I1_e (double x, gsl_sf_result * result)
These routines compute the regular modified cylindrical Bessel function of first order, I_1(x).

Function: double gsl_sf_bessel_In (int n, double x)
Function: int gsl_sf_bessel_In_e (int n, double x, gsl_sf_result * result)
These routines compute the regular modified cylindrical Bessel function of order n, I_n(x).

Function: int gsl_sf_bessel_In_array (int nmin, int nmax, double x, double result_array[])
This routine computes the values of the regular modified cylindrical Bessel functions I_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive or zero. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Function: double gsl_sf_bessel_I0_scaled (double x)
Function: int gsl_sf_bessel_I0_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled regular modified cylindrical Bessel function of zeroth order \exp(-|x|) I_0(x).

Function: double gsl_sf_bessel_I1_scaled (double x)
Function: int gsl_sf_bessel_I1_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled regular modified cylindrical Bessel function of first order \exp(-|x|) I_1(x).

Function: double gsl_sf_bessel_In_scaled (int n, double x)
Function: int gsl_sf_bessel_In_scaled_e (int n, double x, gsl_sf_result * result)
These routines compute the scaled regular modified cylindrical Bessel function of order n, \exp(-|x|) I_n(x)

Function: int gsl_sf_bessel_In_scaled_array (int nmin, int nmax, double x, double result_array[])
This routine computes the values of the scaled regular cylindrical Bessel functions \exp(-|x|) I_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive or zero. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Irregular Modified Cylindrical Bessel Functions

Function: double gsl_sf_bessel_K0 (double x)
Function: int gsl_sf_bessel_K0_e (double x, gsl_sf_result * result)
These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0.

Function: double gsl_sf_bessel_K1 (double x)
Function: int gsl_sf_bessel_K1_e (double x, gsl_sf_result * result)
These routines compute the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0.

Function: double gsl_sf_bessel_Kn (int n, double x)
Function: int gsl_sf_bessel_Kn_e (int n, double x, gsl_sf_result * result)
These routines compute the irregular modified cylindrical Bessel function of order n, K_n(x), for x > 0.

Function: int gsl_sf_bessel_Kn_array (int nmin, int nmax, double x, double result_array[])
This routine computes the values of the irregular modified cylindrical Bessel functions K_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive or zero. The domain of the function is x>0. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Function: double gsl_sf_bessel_K0_scaled (double x)
Function: int gsl_sf_bessel_K0_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order \exp(x) K_0(x) for x>0.

Function: double gsl_sf_bessel_K1_scaled (double x)
Function: int gsl_sf_bessel_K1_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled irregular modified cylindrical Bessel function of first order \exp(x) K_1(x) for x>0.

Function: double gsl_sf_bessel_Kn_scaled (int n, double x)
Function: int gsl_sf_bessel_Kn_scaled_e (int n, double x, gsl_sf_result * result)
These routines compute the scaled irregular modified cylindrical Bessel function of order n, \exp(x) K_n(x), for x>0.

Function: int gsl_sf_bessel_Kn_scaled_array (int nmin, int nmax, double x, double result_array[])
This routine computes the values of the scaled irregular cylindrical Bessel functions \exp(x) K_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive or zero. The domain of the function is x>0. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Regular Spherical Bessel Functions

Function: double gsl_sf_bessel_j0 (double x)
Function: int gsl_sf_bessel_j0_e (double x, gsl_sf_result * result)
These routines compute the regular spherical Bessel function of zeroth order, j_0(x) = \sin(x)/x.

Function: double gsl_sf_bessel_j1 (double x)
Function: int gsl_sf_bessel_j1_e (double x, gsl_sf_result * result)
These routines compute the regular spherical Bessel function of first order, j_1(x) = (\sin(x)/x - \cos(x))/x.

Function: double gsl_sf_bessel_j2 (double x)
Function: int gsl_sf_bessel_j2_e (double x, gsl_sf_result * result)
These routines compute the regular spherical Bessel function of second order, j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x.

Function: double gsl_sf_bessel_jl (int l, double x)
Function: int gsl_sf_bessel_jl_e (int l, double x, gsl_sf_result * result)
These routines compute the regular spherical Bessel function of order l, j_l(x), for @c{$l \geq 0$} l >= 0 and @c{$x \geq 0$} x >= 0.

Function: int gsl_sf_bessel_jl_array (int lmax, double x, double result_array[])
This routine computes the values of the regular spherical Bessel functions j_l(x) for l from 0 to lmax inclusive for @c{$lmax \geq 0$} lmax >= 0 and @c{$x \geq 0$} x >= 0, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Function: int gsl_sf_bessel_jl_steed_array (int lmax, double x, double * jl_x_array)
This routine uses Steed's method to compute the values of the regular spherical Bessel functions j_l(x) for l from 0 to lmax inclusive for @c{$lmax \geq 0$} lmax >= 0 and @c{$x \geq 0$} x >= 0, storing the results in the array result_array. The Steed/Barnett algorithm is described in Comp. Phys. Comm. 21, 297 (1981). Steed's method is more stable than the recurrence used in the other functions but is also slower.

Irregular Spherical Bessel Functions

Function: double gsl_sf_bessel_y0 (double x)
Function: int gsl_sf_bessel_y0_e (double x, gsl_sf_result * result)
These routines compute the irregular spherical Bessel function of zeroth order, y_0(x) = -\cos(x)/x.

Function: double gsl_sf_bessel_y1 (double x)
Function: int gsl_sf_bessel_y1_e (double x, gsl_sf_result * result)
These routines compute the irregular spherical Bessel function of first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.

Function: double gsl_sf_bessel_y2 (double x)
Function: int gsl_sf_bessel_y2_e (double x, gsl_sf_result * result)
These routines compute the irregular spherical Bessel function of second order, y_2(x) = (-3/x^2 + 1/x)\cos(x) - (3/x^2)\sin(x).

Function: double gsl_sf_bessel_yl (int l, double x)
Function: int gsl_sf_bessel_yl_e (int l, double x, gsl_sf_result * result)
These routines compute the irregular spherical Bessel function of order l, y_l(x), for @c{$l \geq 0$} l >= 0.

Function: int gsl_sf_bessel_yl_array (int lmax, double x, double result_array[])
This routine computes the values of the irregular spherical Bessel functions y_l(x) for l from 0 to lmax inclusive for @c{$lmax \geq 0$} lmax >= 0, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Regular Modified Spherical Bessel Functions

The regular modified spherical Bessel functions i_l(x) are related to the modified Bessel functions of fractional order, i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x)

Function: double gsl_sf_bessel_i0_scaled (double x)
Function: int gsl_sf_bessel_i0_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled regular modified spherical Bessel function of zeroth order, \exp(-|x|) i_0(x).

Function: double gsl_sf_bessel_i1_scaled (double x)
Function: int gsl_sf_bessel_i1_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled regular modified spherical Bessel function of first order, \exp(-|x|) i_1(x).

Function: double gsl_sf_bessel_i2_scaled (double x)
Function: int gsl_sf_bessel_i2_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled regular modified spherical Bessel function of second order, \exp(-|x|) i_2(x)

Function: double gsl_sf_bessel_il_scaled (int l, double x)
Function: int gsl_sf_bessel_il_scaled_e (int l, double x, gsl_sf_result * result)
These routines compute the scaled regular modified spherical Bessel function of order l, \exp(-|x|) i_l(x)

Function: int gsl_sf_bessel_il_scaled_array (int lmax, double x, double result_array[])
This routine computes the values of the scaled regular modified cylindrical Bessel functions \exp(-|x|) i_l(x) for l from 0 to lmax inclusive for @c{$lmax \geq 0$} lmax >= 0, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Irregular Modified Spherical Bessel Functions

The irregular modified spherical Bessel functions k_l(x) are related to the irregular modified Bessel functions of fractional order, k_l(x) = \sqrt{\pi/(2x)} K_{l+1/2}(x).

Function: double gsl_sf_bessel_k0_scaled (double x)
Function: int gsl_sf_bessel_k0_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled irregular modified spherical Bessel function of zeroth order, \exp(x) k_0(x), for x>0.

Function: double gsl_sf_bessel_k1_scaled (double x)
Function: int gsl_sf_bessel_k1_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled irregular modified spherical Bessel function of first order, \exp(x) k_1(x), for x>0.

Function: double gsl_sf_bessel_k2_scaled (double x)
Function: int gsl_sf_bessel_k2_scaled_e (double x, gsl_sf_result * result)
These routines compute the scaled irregular modified spherical Bessel function of second order, \exp(x) k_2(x), for x>0.

Function: double gsl_sf_bessel_kl_scaled (int l, double x)
Function: int gsl_sf_bessel_kl_scaled_e (int l, double x, gsl_sf_result * result)
These routines compute the scaled irregular modified spherical Bessel function of order l, \exp(x) k_l(x), for x>0.

Function: int gsl_sf_bessel_kl_scaled_array (int lmax, double x, double result_array[])
This routine computes the values of the scaled irregular modified spherical Bessel functions \exp(x) k_l(x) for l from 0 to lmax inclusive for @c{$lmax \geq 0$} lmax >= 0 and x>0, storing the results in the array result_array. The values are computed using recurrence relations, for efficiency, and therefore may differ slightly from the exact values.

Regular Bessel Function - Fractional Order

Function: double gsl_sf_bessel_Jnu (double nu, double x)
Function: int gsl_sf_bessel_Jnu_e (double nu, double x, gsl_sf_result * result)
These routines compute the regular cylindrical Bessel function of fractional order nu, J_\nu(x).

Function: int gsl_sf_bessel_sequence_Jnu_e (double nu, gsl_mode_t mode, size_t size, double v[])
This function computes the regular cylindrical Bessel function of fractional order \nu, J_\nu(x), evaluated at a series of x values. The array v of length size contains the x values. They are assumed to be strictly ordered and positive. The array is over-written with the values of J_\nu(x_i).

Irregular Bessel Functions - Fractional Order

Function: double gsl_sf_bessel_Ynu (double nu, double x)
Function: int gsl_sf_bessel_Ynu_e (double nu, double x, gsl_sf_result * result)
These routines compute the irregular cylindrical Bessel function of fractional order nu, Y_\nu(x).

Regular Modified Bessel Functions - Fractional Order

Function: double gsl_sf_bessel_Inu (double nu, double x)
Function: int gsl_sf_bessel_Inu_e (double nu, double x, gsl_sf_result * result)
These routines compute the regular modified Bessel function of fractional order nu, I_\nu(x) for x>0, \nu>0.

Function: double gsl_sf_bessel_Inu_scaled (double nu, double x)
Function: int gsl_sf_bessel_Inu_scaled_e (double nu, double x, gsl_sf_result * result)
These routines compute the scaled regular modified Bessel function of fractional order nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.

Irregular Modified Bessel Functions - Fractional Order

Function: double gsl_sf_bessel_Knu (double nu, double x)
Function: int gsl_sf_bessel_Knu_e (double nu, double x, gsl_sf_result * result)
These routines compute the irregular modified Bessel function of fractional order nu, K_\nu(x) for x>0, \nu>0.

Function: double gsl_sf_bessel_lnKnu (double nu, double x)
Function: int gsl_sf_bessel_lnKnu_e (double nu, double x, gsl_sf_result * result)
These routines compute the logarithm of the irregular modified Bessel function of fractional order nu, \ln(K_\nu(x)) for x>0, \nu>0.

Function: double gsl_sf_bessel_Knu_scaled (double nu, double x)
Function: int gsl_sf_bessel_Knu_scaled_e (double nu, double x, gsl_sf_result * result)
These routines compute the scaled irregular modified Bessel function of fractional order nu, \exp(+|x|) K_\nu(x) for x>0, \nu>0.

Zeros of Regular Bessel Functions

Function: double gsl_sf_bessel_zero_J0 (unsigned int s)
Function: int gsl_sf_bessel_zero_J0_e (unsigned int s, gsl_sf_result * result)
These routines compute the location of the s-th positive zero of the Bessel function J_0(x).

Function: double gsl_sf_bessel_zero_J1 (unsigned int s)
Function: int gsl_sf_bessel_zero_J1_e (unsigned int s, gsl_sf_result * result)
These routines compute the location of the s-th positive zero of the Bessel function J_1(x).

Function: double gsl_sf_bessel_zero_Jnu (double nu, unsigned int s)
Function: int gsl_sf_bessel_zero_Jnu_e (double nu, unsigned int s, gsl_sf_result * result)
These routines compute the location of the s-th positive zero of the Bessel function J_\nu(x).

Clausen Functions

The Clausen function is defined by the following integral, It is related to the dilogarithm by Cl_2(\theta) = \Im Li_2(\exp(i \theta)). The Clausen functions are declared in the header file `gsl_sf_clausen.h'.

Function: double gsl_sf_clausen (double x)
Function: int gsl_sf_clausen_e (double x, gsl_sf_result * result)
These routines compute the Clausen integral Cl_2(x).

Coulomb Functions

The Coulomb functions are declared in the header file `gsl_sf_coulomb.h'. Both bound state and scattering solutions are available.

Normalized Hydrogenic Bound States

Function: double gsl_sf_hydrogenicR_1 (double Z, double r)
Function: int gsl_sf_hydrogenicR_1_e (double Z, double r, gsl_sf_result * result)
These routines compute the lowest-order normalized hydrogenic bound state radial wavefunction @c{$R_1 := 2Z \sqrt{Z} \exp(-Z r)$} R_1 := 2Z \sqrt{Z} \exp(-Z r).

Function: double gsl_sf_hydrogenicR (int n, int l, double Z, double r)
Function: int gsl_sf_hydrogenicR_e (int n, int l, double Z, double r, gsl_sf_result * result)
These routines compute the n-th normalized hydrogenic bound state radial wavefunction, The normalization is chosen such that the wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.

Coulomb Wave Functions

The Coulomb wave functions F_L(\eta,x), G_L(\eta,x) are described in Abramowitz & Stegun, Chapter 14. Because there can be a large dynamic range of values for these functions, overflows are handled gracefully. If an overflow occurs, GSL_EOVRFLW is signalled and exponent(s) are returned through the modifiable parameters exp_F, exp_G. The full solution can be reconstructed from the following relations,

Function: int gsl_sf_coulomb_wave_FG_e (double eta, double x, double L_F, int k, gsl_sf_result * F, gsl_sf_result * Fp, gsl_sf_result * G, gsl_sf_result * Gp, double * exp_F, double * exp_G)
This function computes the coulomb wave functions F_L(\eta,x), G_{L-k}(\eta,x) and their derivatives with respect to x, F'_L(\eta,x) G'_{L-k}(\eta,x). The parameters are restricted to L, L-k > -1/2, x > 0 and integer k. Note that L itself is not restricted to being an integer. The results are stored in the parameters F, G for the function values and Fp, Gp for the derivative values. If an overflow occurs, GSL_EOVRFLW is returned and scaling exponents are stored in the modifiable parameters exp_F, exp_G.

Function: int gsl_sf_coulomb_wave_F_array (double L_min, int kmax, double eta, double x, double fc_array[], double * F_exponent)
This function computes the function F_L(eta,x) for L = Lmin \dots Lmin + kmax storing the results in fc_array. In the case of overflow the exponent is stored in F_exponent.

Function: int gsl_sf_coulomb_wave_FG_array (double L_min, int kmax, double eta, double x, double fc_array[], double gc_array[], double * F_exponent, double * G_exponent)
This function computes the functions F_L(\eta,x), G_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the results in fc_array and gc_array. In the case of overflow the exponents are stored in F_exponent and G_exponent.

Function: int gsl_sf_coulomb_wave_FGp_array (double L_min, int kmax, double eta, double x, double fc_array[], double fcp_array[], double gc_array[], double gcp_array[], double * F_exponent, double * G_exponent)
This function computes the functions F_L(\eta,x), G_L(\eta,x) and their derivatives F'_L(\eta,x), G'_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the results in fc_array, gc_array, fcp_array and gcp_array. In the case of overflow the exponents are stored in F_exponent and G_exponent.

Function: int gsl_sf_coulomb_wave_sphF_array (double L_min, int kmax, double eta, double x, double fc_array[], double F_exponent[])
This function computes the Coulomb wave function divided by the argument F_L(\eta, x)/x for L = Lmin \dots Lmin + kmax, storing the results in fc_array. In the case of overflow the exponent is stored in F_exponent. This function reduces to spherical Bessel functions in the limit \eta \to 0.

Coulomb Wave Function Normalization Constant

The Coulomb wave function normalization constant is defined in Abramowitz 14.1.7.

Function: int gsl_sf_coulomb_CL_e (double L, double eta, gsl_sf_result * result)
This function computes the Coulomb wave function normalization constant C_L(\eta) for L > -1.

Function: int gsl_sf_coulomb_CL_array (double Lmin, int kmax, double eta, double cl[])
This function computes the coulomb wave function normalization constant C_L(\eta) for L = Lmin \dots Lmin + kmax, Lmin > -1.

Coupling Coefficients

The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for combined angular momentum vectors. Since the arguments of the standard coupling coefficient functions are integer or half-integer, the arguments of the following functions are, by convention, integers equal to twice the actual spin value. For information on the 3-j coefficients see Abramowitz & Stegun, Section 27.9. The functions described in this section are declared in the header file `gsl_sf_coupling.h'.

3-j Symbols

Function: double gsl_sf_coupling_3j (int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc)
Function: int gsl_sf_coupling_3j_e (int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc, gsl_sf_result * result)
These routines compute the Wigner 3-j coefficient,

where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.

6-j Symbols

Function: double gsl_sf_coupling_6j (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf)
Function: int gsl_sf_coupling_6j_e (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, gsl_sf_result * result)
These routines compute the Wigner 6-j coefficient,

where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.

9-j Symbols

Function: double gsl_sf_coupling_9j (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji)
Function: int gsl_sf_coupling_9j_e (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji, gsl_sf_result * result)
These routines compute the Wigner 9-j coefficient,

where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.

Dawson Function

The Dawson integral is defined by \exp(-x^2) \int_0^x dt \exp(t^2). A table of Dawson's integral can be found in Abramowitz & Stegun, Table 7.5. The Dawson functions are declared in the header file `gsl_sf_dawson.h'.

Function: double gsl_sf_dawson (double x)
Function: int gsl_sf_dawson_e (double x, gsl_sf_result * result)
These routines compute the value of Dawson's integral for x.

Debye Functions

The Debye functions are defined by the integral D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1)). For further information see Abramowitz & Stegun, Section 27.1. The Debye functions are declared in the header file `gsl_sf_debye.h'.

Function: double gsl_sf_debye_1 (double x)
Function: int gsl_sf_debye_1_e (double x, gsl_sf_result * result)
These routines compute the first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1)).

Function: double gsl_sf_debye_2 (double x)
Function: int gsl_sf_debye_2_e (double x, gsl_sf_result * result)
These routines compute the second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1)).

Function: double gsl_sf_debye_3 (double x)
Function: int gsl_sf_debye_3_e (double x, gsl_sf_result * result)
These routines compute the third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1)).

Function: double gsl_sf_debye_4 (double x)
Function: int gsl_sf_debye_4_e (double x, gsl_sf_result * result)
These routines compute the fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1)).

Dilogarithm

The functions described in this section are declared in the header file `gsl_sf_dilog.h'.

Real Argument

Function: double gsl_sf_dilog (double x)
Function: int gsl_sf_dilog_e (double x, gsl_sf_result * result)
These routines compute the dilogarithm for a real argument. In Lewin's notation this is Li_2(x), the real part of the dilogarithm of a real x. It is defined by the integral representation Li_2(x) = - \Re \int_0^x ds \log(1-s) / s. Note that \Im(Li_2(x)) = 0 for @c{$x \le 1$} x <= 1, and -\pi\log(x) for x > 1.

Complex Argument

Function: int gsl_sf_complex_dilog_e (double r, double theta, gsl_sf_result * result_re, gsl_sf_result * result_im)
This function computes the full complex-valued dilogarithm for the complex argument z = r \exp(i \theta). The real and imaginary parts of the result are returned in result_re, result_im.

Elementary Operations

The following functions allow for the propagation of errors when combining quantities by multiplication. The functions are declared in the header file `gsl_sf_elementary.h'.

Function: int gsl_sf_multiply_e (double x, double y, gsl_sf_result * result)
This function multiplies x and y storing the product and its associated error in result.

Function: int gsl_sf_multiply_err_e (double x, double dx, double y, double dy, gsl_sf_result * result)
This function multiplies x and y with associated absolute errors dx and dy. The product xy +/- xy \sqrt((dx/x)^2 +(dy/y)^2) is stored in result.

Elliptic Integrals

The functions described in this section are declared in the header file `gsl_sf_ellint.h'.

Definition of Legendre Forms

The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and P(\phi,k,n) are defined by,

The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k) = E(\pi/2, k). Further information on the Legendre forms of elliptic integrals can be found in Abramowitz & Stegun, Chapter 17. The notation used here is based on Carlson, Numerische Mathematik 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun.

Definition of Carlson Forms

The Carlson symmetric forms of elliptical integrals RC(x,y), RD(x,y,z), RF(x,y,z) and RJ(x,y,z,p) are defined by,

Legendre Form of Complete Elliptic Integrals

Function: double gsl_sf_ellint_Kcomp (double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_Kcomp_e (double k, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the complete elliptic integral K(k) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_Ecomp (double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_Ecomp_e (double k, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode.

Legendre Form of Incomplete Elliptic Integrals

Function: double gsl_sf_ellint_F (double phi, double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_F_e (double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral F(\phi,k) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_E (double phi, double k, gsl_mode_t mode)
Function: int gsl_sf_ellint_E_e (double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral E(\phi,k) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_P (double phi, double k, double n, gsl_mode_t mode)
Function: int gsl_sf_ellint_P_e (double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral P(\phi,k,n) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_D (double phi, double k, double n, gsl_mode_t mode)
Function: int gsl_sf_ellint_D_e (double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
These functions compute the incomplete elliptic integral D(\phi,k,n) which is defined through the Carlson form RD(x,y,z) by the following relation,

Carlson Forms

Function: double gsl_sf_ellint_RC (double x, double y, gsl_mode_t mode)
Function: int gsl_sf_ellint_RC_e (double x, double y, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral RC(x,y) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_RD (double x, double y, double z, gsl_mode_t mode)
Function: int gsl_sf_ellint_RD_e (double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral RD(x,y,z) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_RF (double x, double y, double z, gsl_mode_t mode)
Function: int gsl_sf_ellint_RF_e (double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral RF(x,y,z) to the accuracy specified by the mode variable mode.

Function: double gsl_sf_ellint_RJ (double x, double y, double z, double p, gsl_mode_t mode)
Function: int gsl_sf_ellint_RJ_e (double x, double y, double z, double p, gsl_mode_t mode, gsl_sf_result * result)
These routines compute the incomplete elliptic integral RJ(x,y,z,p) to the accuracy specified by the mode variable mode.

Elliptic Functions (Jacobi)

The Jacobian Elliptic functions are defined in Abramowitz & Stegun, Chapter 16. The functions are declared in the header file `gsl_sf_elljac.h'.

Function: int gsl_sf_elljac_e (double u, double m, double * sn, double * cn, double * dn)
This function computes the Jacobian elliptic functions sn(u|m), cn(u|m), dn(u|m) by descending Landen transformations.

Error Functions

The error function is described in Abramowitz & Stegun, Chapter 7. The functions in this section are declared in the header file `gsl_sf_erf.h'.

Error Function

Function: double gsl_sf_erf (double x)
Function: int gsl_sf_erf_e (double x, gsl_sf_result * result)
These routines compute the error function erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).

Complementary Error Function

Function: double gsl_sf_erfc (double x)
Function: int gsl_sf_erfc_e (double x, gsl_sf_result * result)
These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).

Log Complementary Error Function

Function: double gsl_sf_log_erfc (double x)
Function: int gsl_sf_log_erfc_e (double x, gsl_sf_result * result)
These routines compute the logarithm of the complementary error function \log(\erfc(x)).

Probability functions

The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.

Function: double gsl_sf_erf_Z (double x)
Function: int gsl_sf_erf_Z_e (double x, gsl_sf_result * result)
These routines compute the Gaussian probability function Z(x) = (1/(2\pi)) \exp(-x^2/2).

Function: double gsl_sf_erf_Q (double x)
Function: int gsl_sf_erf_Q_e (double x, gsl_sf_result * result)
These routines compute the upper tail of the Gaussian probability function Q(x) = (1/(2\pi)) \int_x^\infty dt \exp(-t^2/2).

Exponential Functions

The functions described in this section are declared in the header file `gsl_sf_exp.h'.

Exponential Function

Function: double gsl_sf_exp (double x)
Function: int gsl_sf_exp_e (double x, gsl_sf_result * result)
These routines provide an exponential function \exp(x) using GSL semantics and error checking.

Function: int gsl_sf_exp_e10_e (double x, gsl_sf_result_e10 * result)
This function computes the exponential \exp(x) using the gsl_sf_result_e10 type to return a result with extended range. This function may be useful if the value of \exp(x) would overflow the numeric range of double.

Function: double gsl_sf_exp_mult (double x, double y)
Function: int gsl_sf_exp_mult_e (double x, double y, gsl_sf_result * result)
These routines exponentiate x and multiply by the factor y to return the product y \exp(x).

Function: int gsl_sf_exp_mult_e10_e (const double x, const double y, gsl_sf_result_e10 * result)
This function computes the product y \exp(x) using the gsl_sf_result_e10 type to return a result with extended numeric range.

Relative Exponential Functions

Function: double gsl_sf_expm1 (double x)
Function: int gsl_sf_expm1_e (double x, gsl_sf_result * result)
These routines compute the quantity \exp(x)-1 using an algorithm that is accurate for small x.

Function: double gsl_sf_exprel (double x)
Function: int gsl_sf_exprel_e (double x, gsl_sf_result * result)
These routines compute the quantity (\exp(x)-1)/x using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion (\exp(x)-1)/x = 1 + x/2 + x^2/(2*3) + x^3/(2*3*4) + \dots.

Function: double gsl_sf_exprel_2 (double x)
Function: int gsl_sf_exprel_2_e (double x, gsl_sf_result * result)
These routines compute the quantity 2(\exp(x)-1-x)/x^2 using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion 2(\exp(x)-1-x)/x^2 = 1 + x/3 + x^2/(3*4) + x^3/(3*4*5) + \dots.

Function: double gsl_sf_exprel_n (int n, double x)
Function: int gsl_sf_exprel_n_e (int n, double x, gsl_sf_result * result)
These routines compute the N-relative exponential, which is the n-th generalization of the functions gsl_sf_exprel and gsl_sf_exprel2. The N-relative exponential is given by,

Exponentiation With Error Estimate

Function: int gsl_sf_exp_err_e (double x, double dx, gsl_sf_result * result)
This function exponentiates x with an associated absolute error dx.

Function: int gsl_sf_exp_err_e10_e (double x, double dx, gsl_sf_result_e10 * result)
This functions exponentiate a quantity x with an associated absolute error dx using the gsl_sf_result_e10 type to return a result with extended range.

Function: int gsl_sf_exp_mult_err_e (double x, double dx, double y, double dy, gsl_sf_result * result)
This routine computes the product y \exp(x) for the quantities x, y with associated absolute errors dx, dy.

Function: int gsl_sf_exp_mult_err_e10_e (double x, double dx, double y, double dy, gsl_sf_result_e10 * result)
This routine computes the product y \exp(x) for the quantities x, y with associated absolute errors dx, dy using the gsl_sf_result_e10 type to return a result with extended range.

Exponential Integrals

Information on the exponential integrals can be found in Abramowitz & Stegun, Chapter 5. These functions are declared in the header file `gsl_sf_expint.h'.

Exponential Integral

Function: double gsl_sf_expint_E1 (double x)
Function: int gsl_sf_expint_E1_e (double x, gsl_sf_result * result)
These routines compute the exponential integral E_1(x),

Function: double gsl_sf_expint_E2 (double x)
Function: int gsl_sf_expint_E2_e (double x, gsl_sf_result * result)
These routines compute the second-order exponential integral E_2(x),

Ei(x)

Function: double gsl_sf_expint_Ei (double x)
Function: int gsl_sf_expint_Ei_e (double x, gsl_sf_result * result)
These routines compute the exponential integral E_i(x), where PV denotes the principal value of the integral.

Hyperbolic Integrals

Function: double gsl_sf_Shi (double x)
Function: int gsl_sf_Shi_e (double x, gsl_sf_result * result)
These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t.

Function: double gsl_sf_Chi (double x)
Function: int gsl_sf_Chi_e (double x, gsl_sf_result * result)
These routines compute the integral Chi(x) := Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t] , where \gamma_E is the Euler constant (available as the macro M_EULER).

Ei_3(x)

Function: double gsl_sf_expint_3 (double x)
Function: int gsl_sf_expint_3_e (double x, gsl_sf_result * result)
These routines compute the exponential integral Ei_3(x) = \int_0^x dt \exp(-t^3) for @c{$x \ge 0$} x >= 0.

Trigonometric Integrals

Function: double gsl_sf_Si (const double x)
Function: int gsl_sf_Si_e (double x, gsl_sf_result * result)
These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t.

Function: double gsl_sf_Ci (const double x)
Function: int gsl_sf_Ci_e (double x, gsl_sf_result * result)
These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.

Arctangent Integral

Function: double gsl_sf_atanint (double x)
Function: int gsl_sf_atanint_e (double x, gsl_sf_result * result)
These routines compute the Arctangent integral AtanInt(x) = \int_0^x dt \arctan(t)/t.

Fermi-Dirac Function

The functions described in this section are declared in the header file `gsl_sf_fermi_dirac.h'.

Complete Fermi-Dirac Integrals

The complete Fermi-Dirac integral F_j(x) is given by,

Function: double gsl_sf_fermi_dirac_m1 (double x)
Function: int gsl_sf_fermi_dirac_m1_e (double x, gsl_sf_result * result)
These routines compute the complete Fermi-Dirac integral with an index of -1. This integral is given by F_{-1}(x) = e^x / (1 + e^x).

Function: double gsl_sf_fermi_dirac_0 (double x)
Function: int gsl_sf_fermi_dirac_0_e (double x, gsl_sf_result * result)
These routines compute the complete Fermi-Dirac integral with an index of 0. This integral is given by F_0(x) = \ln(1 + e^x).

Function: double gsl_sf_fermi_dirac_1 (double x)
Function: int gsl_sf_fermi_dirac_1_e (double x, gsl_sf_result * result)
These routines compute the complete Fermi-Dirac integral with an index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)).

Function: double gsl_sf_fermi_dirac_2 (double x)
Function: int gsl_sf_fermi_dirac_2_e (double x, gsl_sf_result * result)
These routines compute the complete Fermi-Dirac integral with an index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)).

Function: double gsl_sf_fermi_dirac_int (int j, double x)
Function: int gsl_sf_fermi_dirac_int_e (int j, double x, gsl_sf_result * result)
These routines compute the complete Fermi-Dirac integral with an integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1)).

Function: double gsl_sf_fermi_dirac_mhalf (double x)
Function: int gsl_sf_fermi_dirac_mhalf_e (double x, gsl_sf_result * result)
These routines compute the complete Fermi-Dirac integral F_{-1/2}(x).

Function: double gsl_sf_fermi_dirac_half (double x)
Function: int gsl_sf_fermi_dirac_half_e (double x, gsl_sf_result * result)
These routines compute the complete Fermi-Dirac integral F_{1/2}(x).

Function: double gsl_sf_fermi_dirac_3half (double x)
Function: int gsl_sf_fermi_dirac_3half_e (double x, gsl_sf_result * result)
These routines compute the complete Fermi-Dirac integral F_{3/2}(x).

Incomplete Fermi-Dirac Integrals

The incomplete Fermi-Dirac integral F_j(x,b) is given by,

Function: double gsl_sf_fermi_dirac_inc_0 (double x, double b)
Function: int gsl_sf_fermi_dirac_inc_0_e (double x, double b, gsl_sf_result * result)
These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x).

Gamma Function

The Gamma function is defined by the following integral,

Further information on the Gamma function can be found in Abramowitz & Stegun, Chapter 6. The functions described in this section are declared in the header file `gsl_sf_gamma.h'.

Function: double gsl_sf_gamma (double x)
Function: int gsl_sf_gamma_e (double x, gsl_sf_result * result)
These routines compute the Gamma function \Gamma(x), subject to x not being a negative integer. The function is computed using the real Lanczos method. The maximum value of x such that \Gamma(x) is not considered an overflow is given by the macro GSL_SF_GAMMA_XMAX and is 171.0.

Function: double gsl_sf_lngamma (double x)
Function: int gsl_sf_lngamma_e (double x, gsl_sf_result * result)
These routines compute the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not a being negative integer. For x<0 the real part of \log(\Gamma(x)) is returned, which is equivalent to \log(|\Gamma(x)|). The function is computed using the real Lanczos method.

Function: int gsl_sf_lngamma_sgn_e (double x, gsl_sf_result * result_lg, double * sgn)
This routine computes the sign of the gamma function and the logarithm its magnitude, subject to x not being a negative integer. The function is computed using the real Lanczos method. The value of the gamma function can be reconstructed using the relation \Gamma(x) = sgn * \exp(resultlg).

Function: double gsl_sf_gammastar (double x)
Function: int gsl_sf_gammastar_e (double x, gsl_sf_result * result)
These routines compute the regulated Gamma Function \Gamma^*(x) for x > 0. The regulated gamma function is given by,

and is a useful suggestion of Temme.

Function: double gsl_sf_gammainv (double x)
Function: int gsl_sf_gammainv_e (double x, gsl_sf_result * result)
These routines compute the reciprocal of the gamma function, 1/\Gamma(x) using the real Lanczos method.

Function: int gsl_sf_lngamma_complex_e (double zr, double zi, gsl_sf_result * lnr, gsl_sf_result * arg)
This routine computes \log(\Gamma(z)) for complex z=z_r+i z_i and z not a negative integer, using the complex Lanczos method. The returned parameters are lnr = \log|\Gamma(z)| and arg = \arg(\Gamma(z)) in (-\pi,\pi]. Note that the phase part (arg) is not well-determined when |z| is very large, due to inevitable roundoff in restricting to (-\pi,\pi]. This will result in a GSL_ELOSS error when it occurs. The absolute value part (lnr), however, never suffers from loss of precision.

Function: double gsl_sf_taylorcoeff (int n, double x)
Function: int gsl_sf_taylorcoeff_e (int n, double x, gsl_sf_result * result)
These routines compute the Taylor coefficient x^n / n! for x >= 0, n >= 0.

Function: double gsl_sf_fact (unsigned int n)
Function: int gsl_sf_fact_e (unsigned int n, gsl_sf_result * result)
These routines compute the factorial n!. The factorial is related to the Gamma function by n! = \Gamma(n+1).

Function: double gsl_sf_doublefact (unsigned int n)
Function: int gsl_sf_doublefact_e (unsigned int n, gsl_sf_result * result)
These routines compute the double factorial n!! = n(n-2)(n-4) \dots.

Function: double gsl_sf_lnfact (unsigned int n)
Function: int gsl_sf_lnfact_e (unsigned int n, gsl_sf_result * result)
These routines compute the logarithm of the factorial of n, \log(n!). The algorithm is faster than computing \ln(\Gamma(n+1)) via gsl_sf_lngamma for n < 170, but defers for larger n.

Function: double gsl_sf_lndoublefact (unsigned int n)
Function: int gsl_sf_lndoublefact_e (unsigned int n, gsl_sf_result * result)
These routines compute the logarithm of the double factorial of n, \log(n!!).

Function: double gsl_sf_choose (unsigned int n, unsigned int m)
Function: int gsl_sf_choose_e (unsigned int n, unsigned int m, gsl_sf_result * result)
These routines compute the combinatorial factor n choose m = n!/(m!(n-m)!)

Function: double gsl_sf_lnchoose (unsigned int n, unsigned int m)
Function: int gsl_sf_lnchoose_e (unsigned int n, unsigned int m, gsl_sf_result * result)
These routines compute the logarithm of n choose m. This is equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).

Function: double gsl_sf_poch (double a, double x)
Function: int gsl_sf_poch_e (double a, double x, gsl_sf_result * result)
These routines compute the Pochhammer symbol (a)_x := \Gamma(a + x)/\Gamma(x), subject to a and a+x not being negative integers. The Pochhammer symbol is also known as the Apell symbol.

Function: double gsl_sf_lnpoch (double a, double x)
Function: int gsl_sf_lnpoch_e (double a, double x, gsl_sf_result * result)
These routines compute the logarithm of the Pochhammer symbol, \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)) for a > 0, a+x > 0.

Function: int gsl_sf_lnpoch_sgn_e (double a, double x, gsl_sf_result * result, double * sgn)
These routines compute the sign of the Pochhammer symbol and the logarithm of its magnitude. The computed parameters are result = \log(|(a)_x|) and sgn = sgn((a)_x) where (a)_x := \Gamma(a + x)/\Gamma(a), subject to a, a+x not being negative integers.

Function: double gsl_sf_pochrel (double a, double x)
Function: int gsl_sf_pochrel_e (double a, double x, gsl_sf_result * result)
These routines compute the relative Pochhammer symbol ((a,x) - 1)/x where (a,x) = (a)_x := \Gamma(a + x)/\Gamma(a).

Function: double gsl_sf_gamma_inc_Q (double a, double x)
Function: int gsl_sf_gamma_inc_Q_e (double a, double x, gsl_sf_result * result)
These routines compute the normalized incomplete Gamma Function P(a,x) = 1/\Gamma(a) \int_x\infty dt t^{a-1} \exp(-t) for a > 0, @c{$x \ge 0$} x >= 0.

Function: double gsl_sf_gamma_inc_P (double a, double x)
Function: int gsl_sf_gamma_inc_P_e (double a, double x, gsl_sf_result * result)
These routines compute the complementary normalized incomplete Gamma Function P(a,x) = 1/\Gamma(a) \int_0^x dt t^{a-1} \exp(-t) for a > 0, @c{$x \ge 0$} x >= 0.

Function: double gsl_sf_beta (double a, double b)
Function: int gsl_sf_beta_e (double a, double b, gsl_sf_result * result)
These routines compute the Beta Function, B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b) for a > 0, b > 0.

Function: double gsl_sf_lnbeta (double a, double b)
Function: int gsl_sf_lnbeta_e (double a, double b, gsl_sf_result * result)
These routines compute the logarithm of the Beta Function, \log(B(a,b)) for a > 0, b > 0.

Function: double gsl_sf_beta_inc (double a, double b, double x)
Function: int gsl_sf_beta_inc_e (double a, double b, double x, gsl_sf_result * result)
These routines compute the normalize incomplete Beta function B_x(a,b)/B(a,b) for a > 0, b > 0, and @c{$0 \le x \le 1$} 0 <= x <= 1.

Gegenbauer Functions

The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter 22, where they are known as Ultraspherical polynomials. The functions described in this section are declared in the header file `gsl_sf_gegenbauer.h'.

Function: double gsl_sf_gegenpoly_1 (double lambda, double x)
Function: double gsl_sf_gegenpoly_2 (double lambda, double x)
Function: double gsl_sf_gegenpoly_3 (double lambda, double x)
Function: int gsl_sf_gegenpoly_1_e (double lambda, double x, gsl_sf_result * result)
Function: int gsl_sf_gegenpoly_2_e (double lambda, double x, gsl_sf_result * result)
Function: int gsl_sf_gegenpoly_3_e (double lambda, double x, gsl_sf_result * result)
These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) using explicit representations for n =1, 2, 3.

Function: double gsl_sf_gegenpoly_n (int n, double lambda, double x)
Function: int gsl_sf_gegenpoly_n_e (int n, double lambda, double x, gsl_sf_result * result)
These functions evaluate the Gegenbauer polynomial @c{$C^{(\lambda)}_n(x)$} C^{(\lambda)}_n(x) for a specific value of n, lambda, x subject to \lambda > -1/2, @c{$n \ge 0$} n >= 0.

Function: int gsl_sf_gegenpoly_array (int nmax, double lambda, double x, double result_array[])
This function computes an array of Gegenbauer polynomials C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda > -1/2, @c{$nmax \ge 0$} nmax >= 0.

Hypergeometric Functions

Hypergeometric functions are described in Abramowitz & Stegun, Chapters 13 and 15. These functions are declared in the header file `gsl_sf_hyperg.h'.

Function: double gsl_sf_hyperg_0F1 (double c, double x)
Function: int gsl_sf_hyperg_0F1_e (double c, double x, gsl_sf_result * result)
These routines compute the hypergeometric function @c{${}_0F_1(c,x)$} 0F1(c,x).

Function: double gsl_sf_hyperg_1F1_int (int m, int n, double x)
Function: int gsl_sf_hyperg_1F1_int_e (int m, int n, double x, gsl_sf_result * result)
These routines compute the confluent hypergeometric function 1F1(m,n,x) = M(m,n,x) for integer parameters m, n.

Function: double gsl_sf_hyperg_1F1 (double a, double b, double x)
Function: int gsl_sf_hyperg_1F1_e (double a, double b, double x, gsl_sf_result * result)
These routines compute the confluent hypergeometric function 1F1(a,b,x) = M(a,b,x) for general parameters a, b.

Function: double gsl_sf_hyperg_U_int (int m, int n, double x)
Function: int gsl_sf_hyperg_U_int_e (int m, int n, double x, gsl_sf_result * result)
These routines compute the confluent hypergeometric function U(m,n,x) for integer parameters m, n.

Function: int gsl_sf_hyperg_U_int_e10_e (int m, int n, double x, gsl_sf_result_e10 * result)
This routine computes the confluent hypergeometric function U(m,n,x) for integer parameters m, n using the gsl_sf_result_e10 type to return a result with extended range.

Function: double gsl_sf_hyperg_U (double a, double b, double x)
Function: int gsl_sf_hyperg_U_e (double a, double b, double x)
These routines compute the confluent hypergeometric function U(a,b,x).

Function: int gsl_sf_hyperg_U_e10_e (double a, double b, double x, gsl_sf_result_e10 * result)
This routine computes the confluent hypergeometric function U(a,b,x) using the gsl_sf_result_e10 type to return a result with extended range.

Function: double gsl_sf_hyperg_2F1 (double a, double b, double c, double x)
Function: int gsl_sf_hyperg_2F1_e (double a, double b, double c, double x, gsl_sf_result * result)
These routines compute the Gauss hypergeometric function 2F1(a,b,c,x) for |x| < 1.

Function: double gsl_sf_hyperg_2F1_conj (double aR, double aI, double c, double x)
Function: int gsl_sf_hyperg_2F1_conj_e (double aR, double aI, double c, double x, gsl_sf_result * result)
These routines compute the Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) with complex parameters for |x| < 1. exceptions:

Function: double gsl_sf_hyperg_2F1_renorm (double a, double b, double c, double x)
Function: int gsl_sf_hyperg_2F1_renorm_e (double a, double b, double c, double x, gsl_sf_result * result)
These routines compute the renormalized Gauss hypergeometric function 2F1(a,b,c,x) / \Gamma(c) for |x| < 1.

Function: double gsl_sf_hyperg_2F1_conj_renorm (double aR, double aI, double c, double x)
Function: int gsl_sf_hyperg_2F1_conj_renorm_e (double aR, double aI, double c, double x, gsl_sf_result * result)
These routines compute the renormalized Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c) for |x| < 1.

Function: double gsl_sf_hyperg_2F0 (double a, double b, double x)
Function: int gsl_sf_hyperg_2F0_e (double a, double b, double x, gsl_sf_result * result)
These routines compute the hypergeometric function @c{${}_2F_0(a,b,x)$} 2F0(a,b,x). The series representation is a divergent hypergeometric series. However, for x < 0 we have 2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)

Laguerre Functions

The Laguerre polynomials are defined in terms of confluent hypergeometric functions as L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x). These functions are declared in the header file `gsl_sf_laguerre.h'.

Function: double gsl_sf_laguerre_1 (double a, double x)
Function: double gsl_sf_laguerre_2 (double a, double x)
Function: double gsl_sf_laguerre_3 (double a, double x)
Function: int gsl_sf_laguerre_1_e (double a, double x, gsl_sf_result * result)
Function: int gsl_sf_laguerre_2_e (double a, double x, gsl_sf_result * result)
Function: int gsl_sf_laguerre_3_e (double a, double x, gsl_sf_result * result)
These routines evaluate the generalized Laguerre polynomials L^a_1(x), L^a_2(x), L^a_3(x) using explicit representations.

Function: double gsl_sf_laguerre_n (const int n, const double a, const double x)
Function: int gsl_sf_laguerre_n_e (int n, double a, double x, gsl_sf_result * result)
Thse routines evaluate the generalized Laguerre polynomials L^a_n(x) for a > -1, n >= 0.

Lambert W Functions

Lambert's W functions, W(x), are defined to be solutions of the equation W(x) \exp(W(x)) = x. This function has multiple branches for x < 0; however, it has only two real-valued branches. We define W_0(x) to be the principal branch, where W > -1 for x < 0, and W_{-1}(x) to be the other real branch, where W < -1 for x < 0. The Lambert functions are declared in the header file `gsl_sf_lambert.h'.

Function: double gsl_sf_lambert_W0 (double x)
Function: int gsl_sf_lambert_W0_e (double x, gsl_sf_result * result)
These compute the principal branch of the Lambert W function, W_0(x).

Function: double gsl_sf_lambert_Wm1 (double x)
Function: int gsl_sf_lambert_Wm1_e (double x, gsl_sf_result * result)
These compute the secondary real-valued branch of the Lambert W function, W_{-1}(x).

Legendre Functions and Spherical Harmonics

The Legendre Functions and Legendre Polynomials are described in Abramowitz & Stegun, Chapter 8. These functions are declared in the header file `gsl_sf_legendre.h'.

Legendre Polynomials

Function: double gsl_sf_legendre_P1 (double x)
Function: double gsl_sf_legendre_P2 (double x)
Function: double gsl_sf_legendre_P3 (double x)
Function: int gsl_sf_legendre_P1_e (double x, gsl_sf_result * result)
Function: int gsl_sf_legendre_P2_e (double x, gsl_sf_result * result)
Function: int gsl_sf_legendre_P3_e (double x, gsl_sf_result * result)
These functions evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3.

Function: double gsl_sf_legendre_Pl (int l, double x)
Function: int gsl_sf_legendre_Pl_e (int l, double x, gsl_sf_result * result)
These functions evaluate the Legendre polynomial @c{$P_l(x)$} P_l(x) for a specific value of l, x subject to @c{$l \ge 0$} l >= 0, |x| <= 1

Function: int gsl_sf_legendre_Pl_array (int lmax, double x, double result_array[])
This function computes an array of Legendre polynomials P_l(x) for l = 0, \dots, lmax, |x| <= 1

Function: double gsl_sf_legendre_Q0 (double x)
Function: int gsl_sf_legendre_Q0_e (double x, gsl_sf_result * result)
These routines compute the Legendre function Q_0(x) for x > -1, @c{$x \ne 1$} x != 1.

Function: double gsl_sf_legendre_Q1 (double x)
Function: int gsl_sf_legendre_Q1_e (double x, gsl_sf_result * result)
These routines compute the Legendre function Q_1(x) for x > -1, @c{$x \ne 1$} x != 1.

Function: double gsl_sf_legendre_Ql (int l, double x)
Function: int gsl_sf_legendre_Ql_e (int l, double x, gsl_sf_result * result)
These routines compute the Legendre function Q_l(x) for x > -1, @c{$x \ne 1$} x != 1 and @c{$l \ge 0$} l >= 0.

Associated Legendre Polynomials and Spherical Harmonics

The following functions compute the associated Legendre Polynomials P_l^m(x). Note that this function grows combinatorially with l and can overflow for l larger than about 150. There is no trouble for small m, but overflow occurs when m and l are both large. Rather than allow overflows, these functions refuse to calculate P_l^m(x) and return GSL_EOVRFLW when they can sense that l and m are too big.

If you want to calculate a spherical harmonic, then do not use these functions. Instead use gsl_sf_legendre_sphPlm() below, which uses a similar recursion, but with the normalized functions.

Function: double gsl_sf_legendre_Plm (int l, int m, double x)
Function: int gsl_sf_legendre_Plm_e (int l, int m, double x, gsl_sf_result * result)
These routines compute the associated Legendre polynomial P_l^m(x) for @c{$m \ge 0$} m >= 0, @c{$l \ge m$} l >= m, @c{$|x| \le 1$} |x| <= 1.

Function: int gsl_sf_legendre_Plm_array (int lmax, int m, double x, double result_array[])
This function computes an array of Legendre polynomials P_l^m(x) for @c{$m \ge 0$} m >= 0, @c{$l = |m|, \dots, lmax$} l = |m|, ..., lmax, @c{$|x| \le 1$} |x| <= 1.

Function: double gsl_sf_legendre_sphPlm (int l, int m, double x)
Function: int gsl_sf_legendre_sphPlm_e (int l, int m, double x, gsl_sf_result * result)
These routines compute the normalized associated Legendre polynomial $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ suitable for use in spherical harmonics. The parameters must satisfy @c{$m \ge 0$} m >= 0, @c{$l \ge m$} l >= m, @c{$|x| \le 1$} |x| <= 1. Theses routines avoid the overflows that occur for the standard normalization of P_l^m(x).

Function: int gsl_sf_legendre_sphPlm_array (int lmax, int m, double x, double result_array[])
This function computes an array of normalized associated Legendre functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ for @c{$m \ge 0$} m >= 0, @c{$l = |m|, \dots, lmax$} l = |m|, ..., lmax, @c{$|x| \le 1$} |x| <= 1.0

Function: int gsl_sf_legendre_array_size (const int lmax, const int m)
This functions returns the size of result_array[] needed for the array versions of P_l^m(x), lmax - m + 1.

Conical Functions

The Conical Functions @c{$P^\mu_{-(1/2)+i\lambda}(x)$} P^\mu_{-(1/2)+i\lambda}(x), @c{$Q^\mu_{-(1/2)+i\lambda}$} Q^\mu_{-(1/2)+i\lambda} are described in Abramowitz & Stegun, Section 8.12.

Function: double gsl_sf_conicalP_half (double lambda, double x)
Function: int gsl_sf_conicalP_half_e (double lambda, double x, gsl_sf_result * result)
These routines compute the irregular Spherical Conical Function P^{1/2}_{-1/2 + i \lambda}(x) for x > -1.

Function: double gsl_sf_conicalP_mhalf (double lambda, double x)
Function: int gsl_sf_conicalP_mhalf_e (double lambda, double x, gsl_sf_result * result)
These routines compute the regular Spherical Conical Function P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1.

Function: double gsl_sf_conicalP_0 (double lambda, double x)
Function: int gsl_sf_conicalP_0_e (double lambda, double x, gsl_sf_result * result)
These routines compute the conical function P^0_{-1/2 + i \lambda}(x) for x > -1.

Function: double gsl_sf_conicalP_1 (double lambda, double x)
Function: int gsl_sf_conicalP_1_e (double lambda, double x, gsl_sf_result * result)
These routines compute the conical function P^1_{-1/2 + i \lambda}(x) for x > -1.

Function: double gsl_sf_conicalP_sph_reg (int l, double lambda, double x)
Function: int gsl_sf_conicalP_sph_reg_e (int l, double lambda, double x, gsl_sf_result * result)
These routines compute the Regular Spherical Conical Function P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1, @c{$l \ge -1$} l >= -1.

Function: double gsl_sf_conicalP_cyl_reg (int m, double lambda, double x)
Function: int gsl_sf_conicalP_cyl_reg_e (int m, double lambda, double x, gsl_sf_result * result)
These routines compute the Regular Cylindrical Conical Function P^{-m}_{-1/2 + i \lambda}(x) for x > -1, @c{$m \ge -1$} m >= -1.

Radial Functions for Hyperbolic Space

The following spherical functions are specializations of Legendre functions which give the regular eigenfunctions of the Laplacian on a 3-dimensional hyperbolic space H3d. Of particular interest is the flat limit, \lambda \to \infty, \eta \to 0, \lambda\eta fixed.

Function: double gsl_sf_legendre_H3d_0 (double lambda, double eta)
Function: int gsl_sf_legendre_H3d_0_e (double lambda, double eta, gsl_sf_result * result)
These routines compute the zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta)) for @c{$\eta \ge 0$} \eta >= 0. In the flat limit this takes the form L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta)

Function: double gsl_sf_legendre_H3d_1 (double lambda, double eta)
Function: int gsl_sf_legendre_H3d_1_e (double lambda, double eta, gsl_sf_result * result)
These routines compute the first radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta)) for @c{$\eta \ge 0$} \eta >= 0. In the flat limit this takes the form L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta).

Function: double gsl_sf_legendre_H3d (int l, double lambda, double eta)
Function: int gsl_sf_legendre_H3d_e (int l, double lambda, double eta, gsl_sf_result * result)
These routines compute the l'th radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space @c{$\eta \ge 0$} \eta >= 0, @c{$l \ge 0$} l >= 0. In the flat limit this takes the form L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta).

Function: int gsl_sf_legendre_H3d_array (int lmax, double lambda, double eta, double result_array[])
This function computes an array of radial eigenfunctions L^{H3d}_l(\lambda, \eta) for @c{$0 \le l \le lmax$} 0 <= l <= lmax.

Logarithm and Related Functions

Information on the properties of the Logarithm function can be found in Abramowitz & Stegun, Chapter 4. The functions described in this section are declared in the header file `gsl_sf_log.h'.

Function: double gsl_sf_log (double x)
Function: int gsl_sf_log_e (double x, gsl_sf_result * result)
These routines compute the logarithm of x, \log(x), for x > 0.

Function: double gsl_sf_log_abs (double x)
Function: int gsl_sf_log_abs_e (double x, gsl_sf_result * result)
These routines compute the logarithm of the magnitude of x, \log(|x|), for x \ne 0.

Function: int gsl_sf_complex_log_e (double zr, double zi, gsl_sf_result * lnr, gsl_sf_result * theta)
This routine computes the complex logarithm of z = z_r + i z_i. The results are returned as lnr, theta such that \exp(lnr + i \theta) = z_r + i z_i, where \theta lies in the range [-\pi,\pi].

Function: double gsl_sf_log_1plusx (double x)
Function: int gsl_sf_log_1plusx_e (double x, gsl_sf_result * result)
These routines compute \log(1 + x) for x > -1 using an algorithm that is accurate for small x.

Function: double gsl_sf_log_1plusx_mx (double x)
Function: int gsl_sf_log_1plusx_mx_e (double x, gsl_sf_result * result)
These routines compute \log(1 + x) - x for x > -1 using an algorithm that is accurate for small x.

Power Function

The following functions are equivalent to the function gsl_pow_int (see section Small integer powers) with an error estimate. These functions are declared in the header file `gsl_sf_pow_int.h'.

Function: double gsl_sf_pow_int (double x, int n)
Function: int gsl_sf_pow_int_e (double x, int n, gsl_sf_result * result)
These routines compute the power x^n for integer n. The power is computed using the minimum number of multiplications. For example, x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications. For reasons of efficiency, these functions do not check for overflow or underflow conditions.
#include <gsl/gsl_sf_pow_int.h>
/* compute 3.0**12 */
double y = gsl_sf_pow_int(3.0, 12); 

Psi (Digamma) Function

The polygamma functions of order m defined by \psi^{(m)}(x) = (d/dx)^m \psi(x) = (d/dx)^{m+1} \log(\Gamma(x)), where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function. These functions are declared in the header file `gsl_sf_psi.h'.

Digamma Function

Function: double gsl_sf_psi_int (int n)
Function: int gsl_sf_psi_int_e (int n, gsl_sf_result * result)
These routines compute the digamma function \psi(n) for positive integer n. The digamma function is also called the Psi function.

Function: double gsl_sf_psi (double x)
Function: int gsl_sf_psi_e (double x, gsl_sf_result * result)
These routines compute the digamma function \psi(x) for general x, x \ne 0.

Function: double gsl_sf_psi_1piy (double y)
Function: int gsl_sf_psi_1piy_e (double y, gsl_sf_result * result)
These routines compute the real part of the digamma function on the line 1+i y, Re[\psi(1 + i y)].

Trigamma Function

Function: double gsl_sf_psi_1_int (int n)
Function: int gsl_sf_psi_1_int_e (int n, gsl_sf_result * result)
These routines compute the Trigamma function \psi'(n) for positive integer n.

Polygamma Function

Function: double gsl_sf_psi_n (int m, double x)
Function: int gsl_sf_psi_n_e (int m, double x, gsl_sf_result * result)
These routines compute the polygamma function @c{$\psi^{(m)}(x)$} \psi^{(m)}(x) for m >= 0, x > 0.

Synchrotron Functions

The functions described in this section are declared in the header file `gsl_sf_synchroton.h'.

Function: double gsl_sf_synchrotron_1 (double x)
Function: int gsl_sf_synchrotron_1_e (double x, gsl_sf_result * result)
These routines compute the first synchrotron function x \int_x^\infty dt K_{5/3}(t) for @c{$x \ge 0$} x >= 0.
Function: double gsl_sf_synchrotron_2 (double x)
Function: int gsl_sf_synchrotron_2_e (double x, gsl_sf_result * result)
These routines compute the second synchrotron function x K_{2/3}(x) for @c{$x \ge 0$} x >= 0.

Transport Functions

The transport functions J(n,x) are defined by the integral representations J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2. They are declared in the header file `gsl_sf_transport.h'.

Function: double gsl_sf_transport_2 (double x)
Function: int gsl_sf_transport_2_e (double x, gsl_sf_result * result)
These routines compute the transport function J(2,x).

Function: double gsl_sf_transport_3 (double x)
Function: int gsl_sf_transport_3_e (double x, gsl_sf_result * result)
These routines compute the transport function J(3,x).

Function: double gsl_sf_transport_4 (double x)
Function: int gsl_sf_transport_4_e (double x, gsl_sf_result * result)
These routines compute the transport function J(4,x).

Function: double gsl_sf_transport_5 (double x)
Function: int gsl_sf_transport_5_e (double x, gsl_sf_result * result)
These routines compute the transport function J(5,x).

Trigonometric Functions

The library includes its own trigonometric functions in order to provide consistency across platforms and reliable error estimates. These functions are declared in the header file `gsl_sf_trig.h'.

Circular Trigonometric Functions

Function: double gsl_sf_sin (double x)
Function: int gsl_sf_sin_e (double x, gsl_sf_result * result)
These routines compute the sine function \sin(x).

Function: double gsl_sf_cos (double x)
Function: int gsl_sf_cos_e (double x, gsl_sf_result * result)
These routines compute the cosine function \cos(x).

Function: double gsl_sf_hypot (double x, double y)
Function: int gsl_sf_hypot_e (double x, double y, gsl_sf_result * result)
These routines compute the hypotenuse function @c{$\sqrt{x^2 + y^2}$} \sqrt{x^2 + y^2} avoiding overflow and underflow.

Function: double gsl_sf_sinc (double x)
Function: int gsl_sf_sinc_e (double x, gsl_sf_result * result)
These routines compute \sinc(x) = \sin(\pi x) / (\pi x) for any value of x.

Trigonometric Functions for Complex Arguments

Function: int gsl_sf_complex_sin_e (double zr, double zi, gsl_sf_result * szr, gsl_sf_result * szi)
This function computes the complex sine, \sin(z_r + i z_i) storing the real and imaginary parts in szr, szi.

Function: int gsl_sf_complex_cos_e (double zr, double zi, gsl_sf_result * czr, gsl_sf_result * czi)
This function computes the complex cosine, \cos(z_r + i z_i) storing the real and imaginary parts in szr, szi.

Function: int gsl_sf_complex_logsin_e (double zr, double zi, gsl_sf_result * lszr, gsl_sf_result * lszi)
This function computes the logarithm of the complex sine, \log(\sin(z_r + i z_i)) storing the real and imaginary parts in szr, szi.

Hyperbolic Trigonometric Functions

Function: double gsl_sf_lnsinh (double x)
Function: int gsl_sf_lnsinh_e (double x, gsl_sf_result * result)
These routines compute \log(\sinh(x)) for x > 0.

Function: double gsl_sf_lncosh (double x)
Function: int gsl_sf_lncosh_e (double x, gsl_sf_result * result)
These routines compute \log(\cosh(x)) for any x.

Conversion Functions

Function: int gsl_sf_polar_to_rect (double r, double theta, gsl_sf_result * x, gsl_sf_result * y);
This function converts the polar coordinates (r,theta) to rectilinear coordinates (x,y), x = r\cos(\theta), y = r\sin(\theta).

Function: int gsl_sf_rect_to_polar (double x, double y, gsl_sf_result * r, gsl_sf_result * theta)
This function converts the rectilinear coordinates (x,y) to polar coordinates (r,theta), such that x = r\cos(\theta), y = r\sin(\theta). The argument theta lies in the range [-\pi, \pi].

Restriction Functions

Function: double gsl_sf_angle_restrict_symm (double theta)
Function: int gsl_sf_angle_restrict_symm_e (double * theta)
These routines force the angle theta to lie in the range (-\pi,\pi].

Function: double gsl_sf_angle_restrict_pos (double theta)
Function: int gsl_sf_angle_restrict_pos_e (double * theta)
These routines force the angle theta to lie in the range [0, 2\pi).

Trigonometric Functions With Error Estimates

Function: double gsl_sf_sin_err (double x, double dx)
Function: int gsl_sf_sin_err_e (double x, double dx, gsl_sf_result * result)
These routines compute the sine of an angle x with an associated absolute error dx, \sin(x \pm dx).

Function: double gsl_sf_cos_err (double x, double dx)
Function: int gsl_sf_cos_err_e (double x, double dx, gsl_sf_result * result)
These routines compute the cosine of an angle x with an associated absolute error dx, \cos(x \pm dx).

Zeta Functions

The Riemann zeta function is defined in Abramowitz & Stegun, Section 23.2. The functions described in this section are declared in the header file `gsl_sf_zeta.h'.

Riemann Zeta Function

The Riemann zeta function is defined by the infinite sum \zeta(s) = \sum_{k=1}^\infty k^{-s}.

Function: double gsl_sf_zeta_int (int n)
Function: int gsl_sf_zeta_int_e (int n, gsl_sf_result * result)
These routines compute the Riemann zeta function \zeta(n) for integer n, n \ne 1.

Function: double gsl_sf_zeta (double s)
Function: int gsl_sf_zeta_e (double s, gsl_sf_result * result)
These routines compute the Riemann zeta function \zeta(s) for arbitrary s, s \ne 1.

Hurwitz Zeta Function

The Hurwitz zeta function is defined by \zeta(s,q) = \sum_0^\infty (k+q)^{-s}.

Function: double gsl_sf_hzeta (double s, double q)
Function: int gsl_sf_hzeta_e (double s, double q, gsl_sf_result * result)
These routines compute the Hurwitz zeta function \zeta(s,q) for s > 1, q > 0.

Eta Function

The eta function is defined by \eta(s) = (1-2^{1-s}) \zeta(s).

Function: double gsl_sf_eta_int (int n)
Function: int gsl_sf_eta_int_e (int n, gsl_sf_result * result)
These routines compute the eta function \eta(n) for integer n.

Function: double gsl_sf_eta (double s)
Function: int gsl_sf_eta_e (double s, gsl_sf_result * result)
These routines compute the eta function \eta(s) for arbitrary s.

Examples

The following example demonstrates the use of the error handling form of the special functions, in this case to compute the Bessel function J_0(5.0),

#include <stdio.h>
#include <gsl/gsl_sf_bessel.h>

int
main (void)
{
  double x = 5.0;
  gsl_sf_result result;

  double expected = -0.17759677131433830434739701;
  
  int status = gsl_sf_bessel_J0_e (x, &result);

  printf("status  = %s\n", gsl_strerror(status));
  printf("J0(5.0) = %.18f\n"
         "      +/- % .18f\n", 
         result.val, result.err);
  printf("exact   = %.18f\n", expected);
  return status;
}

Here are the results of running the program,

$ ./a.out 
status  = success
J0(5.0) = -0.177596771314338292 
      +/-  0.000000000000000193
exact   = -0.177596771314338292

The next program computes the same quantity using the natural form of the function. In this case the error term result.err and return status are not accessible.

#include <stdio.h>
#include <gsl/gsl_sf_bessel.h>

int
main (void)
{
  double x = 5.0;
  double expected = -0.17759677131433830434739701;
  
  double y = gsl_sf_bessel_J0 (x);

  printf("J0(5.0) = %.18f\n", y);
  printf("exact   = %.18f\n", expected);
  return 0;
}

The results of the function are the same,

$ ./a.out 
J0(5.0) = -0.177596771314338292
exact   = -0.177596771314338292

References and Further Reading

The library follows the conventions of Abramowitz & Stegun where possible,

The following papers contain information on the algorithms used to compute the special functions,

Vectors and Matrices

The functions described in this chapter provide a simple vector and matrix interface to ordinary C arrays. The memory management of these arrays is implemented using a single underlying type, known as a block. By writing your functions in terms of vectors and matrices you can pass a single structure containing both data and dimensions as an argument without needing additional function parameters. The structures are compatible with the vector and matrix formats used by BLAS routines.

Data types

All the functions are available for each of the standard data-types. The versions for double have the prefix gsl_block, gsl_vector and gsl_matrix. Similarly the versions for single-precision float arrays have the prefix gsl_block_float, gsl_vector_float and gsl_matrix_float. The full list of available types is given below,

gsl_block                       double         
gsl_block_float                 float         
gsl_block_long_double           long double   
gsl_block_int                   int           
gsl_block_uint                  unsigned int  
gsl_block_long                  long          
gsl_block_ulong                 unsigned long 
gsl_block_short                 short         
gsl_block_ushort                unsigned short
gsl_block_char                  char          
gsl_block_uchar                 unsigned char 
gsl_block_complex               complex double        
gsl_block_complex_float         complex float         
gsl_block_complex_long_double   complex long double   

Corresponding types exist for the gsl_vector and gsl_matrix functions.

Blocks

For consistency all memory is allocated through a gsl_block structure. The structure contains two components, the size of an area of memory and a pointer to the memory. The gsl_block structure looks like this,

typedef struct
{
  size_t size;
  double * data;
} gsl_block;

Vectors and matrices are made by slicing an underlying block. A slice is a set of elements formed from an initial offset and a combination of indices and step-sizes. In the case of a matrix the step-size for the column index represents the row-length. The step-size for a vector is known as the stride.

The functions for allocating and deallocating blocks are defined in `gsl_block.h'

Block allocation

The functions for allocating memory to a block follow the style of malloc and free. In addition they also perform their own error checking. If there is insufficient memory available to allocate a block then the functions call the GSL error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every alloc.

Function: gsl_block * gsl_block_alloc (size_t n)
This function allocates memory for a block of n double-precision elements, returning a pointer to the block struct. The block is not initialized and so the values of its elements are undefined. Use the function gsl_block_calloc if you want to ensure that all the elements are initialized to zero.

A null pointer is returned if insufficient memory is available to create the block.

Function: gsl_block * gsl_block_calloc (size_t n)
This function allocates memory for a block and initializes all the elements of the block to zero.

Function: void gsl_block_free (gsl_block * b)
This function frees the memory used by a block b previously allocated with gsl_block_alloc or gsl_block_calloc.

Reading and writing blocks

The library provides functions for reading and writing blocks to a file as binary data or formatted text.

Function: int gsl_block_fwrite (FILE * stream, const gsl_block * b)
This function writes the elements of the block b to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

Function: int gsl_block_fread (FILE * stream, gsl_block * b)
This function reads into the block b from the open stream stream in binary format. The block b must be preallocated with the correct length since the function uses the size of b to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

Function: int gsl_block_fprintf (FILE * stream, const gsl_block * b, const char * format)
This function writes the elements of the block b line-by-line to the stream stream using the format specifier format, which should be one of the %g, %e or %f formats for floating point numbers and %d for integers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file.

Function: int gsl_block_fscanf (FILE * stream, gsl_block * b)
This function reads formatted data from the stream stream into the block b. The block b must be preallocated with the correct length since the function uses the size of b to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

Example programs for blocks

The following program shows how to allocate a block,

#include <stdio.h>
#include <gsl/gsl_block.h>

int
main (void)
{
  gsl_block * b = gsl_block_alloc (100);
  
  printf("length of block = %u\n", b->size);
  printf("block data address = %#x\n", b->data);

  gsl_block_free (b);
  return 0;
}

Here is the output from the program,

length of block = 100
block data address = 0x804b0d8

Vectors

Vectors are defined by a gsl_vector structure which describes a slice of a block. Different vectors can be created which point to the same block. A vector slice is a set of equally-spaced elements of an area of memory.

The gsl_vector structure contains five components, the size, the stride, a pointer to the memory where the elements are stored, data, a pointer to the block owned by the vector, block, if any, and an ownership flag, owner. The structure is very simple and looks like this,

typedef struct
{
  size_t size;
  size_t stride;
  double * data;
  gsl_block * block;
  int owner;
} gsl_vector;

The size is simply the number of vector elements. The range of valid indices runs from 0 to size-1. The stride is the step-size from one element to the next in physical memory, measured in units of the appropriate datatype. The pointer data gives the location of the first element of the vector in memory. The pointer block stores the location of the memory block in which the vector elements are located (if any). If the vector owns this block then the owner field is set to one and the block will be deallocated when the vector is freed. If the vector points to a block owned by another object then the owner field is zero and any underlying block will not be deallocated.

The functions for allocating and accessing vectors are defined in `gsl_vector.h'

Vector allocation

The functions for allocating memory to a vector follow the style of malloc and free. In addition they also perform their own error checking. If there is insufficient memory available to allocate a vector then the functions call the GSL error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every alloc.

Function: gsl_vector * gsl_vector_alloc (size_t n)
This function creates a vector of length n, returning a pointer to a newly initialized vector struct. A new block is allocated for the elements of the vector, and stored in the block component of the vector struct. The block is "owned" by the vector, and will be deallocated when the vector is deallocated.

Function: gsl_vector * gsl_vector_calloc (size_t n)
This function allocates memory for a vector of length n and initializes all the elements of the vector to zero.

Function: void gsl_vector_free (gsl_vector * v)
This function frees a previously allocated vector v. If the vector was created using gsl_vector_alloc then the block underlying the vector will also be deallocated. If the vector has been created from another object then the memory is still owned by that object and will not be deallocated.

Accessing vector elements

Unlike FORTRAN compilers, C compilers do not usually provide support for range checking of vectors and matrices. Range checking is available in the GNU C Compiler extension checkergcc but it is not available on every platform. The functions gsl_vector_get and gsl_vector_set can perform portable range checking for you and report an error if you attempt to access elements outside the allowed range.

The functions for accessing the elements of a vector or matrix are defined in `gsl_vector.h' and declared extern inline to eliminate function-call overhead. If necessary you can turn off range checking completely without modifying any source files by recompiling your program with the preprocessor definition GSL_RANGE_CHECK_OFF. Provided your compiler supports inline functions the effect of turning off range checking is to replace calls to gsl_vector_get(v,i) by v->data[i*v->stride] and and calls to gsl_vector_set(v,i,x) by v->data[i*v->stride]=x. Thus there should be no performance penalty for using the range checking functions when range checking is turned off.

Function: double gsl_vector_get (const gsl_vector * v, size_t i)
This function returns the i-th element of a vector v. If i lies outside the allowed range of 0 to n-1 then the error handler is invoked and 0 is returned.

Function: void gsl_vector_set (gsl_vector * v, size_t i, double x)
This function sets the value of the i-th element of a vector v to x. If i lies outside the allowed range of 0 to n-1 then the error handler is invoked.

Function: double * gsl_vector_ptr (gsl_vector * v, size_t i)
Function: const double * gsl_vector_ptr (const gsl_vector * v, size_t i)
These functions return a pointer to the i-th element of a vector v. If i lies outside the allowed range of 0 to n-1 then the error handler is invoked and a null pointer is returned.

Initializing vector elements

Function: void gsl_vector_set_all (gsl_vector * v, double x)
This function sets all the elements of the vector v to the value x.

Function: void gsl_vector_set_zero (gsl_vector * v)
This function sets all the elements of the vector v to zero.

Function: int gsl_vector_set_basis (gsl_vector * v, size_t i)
This function makes a basis vector by setting all the elements of the vector v to zero except for the i-th element which is set to one.

Reading and writing vectors

The library provides functions for reading and writing vectors to a file as binary data or formatted text.

Function: int gsl_vector_fwrite (FILE * stream, const gsl_vector * v)
This function writes the elements of the vector v to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

Function: int gsl_vector_fread (FILE * stream, gsl_vector * v)
This function reads into the vector v from the open stream stream in binary format. The vector v must be preallocated with the correct length since the function uses the size of v to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

Function: int gsl_vector_fprintf (FILE * stream, const gsl_vector * v, const char * format)
This function writes the elements of the vector v line-by-line to the stream stream using the format specifier format, which should be one of the %g, %e or %f formats for floating point numbers and %d for integers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file.

Function: int gsl_vector_fscanf (FILE * stream, gsl_vector * v)
This function reads formatted data from the stream stream into the vector v. The vector v must be preallocated with the correct length since the function uses the size of v to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

Vector views

In addition to creating vectors from slices of blocks it is also possible to slice vectors and create vector views. For example, a subvector of another vector can be described with a view, or two views can be made which provide access to the even and odd elements of a vector.

A vector view is a temporary object, stored on the stack, which can be used to operate on a subset of vector elements. Vector views can be defined for both constant and non-constant vectors, using separate types that preserve constness. A vector view has the type gsl_vector_view and a constant vector view has the type gsl_vector_const_view. In both cases the elements of the view can be accessed as a gsl_vector using the vector component of the view object. A pointer to a vector of type gsl_vector * or const gsl_vector * can be obtained by taking the address of this component with the & operator.

Function: gsl_vector_view gsl_vector_subvector (gsl_vector *v, size_t offset, size_t n)
Function: gsl_vector_const_view gsl_vector_const_subvector (const gsl_vector * v, size_t offset, size_t n)
These functions return a vector view of a subvector of another vector v. The start of the new vector is offset by offset elements from the start of the original vector. The new vector has n elements. Mathematically, the i-th element of the new vector v' is given by,
v'(i) = v->data[(offset + i)*v->stride]

where the index i runs from 0 to n-1.

The data pointer of the returned vector struct is set to null if the combined parameters (offset,n) overrun the end of the original vector.

The new vector is only a view of the block underlying the original vector, v. The block containing the elements of v is not owned by the new vector. When the view goes out of scope the original vector v and its block will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.

The function gsl_vector_const_subvector is equivalent to gsl_vector_subvector but can be used for vectors which are declared const.

Function: gsl_vector gsl_vector_subvector_with_stride (gsl_vector *v, size_t offset, size_t stride, size_t n)
Function: gsl_vector_const_view gsl_vector_const_subvector_with_stride (const gsl_vector * v, size_t offset, size_t stride, size_t n)
These functions return a vector view of a subvector of another vector v with an additional stride argument. The subvector is formed in the same way as for gsl_vector_subvector but the new vector has n elements with a step-size of stride from one element to the next in the original vector. Mathematically, the i-th element of the new vector v' is given by,
v'(i) = v->data[(offset + i*stride)*v->stride]

where the index i runs from 0 to n-1.

Note that subvector views give direct access to the underlying elements of the original vector. For example, the following code will zero the even elements of the vector v of length n, while leaving the odd elements untouched,

gsl_vector_view v_even 
  = gsl_vector_subvector_with_stride (v, 0, 2, n/2);
gsl_vector_set_zero (&v_even.vector);

A vector view can be passed to any subroutine which takes a vector argument just as a directly allocated vector would be, using &view.vector. For example, the following code computes the norm of odd elements of v using the BLAS routine DNRM2,

gsl_vector_view v_odd 
  = gsl_vector_subvector_with_stride (v, 1, 2, n/2);
double r = gsl_blas_dnrm2 (&v_odd.vector);

The function gsl_vector_const_subvector_with_stride is equivalent to gsl_vector_subvector_with_stride but can be used for vectors which are declared const.

Function: gsl_vector_view gsl_vector_complex_real (gsl_vector_complex *v)
Function: gsl_vector_const_view gsl_vector_complex_const_real (const gsl_vector_complex *v)
These functions return a vector view of the real parts of the complex vector v.

The function gsl_vector_complex_const_real is equivalent to gsl_vector_complex_real but can be used for vectors which are declared const.

Function: gsl_vector_view gsl_vector_complex_imag (gsl_vector_complex *v)
Function: gsl_vector_const_view gsl_vector_complex_const_imag (const gsl_vector_complex *v)
These functions return a vector view of the imaginary parts of the complex vector v.

The function gsl_vector_complex_const_imag is equivalent to gsl_vector_complex_imag but can be used for vectors which are declared const.

Function: gsl_vector_view gsl_vector_view_array (double *base, size_t n)
Function: gsl_vector_const_view gsl_vector_const_view_array (const double *base, size_t n)
These functions return a vector view of an array. The start of the new vector is given by base and has n elements. Mathematically, the i-th element of the new vector v' is given by,
v'(i) = base[i]

where the index i runs from 0 to n-1.

The array containing the elements of v is not owned by the new vector view. When the view goes out of scope the original array will continue to exist. The original memory can only be deallocated by freeing the original pointer base. Of course, the original array should not be deallocated while the view is still in use.

The function gsl_vector_const_view_array is equivalent to gsl_vector_view_array but can be used for arrays which are declared const.

Function: gsl_vector_view gsl_vector_view_array_with_stride (double * base, size_t stride, size_t n)
Function: gsl_vector_const_view gsl_vector_const_view_array_with_stride (const double * base, size_t stride, size_t n)
These functions return a vector view of an array base with an additional stride argument. The subvector is formed in the same way as for gsl_vector_view_array but the new vector has n elements with a step-size of stride from one element to the next in the original array. Mathematically, the i-th element of the new vector v' is given by,
v'(i) = base[i*stride]

where the index i runs from 0 to n-1.

Note that the view gives direct access to the underlying elements of the original array. A vector view can be passed to any subroutine which takes a vector argument just as a directly allocated vector would be, using &view.vector.

The function gsl_vector_const_view_array_with_stride is equivalent to gsl_vector_view_array_with_stride but can be used for arrays which are declared const.

Copying vectors

Common operations on vectors such as addition and multiplication are available in the BLAS part of the library (see section BLAS Support). However, it is useful to have a small number of utility functions which do not require the full BLAS code. The following functions fall into this category.

Function: int gsl_vector_memcpy (gsl_vector * dest, const gsl_vector * src)
This function copies the elements of the vector src into the vector dest. The two vectors must have the same length.

Function: int gsl_vector_swap (gsl_vector * v, gsl_vector * w)
This function exchanges the elements of the vectors v and w by copying. The two vectors must have the same length.

Exchanging elements

The following function can be used to exchange, or permute, the elements of a vector.

Function: int gsl_vector_swap_elements (gsl_vector * v, size_t i, size_t j)
This function exchanges the i-th and j-th elements of the vector v in-place.

Function: int gsl_vector_reverse (gsl_vector * v)
This function reverses the order of the elements of the vector v.

Vector operations

The following operations are only defined for real vectors.

Function: int gsl_vector_add (gsl_vector * a, const gsl_vector * b)
This function adds the elements of vector b to the elements of vector a, a'_i = a_i + b_i. The two vectors must have the same length.

Function: int gsl_vector_sub (gsl_vector * a, const gsl_vector * b)
This function subtracts the elements of vector b from the elements of vector a, a'_i = a_i - b_i. The two vectors must have the same length.

Function: int gsl_vector_mul (gsl_vector * a, const gsl_vector * b)
This function multiplies the elements of vector a by the elements of vector b, a'_i = a_i * b_i. The two vectors must have the same length.

Function: int gsl_vector_div (gsl_vector * a, const gsl_vector * b)
This function divides the elements of vector a by the elements of vector b, a'_i = a_i / b_i. The two vectors must have the same length.

Function: int gsl_vector_scale (gsl_vector * a, const double x)
This function multiplies the elements of vector a by the constant factor x, a'_i = x a_i.

Function: int gsl_vector_add_constant (gsl_vector * a, const double x)
This function adds the constant value x to the elements of the vector a, a'_i = a_i + x.

Finding maximum and minimum elements of vectors

Function: double gsl_vector_max (const gsl_vector * v)
This function returns the maximum value in the vector v.

Function: double gsl_vector_min (const gsl_vector * v)
This function returns the minimum value in the vector v.

Function: void gsl_vector_minmax (const gsl_vector * v, double * min_out, double * max_out)
This function returns the minimum and maximum values in the vector v, storing them in min_out and max_out.

Function: size_t gsl_vector_max_index (const gsl_vector * v)
This function returns the index of the maximum value in the vector v. When there are several equal maximum elements then the lowest index is returned.

Function: size_t gsl_vector_min_index (const gsl_vector * v)
This function returns the index of the minimum value in the vector v. When there are several equal minimum elements then the lowest index is returned.

Function: void gsl_vector_minmax_index (const gsl_vector * v, size_t * imin, size_t * imax)
This function returns the indices of the minimum and maximum values in the vector v, storing them in imin and imax. When there are several equal minimum or maximum elements then the lowest indices are returned.

Vector properties

Function: int gsl_vector_isnull (const gsl_vector * v)
This function returns 1 if all the elements of the vector v are zero, and 0 otherwise.

Example programs for vectors

This program shows how to allocate, initialize and read from a vector using the functions gsl_vector_alloc, gsl_vector_set and gsl_vector_get.

#include <stdio.h>
#include <gsl/gsl_vector.h>

int
main (void)
{
  int i;
  gsl_vector * v = gsl_vector_alloc (3);
  
  for (i = 0; i < 3; i++)
    {
      gsl_vector_set (v, i, 1.23 + i);
    }
  
  for (i = 0; i < 100; i++)
    {
      printf("v_%d = %g\n", i, gsl_vector_get (v, i));
    }

  return 0;
}

Here is the output from the program. The final loop attempts to read outside the range of the vector v, and the error is trapped by the range-checking code in gsl_vector_get.

v_0 = 1.23
v_1 = 2.23
v_2 = 3.23
gsl: vector_source.c:12: ERROR: index out of range
IOT trap/Abort (core dumped)

The next program shows how to write a vector to a file.

#include <stdio.h>
#include <gsl/gsl_vector.h>

int
main (void)
{
  int i; 
  gsl_vector * v = gsl_vector_alloc (100);
  
  for (i = 0; i < 100; i++)
    {
      gsl_vector_set (v, i, 1.23 + i);
    }

  {  
     FILE * f = fopen("test.dat", "w");
     gsl_vector_fprintf (f, v, "%.5g");
     fclose (f);
  }
  return 0;
}

After running this program the file `test.dat' should contain the elements of v, written using the format specifier %.5g. The vector could then be read back in using the function gsl_vector_fscanf (f, v).

Matrices

Matrices are defined by a gsl_matrix structure which describes a generalized slice of a block. Like a vector it represents a set of elements in an area of memory, but uses two indices instead of one.

The gsl_matrix structure contains six components, the two dimensions of the matrix, a physical dimension, a pointer to the memory where the elements of the matrix are stored, data, a pointer to the block owned by the matrix block, if any, and an ownership flag, owner. The physical dimension determines the memory layout and can differ from the matrix dimension to allow the use of submatrices. The gsl_matrix structure is very simple and looks like this,

typedef struct
{
  size_t size1;
  size_t size2;
  size_t tda;
  double * data;
  gsl_block * block;
  int owner;
} gsl_matrix;

Matrices are stored in row-major order, meaning that each row of elements forms a contiguous block in memory. This is the standard "C-language ordering" of two-dimensional arrays. Note that FORTRAN stores arrays in column-major order. The number of rows is size1. The range of valid row indices runs from 0 to size1-1. Similarly size2 is the number of columns. The range of valid column indices runs from 0 to size2-1. The physical row dimension tda, or trailing dimension, specifies the size of a row of the matrix as laid out in memory.

For example, in the following matrix size1 is 3, size2 is 4, and tda is 8. The physical memory layout of the matrix begins in the top left hand-corner and proceeds from left to right along each row in turn.

00 01 02 03 XX XX XX XX
10 11 12 13 XX XX XX XX
20 21 22 23 XX XX XX XX

Each unused memory location is represented by "XX". The pointer data gives the location of the first element of the matrix in memory. The pointer block stores the location of the memory block in which the elements of the matrix are located (if any). If the matrix owns this block then the owner field is set to one and the block will be deallocated when the matrix is freed. If the matrix is only a slice of a block owned by another object then the owner field is zero and any underlying block will not be freed.

The functions for allocating and accessing matrices are defined in `gsl_matrix.h'

Matrix allocation

The functions for allocating memory to a matrix follow the style of malloc and free. They also perform their own error checking. If there is insufficient memory available to allocate a vector then the functions call the GSL error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every alloc.

Function: gsl_matrix * gsl_matrix_alloc (size_t n1, size_t n2)
This function creates a matrix of size n1 rows by n2 columns, returning a pointer to a newly initialized matrix struct. A new block is allocated for the elements of the matrix, and stored in the block component of the matrix struct. The block is "owned" by the matrix, and will be deallocated when the matrix is deallocated.

Function: gsl_matrix * gsl_matrix_calloc (size_t n1, size_t n2)
This function allocates memory for a matrix of size n1 rows by n2 columns and initializes all the elements of the matrix to zero.

Function: void gsl_matrix_free (gsl_matrix * m)
This function frees a previously allocated matrix m. If the matrix was created using gsl_matrix_alloc then the block underlying the matrix will also be deallocated. If the matrix has been created from another object then the memory is still owned by that object and will not be deallocated.

Accessing matrix elements

The functions for accessing the elements of a matrix use the same range checking system as vectors. You turn off range checking by recompiling your program with the preprocessor definition GSL_RANGE_CHECK_OFF.

The elements of the matrix are stored in "C-order", where the second index moves continuously through memory. More precisely, the element accessed by the function gsl_matrix_get(m,i,j) and gsl_matrix_set(m,i,j,x) is

m->data[i * m->tda + j]

where tda is the physical row-length of the matrix.

Function: double gsl_matrix_get (const gsl_matrix * m, size_t i, size_t j)
This function returns the (i,j)th element of a matrix m. If i or j lie outside the allowed range of 0 to n1-1 and 0 to n2-1 then the error handler is invoked and 0 is returned.

Function: void gsl_matrix_set (gsl_matrix * m, size_t i, size_t j, double x)
This function sets the value of the (i,j)th element of a matrix m to x. If i or j lies outside the allowed range of 0 to n1-1 and 0 to n2-1 then the error handler is invoked.

Function: double * gsl_matrix_ptr (gsl_matrix * m, size_t i, size_t j)
Function: const double * gsl_matrix_ptr (const gsl_matrix * m, size_t i, size_t j)
These functions return a pointer to the (i,j)th element of a matrix m. If i or j lie outside the allowed range of 0 to n1-1 and 0 to n2-1 then the error handler is invoked and a null pointer is returned.

Initializing matrix elements

Function: void gsl_matrix_set_all (gsl_matrix * m, double x)
This function sets all the elements of the matrix m to the value x.

Function: void gsl_matrix_set_zero (gsl_matrix * m)
This function sets all the elements of the matrix m to zero.

Function: void gsl_matrix_set_identity (gsl_matrix * m)
This function sets the elements of the matrix m to the corresponding elements of the identity matrix, m(i,j) = \delta(i,j), i.e. a unit diagonal with all off-diagonal elements zero. This applies to both square and rectangular matrices.

Reading and writing matrices

The library provides functions for reading and writing matrices to a file as binary data or formatted text.

Function: int gsl_matrix_fwrite (FILE * stream, const gsl_matrix * m)
This function writes the elements of the matrix m to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

Function: int gsl_matrix_fread (FILE * stream, gsl_matrix * m)
This function reads into the matrix m from the open stream stream in binary format. The matrix m must be preallocated with the correct dimensions since the function uses the size of m to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

Function: int gsl_matrix_fprintf (FILE * stream, const gsl_matrix * m, const char * format)
This function writes the elements of the matrix m line-by-line to the stream stream using the format specifier format, which should be one of the %g, %e or %f formats for floating point numbers and %d for integers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file.

Function: int gsl_matrix_fscanf (FILE * stream, gsl_matrix * m)
This function reads formatted data from the stream stream into the matrix m. The matrix m must be preallocated with the correct dimensions since the function uses the size of m to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

Matrix views

A matrix view is a temporary object, stored on the stack, which can be used to operate on a subset of matrix elements. Matrix views can be defined for both constant and non-constant matrices using separate types that preserve constness. A matrix view has the type gsl_matrix_view and a constant matrix view has the type gsl_matrix_const_view. In both cases the elements of the view can by accessed using the matrix component of the view object. A pointer gsl_matrix * or const gsl_matrix * can be obtained by taking the address of the matrix component with the & operator. In addition to matrix views it is also possible to create vector views of a matrix, such as row or column views.

Function: gsl_matrix_view gsl_matrix_submatrix (gsl_matrix * m, size_t i, size_t j, size_t n1, size_t n2)
Function: gsl_matrix_const_view gsl_matrix_const_submatrix (const gsl_matrix * m, size_t i, size_t j, size_t n1, size_t n2)
These functions return a matrix view of a submatrix of the matrix m. The upper-left element of the submatrix is the element (k1,k2) of the original matrix. The submatrix has n1 rows and n2 columns. The physical number of columns in memory given by tda is unchanged. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1.

The data pointer of the returned matrix struct is set to null if the combined parameters (k1,k2,n1,n2,tda) overrun the ends of the original matrix.

The new matrix view is only a view of the block underlying the existing matrix, m. The block containing the elements of m is not owned by the new matrix view. When the view goes out of scope the original matrix m and its block will continue to exist. The original memory can only be deallocated by freeing the original matrix. Of course, the original matrix should not be deallocated while the view is still in use.

The function gsl_matrix_const_submatrix is equivalent to gsl_matrix_submatrix but can be used for matrices which are declared const.

Function: gsl_matrix_view gsl_matrix_view_array (double * base, size_t n1, size_t n2)
Function: gsl_matrix_const_view gsl_matrix_const_view_array (const double * base, size_t n1, size_t n2)
These functions return a matrix view of the array base. The matrix has n1 rows and n2 columns. The physical number of columns in memory is also given by n2. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = base[i*n2 + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1.

The new matrix is only a view of the array base. When the view goes out of scope the original array base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.

The function gsl_matrix_const_view_array is equivalent to gsl_matrix_view_array but can be used for matrices which are declared const.

Function: gsl_matrix_view gsl_matrix_view_array_with_tda (double * base, size_t n1, size_t n2, size_t tda)
Function: gsl_matrix_const_view gsl_matrix_const_view_array_with_tda (const double * base, size_t n1, size_t n2, size_t tda)
These functions return a matrix view of the array base with a physical number of columns tda which may differ from corresponding the dimension of the matrix. The matrix has n1 rows and n2 columns, and the physical number of columns in memory is given by tda. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = base[i*tda + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1.

The new matrix is only a view of the array base. When the view goes out of scope the original array base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.

The function gsl_matrix_const_view_array_with_tda is equivalent to gsl_matrix_view_array_with_tda but can be used for matrices which are declared const.

Function: gsl_matrix_view gsl_matrix_view_vector (gsl_vector * v, size_t n1, size_t n2)
Function: gsl_matrix_const_view gsl_matrix_const_view_vector (const gsl_vector * v, size_t n1, size_t n2)
These functions return a matrix view of the vector v. The matrix has n1 rows and n2 columns. The vector must have unit stride. The physical number of columns in memory is also given by n2. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = v->data[i*n2 + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1.

The new matrix is only a view of the vector v. When the view goes out of scope the original vector v will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.

The function gsl_matrix_const_view_vector is equivalent to gsl_matrix_view_vector but can be used for matrices which are declared const.

Function: gsl_matrix_view gsl_matrix_view_vector_with_tda (gsl_vector * v, size_t n1, size_t n2, size_t tda)
Function: gsl_matrix_const_view gsl_matrix_const_view_vector_with_tda (const gsl_vector * v, size_t n1, size_t n2, size_t tda)
These functions return a matrix view of the vector v with a physical number of columns tda which may differ from the corresponding matrix dimension. The vector must have unit stride. The matrix has n1 rows and n2 columns, and the physical number of columns in memory is given by tda. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = v->data[i*tda + j]

where the index i runs from 0 to n1-1 and the index j runs from 0 to n2-1.

The new matrix is only a view of the vector v. When the view goes out of scope the original vector v will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.

The function gsl_matrix_const_view_vector_with_tda is equivalent to gsl_matrix_view_vector_with_tda but can be used for matrices which are declared const.

Creating row and column views

In general there are two ways to access an object, by reference or by copying. The functions described in this section create vector views which allow access to a row or column of a matrix by reference. Modifying elements of the view is equivalent to modifying the matrix, since both the vector view and the matrix point to the same memory block.

Function: gsl_vector_view gsl_matrix_row (gsl_matrix * m, size_t i)
Function: gsl_vector_const_view gsl_matrix_const_row (const gsl_matrix * m, size_t i)
These functions return a vector view of the i-th row of the matrix m. The data pointer of the new vector is set to null if i is out of range.

The function gsl_vector_const_row is equivalent to gsl_matrix_row but can be used for matrices which are declared const.

Function: gsl_vector_view gsl_matrix_column (gsl_matrix * m, size_t j)
Function: gsl_vector_const_view gsl_matrix_const_column (const gsl_matrix * m, size_t j)
These functions return a vector view of the j-th column of the matrix m. The data pointer of the new vector is set to null if j is out of range.

The function gsl_vector_const_column equivalent to gsl_matrix_column but can be used for matrices which are declared const.

Function: gsl_vector_view gsl_matrix_diagonal (gsl_matrix * m)
Function: gsl_vector_const_view gsl_matrix_const_diagonal (const gsl_matrix * m)
These functions returns a vector view of the diagonal of the matrix m. The matrix m is not required to be square. For a rectangular matrix the length of the diagonal is the same as the smaller dimension of the matrix.

The function gsl_matrix_const_diagonal is equivalent to gsl_matrix_diagonal but can be used for matrices which are declared const.

Function: gsl_vector_view gsl_matrix_subdiagonal (gsl_matrix * m, size_t k)
Function: gsl_vector_const_view gsl_matrix_const_subdiagonal (const gsl_matrix * m, size_t k)
These functions return a vector view of the k-th subdiagonal of the matrix m. The matrix m is not required to be square. The diagonal of the matrix corresponds to k = 0.

The function gsl_matrix_const_subdiagonal is equivalent to gsl_matrix_subdiagonal but can be used for matrices which are declared const.

Function: gsl_vector_view gsl_matrix_superdiagonal (gsl_matrix * m, size_t k)
Function: gsl_vector_const_view gsl_matrix_const_superdiagonal (const gsl_matrix * m, size_t k)
These functions return a vector view of the k-th superdiagonal of the matrix m. The matrix m is not required to be square. The diagonal of the matrix corresponds to k = 0.

The function gsl_matrix_const_superdiagonal is equivalent to gsl_matrix_superdiagonal but can be used for matrices which are declared const.

Copying matrices

Function: int gsl_matrix_memcpy (gsl_matrix * dest, const gsl_matrix * src)
This function copies the elements of the matrix src into the matrix dest. The two matrices must have the same size.

Function: int gsl_matrix_swap (gsl_matrix * m1, const gsl_matrix * m2)
This function exchanges the elements of the matrices m1 and m2 by copying. The two matrices must have the same size.

Copying rows and columns

The functions described in this section copy a row or column of a matrix into a vector. This allows the elements of the vector and the matrix to be modified independently. Note that if the matrix and the vector point to overlapping regions of memory then the result will be undefined. The same effect can be achieved with more generality using gsl_vector_memcpy with vector views of rows and columns.

Function: int gsl_matrix_get_row (gsl_vector * v, const gsl_matrix * m, size_t i)
This function copies the elements of the i-th row of the matrix m into the vector v. The length of the vector must be the same as the length of the row.

Function: int gsl_matrix_get_col (gsl_vector * v, const gsl_matrix * m, size_t j)
This function copies the elements of the i-th column of the matrix m into the vector v. The length of the vector must be the same as the length of the column.

Function: int gsl_matrix_set_row (gsl_matrix * m, size_t i, const gsl_vector * v)
This function copies the elements of the vector v into the i-th row of the matrix m. The length of the vector must be the same as the length of the row.

Function: int gsl_matrix_set_col (gsl_matrix * m, size_t j, const gsl_vector * v)
This function copies the elements of the vector v into the i-th column of the matrix m. The length of the vector must be the same as the length of the column.

Exchanging rows and columns

The following functions can be used to exchange the rows and columns of a matrix.

Function: int gsl_matrix_swap_rows (gsl_matrix * m, size_t i, size_t j)
This function exchanges the i-th and j-th rows of the matrix m in-place.

Function: int gsl_matrix_swap_columns (gsl_matrix * m, size_t i, size_t j)
This function exchanges the i-th and j-th columns of the matrix m in-place.

Function: int gsl_matrix_swap_rowcol (gsl_matrix * m, size_t i, size_t j)
This function exchanges the i-th row and j-th column of the matrix m in-place. The matrix must be square for this operation to be possible.

Function: int gsl_matrix_transpose_memcpy (gsl_matrix * dest, gsl_matrix * src)
This function makes the matrix dest the transpose of the matrix src by copying the elements of src into dest. This function works for all matrices provided that the dimensions of the matrix dest match the transposed dimensions of the matrix src.

Function: int gsl_matrix_transpose (gsl_matrix * m)
This function replaces the matrix m by its transpose by copying the elements of the matrix in-place. The matrix must be square for this operation to be possible.

Matrix operations

The following operations are only defined for real matrices.

Function: int gsl_matrix_add (gsl_matrix * a, const gsl_matrix * b)
This function adds the elements of matrix b to the elements of matrix a, a'(i,j) = a(i,j) + b(i,j). The two matrices must have the same dimensions.

Function: int gsl_matrix_sub (gsl_matrix * a, const gsl_matrix * b)
This function subtracts the elements of matrix b from the elements of matrix a, a'(i,j) = a(i,j) - b(i,j). The two matrices must have the same dimensions.

Function: int gsl_matrix_mul_elements (gsl_matrix * a, const gsl_matrix * b)
This function multiplies the elements of matrix a by the elements of matrix b, a'(i,j) = a(i,j) * b(i,j). The two matrices must have the same dimensions.

Function: int gsl_matrix_div_elements (gsl_matrix * a, const gsl_matrix * b)
This function divides the elements of matrix a by the elements of matrix b, a'(i,j) = a(i,j) / b(i,j). The two matrices must have the same dimensions.

Function: int gsl_matrix_scale (gsl_matrix * a, const double x)
This function multiplies the elements of matrix a by the constant factor x, a'(i,j) = x a(i,j).

Function: int gsl_matrix_add_constant (gsl_matrix * a, const double x)
This function adds the constant value x to the elements of the matrix a, a'(i,j) = a(i,j) + x.

Finding maximum and minimum elements of matrices

Function: double gsl_matrix_max (const gsl_matrix * m)
This function returns the maximum value in the matrix m.

Function: double gsl_matrix_min (const gsl_matrix * m)
This function returns the minimum value in the matrix m.

Function: void gsl_matrix_minmax (const gsl_matrix * m, double * min_out, double * max_out)
This function returns the minimum and maximum values in the matrix m, storing them in min_out and max_out.

Function: void gsl_matrix_max_index (const gsl_matrix * m, size_t * imax, size_t * jmax)
This function returns the indices of the maximum value in the matrix m, storing them in imax and jmax. When there are several equal maximum elements then the first element found is returned.

Function: void gsl_matrix_min_index (const gsl_matrix * m, size_t * imax, size_t * jmax)
This function returns the indices of the minimum value in the matrix m, storing them in imax and jmax. When there are several equal minimum elements then the first element found is returned.

Function: void gsl_matrix_minmax_index (const gsl_matrix * m, size_t * imin, size_t * imax)
This function returns the indices of the minimum and maximum values in the matrix m, storing them in (imin,jmin) and (imax,jmax). When there are several equal minimum or maximum elements then the first elements found are returned.

Matrix properties

Function: int gsl_matrix_isnull (const gsl_matrix * m)
This function returns 1 if all the elements of the matrix m are zero, and 0 otherwise.

Example programs for matrices

The program below shows how to allocate, initialize and read from a matrix using the functions gsl_matrix_alloc, gsl_matrix_set and gsl_matrix_get.

#include <stdio.h>
#include <gsl/gsl_matrix.h>

int
main (void)
{
  int i, j; 
  gsl_matrix * m = gsl_matrix_alloc (10, 3);
  
  for (i = 0; i < 10; i++)
    for (j = 0; j < 3; j++)
      gsl_matrix_set (m, i, j, 0.23 + 100*i + j);
  
  for (i = 0; i < 100; i++)
    for (j = 0; j < 3; j++)
      printf("m(%d,%d) = %g\n", i, j, 
             gsl_matrix_get (m, i, j));

  return 0;
}

Here is the output from the program. The final loop attempts to read outside the range of the matrix m, and the error is trapped by the range-checking code in gsl_matrix_get.

m(0,0) = 0.23
m(0,1) = 1.23
m(0,2) = 2.23
m(1,0) = 100.23
m(1,1) = 101.23
m(1,2) = 102.23
...
m(9,2) = 902.23
gsl: matrix_source.c:13: ERROR: first index out of range
IOT trap/Abort (core dumped)

The next program shows how to write a matrix to a file.

#include <stdio.h>
#include <gsl/gsl_matrix.h>

int
main (void)
{
  int i, j, k = 0; 
  gsl_matrix * m = gsl_matrix_alloc (100, 100);
  gsl_matrix * a = gsl_matrix_alloc (100, 100);
  
  for (i = 0; i < 100; i++)
    for (j = 0; j < 100; j++)
      gsl_matrix_set (m, i, j, 0.23 + i + j);

  {  
     FILE * f = fopen("test.dat", "w");
     gsl_matrix_fwrite (f, m);
     fclose (f);
  }

  {  
     FILE * f = fopen("test.dat", "r");
     gsl_matrix_fread (f, a);
     fclose (f);
  }

  for (i = 0; i < 100; i++)
    for (j = 0; j < 100; j++)
      {
        double mij = gsl_matrix_get(m, i, j);
        double aij = gsl_matrix_get(a, i, j);
        if (mij != aij) k++;
      }

  printf("differences = %d (should be zero)\n", k);
  return (k > 0);
}

After running this program the file `test.dat' should contain the elements of m, written in binary format. The matrix which is read back in using the function gsl_matrix_fread should be exactly equal to the original matrix.

The following program demonstrates the use of vector views. The program computes the column-norms of a matrix.

#include <math.h>
#include <stdio.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>

int
main (void)
{
  size_t i,j;

  gsl_matrix *m = gsl_matrix_alloc (10, 10);

  for (i = 0; i < 10; i++)
    for (j = 0; j < 10; j++)
      gsl_matrix_set (m, i, j, sin (i) + cos (j));

  for (j = 0; j < 10; j++)
    {
      gsl_vector_view column = gsl_matrix_column (m, j);
      double d;

      d = gsl_blas_dnrm2 (&column.vector);

      printf ("matrix column %d, norm = %g\n", j, d);
    }

  gsl_matrix_free (m);
}

Here is the output of the program, which can be confirmed using GNU OCTAVE,

$ ./a.out
matrix column 0, norm = 4.31461
matrix column 1, norm = 3.1205
matrix column 2, norm = 2.19316
matrix column 3, norm = 3.26114
matrix column 4, norm = 2.53416
matrix column 5, norm = 2.57281
matrix column 6, norm = 4.20469
matrix column 7, norm = 3.65202
matrix column 8, norm = 2.08524
matrix column 9, norm = 3.07313

octave> m = sin(0:9)' * ones(1,10) 
               + ones(10,1) * cos(0:9); 
octave> sqrt(sum(m.^2))
ans =

  4.3146  3.1205  2.1932  3.2611  2.5342  2.5728
  4.2047  3.6520  2.0852  3.0731

References and Further Reading

The block, vector and matrix objects in GSL follow the valarray model of C++. A description of this model can be found in the following reference,

Permutations

This chapter describes functions for creating and manipulating permutations. A permutation p is represented by an array of n integers in the range 0 .. n-1, where each value p_i occurs once and only once. The application of a permutation p to a vector v yields a new vector v' where v'_i = v_{p_i}. For example, the array (0,1,3,2) represents a permutation which exchanges the last two elements of a four element vector. The corresponding identity permutation is (0,1,2,3).

Note that the permutations produced by the linear algebra routines correspond to the exchange of matrix columns, and so should be considered as applying to row-vectors in the form v' = v P rather than column-vectors, when permuting the elements of a vector.

The functions described in this chapter are defined in the header file `gsl_permutation.h'.

The Permutation struct

A permutation is stored by a structure containing two components, the size of the permutation and a pointer to the permutation array. The elements of the permutation array are all of type size_t. The gsl_permutation structure looks like this,

typedef struct
{
  size_t size;
  size_t * data;
} gsl_permutation;

Permutation allocation

Function: gsl_permutation * gsl_permutation_alloc (size_t n)
This function allocates memory for a new permutation of size n. The permutation is not initialized and its elements are undefined. Use the function gsl_permutation_calloc if you want to create a permutation which is initialized to the identity. A null pointer is returned if insufficient memory is available to create the permutation.

Function: gsl_permutation * gsl_permutation_calloc (size_t n)
This function allocates memory for a new permutation of size n and initializes it to the identity. A null pointer is returned if insufficient memory is available to create the permutation.

Function: void gsl_permutation_init (gsl_permutation * p)
This function initializes the permutation p to the identity, i.e. (0,1,2,...,n-1).

Function: void gsl_permutation_free (gsl_permutation * p)
This function frees all the memory used by the permutation p.

Accessing permutation elements

The following functions can be used to access and manipulate permutations.

Function: size_t gsl_permutation_get (const gsl_permutation * p, const size_t i)
This function returns the value of the i-th element of the permutation p. If i lies outside the allowed range of 0 to n-1 then the error handler is invoked and 0 is returned.

Function: int gsl_permutation_swap (gsl_permutation * p, const size_t i, const size_t j)
This function exchanges the i-th and j-th elements of the permutation p.

Permutation properties

Function: size_t gsl_permutation_size (const gsl_permutation * p)
This function returns the size of the permutation p.

Function: size_t * gsl_permutation_data (const gsl_permutation * p)
This function returns a pointer to the array of elements in the permutation p.

Function: int gsl_permutation_valid (gsl_permutation * p)
This function checks that the permutation p is valid. The n elements should contain each of the numbers 0 .. n-1 once and only once.

Permutation functions

Function: void gsl_permutation_reverse (gsl_permutation * p)
This function reverses the elements of the permutation p.

Function: int gsl_permutation_inverse (gsl_permutation * inv, const gsl_permutation * p)
This function computes the inverse of the permutation p, storing the result in inv.

Function: int gsl_permutation_next (gsl_permutation * p)
This function advances the permutation p to the next permutation in lexicographic order and returns GSL_SUCCESS. If no further permutations are available it returns GSL_FAILURE and leaves p unmodified. Starting with the identity permutation and repeatedly applying this function will iterate through all possible permutations of a given order.

Function: int gsl_permutation_prev (gsl_permutation * p)
This function steps backwards from the permutation p to the previous permutation in lexicographic order, returning GSL_SUCCESS. If no previous permutation is available it returns GSL_FAILURE and leaves p unmodified.

Applying Permutations

Function: int gsl_permute (const size_t * p, double * data, size_t stride, size_t n)
This function applies the permutation p to the array data of size n with stride stride.

Function: int gsl_permute_inverse (const size_t * p, double * data, size_t stride, size_t n)
This function applies the inverse of the permutation p to the array data of size n with stride stride.

Function: int gsl_permute_vector (const gsl_permutation * p, gsl_vector * v)
This function applies the permutation p to the elements of the vector v, considered as a row-vector acted on by a permutation matrix from the right, v' = v P. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. The permutation p and the vector v must have the same length.

Function: int gsl_permute_vector_inverse (const gsl_permutation * p, gsl_vector * v)
This function applies the inverse of the permutation p to the elements of the vector v, considered as a row-vector acted on by an inverse permutation matrix from the right, v' = v P^T. Note that for permutation matrices the inverse is the same as the transpose. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. The permutation p and the vector v must have the same length.

Reading and writing permutations

The library provides functions for reading and writing permutations to a file as binary data or formatted text.

Function: int gsl_permutation_fwrite (FILE * stream, const gsl_permutation * p)
This function writes the elements of the permutation p to the stream stream in binary format. The function returns GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

Function: int gsl_permutation_fread (FILE * stream, gsl_permutation * p)
This function reads into the permutation p from the open stream stream in binary format. The permutation p must be preallocated with the correct length since the function uses the size of p to determine how many bytes to read. The function returns GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

Function: int gsl_permutation_fprintf (FILE * stream, const gsl_permutation * p, const char *format)
This function writes the elements of the permutation p line-by-line to the stream stream using the format specifier format, which should be suitable for a type of size_t. On a GNU system the type modifier Z represents size_t, so "%Zu\n" is a suitable format. The function returns GSL_EFAILED if there was a problem writing to the file.

Function: int gsl_permutation_fscanf (FILE * stream, gsl_permutation * p)
This function reads formatted data from the stream stream into the permutation p. The permutation p must be preallocated with the correct length since the function uses the size of p to determine how many numbers to read. The function returns GSL_EFAILED if there was a problem reading from the file.

Examples

The example program below creates a random permutation by shuffling and finds its inverse.

#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_permutation.h>

int
main (void) 
{
  const size_t N = 10;
  const gsl_rng_type * T;
  gsl_rng * r;

  gsl_permutation * p = gsl_permutation_alloc (N);
  gsl_permutation * q = gsl_permutation_alloc (N);

  gsl_rng_env_setup();
  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

  printf("initial permutation:");  
  gsl_permutation_init (p);
  gsl_permutation_fprintf (stdout, p, " %u");
  printf("\n");

  printf(" random permutation:");  
  gsl_ran_shuffle (r, p->data, N, sizeof(size_t));
  gsl_permutation_fprintf (stdout, p, " %u");
  printf("\n");

  printf("inverse permutation:");  
  gsl_permutation_invert (q, p);
  gsl_permutation_fprintf (stdout, q, " %u");
  printf("\n");

  return 0;
}

Here is the output from the program,

bash$ ./a.out 
initial permutation: 0 1 2 3 4 5 6 7 8 9
 random permutation: 1 3 5 2 7 6 0 4 9 8
inverse permutation: 6 0 3 1 7 2 5 4 9 8

The random permutation p[i] and its inverse q[i] are related through the identity p[q[i]] = i, which can be verified from the output.

The next example program steps forwards through all possible 3-rd order permutations, starting from the identity,

#include <stdio.h>
#include <gsl/gsl_permutation.h>

int
main (void) 
{
  gsl_permutation * p = gsl_permutation_alloc (3);

  gsl_permutation_init (p);

  do 
   {
      gsl_permutation_fprintf (stdout, p, " %u");
      printf("\n");
   }
  while (gsl_permutation_next(p) == GSL_SUCCESS);

  return 0;
}

Here is the output from the program,

bash$ ./a.out 
 0 1 2
 0 2 1
 1 0 2
 1 2 0
 2 0 1
 2 1 0

All 6 permutations are generated in lexicographic order. To reverse the sequence, begin with the final permutation (which is the reverse of the identity) and replace gsl_permutation_next with gsl_permutation_prev.

References and Further Reading

The subject of permutations is covered extensively in Knuth's Sorting and Searching,

Sorting

This chapter describes functions for sorting data, both directly and indirectly (using an index). All the functions use the heapsort algorithm. Heapsort is an O(N \log N) algorithm which operates in-place. It does not require any additional storage and provides consistent performance. The running time for its worst-case (ordered data) is not significantly longer than the average and best cases. Note that the heapsort algorithm does not preserve the relative ordering of equal elements -- it is an unstable sort. However the resulting order of equal elements will be consistent across different platforms when using these functions.

Sorting objects

The following function provides a simple alternative to the standard library function qsort. It is intended for systems lacking qsort, not as a replacement for it. The function qsort should be used whenever possible, as it will be faster and can provide stable ordering of equal elements. Documentation for qsort is available in the GNU C Library Reference Manual.

The functions described in this section are defined in the header file `gsl_heapsort.h'.

Function: void gsl_heapsort (void * array, size_t count, size_t size, gsl_comparison_fn_t compare)

This function sorts the count elements of the array array, each of size size, into ascending order using the comparison function compare. The type of the comparison function is defined by,

int (*gsl_comparison_fn_t) (const void * a,
                            const void * b)

A comparison function should return a negative integer if the first argument is less than the second argument, 0 if the two arguments are equal and a positive integer if the first argument is greater than the second argument.

For example, the following function can be used to sort doubles into ascending numerical order.

int
compare_doubles (const double * a,
                 const double * b)
{
    return (int) (*a - *b);
}

The appropriate function call to perform the sort is,

gsl_heapsort (array, count, sizeof(double), 
              compare_doubles);

Note that unlike qsort the heapsort algorithm cannot be made into a stable sort by pointer arithmetic. The trick of comparing pointers for equal elements in the comparison function does not work for the heapsort algorithm. The heapsort algorithm performs an internal rearrangement of the data which destroys its initial ordering.

Function: int gsl_heapsort_index (size_t * p, const void * array, size_t count, size_t size, gsl_comparison_fn_t compare)

This function indirectly sorts the count elements of the array array, each of size size, into ascending order using the comparison function compare. The resulting permutation is stored in p, an array of length n. The elements of p give the index of the array element which would have been stored in that position if the array had been sorted in place. The first element of p gives the index of the least element in array, and the last element of p gives the index of the greatest element in array. The array itself is not changed.

Sorting vectors

The following functions will sort the elements of an array or vector, either directly or indirectly. They are defined for all real and integer types using the normal suffix rules. For example, the float versions of the array functions are gsl_sort_float and gsl_sort_float_index. The corresponding vector functions are gsl_sort_vector_float and gsl_sort_vector_float_index. The prototypes are available in the header files `gsl_sort_float.h' `gsl_sort_vector_float.h'. The complete set of prototypes can be included using the header files `gsl_sort.h' and `gsl_sort_vector.h'.

There are no functions for sorting complex arrays or vectors, since the ordering of complex numbers is not uniquely defined. To sort a complex vector by magnitude compute a real vector containing the the magnitudes of the complex elements, and sort this vector indirectly. The resulting index gives the appropriate ordering of the original complex vector.

Function: void gsl_sort (double * data, size_t stride, size_t n)
This function sorts the n elements of the array data with stride stride into ascending numerical order.

Function: void gsl_sort_vector (gsl_vector * v)
This function sorts the elements of the vector v into ascending numerical order.

Function: int gsl_sort_index (size_t * p, const double * data, size_t stride, size_t n)
This function indirectly sorts the n elements of the array data with stride stride into ascending order, storing the resulting permutation in p. The array p must be allocated to a sufficient length to store the n elements of the permutation. The elements of p give the index of the array element which would have been stored in that position if the array had been sorted in place. The array data is not changed.

Function: int gsl_sort_vector_index (gsl_permutation * p, const gsl_vector * v)
This function indirectly sorts the elements of the vector v into ascending order, storing the resulting permutation in p. The elements of p give the index of the vector element which would have been stored in that position if the vector had been sorted in place. The first element of p gives the index of the least element in v, and the last element of p gives the index of the greatest element in v. The vector v is not changed.

Selecting the k-th smallest or largest elements

The functions described in this section select the k-th smallest or largest elements of a data set of size N. The routines use an O(kN) direct insertion algorithm which is suited to subsets that are small compared with the total size of the dataset. For example, the routines are useful for selecting the 10 largest values from one million data points, but not for selecting the largest 100,000 values. If the subset is a significant part of the total dataset it may be faster to sort all the elements of the dataset directly with an O(N \log N) algorithm and obtain the smallest or largest values that way.

Function: void gsl_sort_smallest (double * dest, size_t k, const double * src, size_t stride, size_t n)
This function copies the k-th smallest elements of the array src, of size n and stride stride, in ascending numerical order in dest. The size of the subset k must be less than or equal to n. The data src is not modified by this operation.

Function: void gsl_sort_largest (double * dest, size_t k, const double * src, size_t stride, size_t n)
This function copies the k-th largest elements of the array src, of size n and stride stride, in descending numerical order in dest. The size of the subset k must be less than or equal to n. The data src is not modified by this operation.

Function: void gsl_sort_vector_smallest (double * dest, size_t k, const gsl_vector * v)
Function: void gsl_sort_vector_largest (double * dest, size_t k, const gsl_vector * v)
These functions copy the k-th smallest or largest elements of the vector v into the array dest. The size of the subset k must be less than or equal to the length of the vector v.

The following functions find the indices of the k-th smallest or largest elements of a dataset,

Function: void gsl_sort_smallest_index (size_t * p, size_t k, const double * src, size_t stride, size_t n)
This function stores the indices of the k-th smallest elements of the array src, of size n and stride stride, in the array p. The indices are chosen so that the corresponding data is in ascending numerical order. The size of the subset k must be less than or equal to n. The data src is not modified by this operation.

Function: void gsl_sort_largest_index (size_t * p, size_t k, const double * src, size_t stride, size_t n)
This function stores the indices of the k-th largest elements of the array src, of size n and stride stride, in the array p. The indices are chosen so that the corresponding data is in descending numerical order. The size of the subset k must be less than or equal to n. The data src is not modified by this operation.

Function: void gsl_sort_vector_smallest_index (size_t * p, size_t k, const gsl_vector * v)
Function: void gsl_sort_vector_largest_index (size_t * p, size_t k, const gsl_vector * v)
These functions store the indices of k-th smallest or largest elements of the vector v in the array p. The size of the subset k must be less than or equal to the length of the vector v.

Computing the rank

The rank of an element is its order in the sorted data. The rank is the inverse of the index permutation, p. It can be computed using the following algorithm,

for (i = 0; i < p->size; i++) 
{
    size_t pi = p->data[i];
    rank->data[pi] = i;
}

This can be computed directly from the function gsl_permutation_invert(rank,p).

The following function will print the rank of each element of the vector v,

void
print_rank (gsl_vector * v)
{
  size_t i;
  size_t n = v->size;
  gsl_permutation * perm = gsl_permutation_alloc(n);
  gsl_permutation * rank = gsl_permutation_alloc(n);

  gsl_sort_vector_index (perm, v);
  gsl_permutation_invert (rank, perm);

  for (i = 0; i < n; i++)
   {
    double vi = gsl_vector_get(v, i);
    printf("element = %d, value = %g, rank = %d\n",
            i, vi, rank->data[i]);
   }

  gsl_permutation_free (perm);
  gsl_permutation_free (rank);
}

Examples

The following example shows how to use the permutation p to print the elements of the vector v in ascending order,

gsl_sort_vector_index (p, v);

for (i = 0; i < v->size; i++)
{
    double vpi = gsl_vector_get(v, p->data[i]);
    printf("order = %d, value = %g\n", i, vpi);
}

The next example uses the function gsl_sort_smallest to select the 5 smallest numbers from 100000 uniform random variates stored in an array,

#include <gsl/gsl_rng.h>
#include <gsl/gsl_sort_double.h>

int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;

  int i, k = 5, N = 100000;

  double * x = malloc (N * sizeof(double));
  double * small = malloc (k * sizeof(double));

  gsl_rng_env_setup();

  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

  for (i = 0; i < N; i++)
    {
      x[i] = gsl_rng_uniform(r);
    }

  gsl_sort_smallest (small, k, x, 1, N);

  printf("%d smallest values from %d\n", k, N);

  for (i = 0; i < k; i++)
    {
      printf ("%d: %.18f\n", i, small[i]);
    }
  return 0;
}

The output lists the 5 smallest values, in ascending order,

$ ./a.out 
5 smallest values from 100000
0: 0.000005466630682349
1: 0.000012384494766593
2: 0.000017581274732947
3: 0.000025131041184068
4: 0.000031369971111417

References and Further Reading

The subject of sorting is covered extensively in Knuth's Sorting and Searching,

The Heapsort algorithm is described in the following book,

BLAS Support

The Basic Linear Algebra Subprograms (BLAS) define a set of fundamental operations on vectors and matrices which can be used to create optimized higher-level linear algebra functionality.

The library provides a low-level layer which corresponds directly to the C-language BLAS standard, referred to here as "CBLAS", and a higher-level interface for operations on GSL vectors and matrices. Users who are interested in simple operations on GSL vector and matrix objects should use the high-level layer, which is declared in the file gsl_blas.h. This should satisfy the needs of most users. Note that GSL matrices are implemented using dense-storage so the interface only includes the corresponding dense-storage BLAS functions. The full BLAS functionality for band-format and packed-format matrices is available through the low-level CBLAS interface.

The interface for the gsl_cblas layer is specified in the file gsl_cblas.h. This interface corresponds the BLAS Technical Forum's draft standard for the C interface to legacy BLAS implementations. Users who have access to other conforming CBLAS implementations can use these in place of the version provided by the library. Note that users who have only a Fortran BLAS library can use a CBLAS conformant wrapper to convert it into a CBLAS library. A reference CBLAS wrapper for legacy Fortran implementations exists as part of the draft CBLAS standard and can be obtained from Netlib. The complete set of CBLAS functions is listed in an appendix (see section GSL CBLAS Library).

There are three levels of BLAS operations,

Level 1
Vector operations, e.g. y = \alpha x + y
Level 2
Matrix-vector operations, e.g. y = \alpha A x + \beta y
Level 3
Matrix-matrix operations, e.g. C = \alpha A B + C

Each routine has a name which specifies the operation, the type of matrices involved and their precisions. Some of the most common operations and their names are given below,

DOT
scalar product, x^T y
AXPY
vector sum, \alpha x + y
MV
matrix-vector product, A x
SV
matrix-vector solve, inv(A) x
MM
matrix-matrix product, A B
SM
matrix-matrix solve, inv(A) B

The type of matrices are,

GE
general
GB
general band
SY
symmetric
SB
symmetric band
SP
symmetric packed
HE
hermitian
HB
hermitian band
HP
hermitian packed
TR
triangular
TB
triangular band
TP
triangular packed

Each operation is defined for four precisions,

S
single real
D
double real
C
single complex
Z
double complex

Thus, for example, the name SGEMM stands for "single-precision general matrix-matrix multiply" and ZGEMM stands for "double-precision complex matrix-matrix multiply".

GSL BLAS Interface

GSL provides dense vector and matrix objects, based on the relevant built-in types. The library provides an interface to the BLAS operations which apply to these objects. The interface to this functionality is given in the file gsl_blas.h.

Level 1

Function: int gsl_blas_sdsdot (float alpha, const gsl_vector_float * x, const gsl_vector_float * y, float * result)
Function: int gsl_blas_dsdot (const gsl_vector_float * x, const gsl_vector_float * y, double * result)
These functions compute the sum \alpha + x^T y for the vectors x and y, returning the result in result.

Function: int gsl_blas_sdot (const gsl_vector_float * x, const gsl_vector_float * y, float * result)
Function: int gsl_blas_ddot (const gsl_vector * x, const gsl_vector * y, double * result)
These functions compute the scalar product x^T y for the vectors x and y, returning the result in result.

Function: int gsl_blas_cdotu (const gsl_vector_complex_float * x, const gsl_vector_complex_float * y, gsl_complex_float * dotu)
Function: int gsl_blas_zdotu (const gsl_vector_complex * x, const gsl_vector_complex * y, gsl_complex * dotu)
These functions compute the complex scalar product x^T y for the vectors x and y, returning the result in result

Function: int gsl_blas_cdotc (const gsl_vector_complex_float * x, const gsl_vector_complex_float * y, gsl_complex_float * dotc)
Function: int gsl_blas_zdotc (const gsl_vector_complex * x, const gsl_vector_complex * y, gsl_complex * dotc)
These functions compute the complex conjugate scalar product x^H y for the vectors x and y, returning the result in result

Function: float gsl_blas_snrm2 (const gsl_vector_float * x)
Function: double gsl_blas_dnrm2 (const gsl_vector * x)
These functions compute the Euclidean norm ||x||_2 = \sqrt {\sum x_i^2} of the vector x.

Function: float gsl_blas_scnrm2 (const gsl_vector_complex_float * x)
Function: double gsl_blas_dznrm2 (const gsl_vector_complex * x)
These functions compute the Euclidean norm of the complex vector x,

Function: float gsl_blas_sasum (const gsl_vector_float * x)
Function: double gsl_blas_dasum (const gsl_vector * x)
These functions compute the absolute sum \sum |x_i| of the elements of the vector x.

Function: float gsl_blas_scasum (const gsl_vector_complex_float * x)
Function: double gsl_blas_dzasum (const gsl_vector_complex * x)
These functions compute the absolute sum \sum |\Re(x_i)| + |\Im(x_i)| of the elements of the vector x.

Function: CBLAS_INDEX_t gsl_blas_isamax (const gsl_vector_float * x)
Function: CBLAS_INDEX_t gsl_blas_idamax (const gsl_vector * x)
Function: CBLAS_INDEX_t gsl_blas_icamax (const gsl_vector_complex_float * x)
Function: CBLAS_INDEX_t gsl_blas_izamax (const gsl_vector_complex * x)
These functions return the index of the largest element of the vector x. The largest element is determined by its absolute magnitude for real vector and by the sum of the magnitudes of the real and imaginary parts |\Re(x_i)| + |\Im(x_i)| for complex vectors. If the largest value occurs several times then the index of the first occurrence is returned.

Function: int gsl_blas_sswap (gsl_vector_float * x, gsl_vector_float * y)
Function: int gsl_blas_dswap (gsl_vector * x, gsl_vector * y)
Function: int gsl_blas_cswap (gsl_vector_complex_float * x, gsl_vector_complex_float * y)
Function: int gsl_blas_zswap (gsl_vector_complex * x, gsl_vector_complex * y)
These functions exchange the elements of the vectors x and y.

Function: int gsl_blas_scopy (const gsl_vector_float * x, gsl_vector_float * y)
Function: int gsl_blas_dcopy (const gsl_vector * x, gsl_vector * y)
Function: int gsl_blas_ccopy (const gsl_vector_complex_float * x, gsl_vector_complex_float * y)
Function: int gsl_blas_zcopy (const gsl_vector_complex * x, gsl_vector_complex * y)
These functions copy the elements of the vector x into the vector y.

Function: int gsl_blas_saxpy (float alpha, const gsl_vector_float * x, gsl_vector_float * y)
Function: int gsl_blas_daxpy (double alpha, const gsl_vector * x, gsl_vector * y)
Function: int gsl_blas_caxpy (const gsl_complex_float alpha, const gsl_vector_complex_float * x, gsl_vector_complex_float * y)
Function: int gsl_blas_zaxpy (const gsl_complex alpha, const gsl_vector_complex * x, gsl_vector_complex * y)
These functions compute the sum y = \alpha x + y for the vectors x and y.

Function: void gsl_blas_sscal (float alpha, gsl_vector_float * x)
Function: void gsl_blas_dscal (double alpha, gsl_vector * x)
Function: void gsl_blas_cscal (const gsl_complex_float alpha, gsl_vector_complex_float * x)
Function: void gsl_blas_zscal (const gsl_complex alpha, gsl_vector_complex * x)
Function: void gsl_blas_csscal (float alpha, gsl_vector_complex_float * x)
Function: void gsl_blas_zdscal (double alpha, gsl_vector_complex * x)
These functions rescale the vector x by the multiplicative factor alpha.

Function: int gsl_blas_srotg (float a[], float b[], float c[], float s[])
Function: int gsl_blas_drotg (double a[], double b[], double c[], double s[])
These functions compute a Givens rotation (c,s) which zeroes the vector (a,b),

The variables a and b are overwritten by the routine.

Function: int gsl_blas_srot (gsl_vector_float * x, gsl_vector_float * y, float c, float s)
Function: int gsl_blas_drot (gsl_vector * x, gsl_vector * y, const double c, const double s)
These functions apply a Givens rotation (x', y') = (c x + s y, -s x + c y) to the vectors x, y.

Function: int gsl_blas_srotmg (float d1[], float d2[], float b1[], float b2, float P[])
Function: int gsl_blas_drotmg (double d1[], double d2[], double b1[], double b2, double P[])
These functions compute a modified Given's transformation.

Function: int gsl_blas_srotm (gsl_vector_float * x, gsl_vector_float * y, const float P[])
Function: int gsl_blas_drotm (gsl_vector * x, gsl_vector * y, const double P[])
These functions apply a modified Given's transformation.

Level 2

Function: int gsl_blas_sgemv (CBLAS_TRANSPOSE_t TransA, float alpha, const gsl_matrix_float * A, const gsl_vector_float * x, float beta, gsl_vector_float * y)
Function: int gsl_blas_dgemv (CBLAS_TRANSPOSE_t TransA, double alpha, const gsl_matrix * A, const gsl_vector * x, double beta, gsl_vector * y)
Function: int gsl_blas_cgemv (CBLAS_TRANSPOSE_t TransA, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, const gsl_vector_complex_float * x, const gsl_complex_float beta, gsl_vector_complex_float * y)
Function: int gsl_blas_zgemv (CBLAS_TRANSPOSE_t TransA, const gsl_complex alpha, const gsl_matrix_complex * A, const gsl_vector_complex * x, const gsl_complex beta, gsl_vector_complex * y)
These functions compute the matrix-vector product and sum y = \alpha op(A) x + \beta y, where op(A) = A, A^T, A^H for TransA = CblasNoTrans, CblasTrans, CblasConjTrans.

Function: int gsl_blas_strmv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix_float * A, gsl_vector_float * x)
Function: int gsl_blas_dtrmv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix * A, gsl_vector * x)
Function: int gsl_blas_ctrmv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix_complex_float * A, gsl_vector_complex_float * x)
Function: int gsl_blas_ztrmv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix_complex * A, gsl_vector_complex * x)
These functions compute the matrix-vector product and sum y =\alpha op(A) x + \beta y for the triangular matrix A, where op(A) = A, A^T, A^H for TransA = CblasNoTrans, CblasTrans, CblasConjTrans. When Uplo is CblasUpper then the upper triangle of A is used, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of the matrix is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are not referenced.

Function: int gsl_blas_strsv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix_float * A, gsl_vector_float * x)
Function: int gsl_blas_dtrsv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix * A, gsl_vector * x)
Function: int gsl_blas_ctrsv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix_complex_float * A, gsl_vector_complex_float * x)
Function: int gsl_blas_ztrsv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix_complex * A, gsl_vector_complex *x)
These functions compute inv(op(A)) x for x, where op(A) = A, A^T, A^H for TransA = CblasNoTrans, CblasTrans, CblasConjTrans. When Uplo is CblasUpper then the upper triangle of A is used, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of the matrix is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are not referenced.

Function: int gsl_blas_ssymv (CBLAS_UPLO_t Uplo, float alpha, const gsl_matrix_float * A, const gsl_vector_float * x, float beta, gsl_vector_float * y)
Function: int gsl_blas_dsymv (CBLAS_UPLO_t Uplo, double alpha, const gsl_matrix * A, const gsl_vector * x, double beta, gsl_vector * y)
These functions compute the matrix-vector product and sum y = \alpha A x + \beta y for the symmetric matrix A. Since the matrix A is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used.

Function: int gsl_blas_chemv (CBLAS_UPLO_t Uplo, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, const gsl_vector_complex_float * x, const gsl_complex_float beta, gsl_vector_complex_float * y)
Function: int gsl_blas_zhemv (CBLAS_UPLO_t Uplo, const gsl_complex alpha, const gsl_matrix_complex * A, const gsl_vector_complex * x, const gsl_complex beta, gsl_vector_complex * y)
These functions compute the matrix-vector product and sum y = \alpha A x + \beta y for the hermitian matrix A. Since the matrix A is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically assumed to be zero and are not referenced.

Function: int gsl_blas_sger (float alpha, const gsl_vector_float * x, const gsl_vector_float * y, gsl_matrix_float * A)
Function: int gsl_blas_dger (double alpha, const gsl_vector * x, const gsl_vector * y, gsl_matrix * A)
Function: int gsl_blas_cgeru (const gsl_complex_float alpha, const gsl_vector_complex_float * x, const gsl_vector_complex_float * y, gsl_matrix_complex_float * A)
Function: int gsl_blas_zgeru (const gsl_complex alpha, const gsl_vector_complex * x, const gsl_vector_complex * y, gsl_matrix_complex * A)
These functions compute the rank-1 update A = \alpha x y^T + A of the matrix A.

Function: int gsl_blas_cgerc (const gsl_complex_float alpha, const gsl_vector_complex_float * x, const gsl_vector_complex_float * y, gsl_matrix_complex_float * A)
Function: int gsl_blas_zgerc (const gsl_complex alpha, const gsl_vector_complex * x, const gsl_vector_complex * y, gsl_matrix_complex * A)
These functions compute the conjugate rank-1 update A = \alpha x y^H + A of the matrix A.

Function: int gsl_blas_ssyr (CBLAS_UPLO_t Uplo, float alpha, const gsl_vector_float * x, gsl_matrix_float * A)
Function: int gsl_blas_dsyr (CBLAS_UPLO_t Uplo, double alpha, const gsl_vector * x, gsl_matrix * A)
These functions compute the symmetric rank-1 update A = \alpha x x^T + A of the symmetric matrix A. Since the matrix A is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used.

Function: int gsl_blas_cher (CBLAS_UPLO_t Uplo, float alpha, const gsl_vector_complex_float * x, gsl_matrix_complex_float * A)
Function: int gsl_blas_zher (CBLAS_UPLO_t Uplo, double alpha, const gsl_vector_complex * x, gsl_matrix_complex * A)
These functions compute the hermitian rank-1 update A = \alpha x x^H + A of the hermitian matrix A. Since the matrix A is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically set to zero.

Function: int gsl_blas_ssyr2 (CBLAS_UPLO_t Uplo, float alpha, const gsl_vector_float * x, const gsl_vector_float * y, gsl_matrix_float * A)
Function: int gsl_blas_dsyr2 (CBLAS_UPLO_t Uplo, double alpha, const gsl_vector * x, const gsl_vector * y, gsl_matrix * A)
These functions compute the symmetric rank-2 update A = \alpha x y^T + \alpha y x^T + A of the symmetric matrix A. Since the matrix A is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used.

Function: int gsl_blas_cher2 (CBLAS_UPLO_t Uplo, const gsl_complex_float alpha, const gsl_vector_complex_float * x, const gsl_vector_complex_float * y, gsl_matrix_complex_float * A)
Function: int gsl_blas_zher2 (CBLAS_UPLO_t Uplo, const gsl_complex alpha, const gsl_vector_complex * x, const gsl_vector_complex * y, gsl_matrix_complex * A)
These functions compute the hermitian rank-2 update A = \alpha x y^H + \alpha^* y x^H A of the hermitian matrix A. Since the matrix A is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically set to zero.

Level 3

Function: int gsl_blas_sgemm (CBLAS_TRANSPOSE_t TransA, CBLAS_TRANSPOSE_t TransB, float alpha, const gsl_matrix_float * A, const gsl_matrix_float * B, float beta, gsl_matrix_float * C)
Function: int gsl_blas_dgemm (CBLAS_TRANSPOSE_t TransA, CBLAS_TRANSPOSE_t TransB, double alpha, const gsl_matrix * A, const gsl_matrix * B, double beta, gsl_matrix * C)
Function: int gsl_blas_cgemm (CBLAS_TRANSPOSE_t TransA, CBLAS_TRANSPOSE_t TransB, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, const gsl_matrix_complex_float * B, const gsl_complex_float beta, gsl_matrix_complex_float * C)
Function: int gsl_blas_zgemm (CBLAS_TRANSPOSE_t TransA, CBLAS_TRANSPOSE_t TransB, const gsl_complex alpha, const gsl_matrix_complex * A, const gsl_matrix_complex * B, const gsl_complex beta, gsl_matrix_complex * C)
These functions compute the matrix-matrix product and sum C = \alpha op(A) op(B) + \beta C where op(A) = A, A^T, A^H for TransA = CblasNoTrans, CblasTrans, CblasConjTrans and similarly for the parameter TransB.

Function: int gsl_blas_ssymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, float alpha, const gsl_matrix_float * A, const gsl_matrix_float * B, float beta, gsl_matrix_float * C)
Function: int gsl_blas_dsymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, double alpha, const gsl_matrix * A, const gsl_matrix * B, double beta, gsl_matrix * C)
Function: int gsl_blas_csymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, const gsl_matrix_complex_float * B, const gsl_complex_float beta, gsl_matrix_complex_float * C)
Function: int gsl_blas_zsymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, const gsl_complex alpha, const gsl_matrix_complex * A, const gsl_matrix_complex * B, const gsl_complex beta, gsl_matrix_complex * C)
These functions compute the matrix-matrix product and sum C = \alpha A B + \beta C for Side is CblasLeft and C = \alpha B A + \beta C for Side is CblasRight, where the matrix A is symmetric. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used.

Function: int gsl_blas_chemm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, const gsl_matrix_complex_float * B, const gsl_complex_float beta, gsl_matrix_complex_float * C)
Function: int gsl_blas_zhemm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, const gsl_complex alpha, const gsl_matrix_complex * A, const gsl_matrix_complex * B, const gsl_complex beta, gsl_matrix_complex * C)
These functions compute the matrix-matrix product and sum C = \alpha A B + \beta C for Side is CblasLeft and C = \alpha B A + \beta C for Side is CblasRight, where the matrix A is hermitian. When Uplo is CblasUpper then the upper triangle and diagonal of A are used, and when Uplo is CblasLower then the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically set to zero.

Function: int gsl_blas_strmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, float alpha, const gsl_matrix_float * A, gsl_matrix_float * B)
Function: int gsl_blas_dtrmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, double alpha, const gsl_matrix * A, gsl_matrix * B)
Function: int gsl_blas_ctrmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, gsl_matrix_complex_float * B)
Function: int gsl_blas_ztrmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_complex alpha, const gsl_matrix_complex * A, gsl_matrix_complex * B)
These functions compute the matrix-matrix product B = \alpha op(A) B for Side is CblasLeft and B = \alpha B op(A) for Side is CblasRight. The matrix A is triangular and op(A) = A, A^T, A^H for TransA = CblasNoTrans, CblasTrans, CblasConjTrans When Uplo is CblasUpper then the upper triangle of A is used, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of A is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are not referenced.

Function: int gsl_blas_strsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, float alpha, const gsl_matrix_float * A, gsl_matrix_float * B)
Function: int gsl_blas_dtrsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, double alpha, const gsl_matrix * A, gsl_matrix * B)
Function: int gsl_blas_ctrsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, gsl_matrix_complex_float * B)
Function: int gsl_blas_ztrsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_complex alpha, const gsl_matrix_complex * A, gsl_matrix_complex * B)
These functions compute the matrix-matrix product B = \alpha op(inv(A)) B for Side is CblasLeft and B = \alpha B op(inv(A)) for Side is CblasRight. The matrix A is triangular and op(A) = A, A^T, A^H for TransA = CblasNoTrans, CblasTrans, CblasConjTrans When Uplo is CblasUpper then the upper triangle of A is used, and when Uplo is CblasLower then the lower triangle of A is used. If Diag is CblasNonUnit then the diagonal of A is used, but if Diag is CblasUnit then the diagonal elements of the matrix A are taken as unity and are not referenced.

Function: int gsl_blas_ssyrk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, float alpha, const gsl_matrix_float * A, float beta, gsl_matrix_float * C)
Function: int gsl_blas_dsyrk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, double alpha, const gsl_matrix * A, double beta, gsl_matrix * C)
Function: int gsl_blas_csyrk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, const gsl_complex_float beta, gsl_matrix_complex_float * C)
Function: int gsl_blas_zsyrk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex alpha, const gsl_matrix_complex * A, const gsl_complex beta, gsl_matrix_complex * C)
These functions compute a rank-k update of the symmetric matrix C, C = \alpha A A^T + \beta C when Trans is CblasNoTrans and C = \alpha A^T A + \beta C when Trans is CblasTrans. Since the matrix C is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used.

Function: int gsl_blas_cherk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, float alpha, const gsl_matrix_complex_float * A, float beta, gsl_matrix_complex_float * C)
Function: int gsl_blas_zherk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, double alpha, const gsl_matrix_complex * A, double beta, gsl_matrix_complex * C)
These functions compute a rank-k update of the hermitian matrix C, C = \alpha A A^H + \beta C when Trans is CblasNoTrans and C = \alpha A^H A + \beta C when Trans is CblasTrans. Since the matrix C is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used. The imaginary elements of the diagonal are automatically set to zero.

Function: int gsl_blas_ssyr2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, float alpha, const gsl_matrix_float * A, const gsl_matrix_float * B, float beta, gsl_matrix_float * C)
Function: int gsl_blas_dsyr2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, double alpha, const gsl_matrix * A, const gsl_matrix * B, double beta, gsl_matrix * C)
Function: int gsl_blas_csyr2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, const gsl_matrix_complex_float * B, const gsl_complex_float beta, gsl_matrix_complex_float * C)
Function: int gsl_blas_zsyr2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex alpha, const gsl_matrix_complex * A, const gsl_matrix_complex * B, const gsl_complex beta, gsl_matrix_complex *C)
These functions compute a rank-2k update of the symmetric matrix C, C = \alpha A B^T + \alpha B A^T + \beta C when Trans is CblasNoTrans and C = \alpha A^T B + \alpha B^T A + \beta C when Trans is CblasTrans. Since the matrix C is symmetric only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used.

Function: int gsl_blas_cher2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex_float alpha, const gsl_matrix_complex_float * A, const gsl_matrix_complex_float * B, float beta, gsl_matrix_complex_float * C)
Function: int gsl_blas_zher2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t Trans, const gsl_complex alpha, const gsl_matrix_complex * A, const gsl_matrix_complex * B, double beta, gsl_matrix_complex * C)
These functions compute a rank-2k update of the hermitian matrix C, C = \alpha A B^H + \alpha^* B A^H + \beta C when Trans is CblasNoTrans and C = \alpha A^H B + \alpha^* B^H A + \beta C when Trans is CblasTrans. Since the matrix C is hermitian only its upper half or lower half need to be stored. When Uplo is CblasUpper then the upper triangle and diagonal of C are used, and when Uplo is CblasLower then the lower triangle and diagonal of C are used. The imaginary elements of the diagonal are automatically set to zero.

Examples

The following program computes the product of two matrices using the Level-3 BLAS function DGEMM,

The matrices are stored in row major order, according to the C convention for arrays.

#include <stdio.h>
#include <gsl/gsl_blas.h>

int
main (void)
{
  double a[] = { 0.11, 0.12, 0.13,
                 0.21, 0.22, 0.23 };

  double b[] = { 1011, 1012,
                 1021, 1022,
                 1031, 1032 };

  double c[] = { 0.00, 0.00,
                 0.00, 0.00 };

  gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
  gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
  gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);

  /* Compute C = A B */

  gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
                  1.0, &A.matrix, &B.matrix,
                  0.0, &C.matrix);

  printf("[ %g, %g\n", c[0], c[1]);
  printf("  %g, %g ]\n", c[2], c[3]);

  return 0;  
}

Here is the output from the program,

$ ./a.out
[ 367.76, 368.12
  674.06, 674.72 ]

References and Further Reading

Information on the BLAS standards, including both the legacy and draft interface standards, is available online from the BLAS Homepage and BLAS Technical Forum web-site.

The following papers contain the specifications for Level 1, Level 2 and Level 3 BLAS.

Postscript versions of the latter two papers are available from http://www.netlib.org/blas/. A CBLAS wrapper for Fortran BLAS libraries is available from the same location.

Linear Algebra

This chapter describes functions for solving linear systems. The library provides simple linear algebra operations which operate directly on the gsl_vector and gsl_matrix objects. These are intended for use with "small" systems where simple algorithms are acceptable.

Anyone interested in large systems will want to use the sophisticated routines found in LAPACK. The Fortran version of LAPACK is recommended as the standard package for linear algebra. It supports blocked algorithms, specialized data representations and other optimizations.

The functions described in this chapter are declared in the header file `gsl_linalg.h'.

LU Decomposition

A general square matrix A has an LU decomposition into upper and lower triangular matrices,

where P is a permutation matrix, L is unit lower triangular matrix and U is upper triangular matrix. For square matrices this decomposition can be used to convert the linear system A x = b into a pair of triangular systems (L y = P b, U x = y), which can be solved by forward and back-substitution.

Function: int gsl_linalg_LU_decomp (gsl_matrix * A, gsl_permutation * p, int *signum)
Function: int gsl_linalg_complex_LU_decomp (gsl_matrix_complex * A, gsl_permutation * p, int *signum)
These functions factorize the square matrix A into the LU decomposition PA = LU. On output the diagonal and upper triangular part of the input matrix A contain the matrix U. The lower triangular part of the input matrix (excluding the diagonal) contains L. The diagonal elements of L are unity, and are not stored.

The permutation matrix P is encoded in the permutation p. The j-th column of the matrix P is given by the k-th column of the identity matrix, where k = p_j the j-th element of the permutation vector. The sign of the permutation is given by signum. It has the value (-1)^n, where n is the number of interchanges in the permutation.

The algorithm used in the decomposition is Gaussian Elimination with partial pivoting (Golub & Van Loan, Matrix Computations, Algorithm 3.4.1).

Function: int gsl_linalg_LU_solve (const gsl_matrix * LU, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)
Function: int gsl_linalg_complex_LU_solve (const gsl_matrix_complex * LU, const gsl_permutation * p, const gsl_vector_complex * b, gsl_vector_complex * x)
These functions solve the system A x = b using the LU decomposition of A into (LU, p) given by gsl_linalg_LU_decomp or gsl_linalg_complex_LU_decomp.

Function: int gsl_linalg_LU_svx (const gsl_matrix * LU, const gsl_permutation * p, gsl_vector * x)
Function: int gsl_linalg_complex_LU_svx (const gsl_matrix_complex * LU, const gsl_permutation * p, gsl_vector_complex * x)
These functions solve the system A x = b in-place using the LU decomposition of A into (LU,p). On input x should contain the right-hand side b, which is replaced by the solution on output.

Function: int gsl_linalg_LU_refine (const gsl_matrix * A, const gsl_matrix * LU, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x, gsl_vector * residual)
Function: int gsl_linalg_complex_LU_refine (const gsl_matrix_complex * A, const gsl_matrix_complex * LU, const gsl_permutation * p, const gsl_vector_complex * b, gsl_vector_complex * x, gsl_vector_complex * residual)
These functions apply an iterative improvement to x, the solution of A x = b, using the LU decomposition of A into (LU,p). The initial residual r = A x - b is also computed and stored in residual.

Function: int gsl_linalg_LU_invert (const gsl_matrix * LU, const gsl_permutation * p, gsl_matrix * inverse)
Function: int gsl_complex_linalg_LU_invert (const gsl_matrix_complex * LU, const gsl_permutation * p, gsl_matrix_complex * inverse)
These functions compute the inverse of a matrix A from its LU decomposition (LU,p), storing the result in the matrix inverse. The inverse is computed by solving the system A x = b for each column of the identity matrix.

Function: double gsl_linalg_LU_det (gsl_matrix * LU, int signum)
Function: gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex * LU, int signum)
These functions compute the determinant of a matrix A from its LU decomposition, LU. The determinant is computed as the product of the diagonal elements of U and the sign of the row permutation signum.

Function: double gsl_linalg_LU_lndet (gsl_matrix * LU)
Function: double gsl_linalg_complex_LU_lndet (gsl_matrix_complex * LU)
These functions compute the logarithm of the absolute value of the determinant of a matrix A, \ln|det(A)|, from its LU decomposition, LU. This function may be useful if the direct computation of the determinant would overflow or underflow.

Function: int gsl_linalg_LU_sgndet (gsl_matrix * LU, int signum)
Function: gsl_complex gsl_linalg_complex_LU_sgndet (gsl_matrix_complex * LU, int signum)
These functions compute the sign or phase factor of the determinant of a matrix A, det(A)/|det(A)|, from its LU decomposition, LU.

QR Decomposition

A general rectangular M-by-N matrix A has a QR decomposition into the product of an orthogonal M-by-M square matrix Q (where Q^T Q = I) and an M-by-N right-triangular matrix R,

This decomposition can be used to convert the linear system A x = b into the triangular system R x = Q^T b, which can be solved by back-substitution. Another use of the QR decomposition is to compute an orthonormal basis for a set of vectors. The first N columns of Q form an orthonormal basis for the range of A, ran(A), when A has full column rank.

Function: int gsl_linalg_QR_decomp (gsl_matrix * A, gsl_vector * tau)
This function factorizes the M-by-N matrix A into the QR decomposition A = Q R. On output the diagonal and upper triangular part of the input matrix contain the matrix R. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and Householder vectors which encode the orthogonal matrix Q. The vector tau must be of length k=\min(M,N). The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the same storage scheme as used by LAPACK.

The algorithm used to perform the decomposition is Householder QR (Golub & Van Loan, Matrix Computations, Algorithm 5.2.1).

Function: int gsl_linalg_QR_solve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x)
This function solves the system A x = b using the QR decomposition of A into (QR, tau) given by gsl_linalg_QR_decomp.

Function: int gsl_linalg_QR_svx (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * x)
This function solves the system A x = b in-place using the QR decomposition of A into (QR,tau) given by gsl_linalg_QR_decomp. On input x should contain the right-hand side b, which is replaced by the solution on output.

Function: int gsl_linalg_QR_lssolve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x, gsl_vector * residual)
This function finds the least squares solution to the overdetermined system A x = b where the matrix A has more rows than columns. The least squares solution minimizes the Euclidean norm of the residual, ||Ax - b||.The routine uses the QR decomposition of A into (QR, tau) given by gsl_linalg_QR_decomp. The solution is returned in x. The residual is computed as a by-product and stored in residual.

Function: int gsl_linalg_QR_QTvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v)
This function applies the matrix Q^T encoded in the decomposition (QR,tau) to the vector v, storing the result Q^T v in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^T.

Function: int gsl_linalg_QR_Qvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v)
This function applies the matrix Q encoded in the decomposition (QR,tau) to the vector v, storing the result Q v in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q.

Function: int gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const gsl_vector * b, gsl_vector * x)
This function solves the triangular system R x = b for x. It may be useful if the product b' = Q^T b has already been computed using gsl_linalg_QR_QTvec.

Function: int gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector * x)
This function solves the triangular system R x = b for x in-place. On input x should contain the right-hand side b and is replaced by the solution on output. This function may be useful if the product b' = Q^T b has already been computed using gsl_linalg_QR_QTvec.

Function: int gsl_linalg_QR_unpack (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * R)
This function unpacks the encoded QR decomposition (QR,tau) into the matrices Q and R, where Q is M-by-M and R is M-by-N.

Function: int gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix * R, const gsl_vector * b, gsl_vector * x)
This function solves the system R x = Q^T b for x. It can be used when the QR decomposition of a matrix is available in unpacked form as (Q,R).

Function: int gsl_linalg_QR_update (gsl_matrix * Q, gsl_matrix * R, gsl_vector * w, const gsl_vector * v)
This function performs a rank-1 update w v^T of the QR decomposition (Q, R). The update is given by Q'R' = Q R + w v^T where the output matrices Q' and R' are also orthogonal and right triangular. Note that w is destroyed by the update.

Function: int gsl_linalg_R_solve (const gsl_matrix * R, const gsl_vector * b, gsl_vector * x)
This function solves the triangular system R x = b for the N-by-N matrix R.

Function: int gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector * x)
This function solves the triangular system R x = b in-place. On input x should contain the right-hand side b, which is replaced by the solution on output.

QR Decomposition with Column Pivoting

The QR decomposition can be extended to the rank deficient case by introducing a column permutation P,

The first r columns of this Q form an orthonormal basis for the range of A for a matrix with column rank r. This decomposition can also be used to convert the linear system A x = b into the triangular system R y = Q^T b, x = P y, which can be solved by back-substitution and permutation. We denote the QR decomposition with column pivoting by QRP^T since A = Q R P^T.

Function: int gsl_linalg_QRPT_decomp (gsl_matrix * A, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm)
This function factorizes the M-by-N matrix A into the QRP^T decomposition A = Q R P^T. On output the diagonal and upper triangular part of the input matrix contain the matrix R. The permutation matrix P is stored in the permutation p. The sign of the permutation is given by signum. It has the value (-1)^n, where n is the number of interchanges in the permutation. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and vectors which encode the orthogonal matrix Q. The vector tau must be of length k=\min(M,N). The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the same storage scheme as used by LAPACK. On output the norms of each column of R are stored in the vector norm.

The algorithm used to perform the decomposition is Householder QR with column pivoting (Golub & Van Loan, Matrix Computations, Algorithm 5.4.1).

Function: int gsl_linalg_QRPT_decomp2 (const gsl_matrix * A, gsl_matrix * q, gsl_matrix * r, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm)
This function factorizes the matrix A into the decomposition A = Q R P^T without modifying A itself and storing the output in the separate matrices q and r.

Function: int gsl_linalg_QRPT_solve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)
This function solves the system A x = b using the QRP^T decomposition of A into (QR, tau, p) given by gsl_linalg_QRPT_decomp.

Function: int gsl_linalg_QRPT_svx (const gsl_matrix * QR, const gsl_vector * tau, const gsl_permutation * p, gsl_vector * x)
This function solves the system A x = b in-place using the QRP^T decomposition of A into (QR,tau,p). On input x should contain the right-hand side b, which is replaced by the solution on output.

Function: int gsl_linalg_QRPT_QRsolve (const gsl_matrix * Q, const gsl_matrix * R, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)
This function solves the system R P^T x = Q^T b for x. It can be used when the QR decomposition of a matrix is available in unpacked form as (Q,R).

Function: int gsl_linalg_QRPT_update (gsl_matrix * Q, gsl_matrix * R, const gsl_permutation * p, gsl_vector * u, const gsl_vector * v)
This function performs a rank-1 update w v^T of the QRP^T decomposition (Q, R,p). The update is given by Q'R' = Q R + w v^T where the output matrices Q' and R' are also orthogonal and right triangular. Note that w is destroyed by the update. The permutation p is not changed.

Function: int gsl_linalg_QRPT_Rsolve (const gsl_matrix * QR, const gsl_permutation * p, const gsl_vector * b, gsl_vector * x)
This function solves the triangular system R P^T x = b for the N-by-N matrix R contained in QR.

Function: int gsl_linalg_QRPT_Rsvx (const gsl_matrix * QR, const gsl_permutation * p, gsl_vector * x)
This function solves the triangular system R P^T x = b in-place for the N-by-N matrix R contained in QR. On input x should contain the right-hand side b, which is replaced by the solution on output.

Singular Value Decomposition

A general rectangular M-by-N matrix A has a singular value decomposition (SVD) into the product of an M-by-N orthogonal matrix U, an N-by-N diagonal matrix of singular values S and the transpose of an M-by-M orthogonal square matrix V,

The singular values \sigma_i = S_{ii} are all non-negative and are generally chosen to form a non-increasing sequence \sigma_1 >= \sigma_2 >= ... >= \sigma_N >= 0.

The singular value decomposition of a matrix has many practical uses. The condition number of the matrix is given by the ratio of the largest singular value to the smallest singular value. The presence of a zero singular value indicates that the matrix is singular. The number of non-zero singular values indicates the rank of the matrix. In practice singular value decomposition of a rank-deficient matrix will not produce exact zeroes for singular values, due to finite numerical precision. Small singular values should be edited by choosing a suitable tolerance.

Function: int gsl_linalg_SV_decomp (gsl_matrix * A, gsl_matrix * V, gsl_vector * S, gsl_vector * work)
This function factorizes the M-by-N matrix A into the singular value decomposition A = U S V^T. On output the matrix A is replaced by U. The diagonal elements of the singular value matrix S are stored in the vector S. The singular values are non-negative and form a non-increasing sequence from S_1 to S_N. The matrix V contains the elements of V in untransposed form. To form the product U S V^T it is necessary to take the transpose of V. A workspace of length N is required in work.

This routine uses the Golub-Reinsch SVD algorithm.

Function: int gsl_linalg_SV_decomp_mod (gsl_matrix * A, gsl_matrix * X, gsl_matrix * V, gsl_vector * S, gsl_vector * work)
This function computes the SVD using the modified Golub-Reinsch algorithm, which is faster for M>>N. It requires the vector work and the N-by-N matrix X as additional working space.

Function: int gsl_linalg_SV_decomp_jacobi (gsl_matrix * A, gsl_matrix * V, gsl_vector * S)
This function computes the SVD using one-sided Jacobi orthogonalization (see references for details). The Jacobi method can compute singular values to higher relative accuracy than Golub-Reinsch algorithms.

Function: int gsl_linalg_SV_solve (gsl_matrix * U, gsl_matrix * V, gsl_vector * S, const gsl_vector * b, gsl_vector * x)
This function solves the system A x = b using the singular value decomposition (U, S, V) of A given by gsl_linalg_SV_decomp.

Only non-zero singular values are used in computing the solution. The parts of the solution corresponding to singular values of zero are ignored. Other singular values can be edited out by setting them to zero before calling this function.

In the over-determined case where A has more rows than columns the system is solved in the least squares sense, returning the solution x which minimizes ||A x - b||_2.

Cholesky Decomposition

A symmetric, positive definite square matrix A has a Cholesky decomposition into a product of a lower triangular matrix L and its transpose L^T,

This is sometimes referred to as taking the square-root of a matrix. The Cholesky decomposition can only be carried out when all the eigenvalues of the matrix are positive. This decomposition can be used to convert the linear system A x = b into a pair of triangular systems (L y = b, L^T x = y), which can be solved by forward and back-substitution.

Function: int gsl_linalg_cholesky_decomp (gsl_matrix * A)
This function factorizes the positive-definite square matrix A into the Cholesky decomposition A = L L^T. On output the diagonal and lower triangular part of the input matrix A contain the matrix L. The upper triangular part of the input matrix contains L^T, the diagonal terms being identical for both L and L^T. If the matrix is not positive-definite then the decomposition will fail, returning the error code GSL_EDOM.

Function: int gsl_linalg_cholesky_solve (const gsl_matrix * cholesky, const gsl_vector * b, gsl_vector * x)
This function solves the system A x = b using the Cholesky decomposition of A into the matrix cholesky given by gsl_linalg_cholesky_decomp.

Function: int gsl_linalg_cholesky_svx (const gsl_matrix * cholesky, gsl_vector * x)
This function solves the system A x = b in-place using the Cholesky decomposition of A into the matrix cholesky given by gsl_linalg_cholesky_decomp. On input x should contain the right-hand side b, which is replaced by the solution on output.

Tridiagonal Decomposition of Real Symmetric Matrices

A symmetric matrix A can be factorized by similarity transformations into the form,

where Q is an orthogonal matrix and T is a symmetric tridiagonal matrix.

Function: int gsl_linalg_symmtd_decomp (gsl_matrix * A, gsl_vector * tau)
This function factorizes the symmetric square matrix A into the symmetric tridiagonal decomposition Q T Q^T. On output the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients tau, encode the orthogonal matrix Q. This storage scheme is the same as used by LAPACK. The upper triangular part of A is not referenced.

Function: int gsl_linalg_symmtd_unpack (const gsl_matrix * A, const gsl_vector * tau, gsl_matrix * Q, gsl_vector * d, gsl_vector * sd)
This function unpacks the encoded symmetric tridiagonal decomposition (A, tau) obtained from gsl_linalg_symmtd_decomp into the orthogonal matrix Q, the vector of diagonal elements d and the vector of subdiagonal elements sd.

Function: int gsl_linalg_symmtd_unpack_dsd (const gsl_matrix * A, gsl_vector * d, gsl_vector * sd)
This function unpacks the diagonal and subdiagonal of the encoded symmetric tridiagonal decomposition (A, tau) obtained from gsl_linalg_symmtd_decomp into the vectors d and sd.

Tridiagonal Decomposition of Hermitian Matrices

A hermitian matrix A can be factorized by similarity transformations into the form,

where U is an unitary matrix and T is a real symmetric tridiagonal matrix.

Function: int gsl_linalg_hermtd_decomp (gsl_matrix_complex * A, gsl_vector_complex * tau)
This function factorizes the hermitian matrix A into the symmetric tridiagonal decomposition U T U^T. On output the real parts of the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients tau, encode the orthogonal matrix Q. This storage scheme is the same as used by LAPACK. The upper triangular part of A and imaginary parts of the diagonal are not referenced.

Function: int gsl_linalg_hermtd_unpack (const gsl_matrix_complex * A, const gsl_vector_complex * tau, gsl_matrix_complex * Q, gsl_vector * d, gsl_vector * sd)
This function unpacks the encoded tridiagonal decomposition (A, tau) obtained from gsl_linalg_hermtd_decomp into the unitary matrix U, the real vector of diagonal elements d and the real vector of subdiagonal elements sd.

Function: int gsl_linalg_hermtd_unpack_dsd (const gsl_matrix_complex * A, gsl_vector * d, gsl_vector * sd)
This function unpacks the diagonal and subdiagonal of the encoded tridiagonal decomposition (A, tau) obtained from gsl_linalg_hermtd_decomp into the real vectors d and sd.

Bidiagonalization

A general matrix A can be factorized by similarity transformations into the form,

where U and V are orthogonal matrices and B is a N-by-N bidiagonal matrix with non-zero entries only on the diagonal and superdiagonal. The size of U is M-by-N and the size of V is N-by-N.

Function: int gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V)
This function factorizes the M-by-N matrix A into bidiagonal form U B V^T. The diagonal and superdiagonal of the matrix B are stored in the diagonal and superdiagonal of A. The orthogonal matrices U and V are stored as compressed Householder vectors in the remaining elements of A. The Householder coefficients are stored in the vectors tau_U and tau_V. The length of tau_U must equal the number of elements in the diagonal of A and the length of tau_V should be one element shorter.

Function: int gsl_linalg_bidiag_unpack (const gsl_matrix * A, const gsl_vector * tau_U, gsl_matrix * U, const gsl_vector * tau_V, gsl_matrix * V, gsl_vector * diag, gsl_vector * superdiag)
This function unpacks the bidiagonal decomposition of A given by gsl_linalg_bidiag_decomp, (A, tau_U, tau_V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal superdiag.

Function: int gsl_linalg_bidiag_unpack2 (gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V, gsl_matrix * V)
This function unpacks the bidiagonal decomposition of A given by gsl_linalg_bidiag_decomp, (A, tau_U, tau_V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal superdiag. The matrix U is stored in-place in A.

Function: int gsl_linalg_bidiag_unpack_B (const gsl_matrix * A, gsl_vector * diag, gsl_vector * superdiag)
This function unpacks the diagonal and superdiagonal of the bidiagonal decomposition of A given by gsl_linalg_bidiag_decomp, into the diagonal vector diag and superdiagonal vector superdiag.

Householder solver for linear systems

Function: int gsl_linalg_HH_solve (gsl_matrix * A, const gsl_vector * b, gsl_vector * x)
This function solves the system A x = b directly using Householder transformations. On output the solution is stored in x and b is not modified. The matrix A is destroyed by the Householder transformations.

Function: int gsl_linalg_HH_svx (gsl_matrix * A, gsl_vector * x)
This function solves the system A x = b in-place using Householder transformations. On input x should contain the right-hand side b, which is replaced by the solution on output. The matrix A is destroyed by the Householder transformations.

Tridiagonal Systems

Function: int gsl_linalg_solve_symm_tridiag (const gsl_vector * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector * x)
This function solves the general N-by-N system A x = b where A is symmetric tridiagonal. The form of A for the 4-by-4 case is shown below,

Function: int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector * x)
This function solves the general N-by-N system A x = b where A is symmetric cyclic tridiagonal. The form of A for the 4-by-4 case is shown below,

Examples

The following program solves the linear system A x = b. The system to be solved is,

and the solution is found using LU decomposition of the matrix A.

#include <stdio.h>
#include <gsl/gsl_linalg.h>

int
main (void)
{
  double a_data[] = { 0.18, 0.60, 0.57, 0.96,
                      0.41, 0.24, 0.99, 0.58,
                      0.14, 0.30, 0.97, 0.66,
                      0.51, 0.13, 0.19, 0.85 };

  double b_data[] = { 1.0, 2.0, 3.0, 4.0 };

  gsl_matrix_view m 
    = gsl_matrix_view_array(a_data, 4, 4);

  gsl_vector_view b
    = gsl_vector_view_array(b_data, 4);

  gsl_vector *x = gsl_vector_alloc (4);
  
  int s;

  gsl_permutation * p = gsl_permutation_alloc (4);

  gsl_linalg_LU_decomp (&m.matrix, p, &s);

  gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);

  printf ("x = \n");
  gsl_vector_fprintf(stdout, x, "%g");

  gsl_permutation_free (p);
  return 0;
}

Here is the output from the program,

x = -4.05205
-12.6056
1.66091
8.69377

This can be verified by multiplying the solution x by the original matrix A using GNU OCTAVE,

octave> A = [ 0.18, 0.60, 0.57, 0.96;
              0.41, 0.24, 0.99, 0.58; 
              0.14, 0.30, 0.97, 0.66; 
              0.51, 0.13, 0.19, 0.85 ];

octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];

octave> A * x
ans =

  1.0000
  2.0000
  3.0000
  4.0000

This reproduces the original right-hand side vector, b, in accordance with the equation A x = b.

References and Further Reading

Further information on the algorithms described in this section can be found in the following book,

The LAPACK library is described in,

The LAPACK source code can be found at the website above, along with an online copy of the users guide.

The Modified Golub-Reinsch algorithm is described in the following paper,

The Jacobi algorithm for singular value decomposition is described in the following papers,

Eigensystems

This chapter describes functions for computing eigenvalues and eigenvectors of matrices. There are routines for real symmetric and complex hermitian matrices, and eigenvalues can be computed with or without eigenvectors. The algorithms used are symmetric bidiagonalization followed by QR reduction.

These routines are intended for "small" systems where simple algorithms are acceptable. Anyone interested finding eigenvalues and eigenvectors of large matrices will want to use the sophisticated routines found in LAPACK. The Fortran version of LAPACK is recommended as the standard package for linear algebra.

The functions described in this chapter are declared in the header file `gsl_eigen.h'.

Real Symmetric Matrices

Function: gsl_eigen_symm_workspace * gsl_eigen_symm_alloc (const size_t n)
This function allocates a workspace for computing eigenvalues of n-by-n real symmetric matrices. The size of the workspace is O(2n).

Function: void gsl_eigen_symm_free (gsl_eigen_symm_workspace * w)
This function frees the memory associated with the workspace w.

Function: int gsl_eigen_symm (gsl_matrix * A, gsl_vector * eval, gsl_eigen_symm_workspace * w)
This function computes the eigenvalues of the real symmetric matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The eigenvalues are stored in the vector eval and are unordered.

Function: gsl_eigen_symmv_workspace * gsl_eigen_symmv_alloc (const size_t n)
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real symmetric matrices. The size of the workspace is O(4n).

Function: void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * w)
This function frees the memory associated with the workspace w.

Function: int gsl_eigen_symmv (gsl_matrix * A, gsl_vector * eval, gsl_matrix * evec, gsl_eigen_symmv_workspace * w)
This function computes the eigenvalues and eigenvectors of the real symmetric matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The eigenvalues are stored in the vector eval and are unordered. The corresponding eigenvectors are stored in the columns of the matrix evec. For example, the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed to be mutually orthogonal and normalised to unit magnitude.

Complex Hermitian Matrices

Function: gsl_eigen_herm_workspace * gsl_eigen_herm_alloc (const size_t n)
This function allocates a workspace for computing eigenvalues of n-by-n complex hermitian matrices. The size of the workspace is O(3n).

Function: void gsl_eigen_herm_free (gsl_eigen_herm_workspace * w)
This function frees the memory associated with the workspace w.

Function: int gsl_eigen_herm (gsl_matrix_complex * A, gsl_vector * eval, gsl_eigen_herm_workspace * w)
This function computes the eigenvalues of the complex hermitian matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The imaginary parts of the diagonal are assumed to be zero and are not referenced. The eigenvalues are stored in the vector eval and are unordered.

Function: gsl_eigen_hermv_workspace * gsl_eigen_hermv_alloc (const size_t n)
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n complex hermitian matrices. The size of the workspace is O(5n).

Function: void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * w)
This function frees the memory associated with the workspace w.

Function: int gsl_eigen_hermv (gsl_matrix_complex * A, gsl_vector * eval, gsl_matrix_complex * evec, gsl_eigen_hermv_workspace * w)
This function computes the eigenvalues and eigenvectors of the complex hermitian matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The imaginary parts of the diagonal are assumed to be zero and are not referenced. The eigenvalues are stored in the vector eval and are unordered. The corresponding complex eigenvectors are stored in the columns of the matrix evec. For example, the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed to be mutually orthogonal and normalised to unit magnitude.

Sorting Eigenvalues and Eigenvectors

Function: int gsl_eigen_symmv_sort (gsl_vector * eval, gsl_matrix * evec, gsl_eigen_sort_t sort_type)
This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding real eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort_type,
GSL_EIGEN_SORT_VAL_ASC
ascending order in numerical value
GSL_EIGEN_SORT_VAL_DESC
descending order in numerical value
GSL_EIGEN_SORT_ABS_ASC
ascending order in magnitude
GSL_EIGEN_SORT_ABS_DESC
descending order in magnitude

Function: int gsl_eigen_hermv_sort (gsl_vector * eval, gsl_matrix_complex * evec, gsl_eigen_sort_t sort_type)
This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding complex eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort_type as shown above.

Examples

The following program computes the eigenvalues and eigenvectors of the 4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>

int
main (void)
{
  double data[] = { 1.0  , 1/2.0, 1/3.0, 1/4.0,
                    1/2.0, 1/3.0, 1/4.0, 1/5.0,
                    1/3.0, 1/4.0, 1/5.0, 1/6.0,
                    1/4.0, 1/5.0, 1/6.0, 1/7.0 };

  gsl_matrix_view m 
    = gsl_matrix_view_array(data, 4, 4);

  gsl_vector *eval = gsl_vector_alloc (4);
  gsl_matrix *evec = gsl_matrix_alloc (4, 4);

  gsl_eigen_symmv_workspace * w = 
    gsl_eigen_symmv_alloc (4);
  
  gsl_eigen_symmv (&m.matrix, eval, evec, w);

  gsl_eigen_symmv_free(w);

  gsl_eigen_symmv_sort (eval, evec, 
                        GSL_EIGEN_SORT_ABS_ASC);
  
  {
    int i;

    for (i = 0; i < 4; i++)
      {
        double eval_i 
           = gsl_vector_get(eval, i);
        gsl_vector_view evec_i 
           = gsl_matrix_column(evec, i);

        printf("eigenvalue = %g\n", eval_i);
        printf("eigenvector = \n");
        gsl_vector_fprintf(stdout, 
                           &evec_i.vector, "%g");
      }
  }

  return 0;
}

Here is the beginning of the output from the program,

$ ./a.out 
eigenvalue = 9.67023e-05
eigenvector = 
-0.0291933
0.328712
-0.791411
0.514553
...

This can be compared with the corresponding output from GNU OCTAVE,

octave> [v,d] = eig(hilb(4));
octave> diag(d)  
ans =

   9.6702e-05
   6.7383e-03
   1.6914e-01
   1.5002e+00

octave> v 
v =

   0.029193   0.179186  -0.582076   0.792608
  -0.328712  -0.741918   0.370502   0.451923
   0.791411   0.100228   0.509579   0.322416
  -0.514553   0.638283   0.514048   0.252161

Note that the eigenvectors can differ by a change of sign, since the sign of an eigenvector is arbitrary.

References and Further Reading

Further information on the algorithms described in this section can be found in the following book,

The LAPACK library is described in,

The LAPACK source code can be found at the website above along with an online copy of the users guide.

Fast Fourier Transforms (FFTs)

This chapter describes functions for performing Fast Fourier Transforms (FFTs). The library includes radix-2 routines (for lengths which are a power of two) and mixed-radix routines (which work for any length). For efficiency there are separate versions of the routines for real data and for complex data. The mixed-radix routines are a reimplementation of the FFTPACK library by Paul Swarztrauber. Fortran code for FFTPACK is available on Netlib (FFTPACK also includes some routines for sine and cosine transforms but these are currently not available in GSL). For details and derivations of the underlying algorithms consult the document GSL FFT Algorithms (see section References and Further Reading)

Mathematical Definitions

Fast Fourier Transforms are efficient algorithms for calculating the discrete fourier transform (DFT),

The DFT usually arises as an approximation to the continuous fourier transform when functions are sampled at discrete intervals in space or time. The naive evaluation of the discrete fourier transform is a matrix-vector multiplication W\vec{z}. A general matrix-vector multiplication takes O(N^2) operations for N data-points. Fast fourier transform algorithms use a divide-and-conquer strategy to factorize the matrix W into smaller sub-matrices, corresponding to the integer factors of the length N. If N can be factorized into a product of integers f_1 f_2 ... f_n then the DFT can be computed in O(N \sum f_i) operations. For a radix-2 FFT this gives an operation count of O(N \log_2 N).

All the FFT functions offer three types of transform: forwards, inverse and backwards, based on the same mathematical definitions. The definition of the forward fourier transform, x = FFT(z), is,

and the definition of the inverse fourier transform, x = IFFT(z), is,

The factor of 1/N makes this a true inverse. For example, a call to gsl_fft_complex_forward followed by a call to gsl_fft_complex_inverse should return the original data (within numerical errors).

In general there are two possible choices for the sign of the exponential in the transform/ inverse-transform pair. GSL follows the same convention as FFTPACK, using a negative exponential for the forward transform. The advantage of this convention is that the inverse transform recreates the original function with simple fourier synthesis. Numerical Recipes uses the opposite convention, a positive exponential in the forward transform.

The backwards FFT is simply our terminology for an unscaled version of the inverse FFT,

When the overall scale of the result is unimportant it is often convenient to use the backwards FFT instead of the inverse to save unnecessary divisions.

Overview of complex data FFTs

The inputs and outputs for the complex FFT routines are packed arrays of floating point numbers. In a packed array the real and imaginary parts of each complex number are placed in alternate neighboring elements. For example, the following definition of a packed array of length 6,

gsl_complex_packed_array data[6];

can be used to hold an array of three complex numbers, z[3], in the following way,

data[0] = Re(z[0])
data[1] = Im(z[0])
data[2] = Re(z[1])
data[3] = Im(z[1])
data[4] = Re(z[2])
data[5] = Im(z[2])

A stride parameter allows the user to perform transforms on the elements z[stride*i] instead of z[i]. A stride greater than 1 can be used to take an in-place FFT of the column of a matrix. A stride of 1 accesses the array without any additional spacing between elements.

The array indices have the same ordering as those in the definition of the DFT -- i.e. there are no index transformations or permutations of the data.

For physical applications it is important to remember that the index appearing in the DFT does not correspond directly to a physical frequency. If the time-step of the DFT is \Delta then the frequency-domain includes both positive and negative frequencies, ranging from -1/(2\Delta) through 0 to +1/(2\Delta). The positive frequencies are stored from the beginning of the array up to the middle, and the negative frequencies are stored backwards from the end of the array.

Here is a table which shows the layout of the array data, and the correspondence between the time-domain data z, and the frequency-domain data x.

index    z               x = FFT(z)

0        z(t = 0)        x(f = 0)
1        z(t = 1)        x(f = 1/(N Delta))
2        z(t = 2)        x(f = 2/(N Delta))
.        ........        ..................
N/2      z(t = N/2)      x(f = +1/(2 Delta),
                               -1/(2 Delta))
.        ........        ..................
N-3      z(t = N-3)      x(f = -3/(N Delta))
N-2      z(t = N-2)      x(f = -2/(N Delta))
N-1      z(t = N-1)      x(f = -1/(N Delta))

When N is even the location N/2 contains the most positive and negative frequencies +1/(2 \Delta), -1/(2 \Delta)) which are equivalent. If N is odd then general structure of the table above still applies, but N/2 does not appear.

Radix-2 FFT routines for complex data

The radix-2 algorithms described in this section are simple and compact, although not necessarily the most efficient. They use the Cooley-Tukey algorithm to compute in-place complex FFTs for lengths which are a power of 2 -- no additional storage is required. The corresponding self-sorting mixed-radix routines offer better performance at the expense of requiring additional working space.

All these functions are declared in the header file `gsl_fft_complex.h'.

Function: int gsl_fft_complex_radix2_forward (gsl_complex_packed_array data[], size_t stride, size_t n)

Function: int gsl_fft_complex_radix2_transform (gsl_complex_packed_array data[], size_t stride, size_t n)

Function: int gsl_fft_complex_radix2_backward (gsl_complex_packed_array data[], size_t stride, size_t n)

Function: int gsl_fft_complex_radix2_inverse (gsl_complex_packed_array data[], size_t stride, size_t n)

These functions compute forward, backward and inverse FFTs of length n with stride stride, on the packed complex array data using an in-place radix-2 decimation-in-time algorithm. The length of the transform n is restricted to powers of two.

The functions return a value of GSL_SUCCESS if no errors were detected, or GSL_EDOM if the length of the data n is not a power of two.

Function: int gsl_fft_complex_radix2_dif_forward (gsl_complex_packed_array data[], size_t stride, size_t n)

Function: int gsl_fft_complex_radix2_dif_transform (gsl_complex_packed_array data[], size_t stride, size_t n)

Function: int gsl_fft_complex_radix2_dif_backward (gsl_complex_packed_array data[], size_t stride, size_t n)

Function: int gsl_fft_complex_radix2_dif_inverse (gsl_complex_packed_array data[], size_t stride, size_t n)

These are decimation-in-frequency versions of the radix-2 FFT functions.

Here is an example program which computes the FFT of a short pulse in a sample of length 128. To make the resulting fourier transform real the pulse is defined for equal positive and negative times (-10 ... 10), where the negative times wrap around the end of the array.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>

#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])

int
main (void)
{
  int i;
  double data[2*128];

  for (i = 0; i < 128; i++)
    {
       REAL(data,i) = 0.0;
       IMAG(data,i) = 0.0;
    }

  REAL(data,0) = 1.0;

  for (i = 1; i <= 10; i++)
    {
       REAL(data,i) = REAL(data,128-i) = 1.0;
    }

  for (i = 0; i < 128; i++)
    {
      printf ("%d %e %e\n", i, 
              REAL(data,i), IMAG(data,i));
    }
  printf ("\n");

  gsl_fft_complex_radix2_forward (data, 1, 128);

  for (i = 0; i < 128; i++)
    {
      printf ("%d %e %e\n", i, 
              REAL(data,i)/sqrt(128), 
              IMAG(data,i)/sqrt(128));
    }

  return 0;
}

Note that we have assumed that the program is using the default error handler (which calls abort for any errors). If you are not using a safe error handler you would need to check the return status of gsl_fft_complex_radix2_forward.

The transformed data is rescaled by 1/\sqrt N so that it fits on the same plot as the input. Only the real part is shown, by the choice of the input data the imaginary part is zero. Allowing for the wrap-around of negative times at t=128, and working in units of k/N, the DFT approximates the continuum fourier transform, giving a modulated \sin function.

fft-complex-radix2-tfft-complex-radix2-f

A pulse and its discrete fourier transform, output from
the example program.

Mixed-radix FFT routines for complex data

This section describes mixed-radix FFT algorithms for complex data. The mixed-radix functions work for FFTs of any length. They are a reimplementation of the Fortran FFTPACK library by Paul Swarztrauber. The theory is explained in the review article Self-sorting Mixed-radix FFTs by Clive Temperton. The routines here use the same indexing scheme and basic algorithms as FFTPACK.

The mixed-radix algorithm is based on sub-transform modules -- highly optimized small length FFTs which are combined to create larger FFTs. There are efficient modules for factors of 2, 3, 4, 5, 6 and 7. The modules for the composite factors of 4 and 6 are faster than combining the modules for 2*2 and 2*3.

For factors which are not implemented as modules there is a fall-back to a general length-n module which uses Singleton's method for efficiently computing a DFT. This module is O(n^2), and slower than a dedicated module would be but works for any length n. Of course, lengths which use the general length-n module will still be factorized as much as possible. For example, a length of 143 will be factorized into 11*13. Large prime factors are the worst case scenario, e.g. as found in n=2*3*99991, and should be avoided because their O(n^2) scaling will dominate the run-time (consult the document GSL FFT Algorithms included in the GSL distribution if you encounter this problem).

The mixed-radix initialization function gsl_fft_complex_wavetable_alloc returns the list of factors chosen by the library for a given length N. It can be used to check how well the length has been factorized, and estimate the run-time. To a first approximation the run-time scales as N \sum f_i, where the f_i are the factors of N. For programs under user control you may wish to issue a warning that the transform will be slow when the length is poorly factorized. If you frequently encounter data lengths which cannot be factorized using the existing small-prime modules consult GSL FFT Algorithms for details on adding support for other factors.

All these functions are declared in the header file `gsl_fft_complex.h'.

Function: gsl_fft_complex_wavetable * gsl_fft_complex_wavetable_alloc (size_t n)
This function prepares a trigonometric lookup table for a complex FFT of length n. The function returns a pointer to the newly allocated gsl_fft_complex_wavetable if no errors were detected, and a null pointer in the case of error. The length n is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls to sin and cos, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTs of the same length then this computation is a one-off overhead which does not affect the final throughput.

The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The same wavetable can be used for both forward and backward (or inverse) transforms of a given length.

Function: void gsl_fft_complex_wavetable_free (gsl_fft_complex_wavetable * wavetable)
This function frees the memory associated with the wavetable wavetable. The wavetable can be freed if no further FFTs of the same length will be needed.
These functions operate on a gsl_fft_complex_wavetable structure which contains internal parameters for the FFT. It is not necessary to set any of the components directly but it can sometimes be useful to examine them. For example, the chosen factorization of the FFT length is given and can be used to provide an estimate of the run-time or numerical error.

The wavetable structure is declared in the header file `gsl_fft_complex.h'.

Data Type: gsl_fft_complex_wavetable
This is a structure that holds the factorization and trigonometric lookup tables for the mixed radix fft algorithm. It has the following components:
size_t n
This is the number of complex data points
size_t nf
This is the number of factors that the length n was decomposed into.
size_t factor[64]
This is the array of factors. Only the first nf elements are used.
gsl_complex * trig
This is a pointer to a preallocated trigonometric lookup table of n complex elements.
gsl_complex * twiddle[64]
This is an array of pointers into trig, giving the twiddle factors for each pass.

The mixed radix algorithms require an additional working space to hold the intermediate steps of the transform.

Function: gsl_fft_complex_workspace * gsl_fft_complex_workspace_alloc (size_t n)
This function allocates a workspace for a complex transform of length n.

Function: void gsl_fft_complex_workspace_free (gsl_fft_complex_workspace * workspace)
This function frees the memory associated with the workspace workspace. The workspace can be freed if no further FFTs of the same length will be needed.

The following functions compute the transform,

Function: int gsl_fft_complex_forward (gsl_complex_packed_array data[], size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)
Function: int gsl_fft_complex_transform (gsl_complex_packed_array data[], size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)
Function: int gsl_fft_complex_backward (gsl_complex_packed_array data[], size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)
Function: int gsl_fft_complex_inverse (gsl_complex_packed_array data[], size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)

These functions compute forward, backward and inverse FFTs of length n with stride stride, on the packed complex array data, using a mixed radix decimation-in-frequency algorithm. There is no restriction on the length n. Efficient modules are provided for subtransforms of length 2, 3, 4, 5, 6 and 7. Any remaining factors are computed with a slow, O(n^2), general-n module. The caller must supply a wavetable containing the trigonometric lookup tables and a workspace work.

The functions return a value of 0 if no errors were detected. The following gsl_errno conditions are defined for these functions:

GSL_EDOM
The length of the data n is not a positive integer (i.e. n is zero).
GSL_EINVAL
The length of the data n and the length used to compute the given wavetable do not match.

Here is an example program which computes the FFT of a short pulse in a sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>

#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])

int
main (void)
{
  int i;
  const int n = 630;
  double data[2*n];

  gsl_fft_complex_wavetable * wavetable;
  gsl_fft_complex_workspace * workspace;

  for (i = 0; i < n; i++)
    {
      REAL(data,i) = 0.0;
      IMAG(data,i) = 0.0;
    }

  data[0].real = 1.0;

  for (i = 1; i <= 10; i++)
    {
      REAL(data,i) = REAL(data,n-i) = 1.0;
    }

  for (i = 0; i < n; i++)
    {
      printf ("%d: %e %e\n", i, REAL(data,i), 
                                IMAG(data,i));
    }
  printf ("\n");

  wavetable = gsl_fft_complex_wavetable_alloc (n);
  workspace = gsl_fft_complex_workspace_alloc (n);

  for (i = 0; i < wavetable->nf; i++)
    {
       printf("# factor %d: %d\n", i, 
              wavetable->factor[i]);
    }

  gsl_fft_complex_forward (data, 1, n, 
                           wavetable, workspace);

  for (i = 0; i < n; i++)
    {
      printf ("%d: %e %e\n", i, REAL(data,i), 
                                IMAG(data,i));
    }

  gsl_fft_complex_wavetable_free (wavetable);
  gsl_fft_complex_workspace_free (workspace);
  return 0;
}

Note that we have assumed that the program is using the default gsl error handler (which calls abort for any errors). If you are not using a safe error handler you would need to check the return status of all the gsl routines.

Overview of real data FFTs

The functions for real data are similar to those for complex data. However, there is an important difference between forward and inverse transforms. The fourier transform of a real sequence is not real. It is a complex sequence with a special symmetry:

A sequence with this symmetry is called conjugate-complex or half-complex. This different structure requires different storage layouts for the forward transform (from real to half-complex) and inverse transform (from half-complex back to real). As a consequence the routines are divided into two sets: functions in gsl_fft_real which operate on real sequences and functions in gsl_fft_halfcomplex which operate on half-complex sequences.

Functions in gsl_fft_real compute the frequency coefficients of a real sequence. The half-complex coefficients c of a real sequence x are given by fourier analysis,

Functions in gsl_fft_halfcomplex compute inverse or backwards transforms. They reconstruct real sequences by fourier synthesis from their half-complex frequency coefficients, c,

The symmetry of the half-complex sequence implies that only half of the complex numbers in the output need to be stored. The remaining half can be reconstructed using the half-complex symmetry condition. (This works for all lengths, even and odd. When the length is even the middle value, where k=N/2, is also real). Thus only N real numbers are required to store the half-complex sequence, and the transform of a real sequence can be stored in the same size array as the original data.

The precise storage arrangements depend on the algorithm, and are different for radix-2 and mixed-radix routines. The radix-2 function operates in-place, which constrain the locations where each element can be stored. The restriction forces real and imaginary parts to be stored far apart. The mixed-radix algorithm does not have this restriction, and it stores the real and imaginary parts of a given term in neighboring locations. This is desirable for better locality of memory accesses.

Radix-2 FFT routines for real data

This section describes radix-2 FFT algorithms for real data. They use the Cooley-Tukey algorithm to compute in-place FFTs for lengths which are a power of 2.

The radix-2 FFT functions for real data are declared in the header files `gsl_fft_real.h'

Function: int gsl_fft_real_radix2_transform (double data[], size_t stride, size_t n)

This function computes an in-place radix-2 FFT of length n and stride stride on the real array data. The output is a half-complex sequence, which is stored in-place. The arrangement of the half-complex terms uses the following scheme: for k < N/2 the real part of the k-th term is stored in location k, and the corresponding imaginary part is stored in location N-k. Terms with k > N/2 can be reconstructed using the symmetry z_k = z^*_{N-k}. The terms for k=0 and k=N/2 are both purely real, and count as a special case. Their real parts are stored in locations 0 and N/2 respectively, while their imaginary parts which are zero are not stored.

The following table shows the correspondence between the output data and the equivalent results obtained by considering the input data as a complex sequence with zero imaginary part,

complex[0].real    =    data[0] 
complex[0].imag    =    0 
complex[1].real    =    data[1] 
complex[1].imag    =    data[N-1]
...............         ................
complex[k].real    =    data[k]
complex[k].imag    =    data[N-k] 
...............         ................
complex[N/2].real  =    data[N/2]
complex[N/2].real  =    0
...............         ................
complex[k'].real   =    data[k]        k' = N - k
complex[k'].imag   =   -data[N-k] 
...............         ................
complex[N-1].real  =    data[1]
complex[N-1].imag  =   -data[N-1]

The radix-2 FFT functions for halfcomplex data are declared in the header file `gsl_fft_halfcomplex.h'.

Function: int gsl_fft_halfcomplex_radix2_inverse (double data[], size_t stride, size_t n)
Function: int gsl_fft_halfcomplex_radix2_backward (double data[], size_t stride, size_t n)

These functions compute the inverse or backwards in-place radix-2 FFT of length n and stride stride on the half-complex sequence data stored according the output scheme used by gsl_fft_real_radix2. The result is a real array stored in natural order.

Mixed-radix FFT routines for real data

This section describes mixed-radix FFT algorithms for real data. The mixed-radix functions work for FFTs of any length. They are a reimplementation of the real-FFT routines in the Fortran FFTPACK library by Paul Swarztrauber. The theory behind the algorithm is explained in the article Fast Mixed-Radix Real Fourier Transforms by Clive Temperton. The routines here use the same indexing scheme and basic algorithms as FFTPACK.

The functions use the FFTPACK storage convention for half-complex sequences. In this convention the half-complex transform of a real sequence is stored with frequencies in increasing order, starting at zero, with the real and imaginary parts of each frequency in neighboring locations. When a value is known to be real the imaginary part is not stored. The imaginary part of the zero-frequency component is never stored. It is known to be zero (since the zero frequency component is simply the sum of the input data (all real)). For a sequence of even length the imaginary part of the frequency n/2 is not stored either, since the symmetry z_k = z_{N-k}^* implies that this is purely real too.

The storage scheme is best shown by some examples. The table below shows the output for an odd-length sequence, n=5. The two columns give the correspondence between the 5 values in the half-complex sequence returned by gsl_fft_real_transform, halfcomplex[] and the values complex[] that would be returned if the same real input sequence were passed to gsl_fft_complex_backward as a complex sequence (with imaginary parts set to 0),

complex[0].real  =  halfcomplex[0] 
complex[0].imag  =  0
complex[1].real  =  halfcomplex[1] 
complex[1].imag  =  halfcomplex[2]
complex[2].real  =  halfcomplex[3]
complex[2].imag  =  halfcomplex[4]
complex[3].real  =  halfcomplex[3]
complex[3].imag  = -halfcomplex[4]
complex[4].real  =  halfcomplex[1]
complex[4].imag  = -halfcomplex[2]

The upper elements of the complex array, complex[3] and complex[4] are filled in using the symmetry condition. The imaginary part of the zero-frequency term complex[0].imag is known to be zero by the symmetry.

The next table shows the output for an even-length sequence, n=5 In the even case both the there are two values which are purely real,

complex[0].real  =  halfcomplex[0]
complex[0].imag  =  0
complex[1].real  =  halfcomplex[1] 
complex[1].imag  =  halfcomplex[2] 
complex[2].real  =  halfcomplex[3] 
complex[2].imag  =  halfcomplex[4] 
complex[3].real  =  halfcomplex[5] 
complex[3].imag  =  0 
complex[4].real  =  halfcomplex[3] 
complex[4].imag  = -halfcomplex[4]
complex[5].real  =  halfcomplex[1] 
complex[5].imag  = -halfcomplex[2] 

The upper elements of the complex array, complex[4] and complex[5] are filled in using the symmetry condition. Both complex[0].imag and complex[3].imag are known to be zero.

All these functions are declared in the header files `gsl_fft_real.h' and `gsl_fft_halfcomplex.h'.

Function: gsl_fft_real_wavetable * gsl_fft_real_wavetable_alloc (size_t n)
Function: gsl_fft_halfcomplex_wavetable * gsl_fft_halfcomplex_wavetable_alloc (size_t n)
These functions prepare trigonometric lookup tables for an FFT of size n real elements. The functions return a pointer to the newly allocated struct if no errors were detected, and a null pointer in the case of error. The length n is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls to sin and cos, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTs of the same length then computing the wavetable is a one-off overhead which does not affect the final throughput.

The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The appropriate type of wavetable must be used for forward real or inverse half-complex transforms.

Function: void gsl_fft_real_wavetable_free (gsl_fft_real_wavetable * wavetable)
Function: void gsl_fft_halfcomplex_wavetable_free (gsl_fft_halfcomplex_wavetable * wavetable)
These functions free the memory associated with the wavetable wavetable. The wavetable can be freed if no further FFTs of the same length will be needed.
The mixed radix algorithms require an additional working space to hold the intermediate steps of the transform,

Function: gsl_fft_real_workspace * gsl_fft_real_workspace_alloc (size_t n)
This function allocates a workspace for a real transform of length n. The same workspace is used for both forward real and inverse halfcomplex transforms.

Function: void gsl_fft_real_workspace_free (gsl_fft_real_workspace * workspace)
This function frees the memory associated with the workspace workspace. The workspace can be freed if no further FFTs of the same length will be needed.
The following functions compute the transforms of real and half-complex data,

Function: int gsl_fft_real_transform (double data[], size_t stride, size_t n, const gsl_fft_real_wavetable * wavetable, gsl_fft_real_workspace * work)
Function: int gsl_fft_halfcomplex_transform (double data[], size_t stride, size_t n, const gsl_fft_halfcomplex_wavetable * wavetable, gsl_fft_real_workspace * work)
These functions compute the FFT of data, a real or half-complex array of length n, using a mixed radix decimation-in-frequency algorithm. For gsl_fft_real_transform data is an array of time-ordered real data. For gsl_fft_halfcomplex_transform data contains fourier coefficients in the half-complex ordering described above. There is no restriction on the length n. Efficient modules are provided for subtransforms of length 2, 3, 4 and 5. Any remaining factors are computed with a slow, O(n^2), general-n module. The caller must supply a wavetable containing trigonometric lookup tables and a workspace work.

Function: int gsl_fft_real_unpack (const double real_coefficient[], gsl_complex_packed_array complex_coefficient[], size_t stride, size_t n)

This function converts a single real array, real_coefficient into an equivalent complex array, complex_coefficient, (with imaginary part set to zero), suitable for gsl_fft_complex routines. The algorithm for the conversion is simply,

for (i = 0; i < n; i++)
  {
    complex_coefficient[i].real 
      = real_coefficient[i];
    complex_coefficient[i].imag 
      = 0.0;
  }

Function: int gsl_fft_halfcomplex_unpack (const double halfcomplex_coefficient[], gsl_complex_packed_array complex_coefficient[], size_t stride, size_t n)

This function converts halfcomplex_coefficient, an array of half-complex coefficients as returned by gsl_fft_real_transform, into an ordinary complex array, complex_coefficient. It fills in the complex array using the symmetry z_k = z_{N-k}^* to reconstruct the redundant elements. The algorithm for the conversion is,

complex_coefficient[0].real 
  = halfcomplex_coefficient[0];
complex_coefficient[0].imag 
  = 0.0;

for (i = 1; i < n - i; i++)
  {
    double hc_real 
      = halfcomplex_coefficient[2 * i - 1];
    double hc_imag 
      = halfcomplex_coefficient[2 * i];
    complex_coefficient[i].real = hc_real;
    complex_coefficient[i].imag = hc_imag;
    complex_coefficient[n - i].real = hc_real;
    complex_coefficient[n - i].imag = -hc_imag;
  }

if (i == n - i)
  {
    complex_coefficient[i].real 
      = halfcomplex_coefficient[n - 1];
    complex_coefficient[i].imag 
      = 0.0;
  }

Here is an example program using gsl_fft_real_transform and gsl_fft_halfcomplex_inverse. It generates a real signal in the shape of a square pulse. The pulse is fourier transformed to frequency space, and all but the lowest ten frequency components are removed from the array of fourier coefficients returned by gsl_fft_real_transform.

The remaining fourier coefficients are transformed back to the time-domain, to give a filtered version of the square pulse. Since fourier coefficients are stored using the half-complex symmetry both positive and negative frequencies are removed and the final filtered signal is also real.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_real.h>
#include <gsl/gsl_fft_halfcomplex.h>

int
main (void)
{
  int i, n = 100;
  double data[n];

  gsl_fft_real_wavetable * real;
  gsl_fft_halfcomplex_wavetable * hc;
  gsl_fft_real_workspace * work;

  for (i = 0; i < n; i++)
    {
      data[i] = 0.0;
    }

  for (i = n / 3; i < 2 * n / 3; i++)
    {
      data[i] = 1.0;
    }

  for (i = 0; i < n; i++)
    {
      printf ("%d: %e\n", i, data[i]);
    }
  printf ("\n");

  work = gsl_fft_real_workspace_alloc (n);
  real = gsl_fft_real_wavetable_alloc (n);

  gsl_fft_real_transform (data, 1, n, 
                          real, work);

  gsl_fft_real_wavetable_free (real);

  for (i = 11; i < n; i++)
    {
      data[i] = 0;
    }

  hc = gsl_fft_halfcomplex_wavetable_alloc (n);

  gsl_fft_halfcomplex_inverse (data, 1, n, 
                               hc, work);
  gsl_fft_halfcomplex_wavetable_free (hc);

  for (i = 0; i < n; i++)
    {
      printf ("%d: %e\n", i, data[i]);
    }

  gsl_fft_real_workspace_free (work);
  return 0;
}

fft-real-mixedradix}

Low-pass filtered version of a real pulse, output from the example program.

References and Further Reading

A good starting point for learning more about the FFT is the review article Fast Fourier Transforms: A Tutorial Review and A State of the Art by Duhamel and Vetterli,

To find out about the algorithms used in the GSL routines you may want to consult the latex document GSL FFT Algorithms (it is included in GSL, as `doc/fftalgorithms.tex'). This has general information on FFTs and explicit derivations of the implementation for each routine. There are also references to the relevant literature. For convenience some of the more important references are reproduced below.

There are several introductory books on the FFT with example programs, such as The Fast Fourier Transform by Brigham and DFT/FFT and Convolution Algorithms by Burrus and Parks,

Both these introductory books cover the radix-2 FFT in some detail. The mixed-radix algorithm at the heart of the FFTPACK routines is reviewed in Clive Temperton's paper,

The derivation of FFTs for real-valued data is explained in the following two articles,

In 1979 the IEEE published a compendium of carefully-reviewed Fortran FFT programs in Programs for Digital Signal Processing. It is a useful reference for implementations of many different FFT algorithms,

For serious FFT work we recommend the use of the dedicated FFTW library by Frigo and Johnson. The FFTW library is self-optimizing -- it automatically tunes itself for each hardware platform in order to achieve maximum performance. It is available under the GNU GPL.

Numerical Integration

この章では1次元関数の数値積分(求積)を実行するルーチンについて述べる. 一般 の関数の適応的および非適応的積分ルーチンが実装されている. 無限領域および 半無限領域での積分, 対数特異点を含む特異点の積分, コーシー主値の計算, そ して振動積分も実装されている. このライブラリはQUADPACKで使われてい るPiessens, Doncker-Kapenga, Uberhuber, Kahanerにより実装された数値積分 パッケージを移植したものである. QUADPACKのFortranコードはNetlibで利 用できる.

この章で説明される関数はヘッダファイル`gsl_integration.h'で宣言され ている.

Introduction

各アルゴリズムでは次の形の有界積分の近似値を計算する:

ここで は重み関数(一般の被積分関数では )である. ユーザーは次の精度要求を特定する相対誤差範囲 (epsabs, epsrel)を指定できる:

ここで はアルゴリズムにより得られた数値的近似値である. アルゴリズムは絶対誤差 を次の不等式をみたす形で評価する:

ルーチンは誤差範囲が厳密すぎると収束しなくなる. その場合途中で得られる最 良の近似値が返される.

QUADPACKにあるアルゴリズムは次のような名前付け規則に従っている:

Q - 求積ルーチン

N - 非適応的積分
A - 適応的積分

G - 一般積分(ユーザー定義)
W - 被積分関数に重みづけ関数をかける

S - 発散を含む積分
P - 特異点を含む積分
I - 無限範囲での積分
O - cosまたはsinによる振動関数での重みづけ
F - フーリエ積分
C - コーシー主値

アルゴリズムには, 高次および低次の規則による1対の積分則が使われている. 高次規則は狭い範囲の積分の近似値を計算するのに使われる. 高次および低次の 結果の差は近似の誤差評価に用いられる.

一般の(重みのない)関数の積分アルゴリズムはGauss-Kronrod則に基づいている. Gauss-Kronrod則はm次の古典的ガウス積分から始まる. これに 2m+1次の高次Kronrod則を与えるように横軸に点を付加する. Kornrod則 はガウス則で評価した関数値を再利用するので効果的である. 高次Kronrod則は 積分の最良近似値として用いられ, これら2つの規則の差は近似誤差として評価 される.

重み関数のある被積分関数の場合はClenshaw-Curtis求積法が用いられる. Clenshaw-Curtis則はn次Chebyschev多項式近似を被積分関数に適用する. この多項式は厳密に積分できるので, 元の被積分関数の積分近似値が求められる. Chebyschev展開は近似を改良するために高次に拡張することができる. 被積分関 数の特異点(や他の特徴)はChebyschev近似では収束を遅くする. QUADPACK で使われている改良Clenshaw-Curtis則は収束を遅くするいくつかの汎用重み関 数を分離している. これらの重み関数はChebyschev多項式に対して変形 Chebyschevモーメントとしてあらかじめ解析的に積分されている. このモーメ ントと関数のChebyschev近似を組みあわせることにより好きな積分を実行できる. 関数の特異部分に解析的積分を使うことで厳密に相殺することができ, 積分全体 の収束性をかなり改善できる.

QNG non-adaptive Gauss-Kronrod integration

QNGアルゴリズムは, 最大で87点で被積分関数のサンプリングを行う固定 Gauss-Kronrod則による非適応プロシージャである. これにより滑らかな関数を 高速に積分できる.

Function: int gsl_integration_qng (const gsl_function *f, double a, double b, double epsabs, double epsrel, double * result, double * abserr, size_t * neval)

この関数は, (a,b)上のfの積分の近似値が要求される絶対および 相対誤差epsabsおよびepsrelの範囲内にある限り10点, 21点, 43点 および87点のGauss-Kronrod積分を実行する. この関数は最終的な近似値 result, 絶対誤差の見積りabserr, 用いられた関数評価数 nevalを返す. Gauss-Kronrod則は, 関数評価の総数を減らすため, 各段で 前段の結果を利用するよう設計されている.

QAG adaptive integration

QAGアルゴリズムは簡単な適応的積分プロシージャである. 積分区間を分割し, 各段で最大見積り誤差を与える区間を分割する. これにより全体の誤差が急減し, 区間の分割は被積分関数の局所的難点に集中することになる. この分割区間は gsl_integration_workspace構造体で管理され, 区間, 結果, そして評価 誤差が格納される.

Function: gsl_integration_workspace * gsl_integration_workspace_alloc (size_t n)

この関数はn個の倍精度の区間, 積分結果, 評価誤差を格納するのに十分 な作業空間を確保する.

Function: void gsl_integration_workspace_free (gsl_integration_workspace * w)

この関数は作業空間wに割りあてられていたメモリを解放する.

Function: int gsl_integration_qag (const gsl_function *f, double a, double b, double epsabs, double epsrel, size_t limit, int key, gsl_integration_workspace * workspace, double * result, double * abserr)

この関数は(a,b)上のfの積分の近似値が要求される絶対および相 対誤差epsabsおよびepsrelの範囲内にある限り適応的に積分を実行 する. この関数は最終的な近似値result, 絶対誤差の評価値abserr を返す. 積分則はkeyの値により決定される. keyは次のシンボル名 から選ぶ:

GSL_INTEG_GAUSS15  (key = 1)
GSL_INTEG_GAUSS21  (key = 2)
GSL_INTEG_GAUSS31  (key = 3)
GSL_INTEG_GAUSS41  (key = 4)
GSL_INTEG_GAUSS51  (key = 5)
GSL_INTEG_GAUSS61  (key = 6)

これは15, 21, 31, 41, 51, 61点Gauss-Kronrod則に相当する. 高次則は滑らか な関数であれば精度がよくなる. 低次則は不連続のような局所的難点を含む関数 で時間を節約できる.

積分の各段で, 適応的積分ストラテジに従い評価誤差が最大の区間を分割する. 区間分割とその結果はworkspaceで割りあてられるメモリに格納される. 区間分割数の最大値はlimitで与えられる. これは割りあてた作業領域の サイズを越えてはならない.

QAGS adaptive integration with singularities

積分区間に可積分の特異点が存在すると, 適応的ルーチンの区間分割が特異点の まわりに集中してしまう. 分割された区間の幅が減少するとそれによる積分の近 似値は限られた形でしか収束しない. この収束を外挿により加速させる. QAGSア ルゴリズムは適応的区間分割にWynnεアルゴリズムを融合させ, 様々な可積分特 異点の積分をスピードアップさせる.

Function: int gsl_integration_qags (const gsl_function * f, double a, double b, double epsabs, double epsrel, size_t limit, gsl_integration_workspace * workspace, double *result, double *abserr)

この関数は(a,b)上のfの積分の近似値が要求される絶対および相 対誤差epsabsおよびepsrelの範囲内にある限り21点Gauss-Kronrod 積分則を適応的に実行する. 結果は アルゴリズムにより外挿され, 不連続や可積分特異点の存在する積分の収束を加 速させる. この関数は外挿による最終近似値result, 絶対誤差の評価値 abserrを返す. 区間分割とその結果はworkspaceで割りあてられる メモリに格納される. 分割区間の最大数はlimitで指定する. 作業空間の 割りあてサイズを越えてはならない.

QAGP adaptive integration with known singular points

Function: int gsl_integration_qagp (const gsl_function * f, double *pts, size_t npts, double epsabs, double epsrel, size_t limit, gsl_integration_workspace * workspace, double *result, double *abserr)

この関数はユーザーが提供する特異点の場所を考慮しながら適応的に積分を実行 するアルゴリズムQAGSの実装である. 長さnptsの配列ptsには積分 区間の端点と特異点の位置を格納する. 例えば, 区間(a,b)上で, に特異点をもつ積分を実行する場合(ただし ), 次のようなpts配列を与える:

pts[0] = a
pts[1] = x_1
pts[2] = x_2
pts[3] = x_3
pts[4] = b

ここで npts = 5.

積分区間での特異点の位置を知っている場合は, このルーチンのほうが QAGSより早い.

QAGI adaptive integration on infinite intervals

Function: int gsl_integration_qagi (gsl_function * f, double epsabs, double epsrel, size_t limit, gsl_integration_workspace * workspace, double *result, double *abserr)

この関数は非有界区間 で関数fを積分する. 積分は変数変換 により区間 に写像される:

そしてQAGSアルゴリズムを用いて積分される. 変数変換により原点に可積分の特 異点ができてしまうので, 通常のQAGSの21点Gauss-Kronrod則を15点則におきか える. この場合低次則のほうがより効果的だからである.

Function: int gsl_integration_qagiu (gsl_function * f, double a, double epsabs, double epsrel, size_t limit, gsl_integration_workspace * workspace, double *result, double *abserr)

この関数は半非有界区間 上で関数fを積分する. 積分は変数変換 により区間 に写像される.

そしてQAGSアルゴリズムを用いて積分される.

Function: int gsl_integration_qagil (gsl_function * f, double b, double epsabs, double epsrel, size_t limit, gsl_integration_workspace * workspace, double *result, double *abserr)

この関数は半非有界区間 上で関数fを積分する. 積分は変数変換 により区間 に写像される.

そしてQAGSアルゴリズムを用いて積分される.

QAWC adaptive integration for Cauchy principal values

Function: int gsl_integration_qawc (gsl_function *f, double a, double b, double c, double epsabs, double epsrel, size_t limit, gsl_integration_workspace * workspace, double * result, double * abserr)

この関数は 上のfの積分のcでの特異点のコーシー主値を求める.

QAGの適応的分割アルゴリズムが使われるが, 特異点 で分割されないように工夫されている. 分割区間が点 を含んでいたり, その点に近い場合は特別な25点変形Clenshaw-Curtis則が特異 点を避けるために使われる. 特異点から離れた場所では通常の15点 Gauss-Kronrod積分則が使われる.

QAWS adaptive integration for singular functions

QAWSアルゴリズムは, 被積分関数が積分領域の端点で対数的な発散をするときに 用いられる. 効果的に計算するため, Chebyschevモーメントをあらかじめ計算し ておく.

Function: gsl_integration_qaws_table * gsl_integration_qaws_table_alloc (double alpha, double beta, int mu, int nu)

この関数はgsl_integration_gaws_tableの領域を確保し, 特異点の重み 関数 をパラメータ で表現するための作業領域を割りあてる:

ここで である. 重み関数は , の値により以下の4つの形をとる:

特異点 は積分が計算されるまで特定されなくてもよい. これらは積分領域の端点である.

この関数は, エラーが検出されなければ, 新しく割りあてられた gsl_integration_qaws_tableへのポインタを返す. エラーの場合には0を 返す.

Function: int gsl_integration_qaws_table_set (gsl_integration_qaws_table * t, double alpha, double beta, int mu, int nu)
この関数はgsl_integration_qaws_table構造体tのパラメータ を変更する.

Function: void gsl_integration_qaws_table_free (gsl_integration_qaws_table * t)
この関数はgsl_integration_qaws_table構造体tに割りあてられた メモリを解放する.

Function: int gsl_integration_qaws (gsl_function * f, const double a, const double b, gsl_integration_qaws_table * t, const double epsabs, const double epsrel, const size_t limit, gsl_integration_workspace * workspace, double *result, double *abserr)

この関数は特異点重み関数 を用いて区間 上で関数f(x)を積分する. パラメータ はテーブルtから取得する.

QAGの適応的分割アルゴリズムが用いられる. 分割区間に端点が含まれるときは 特殊25点変形Clenshaw-Curtis則が特異点を避けるために用いられる. 端点を含 まない区間では通常の15点Gauss-Kronrod則が用いられる.

QAWO adaptive integration for oscillatory functions

QAWOアルゴリズムは振動因子 をもつ積分を計算する. 効果的に計算するため, この因子をあらかじめ計算した Chebyschevモーメントのテーブルが必要となる.

Function: gsl_integration_qawo_table * gsl_integration_qawo_table_alloc (double omega, double L, enum gsl_integration_qawo_enum sine, size_t n)

この関数はgsl_integration_qawo_table構造体のために空間を割りあて, パラメータ をもつサインまたはコサインの重み関数 のための作業領域となる.

パラメータLは関数の積分領域の長さ を与える. サインかコサインの選択はパラメータsineで行われる. 値には 以下のシンボル値を用いる:

GSL_INTEG_COSINE
GSL_INTEG_SINE

gsl_integration_qawo_tableは積分過程で必要となる三角関数表である. パラメータnにより計算される係数のレベルが指定される. 各レベルは間 隔Lの分割に相当するので, nレベルは長さ までの分割区間に対応できる. 積分ルーチンgsl_integration_qawoは, 要求された精度に対してレベル数が不足している場合にはGSL_ETABLEを 返す.

Function: int gsl_integration_qawo_table_set (gsl_integration_qawo_table * t, double omega, double L, enum gsl_integration_qawo_enum sine)
この関数は存在する作業領域tのパラメータomega, L, sineを変更する.

Function: int gsl_integration_qawo_table_set_length (gsl_integration_qawo_table * t, double L)
この関数は作業領域tの長さのパラメータLを変更する.

Function: void gsl_integration_qawo_table_free (gsl_integration_qawo_table * t)
この関数は作業領域tに割りあてられたメモリを解放する.

Function: int gsl_integration_qawo (gsl_function * f, const double a, const double epsabs, const double epsrel, const size_t limit, gsl_integration_workspace * workspace, gsl_integration_qawo_table * wf, double *result, double *abserr)

この関数は, テーブルwfで定義される重み関数 を用い, 区間 上で関数fの積分を計算する適応的アルゴリズムである.

積分の収束加速するため, 結果は アルゴリズムを用いて外挿される. 関数は最終的な推定値result, 絶対誤 差の評価値abserrを返す. 分割区間とその結果はworkspaceのメモ リに格納される. 分割区間の最大数はlimitで与えられる. これは作業領 域の割りあてサイズを越えてはならない.

「大きな」幅 をもつ分割区間は25点Clenshaw-Curtis積分則を用いて計算し, 振動を処理する. 「小さな」幅 をもつ分割区間では15点Gauss-Kronrod積分を用いる.

QAWF adaptive integration for Fourier integrals

Function: int gsl_integration_qawf (gsl_function * f, const double a, const double epsabs, const size_t limit, gsl_integration_workspace * workspace, gsl_integration_workspace * cycle_workspace, gsl_integration_qawo_table * wf, double *result, double *abserr)

この関数は半非有界区間 での関数fのFourier積分を計算する.

パラメータ はテーブルwf(長さLは好きな値をとることができる. Fourier積分 に適切な値となるようにこの関数によりオーバーライドされるからである)から とられる. 積分はQAWOアルゴリズムを使って各分割区間で計算される.

ここで は周期の奇数倍をカバーするように選ばれる. 各区間からの寄与は符号が交代し, fが正で単調減少する場合には単調減少する. この数列の和は アルゴリズムで加速する.

この関数は全体の絶対誤差abserrで押さえられる. 以下のストラテジが用 いられる: 各区間 でアルゴリズムは許容誤差を達成しようとする:

ここで および である. 各区間の三角関数列の寄与の和は全体の許容誤差abserrを与える.

分割区間の積分に困難が発生した場合は分割区間に対する精度要求を緩和する.

ここで は区間 での評価誤差である.

分割区間とその結果はworkspaceのメモリに格納される. 分割区間の最大 数はlimitで与えられる. 作業領域の割りあてサイズより大きくなくては ならない. 各分割区間での積分にはcycle_workspaceのメモリがQAWOアル ゴリズムのための作業領域として用いられる.

Error codes

不正引数に関する標準エラーコードの他, これらの関数は次の値を返す:

GSL_EMAXITER
区間分割の最大数を越えた.
GSL_EROUND
打ちきり誤差により許容範囲に届かなかった, もしくは外挿テーブルに打ちきり 誤差が検出された.
GSL_ESING
可積分でない特異点もしくはその他の被積分関数の悪い振舞いが積分区間内に見 つかった.
GSL_EDIVERGE
積分が発散した, もしくは積分の収束が遅すぎる.

Examples

積分ルーチンQAGSは有界積分の大くのクラスで計算できる. 例えば, 次 のような積分を考える. これは原点に対数的な特異点をもつ:

以下のプログラムはこの積分を相対誤差1e-7で積分するものである.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_integration.h>

double f (double x, void * params) {
  double alpha = *(double *) params;
  double f = log(alpha*x) / sqrt(x);
  return f;
}

int 
main (void)
{
  gsl_integration_workspace * w 
    = gsl_integration_workspace_alloc(1000);
  
  double result, error;
  double expected = -4.0;
  double alpha = 1.0;

  gsl_function F;
  F.function = &f;
  F.params = &alpha;

  gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
                        w, &result, &error); 

  printf("result          = % .18f\n", result);
  printf("exact result    = % .18f\n", expected);
  printf("estimated error = % .18f\n", error);
  printf("actual error    = % .18f\n", result - expected);
  printf("intervals =  %d\n", w->size);

  return 0;
}

以下に示す結果は, 8回分割することにより望んだ精度が達成できたことを示し ている.

bash$ ./a.out 
result          = -3.999999999999973799
exact result    = -4.000000000000000000
estimated error =  0.000000000000246025
actual error    =  0.000000000000026201
intervals =  8

実際, QAGSで用いられる外挿プロシージャは要求精度の倍近い精度をも つ. 外挿プロシージャの返す評価誤差は実際の誤差よりも大きい. 安全のため1 桁のマージンをとっているからである.

References and Further Reading

以下の本はQUADPACKの参考書の決定版であり, 作者により書かれたもので ある. アルゴリズム, プログラム一覧, テストプログラム, そして例題が載せら れている. 数値積分に関する有用なアドバイスや, QUADPACKの開発に用い られた数値積分に関する参考書も載せれている.

Random Number Generation

The library provides a large collection of random number generators which can be accessed through a uniform interface. Environment variables allow you to select different generators and seeds at runtime, so that you can easily switch between generators without needing to recompile your program. Each instance of a generator keeps track of its own state, allowing the generators to be used in multi-threaded programs. Additional functions are available for transforming uniform random numbers into samples from continuous or discrete probability distributions such as the Gaussian, log-normal or Poisson distributions.

These functions are declared in the header file `gsl_rng.h'.

General comments on random numbers

In 1988, Park and Miller wrote a paper entitled "Random number generators: good ones are hard to find." [Commun. ACM, 31, 1192--1201]. Fortunately, some excellent random number generators are available, though poor ones are still in common use. You may be happy with the system-supplied random number generator on your computer, but you should be aware that as computers get faster, requirements on random number generators increase. Nowadays, a simulation that calls a random number generator millions of times can often finish before you can make it down the hall to the coffee machine and back.

A very nice review of random number generators was written by Pierre L'Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks, ed. (Wiley, 1997). The chapter is available in postscript from from L'Ecuyer's ftp site (see references). Knuth's volume on Seminumerical Algorithms (originally published in 1968) devotes 170 pages to random number generators, and has recently been updated in its 3rd edition (1997). It is brilliant, a classic. If you don't own it, you should stop reading right now, run to the nearest bookstore, and buy it.

A good random number generator will satisfy both theoretical and statistical properties. Theoretical properties are often hard to obtain (they require real math!), but one prefers a random number generator with a long period, low serial correlation, and a tendency not to "fall mainly on the planes." Statistical tests are performed with numerical simulations. Generally, a random number generator is used to estimate some quantity for which the theory of probability provides an exact answer. Comparison to this exact answer provides a measure of "randomness".

The Random Number Generator Interface

It is important to remember that a random number generator is not a "real" function like sine or cosine. Unlike real functions, successive calls to a random number generator yield different return values. Of course that is just what you want for a random number generator, but to achieve this effect, the generator must keep track of some kind of "state" variable. Sometimes this state is just an integer (sometimes just the value of the previously generated random number), but often it is more complicated than that and may involve a whole array of numbers, possibly with some indices thrown in. To use the random number generators, you do not need to know the details of what comprises the state, and besides that varies from algorithm to algorithm.

The random number generator library uses two special structs, gsl_rng_type which holds static information about each type of generator and gsl_rng which describes an instance of a generator created from a given gsl_rng_type.

The functions described in this section are declared in the header file `gsl_rng.h'.

Random number generator initialization

Random: gsl_rng * gsl_rng_alloc (const gsl_rng_type * T)
This function returns a pointer to a newly-created instance of a random number generator of type T. For example, the following code creates an instance of the Tausworthe generator,
gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);

If there is insufficient memory to create the generator then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

The generator is automatically initialized with the default seed, gsl_rng_default_seed. This is zero by default but can be changed either directly or by using the environment variable GSL_RNG_SEED (see section Random number environment variables).

The details of the available generator types are described later in this chapter.

Random: void gsl_rng_set (const gsl_rng * r, unsigned long int s)
This function initializes (or `seeds') the random number generator. If the generator is seeded with the same value of s on two different runs, the same stream of random numbers will be generated by successive calls to the routines below. If different values of s are supplied, then the generated streams of random numbers should be completely different. If the seed s is zero then the standard seed from the original implementation is used instead. For example, the original Fortran source code for the ranlux generator used a seed of 314159265, and so choosing s equal to zero reproduces this when using gsl_rng_ranlux.

Random: void gsl_rng_free (gsl_rng * r)
This function frees all the memory associated with the generator r.

Sampling from a random number generator

The following functions return uniformly distributed random numbers, either as integers or double precision floating point numbers. To obtain non-uniform distributions see section Random Number Distributions.

Random: unsigned long int gsl_rng_get (const gsl_rng * r)
This function returns a random integer from the generator r. The minimum and maximum values depend on the algorithm used, but all integers in the range [min,max] are equally likely. The values of min and max can determined using the auxiliary functions gsl_rng_max (r) and gsl_rng_min (r).

Random: double gsl_rng_uniform (const gsl_rng * r)
This function returns a double precision floating point number uniformly distributed in the range [0,1). The range includes 0.0 but excludes 1.0. The value is typically obtained by dividing the result of gsl_rng_get(r) by gsl_rng_max(r) + 1.0 in double precision. Some generators compute this ratio internally so that they can provide floating point numbers with more than 32 bits of randomness (the maximum number of bits that can be portably represented in a single unsigned long int).

Random: double gsl_rng_uniform_pos (const gsl_rng * r)
This function returns a positive double precision floating point number uniformly distributed in the range (0,1), excluding both 0.0 and 1.0. The number is obtained by sampling the generator with the algorithm of gsl_rng_uniform until a non-zero value is obtained. You can use this function if you need to avoid a singularity at 0.0.

Random: unsigned long int gsl_rng_uniform_int (const gsl_rng * r, unsigned long int n)
This function returns a random integer from 0 to n-1 inclusive. All integers in the range [0,n-1] are equally likely, regardless of the generator used. An offset correction is applied so that zero is always returned with the correct probability, for any minimum value of the underlying generator.

If n is larger than the range of the generator then the function calls the error handler with an error code of GSL_EINVAL and returns zero.

Auxiliary random number generator functions

The following functions provide information about an existing generator. You should use them in preference to hard-coding the generator parameters into your own code.

Random: const char * gsl_rng_name (const gsl_rng * r)
This function returns a pointer to the name of the generator. For example,
printf("r is a '%s' generator\n", 
       gsl_rng_name (r));

would print something like r is a 'taus' generator.

Random: unsigned long int gsl_rng_max (const gsl_rng * r)
gsl_rng_max returns the largest value that gsl_rng_get can return.

Random: unsigned long int gsl_rng_min (const gsl_rng * r)
gsl_rng_min returns the smallest value that gsl_rng_get can return. Usually this value is zero. There are some generators with algorithms that cannot return zero, and for these generators the minimum value is 1.

Random: void * gsl_rng_state (const gsl_rng * r)
Random: size_t gsl_rng_size (const gsl_rng * r)
These function return a pointer to the state of generator r and its size. You can use this information to access the state directly. For example, the following code will write the state of a generator to a stream,
void * state = gsl_rng_state (r);
size_t n = gsl_rng_size (r);
fwrite (state, n, 1, stream);

Random: const gsl_rng_type ** gsl_rng_types_setup (void)
This function returns a pointer to an array of all the available generator types, terminated by a null pointer. The function should be called once at the start of the program, if needed. The following code fragment shows how to iterate over the array of generator types to print the names of the available algorithms,
const gsl_rng_type **t, **t0;

t0 = gsl_rng_types_setup ();

printf("Available generators:\n");

for (t = t0; *t != 0; t++)
  {
    printf("%s\n", (*t)->name);
  }

Random number environment variables

The library allows you to choose a default generator and seed from the environment variables GSL_RNG_TYPE and GSL_RNG_SEED and the function gsl_rng_env_setup. This makes it easy try out different generators and seeds without having to recompile your program.

Function: const gsl_rng_type * gsl_rng_env_setup (void)
This function reads the environment variables GSL_RNG_TYPE and GSL_RNG_SEED and uses their values to set the corresponding library variables gsl_rng_default and gsl_rng_default_seed. These global variables are defined as follows,
extern const gsl_rng_type *gsl_rng_default
extern unsigned long int gsl_rng_default_seed

The environment variable GSL_RNG_TYPE should be the name of a generator, such as taus or mt19937. The environment variable GSL_RNG_SEED should contain the desired seed value. It is converted to an unsigned long int using the C library function strtoul.

If you don't specify a generator for GSL_RNG_TYPE then gsl_rng_mt19937 is used as the default. The initial value of gsl_rng_default_seed is zero.

Here is a short program which shows how to create a global generator using the environment variables GSL_RNG_TYPE and GSL_RNG_SEED,

#include <stdio.h>
#include <gsl/gsl_rng.h>

gsl_rng * r;  /* global generator */

int
main (void)
{
  const gsl_rng_type * T;

  gsl_rng_env_setup();

  T = gsl_rng_default;
  r = gsl_rng_alloc (T);
  
  printf("generator type: %s\n", gsl_rng_name (r));
  printf("seed = %u\n", gsl_rng_default_seed);
  printf("first value = %u\n", gsl_rng_get (r));
  return 0;
}

Running the program without any environment variables uses the initial defaults, an mt19937 generator with a seed of 0,

bash$ ./a.out 
generator type: mt19937
seed = 0
first value = 2867219139

By setting the two variables on the command line we can change the default generator and the seed,

bash$ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out 
GSL_RNG_TYPE=taus
GSL_RNG_SEED=123
generator type: taus
seed = 123
first value = 2720986350

Saving and restoring random number generator state

The above methods ignore the random number `state' which changes from call to call. It is often useful to be able to save and restore the state. To permit these practices, a few somewhat more advanced functions are supplied. These include:

Random: int gsl_rng_memcpy (gsl_rng * dest, const gsl_rng * src)
This function copies the random number generator src into the pre-existing generator dest, making dest into an exact copy of src. The two generators must be of the same type.

Random: gsl_rng * gsl_rng_clone (const gsl_rng * r)
This function returns a pointer to a newly created generator which is an exact copy of the generator r.

Random: void gsl_rng_print_state (const gsl_rng * r)
This function prints a hex-dump of the state of the generator r to stdout. At the moment its only use is for debugging.

Random number generator algorithms

The functions described above make no reference to the actual algorithm used. This is deliberate so that you can switch algorithms without having to change any of your application source code. The library provides a large number of generators of different types, including simulation quality generators, generators provided for compatibility with other libraries and historical generators from the past.

The following generators are recommended for use in simulation. They have extremely long periods, low correlation and pass most statistical tests.

Generator: gsl_rng_mt19937
The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a variant of the twisted generalized feedback shift-register algorithm, and is known as the "Mersenne Twister" generator. It has a Mersenne prime period of 2^19937 - 1 (about 10^6000) and is equi-distributed in 623 dimensions. It has passed the DIEHARD statistical tests. It uses 624 words of state per generator and is comparable in speed to the other generators. The original generator used a default seed of 4357 and choosing s equal to zero in gsl_rng_set reproduces this.

For more information see,

The generator gsl_rng_19937 uses the corrected version of the seeding procedure published later by the two authors above. The original seeding procedure suffered from low-order periodicity, but can be used by selecting the alternate generator gsl_rng_mt19937_1998.

Generator: gsl_rng_ranlxs0
Generator: gsl_rng_ranlxs1
Generator: gsl_rng_ranlxs2

The generator ranlxs0 is a second-generation version of the RANLUX algorithm of L@"uscher, which produces "luxury random numbers". This generator provides single precision output (24 bits) at three luxury levels ranlxs0, ranlxs1 and ranlxs2. It uses double-precision floating point arithmetic internally and can be significantly faster than the integer version of ranlux, particularly on 64-bit architectures. The period of the generator is about @c{$10^{171}$} 10^171. The algorithm has mathematically proven properties and can provide truly decorrelated numbers at a known level of randomness. The higher luxury levels provide additional decorrelation between samples as an additional safety margin.

Generator: gsl_rng_ranlxd1
Generator: gsl_rng_ranlxd2

These generators produce double precision output (48 bits) from the RANLXS generator. The library provides two luxury levels ranlxd1 and ranlxd2.

Generator: gsl_rng_ranlux
Generator: gsl_rng_ranlux389

The ranlux generator is an implementation of the original algorithm developed by L@"uscher. It uses a lagged-fibonacci-with-skipping algorithm to produce "luxury random numbers". It is a 24-bit generator, originally designed for single-precision IEEE floating point numbers. This implementation is based on integer arithmetic, while the second-generation versions RANLXS and RANLXD described above provide floating-point implementations which will be faster on many platforms. The period of the generator is about @c{$10^{171}$} 10^171. The algorithm has mathematically proven properties and it can provide truly decorrelated numbers at a known level of randomness. The default level of decorrelation recommended by L@"uscher is provided by gsl_rng_ranlux, while gsl_rng_ranlux389 gives the highest level of randomness, with all 24 bits decorrelated. Both types of generator use 24 words of state per generator.

For more information see,

Generator: gsl_rng_cmrg
This is a combined multiple recursive generator by L'Ecuyer. Its sequence is,

where the two underlying generators x_n and y_n are,

with coefficients a_1 = 0, a_2 = 63308, a_3 = -183326, b_1 = 86098, b_2 = 0, b_3 = -539608, and moduli m_1 = 2^31 - 1 = 2147483647 and m_2 = 2145483479.

The period of this generator is 2^205 (about 10^61). It uses 6 words of state per generator. For more information see,

Generator: gsl_rng_mrg
This is a fifth-order multiple recursive generator by L'Ecuyer, Blouin and Coutre. Its sequence is,

with a_1 = 107374182, a_2 = a_3 = a_4 = 0, a_5 = 104480 and m = 2^31 - 1.

The period of this generator is about 10^46. It uses 5 words of state per generator. More information can be found in the following paper,

Generator: gsl_rng_taus
This is a maximally equidistributed combined Tausworthe generator by L'Ecuyer. The sequence is,

where,

computed modulo 2^32. In the formulas above ^^ denotes "exclusive-or". Note that the algorithm relies on the properties of 32-bit unsigned integers and has been implemented using a bitmask of 0xFFFFFFFF to make it work on 64 bit machines.

The period of this generator is @c{$2^{88}$} 2^88 (about 10^26). It uses 3 words of state per generator. For more information see,

Generator: gsl_rng_gfsr4
The gfsr4 generator is like a lagged-fibonacci generator, and produces each number as an xor'd sum of four previous values.

Ziff (ref below) notes that "it is now widely known" that two-tap registers (such as R250, which is described below) have serious flaws, the most obvious one being the three-point correlation that comes from the definition of the generator. Nice mathematical properties can be derived for GFSR's, and numerics bears out the claim that 4-tap GFSR's with appropriately chosen offsets are as random as can be measured, using the author's test.

This implementation uses the values suggested the the example on p392 of Ziff's article: A=471, B=1586, C=6988, D=9689.

If the offsets are appropriately chosen (such the one ones in this implementation), then the sequence is said to be maximal. I'm not sure what that means, but I would guess that means all states are part of the same cycle, which would mean that the period for this generator is astronomical; it is (2^K)^D \approx 10^{93334} where K=32 is the number of bits in the word, and D is the longest lag. This would also mean that any one random number could easily be zero; ie 0 <= r < 2^32.

Ziff doesn't say so, but it seems to me that the bits are completely independent here, so one could use this as an efficient bit generator; each number supplying 32 random bits. The quality of the generated bits depends on the underlying seeding procedure, which may need to be improved in some circumstances.

For more information see,

Unix random number generators

The standard Unix random number generators rand, random and rand48 are provided as part of GSL. Although these generators are widely available individually often they aren't all available on the same platform. This makes it difficult to write portable code using them and so we have included the complete set of Unix generators in GSL for convenience. Note that these generators don't produce high-quality randomness and aren't suitable for work requiring accurate statistics. However, if you won't be measuring statistical quantities and just want to introduce some variation into your program then these generators are quite acceptable.

Generator: gsl_rng_rand
This is the BSD rand() generator. Its sequence is

with a = 1103515245, c = 12345 and m = 2^31. The seed specifies the initial value, x_1. The period of this generator is 2^31, and it uses 1 word of storage per generator.

Generator: gsl_rng_random_bsd
Generator: gsl_rng_random_libc5
Generator: gsl_rng_random_glibc2
These generators implement the random() family of functions, a set of linear feedback shift register generators originally used in BSD Unix. There are several versions of random() in use today: the original BSD version (e.g. on SunOS4), a libc5 version (found on older GNU/Linux systems) and a glibc2 version. Each version uses a different seeding procedure, and thus produces different sequences.

The original BSD routines accepted a variable length buffer for the generator state, with longer buffers providing higher-quality randomness. The random() function implemented algorithms for buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm with the largest length that would fit into the user-supplied buffer was used. To support these algorithms additional generators are available with the following names,

gsl_rng_random8_bsd
gsl_rng_random32_bsd
gsl_rng_random64_bsd
gsl_rng_random128_bsd
gsl_rng_random256_bsd

where the numeric suffix indicates the buffer length. The original BSD random function used a 128-byte default buffer and so gsl_rng_random_bsd has been made equivalent to gsl_rng_random128_bsd. Corresponding versions of the libc5 and glibc2 generators are also available, with the names gsl_rng_random8_libc5, gsl_rng_random8_glibc2, etc.

Generator: gsl_rng_rand48
This is the Unix rand48 generator. Its sequence is

defined on 48-bit unsigned integers with a = 25214903917, c = 11 and m = 2^48. The seed specifies the upper 32 bits of the initial value, x_1, with the lower 16 bits set to 0x330E. The function gsl_rng_get returns the upper 32 bits from each term of the sequence. This does not have a direct parallel in the original rand48 functions, but forcing the result to type long int reproduces the output of mrand48. The function gsl_rng_uniform uses the full 48 bits of internal state to return the double precision number x_n/m, which is equivalent to the function drand48. Note that some versions of the GNU C Library contained a bug in mrand48 function which caused it to produce different results (only the lower 16-bits of the return value were set).

Numerical Recipes generators

The following generators are provided for compatibility with Numerical Recipes. Note that the original Numerical Recipes functions used single precision while we use double precision. This will lead to minor discrepancies, but only at the level of single-precision rounding error. If necessary you can force the returned values to single precision by storing them in a volatile float, which prevents the value being held in a register with double or extended precision. Apart from this difference the underlying algorithms for the integer part of the generators are the same.

Generator: gsl_rng_ran0
Numerical recipes ran0 implements Park and Miller's MINSTD algorithm with a modified seeding procedure.

Generator: gsl_rng_ran1
Numerical recipes ran1 implements Park and Miller's MINSTD algorithm with a 32-element Bayes-Durham shuffle box.

Generator: gsl_rng_ran2
Numerical recipes ran2 implements a L'Ecuyer combined recursive generator with a 32-element Bayes-Durham shuffle-box.

Generator: gsl_rng_ran3
Numerical recipes ran3 implements Knuth's portable subtractive generator.

Other random number generators

The generators in this section are provided for compatibility with existing libraries. If you are converting an existing program to use GSL then you can select these generators to check your new implementation against the original one, using the same random number generator. After verifying that your new program reproduces the original results you can then switch to a higher-quality generator.

Note that most of the generators in this section are based on single linear congruence relations, which are the least sophisticated type of generator. In particular, linear congruences have poor properties when used with a non-prime modulus, as several of these routines do (e.g. with a power of two modulus, 2^31 or 2^32). This leads to periodicity in the least significant bits of each number, with only the higher bits having any randomness. Thus if you want to produce a random bitstream it is best to avoid using the least significant bits.

Generator: gsl_rng_ranf
This is the CRAY random number generator RANF. Its sequence is

defined on 48-bit unsigned integers with a = 44485709377909 and m = 2^48. The seed specifies the lower 32 bits of the initial value, x_1, with the lowest bit set to prevent the seed taking an even value. The upper 16 bits of x_1 are set to 0. A consequence of this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc produce the same sequences.

The generator compatibile with the CRAY MATHLIB routine RANF. It produces double precision floating point numbers which should be identical to those from the original RANF.

There is a subtlety in the implementation of the seeding. The initial state is reversed through one step, by multiplying by the modular inverse of a mod m. This is done for compatibility with the original CRAY implementation.

Note that you can only seed the generator with integers up to 2^32, while the original CRAY implementation uses non-portable wide integers which can cover all 2^48 states of the generator.

The function gsl_rng_get returns the upper 32 bits from each term of the sequence. The function gsl_rng_uniform uses the full 48 bits to return the double precision number x_n/m.

The period of this generator is @c{$2^{46}$} 2^46.

Generator: gsl_rng_ranmar
This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman and Tsang. It is a 24-bit generator, originally designed for single-precision IEEE floating point numbers. It was included in the CERNLIB high-energy physics library.

Generator: gsl_rng_r250
This is the shift-register generator of Kirkpatrick and Stoll. The sequence is

where ^^ denote "exclusive-or", defined on 32-bit words. The period of this generator is about @c{$2^{250}$} 2^250 and it uses 250 words of state per generator.

For more information see,

Generator: gsl_rng_tt800
This is an earlier version of the twisted generalized feedback shift-register generator, and has been superseded by the development of MT19937. However, it is still an acceptable generator in its own right. It has a period of 2^800 and uses 33 words of storage per generator.

For more information see,

Generator: gsl_rng_vax
This is the VAX generator MTH$RANDOM. Its sequence is,

with a = 69069, c = 1 and m = 2^32. The seed specifies the initial value, x_1. The period of this generator is 2^32 and it uses 1 word of storage per generator.

Generator: gsl_rng_transputer
This is the random number generator from the INMOS Transputer Development system. Its sequence is,

with a = 1664525 and m = 2^32. The seed specifies the initial value, x_1.

Generator: gsl_rng_randu
This is the IBM RANDU generator. Its sequence is

with a = 65539 and m = 2^31. The seed specifies the initial value, x_1. The period of this generator was only 2^29. It has become a textbook example of a poor generator.

Generator: gsl_rng_minstd
This is Park and Miller's "minimal standard" MINSTD generator, a simple linear congruence which takes care to avoid the major pitfalls of such algorithms. Its sequence is,

with a = 16807 and m = 2^31 - 1 = 2147483647. The seed specifies the initial value, x_1. The period of this generator is about 2^31.

This generator is used in the IMSL Library (subroutine RNUN) and in MATLAB (the RAND function). It is also sometimes known by the acronym "GGL" (I'm not sure what that stands for).

For more information see,

Generator: gsl_rng_uni
Generator: gsl_rng_uni32
This is a reimplementation of the 16-bit SLATEC random number generator RUNIF. A generalization of the generator to 32 bits is provided by gsl_rng_uni32. The original source code is available from NETLIB.

Generator: gsl_rng_slatec
This is the SLATEC random number generator RAND. It is ancient. The original source code is available from NETLIB.

Generator: gsl_rng_zuf
This is the ZUFALL lagged Fibonacci series generator of Peterson. Its sequence is,

The original source code is available from NETLIB. For more information see,

Random Number Generator Performance

The following table shows the relative performance of a selection the available random number generators. The simulation quality generators which offer the best performance are taus, gfsr4 and mt19937.

1754 k ints/sec,    870 k doubles/sec, taus
1613 k ints/sec,    855 k doubles/sec, gfsr4
1370 k ints/sec,    769 k doubles/sec, mt19937
 565 k ints/sec,    571 k doubles/sec, ranlxs0
 400 k ints/sec,    405 k doubles/sec, ranlxs1
 490 k ints/sec,    389 k doubles/sec, mrg
 407 k ints/sec,    297 k doubles/sec, ranlux
 243 k ints/sec,    254 k doubles/sec, ranlxd1
 251 k ints/sec,    253 k doubles/sec, ranlxs2
 238 k ints/sec,    215 k doubles/sec, cmrg
 247 k ints/sec,    198 k doubles/sec, ranlux389
 141 k ints/sec,    140 k doubles/sec, ranlxd2

1852 k ints/sec,    935 k doubles/sec, ran3
 813 k ints/sec,    575 k doubles/sec, ran0
 787 k ints/sec,    476 k doubles/sec, ran1
 379 k ints/sec,    292 k doubles/sec, ran2

References and Further Reading

The subject of random number generation and testing is reviewed extensively in Knuth's Seminumerical Algorithms.

Further information is available in the review paper written by Pierre L'Ecuyer,

On the World Wide Web, see the pLab home page (http://random.mat.sbg.ac.at/) for a lot of information on the state-of-the-art in random number generation, and for numerous links to various "random" WWW sites.

The source code for the DIEHARD random number generator tests is also available online.

Acknowledgements

Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for making the source code to their generators (MT19937, MM&TN; TT800, MM&YK) available under the GNU General Public License. Thanks to Martin L@"uscher for providing notes and source code for the RANLXS and RANLXD generators.

Quasi-Random Sequences

This chapter describes functions for generating quasi-random sequences in arbitrary dimensions. A quasi-random sequence progressively covers a d-dimensional space with a set of points that are uniformly distributed. Quasi-random sequences are also known as low-discrepancy sequences. The quasi-random sequence generators use an interface that is similar to the interface for random number generators.

The functions described in this section are declared in the header file `gsl_qrng.h'.

Quasi-random number generator initialization

Function: gsl_qrng * gsl_qrng_alloc (const gsl_qrng_type * T, unsigned int d)
This function returns a pointer to a newly-created instance of a quasi-random sequence generator of type T and dimension d. If there is insufficient memory to create the generator then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

Function: void gsl_qrng_free (gsl_qrng * q)
This function frees all the memory associated with the generator q.

Function: void gsl_qrng_init (const gsl_qrng * q)
This function reinitializes the generator q to its starting point.

Sampling from a quasi-random number generator

Function: int gsl_qrng_get (const gsl_qrng * q, double x[])
This function returns the next point x from the sequence generator q. The space available for x must match the dimension of the generator. The point x will lie in the range 0 < x_i < 1 for each x_i.

Auxiliary quasi-random number generator functions

Function: const char * gsl_qrng_name (const gsl_qrng * q)
This function returns a pointer to the name of the generator.

Function: size_t gsl_qrng_size (const gsl_qrng * q)
Function: void * gsl_qrng_state (const gsl_qrng * q)
These function return a pointer to the state of generator r and its size. You can use this information to access the state directly. For example, the following code will write the state of a generator to a stream,
void * state = gsl_qrng_state (q);
size_t n = gsl_qrng_size (q);
fwrite (state, n, 1, stream);

Saving and resorting quasi-random number generator state

Function: int gsl_qrng_memcpy (gsl_qrng * dest, const gsl_qrng * src)
This function copies the quasi-random sequence generator src into the pre-existing generator dest, making dest into an exact copy of src. The two generators must be of the same type.

Function: gsl_qrng * gsl_qrng_clone (const gsl_qrng * q)
This function returns a pointer to a newly created generator which is an exact copy of the generator r.

Quasi-random number generator algorithms

The following quasi-random sequence algorithms are available,

Generator: gsl_qrng_niederreiter_2
This generator uses the algorithm described in Bratley, Fox, Niederreiter, ACM Trans. Model. Comp. Sim. 2, 195 (1992). It is valid up to 12 dimensions.

Generator: gsl_qrng_sobol
This generator uses the Sobol sequence described in Antonov, Saleev, USSR Comput. Maths. Math. Phys. 19, 252 (1980). It is valid up to 40 dimensions.

Examples

The following program prints the first 1024 points of the 2-dimensional Sobol sequence.

#include <stdio.h>
#include <gsl/gsl_qrng.h>

int
main (void)
{
  int i;
  gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);

  for (i = 0; i < 1024; i++)
    {
      double v[2];
      gsl_qrng_get(q, v);
      printf("%.5f %.5f\n", v[0], v[1]);
    }

  gsl_qrng_free(q);
  return 0;
}

Here is the output from the program,

$ ./a.out
0.50000 0.50000
0.75000 0.25000
0.25000 0.75000
0.37500 0.37500
0.87500 0.87500
0.62500 0.12500
0.12500 0.62500
....

It can be seen that successive points progressively fill-in the spaces between previous points. The following plot shows the distribution in the x-y plane of the first 1024 points from the Sobol sequence, qrng}
Distribution of the first 1024 points from the quasi-random Sobol sequence

References

The implementations of the quasi-random sequence routines are based on the algorithms described in the following paper,

Random Number Distributions

This chapter describes functions for generating random variates and computing their probability distributions. Samples from the distributions described in this chapter can be obtained using any of the random number generators in the library as an underlying source of randomness. In the simplest cases a non-uniform distribution can be obtained analytically from the uniform distribution of a random number generator by applying an appropriate transformation. This method uses one call to the random number generator.

More complicated distributions are created by the acceptance-rejection method, which compares the desired distribution against a distribution which is similar and known analytically. This usually requires several samples from the generator.

The functions described in this section are declared in `gsl_randist.h'.

The Gaussian Distribution

Random: double gsl_ran_gaussian (const gsl_rng * r, double sigma)
This function returns a Gaussian random variate, with mean zero and standard deviation sigma. The probability distribution for Gaussian random variates is,

for x in the range -\infty to +\infty. Use the transformation z = \mu + x on the numbers returned by gsl_ran_gaussian to obtain a Gaussian distribution with mean \mu. This function uses the Box-Mueller algorithm which requires two calls the random number generator r.

Function: double gsl_ran_gaussian_pdf (double x, double sigma)
This function computes the probability density p(x) at x for a Gaussian distribution with standard deviation sigma, using the formula given above.

Function: double gsl_ran_gaussian_ratio_method (const gsl_rng * r, const double sigma)
This function computes a gaussian random variate using the Kinderman-Monahan ratio method.

Random: double gsl_ran_ugaussian (const gsl_rng * r)
Random: double gsl_ran_ugaussian_pdf (double x)
Random: double gsl_ran_ugaussian_ratio_method (const gsl_rng * r, const double sigma)
These functions compute results for the unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one, sigma = 1.

The Gaussian Tail Distribution

Random: double gsl_ran_gaussian_tail (const gsl_rng * r, double a, double sigma)
This function provides random variates from the upper tail of a Gaussian distribution with standard deviation sigma. The values returned are larger than the lower limit a, which must be positive. The method is based on Marsaglia's famous rectangle-wedge-tail algorithm (Ann Math Stat 32, 894-899 (1961)), with this aspect explained in Knuth, v2, 3rd ed, p139,586 (exercise 11).

The probability distribution for Gaussian tail random variates is,

for x > a where N(a;\sigma) is the normalization constant,

Function: double gsl_ran_gaussian_tail_pdf (double x, double a, double sigma)
This function computes the probability density p(x) at x for a Gaussian tail distribution with standard deviation sigma and lower limit a, using the formula given above.

Random: double gsl_ran_ugaussian_tail (const gsl_rng * r, double a)
Random: double gsl_ran_ugaussian_tail_pdf (double x, double a)
These functions compute results for the tail of a unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one, sigma = 1.

The Bivariate Gaussian Distribution

Random: void gsl_ran_bivariate_gaussian (const gsl_rng * r, double sigma_x, double sigma_y, double rho, double * x, double * y)
This function generates a pair of correlated gaussian variates, with mean zero, correlation coefficient rho and standard deviations sigma_x and sigma_y in the x and y directions. The probability distribution for bivariate gaussian random variates is,

for x,y in the range -\infty to +\infty. The correlation coefficient rho should lie between 1 and -1.

Function: double gsl_ran_bivariate_gaussian_pdf (double x, double y, double sigma_x, double sigma_y, double rho)
This function computes the probability density p(x,y) at (x,y) for a bivariate gaussian distribution with standard deviations sigma_x, sigma_y and correlation coefficient rho, using the formula given above.

The Exponential Distribution

Random: double gsl_ran_exponential (const gsl_rng * r, double mu)
This function returns a random variate from the exponential distribution with mean mu. The distribution is,

for @c{$x \ge 0$} x >= 0.

Function: double gsl_ran_exponential_pdf (double x, double mu)
This function computes the probability density p(x) at x for an exponential distribution with mean mu, using the formula given above.

The Laplace Distribution

Random: double gsl_ran_laplace (const gsl_rng * r, double a)
This function returns a random variate from the the Laplace distribution with width a. The distribution is,

for -\infty < x < \infty.

Function: double gsl_ran_laplace_pdf (double x, double a)
This function computes the probability density p(x) at x for a Laplace distribution with mean a, using the formula given above.

The Exponential Power Distribution

Random: double gsl_ran_exppow (const gsl_rng * r, double a, double b)
This function returns a random variate from the exponential power distribution with scale parameter a and exponent b. The distribution is,

for @c{$x \ge 0$} x >= 0. For b = 1 this reduces to the Laplace distribution. For b = 2 it has the same form as a gaussian distribution, but with @c{$a = \sqrt{2} \sigma$} a = \sqrt{2} \sigma.

Function: double gsl_ran_exppow_pdf (double x, double a, double b)
This function computes the probability density p(x) at x for an exponential power distribution with scale parameter a and exponent b, using the formula given above.

The Cauchy Distribution

Random: double gsl_ran_cauchy (const gsl_rng * r, double a)
This function returns a random variate from the Cauchy distribution with scale parameter a. The probability distribution for Cauchy random variates is,

for x in the range -\infty to +\infty. The Cauchy distribution is also known as the Lorentz distribution.

Function: double gsl_ran_cauchy_pdf (double x, double a)
This function computes the probability density p(x) at x for a Cauchy distribution with scale parameter a, using the formula given above.

The Rayleigh Distribution

Random: double gsl_ran_rayleigh (const gsl_rng * r, double sigma)
This function returns a random variate from the Rayleigh distribution with scale parameter sigma. The distribution is,

for x > 0.

Function: double gsl_ran_rayleigh_pdf (double x, double sigma)
This function computes the probability density p(x) at x for a Rayleigh distribution with scale parameter sigma, using the formula given above.

The Rayleigh Tail Distribution

Random: double gsl_ran_rayleigh_tail (const gsl_rng * r, double a double sigma)
This function returns a random variate from the tail of the Rayleigh distribution with scale parameter sigma and a lower limit of a. The distribution is,

for x > a.

Function: double gsl_ran_rayleigh_tail_pdf (double x, double a, double sigma)
This function computes the probability density p(x) at x for a Rayleigh tail distribution with scale parameter sigma and lower limit a, using the formula given above.

The Landau Distribution

Random: double gsl_ran_landau (const gsl_rng * r)
This function returns a random variate from the Landau distribution. The probability distribution for Landau random variates is defined analytically by the complex integral,

For numerical purposes it is more convenient to use the following equivalent form of the integral,

Function: double gsl_ran_landau_pdf (double x)
This function computes the probability density p(x) at x for the Landau distribution using an approximation to the formula given above.

The Levy alpha-Stable Distributions

Random: double gsl_ran_levy (const gsl_rng * r, double c, double alpha)
This function returns a random variate from the Levy symmetric stable distribution with scale c and exponent alpha. The symmetric stable probability distribution is defined by a fourier transform,

There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function. For \alpha = 1 the distribution reduces to the Cauchy distribution. For \alpha = 2 it is a Gaussian distribution with @c{$\sigma = \sqrt{2} c$} \sigma = \sqrt{2} c. For \alpha < 1 the tails of the distribution become extremely wide.

The algorithm only works for @c{$0 < \alpha \le 2$} 0 < alpha <= 2.

The Levy skew alpha-Stable Distribution

Random: double gsl_ran_levy_skew (const gsl_rng * r, double c, double alpha, double beta)
This function returns a random variate from the Levy skew stable distribution with scale c, exponent alpha and skewness parameter beta. The skewness parameter must lie in the range [-1,1]. The Levy skew stable probability distribution is defined by a fourier transform,

When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by -(2/\pi)\log|t|. There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function. For \alpha = 2 the distribution reduces to a Gaussian distribution with @c{$\sigma = \sqrt{2} c$} \sigma = \sqrt{2} c and the skewness parameter has no effect. For \alpha < 1 the tails of the distribution become extremely wide. The symmetric distribution corresponds to \beta = 0.

The algorithm only works for @c{$0 < \alpha \le 2$} 0 < alpha <= 2.

The Levy alpha-stable distributions have the property that if N alpha-stable variates are drawn from the distribution p(c, \alpha, \beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be distributed as an alpha-stable variate, p(N^(1/\alpha) c, \alpha, \beta).

The Gamma Distribution

Random: double gsl_ran_gamma (const gsl_rng * r, double a, double b)
This function returns a random variate from the gamma distribution. The distribution function is,

for x > 0.

Function: double gsl_ran_gamma_pdf (double x, double a, double b)
This function computes the probability density p(x) at x for a gamma distribution with parameters a and b, using the formula given above.

The Flat (Uniform) Distribution

Random: double gsl_ran_flat (const gsl_rng * r, double a, double b)
This function returns a random variate from the flat (uniform) distribution from a to b. The distribution is,

if @c{$a \le x < b$} a <= x < b and 0 otherwise.

Function: double gsl_ran_flat_pdf (double x, double a, double b)
This function computes the probability density p(x) at x for a uniform distribution from a to b, using the formula given above.

The Lognormal Distribution

Random: double gsl_ran_lognormal (const gsl_rng * r, double zeta, double sigma)
This function returns a random variate from the lognormal distribution. The distribution function is,

for x > 0.

Function: double gsl_ran_lognormal_pdf (double x, double zeta, double sigma)
This function computes the probability density p(x) at x for a lognormal distribution with parameters zeta and sigma, using the formula given above.

The Chi-squared Distribution

The chi-squared distribution arises in statistics If Y_i are n independent gaussian random variates with unit variance then the sum-of-squares,

has a chi-squared distribution with n degrees of freedom.

Random: double gsl_ran_chisq (const gsl_rng * r, double nu)
This function returns a random variate from the chi-squared distribution with nu degrees of freedom. The distribution function is,

for @c{$x \ge 0$} x >= 0.

Function: double gsl_ran_chisq_pdf (double x, double nu)
This function computes the probability density p(x) at x for a chi-squared distribution with nu degrees of freedom, using the formula given above.

The F-distribution

The F-distribution arises in statistics. If Y_1 and Y_2 are chi-squared deviates with \nu_1 and \nu_2 degrees of freedom then the ratio,

has an F-distribution F(x;\nu_1,\nu_2).

Random: double gsl_ran_fdist (const gsl_rng * r, double nu1, double nu2)
This function returns a random variate from the F-distribution with degrees of freedom nu1 and nu2. The distribution function is,

for @c{$x \ge 0$} x >= 0.

Function: double gsl_ran_fdist_pdf (double x, double nu1, double nu2)
This function computes the probability density p(x) at x for an F-distribution with nu1 and nu2 degrees of freedom, using the formula given above.

The t-distribution

The t-distribution arises in statistics. If Y_1 has a normal distribution and Y_2 has a chi-squared distribution with \nu degrees of freedom then the ratio,

has a t-distribution t(x;\nu) with \nu degrees of freedom.

Random: double gsl_ran_tdist (const gsl_rng * r, double nu)
This function returns a random variate from the t-distribution. The distribution function is,

for -\infty < x < +\infty.

Function: double gsl_ran_tdist_pdf (double x, double nu)
This function computes the probability density p(x) at x for a t-distribution with nu degrees of freedom, using the formula given above.

The Beta Distribution

Random: double gsl_ran_beta (const gsl_rng * r, double a, double b)
This function returns a random variate from the beta distribution. The distribution function is,

for @c{$0 \le x \le 1$} 0 <= x <= 1.

Function: double gsl_ran_beta_pdf (double x, double a, double b)
This function computes the probability density p(x) at x for a beta distribution with parameters a and b, using the formula given above.

The Logistic Distribution

Random: double gsl_ran_logistic (const gsl_rng * r, double a)
This function returns a random variate from the logistic distribution. The distribution function is,

for -\infty < x < +\infty.

Function: double gsl_ran_logistic_pdf (double x, double a)
This function computes the probability density p(x) at x for a logistic distribution with scale parameter a, using the formula given above.

The Pareto Distribution

Random: double gsl_ran_pareto (const gsl_rng * r, double a, double b)
This function returns a random variate from the Pareto distribution of order a. The distribution function is,

for @c{$x \ge b$} x >= b.

Function: double gsl_ran_pareto_pdf (double x, double a, double b)
This function computes the probability density p(x) at x for a Pareto distribution with exponent a and scale b, using the formula given above.

The Spherical Distribution (2D & 3D)

The spherical distributions generate random vectors, located on a spherical surface. They can be used as random directions, for example in the steps of a random walk.

Random: void gsl_ran_dir_2d (const gsl_rng * r, double *x, double *y)
Random: void gsl_ran_dir_2d_trig_method (const gsl_rng * r, double *x, double *y)
This function returns a random direction vector v = (x,y) in two dimensions. The vector is normalized such that |v|^2 = x^2 + y^2 = 1. The obvious way to do this is to take a uniform random number between 0 and 2\pi and let x and y be the sine and cosine respectively. Two trig functions would have been expensive in the old days, but with modern hardware implementations, this is sometimes the fastest way to go. This is the case for my home Pentium (but not the case for my Sun Sparcstation 20 at work). Once can avoid the trig evaluations by choosing x and y in the interior of a unit circle (choose them at random from the interior of the enclosing square, and then reject those that are outside the unit circle), and then dividing by @c{$\sqrt{x^2 + y^2}$} \sqrt{x^2 + y^2}. A much cleverer approach, attributed to von Neumann (See Knuth, v2, 3rd ed, p140, exercise 23), requires neither trig nor a square root. In this approach, u and v are chosen at random from the interior of a unit circle, and then x=(u^2-v^2)/(u^2+v^2) and y=uv/(u^2+v^2).

Random: void gsl_ran_dir_3d (const gsl_rng * r, double *x, double *y, double * z)
This function returns a random direction vector v = (x,y,z) in three dimensions. The vector is normalized such that |v|^2 = x^2 + y^2 + z^2 = 1. The method employed is due to Robert E. Knop (CACM 13, 326 (1970)), and explained in Knuth, v2, 3rd ed, p136. It uses the surprising fact that the distribution projected along any axis is actually uniform (this is only true for 3d).

Random: void gsl_ran_dir_nd (const gsl_rng * r, int n, double *x)

This function returns a random direction vector v = (x_1,x_2,...,x_n) in n dimensions. The vector is normalized such that |v|^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1. The method uses the fact that a multivariate gaussian distribution is spherically symmetric. Each component is generated to have a gaussian distribution, and then the components are normalized. The method is described by Knuth, v2, 3rd ed, p135-136, and attributed to G. W. Brown, Modern Mathematics for the Engineer (1956).

The Weibull Distribution

Random: double gsl_ran_weibull (const gsl_rng * r, double a, double b)
This function returns a random variate from the Weibull distribution. The distribution function is,

for @c{$x \ge 0$} x >= 0.

Function: double gsl_ran_weibull_pdf (double x, double a, double b)
This function computes the probability density p(x) at x for a Weibull distribution with scale a and exponent b, using the formula given above.

The Type-1 Gumbel Distribution

Random: double gsl_ran_gumbel1 (const gsl_rng * r, double a, double b)
This function returns a random variate from the Type-1 Gumbel distribution. The Type-1 Gumbel distribution function is,

for -\infty < x < \infty.

Function: double gsl_ran_gumbel1_pdf (double x, double a, double b)
This function computes the probability density p(x) at x for a Type-1 Gumbel distribution with parameters a and b, using the formula given above.

The Type-2 Gumbel Distribution

Random: double gsl_ran_gumbel2 (const gsl_rng * r, double a, double b)
This function returns a random variate from the Type-2 Gumbel distribution. The Type-2 Gumbel distribution function is,

for 0 < x < \infty.

Function: double gsl_ran_gumbel2_pdf (double x, double a, double b)
This function computes the probability density p(x) at x for a Type-2 Gumbel distribution with parameters a and b, using the formula given above.

General Discrete Distributions

Given K discrete events with different probabilities P[k], produce a random value k consistent with its probability.

The obvious way to do this is to preprocess the probability list by generating a cumulative probability array with K+1 elements:

Note that this construction produces C[K]=1. Now choose a uniform deviate u between 0 and 1, and find the value of k such that @c{$C[k] \le u < C[k+1]$} C[k] <= u < C[k+1]. Although this in principle requires of order \log K steps per random number generation, they are fast steps, and if you use something like \lfloor uK \rfloor as a starting point, you can often do pretty well.

But faster methods have been devised. Again, the idea is to preprocess the probability list, and save the result in some form of lookup table; then the individual calls for a random discrete event can go rapidly. An approach invented by G. Marsaglia (Generating discrete random numbers in a computer, Comm ACM 6, 37-38 (1963)) is very clever, and readers interested in examples of good algorithm design are directed to this short and well-written paper. Unfortunately, for large K, Marsaglia's lookup table can be quite large.

A much better approach is due to Alastair J. Walker (An efficient method for generating discrete random variables with general distributions, ACM Trans on Mathematical Software 3, 253-256 (1977); see also Knuth, v2, 3rd ed, p120-121,139). This requires two lookup tables, one floating point and one integer, but both only of size K. After preprocessing, the random numbers are generated in O(1) time, even for large K. The preprocessing suggested by Walker requires O(K^2) effort, but that is not actually necessary, and the implementation provided here only takes O(K) effort. In general, more preprocessing leads to faster generation of the individual random numbers, but a diminishing return is reached pretty early. Knuth points out that the optimal preprocessing is combinatorially difficult for large K.

This method can be used to speed up some of the discrete random number generators below, such as the binomial distribution. To use if for something like the Poisson Distribution, a modification would have to be made, since it only takes a finite set of K outcomes.

Random: gsl_ran_discrete_t * gsl_ran_discrete_preproc (size_t K, const double * P)
This function returns a pointer to a structure that contains the lookup table for the discrete random number generator. The array P[] contains the probabilities of the discrete events; these array elements must all be positive, but they needn't add up to one (so you can think of them more generally as "weights") -- the preprocessor will normalize appropriately. This return value is used as an argument for the gsl_ran_discrete function below.

Random: size_t gsl_ran_discrete (const gsl_rng * r, const gsl_ran_discrete_t * g)
After the preprocessor, above, has been called, you use this function to get the discrete random numbers.

Random: double gsl_ran_discrete_pdf (size_t k, const gsl_ran_discrete_t * g)
Returns the probability P[k] of observing the variable k. Since P[k] is not stored as part of the lookup table, it must be recomputed; this computation takes O(K), so if K is large and you care about the original array P[k] used to create the lookup table, then you should just keep this original array P[k] around.

Random: void gsl_ran_discrete_free (gsl_ran_discrete_t * g)
De-allocates the lookup table pointed to by g.

The Poisson Distribution

Random: unsigned int gsl_ran_poisson (const gsl_rng * r, double mu)
This function returns a random integer from the Poisson distribution with mean mu. The probability distribution for Poisson variates is,

for @c{$k \ge 0$} k >= 0.

Function: double gsl_ran_poisson_pdf (unsigned int k, double mu)
This function computes the probability p(k) of obtaining k from a Poisson distribution with mean mu, using the formula given above.

The Bernoulli Distribution

Random: unsigned int gsl_ran_bernoulli (const gsl_rng * r, double p)
This function returns either 0 or 1, the result of a Bernoulli trial with probability p. The probability distribution for a Bernoulli trial is,

Function: double gsl_ran_bernoulli_pdf (unsigned int k, double p)
This function computes the probability p(k) of obtaining k from a Bernoulli distribution with probability parameter p, using the formula given above.

The Binomial Distribution

Random: unsigned int gsl_ran_binomial (const gsl_rng * r, double p, unsigned int n)
This function returns a random integer from the binomial distribution, the number of successes in n independent trials with probability p. The probability distribution for binomial variates is,

for @c{$0 \le k \le n$} 0 <= k <= n.

Function: double gsl_ran_binomial_pdf (unsigned int k, double p, unsigned int n)
This function computes the probability p(k) of obtaining k from a binomial distribution with parameters p and n, using the formula given above.

The Negative Binomial Distribution

Random: unsigned int gsl_ran_negative_binomial (const gsl_rng * r, double p, double n)
This function returns a random integer from the negative binomial distribution, the number of failures occurring before n successes in independent trials with probability p of success. The probability distribution for negative binomial variates is,

Note that n is not required to be an integer.

Function: double gsl_ran_negative_binomial_pdf (unsigned int k, double p, double n)
This function computes the probability p(k) of obtaining k from a negative binomial distribution with parameters p and n, using the formula given above.

The Pascal Distribution

Random: unsigned int gsl_ran_pascal (const gsl_rng * r, double p, unsigned int k)
This function returns a random integer from the Pascal distribution. The Pascal distribution is simply a negative binomial distribution with an integer value of n.

for @c{$k \ge 0$} k >= 0

Function: double gsl_ran_pascal_pdf (unsigned int k, double p, unsigned int n)
This function computes the probability p(k) of obtaining k from a Pascal distribution with parameters p and n, using the formula given above.

The Geometric Distribution

Random: unsigned int gsl_ran_geometric (const gsl_rng * r, double p)
This function returns a random integer from the geometric distribution, the number of independent trials with probability p until the first success. The probability distribution for geometric variates is,

for @c{$k \ge 1$} k >= 1.

Function: double gsl_ran_geometric_pdf (unsigned int k, double p)
This function computes the probability p(k) of obtaining k from a geometric distribution with probability parameter p, using the formula given above.

The Hypergeometric Distribution

Random: unsigned int gsl_ran_hypergeometric (const gsl_rng * r, unsigned int n1, unsigned int n2, unsigned int t)
This function returns a random integer from the hypergeometric distribution. The probability distribution for hypergeometric random variates is,

where C(a,b) = a!/(b!(a-b)!). The domain of k is max(0,t-n_2), ..., max(t,n_1).

Function: double gsl_ran_hypergeometric_pdf (unsigned int k, unsigned int n1, unsigned int n2, unsigned int t)
This function computes the probability p(k) of obtaining k from a hypergeometric distribution with parameters n1, n2, n3, using the formula given above.

The Logarithmic Distribution

Random: unsigned int gsl_ran_logarithmic (const gsl_rng * r, double p)
This function returns a random integer from the logarithmic distribution. The probability distribution for logarithmic random variates is,

for @c{$k \ge 1$} k >= 1.

Function: double gsl_ran_logarithmic_pdf (unsigned int k, double p)
This function computes the probability p(k) of obtaining k from a logarithmic distribution with probability parameter p, using the formula given above.

Shuffling and Sampling

The following functions allow the shuffling and sampling of a set of objects. The algorithms rely on a random number generator as source of randomness and a poor quality generator can lead to correlations in the output. In particular it is important to avoid generators with a short period. For more information see Knuth, v2, 3rd ed, Section 3.4.2, "Random Sampling and Shuffling".

Random: void gsl_ran_shuffle (const gsl_rng * r, void * base, size_t n, size_t size)

This function randomly shuffles the order of n objects, each of size size, stored in the array base[0..n-1]. The output of the random number generator r is used to produce the permutation. The algorithm generates all possible n! permutations with equal probability, assuming a perfect source of random numbers.

The following code shows how to shuffle the numbers from 0 to 51,

int a[52];

for (i = 0; i < 52; i++)
  {
    a[i] = i;
  }

gsl_ran_shuffle (r, a, 52, sizeof (int));

Random: int gsl_ran_choose (const gsl_rng * r, void * dest, size_t k, void * src, size_t n, size_t size)
This function fills the array dest[k] with k objects taken randomly from the n elements of the array src[0..n-1]. The objects are each of size size. The output of the random number generator r is used to make the selection. The algorithm ensures all possible samples are equally likely, assuming a perfect source of randomness.

The objects are sampled without replacement, thus each object can only appear once in dest[k]. It is required that k be less than or equal to n. The objects in dest will be in the same relative order as those in src. You will need to call gsl_ran_shuffle(r, dest, n, size) if you want to randomize the order.

The following code shows how to select a random sample of three unique numbers from the set 0 to 99,

double a[3], b[100];

for (i = 0; i < 100; i++)
  {
    b[i] = (double) i;
  }

gsl_ran_choose (r, a, 3, b, 100, sizeof (double));

Random: void gsl_ran_sample (const gsl_rng * r, void * dest, size_t k, void * src, size_t n, size_t size)
This function is like gsl_ran_choose but samples k items from the original array of n items src with replacement, so the same object can appear more than once in the output sequence dest. There is no requirement that k be less than n in this case.

Examples

The following program demonstrates the use of a random number generator to produce variates from a distribution. It prints 10 samples from the Poisson distribution with a mean of 3.

#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>

int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;

  int i, n = 10;
  double mu = 3.0;

  /* create a generator chosen by the 
     environment variable GSL_RNG_TYPE */

  gsl_rng_env_setup();

  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

  /* print n random variates chosen from 
     the poisson distribution with mean 
     parameter mu */

  for (i = 0; i < n; i++) 
    {
      unsigned int k = gsl_ran_poisson (r, mu);
      printf(" %u", k);
    }

  printf("\n");
  return 0;
}

If the library and header files are installed under `/usr/local' (the default location) then the program can be compiled with these options,

gcc demo.c -lgsl -lgslcblas -lm

Here is the output of the program,

$ ./a.out 
 4 2 3 3 1 3 4 1 3 5

The variates depend on the seed used by the generator. The seed for the default generator type gsl_rng_default can be changed with the GSL_RNG_SEED environment variable to produce a different stream of variates,

$ GSL_RNG_SEED=123 ./a.out 
GSL_RNG_SEED=123
 1 1 2 1 2 6 2 1 8 7

The following program generates a random walk in two dimensions.

#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>

int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;

  gsl_rng_env_setup();
  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

  int i;
  double x = 0, y = 0, dx, dy;

  printf("%g %g\n", x, y);

  for (i = 0; i < 10; i++)
    {
      gsl_ran_dir_2d (r, &dx, &dy);
      x += dx; y += dy; 
      printf("%g %g\n", x, y);
    }
  return 0;
}

Example output from the program, three 10-step random walks from the origin.

References and Further Reading

For an encyclopaedic coverage of the subject readers are advised to consult the book Non-Uniform Random Variate Generation by Luc Devroye. It covers every imaginable distribution and provides hundreds of algorithms.

The subject of random variate generation is also reviewed by Knuth, who describes algorithms for all the major distributions.

The Particle Data Group provides a short review of techniques for generating distributions of random numbers in the "Monte Carlo" section of its Annual Review of Particle Physics.

The Review of Particle Physics is available online in postscript and pdf format.

Statistics

This chapter describes the statistical functions in the library. The basic statistical functions include routines to compute the mean, variance and standard deviation. More advanced functions allow you to calculate absolute deviations, skewness, and kurtosis as well as the median and arbitrary percentiles. The algorithms use recurrence relations to compute average quantities in a stable way, without large intermediate values that might overflow.

The functions are available in versions for datasets in the standard floating-point and integer types. The versions for double precision floating-point data have the prefix gsl_stats and are declared in the header file `gsl_stats_double.h'. The versions for integer data have the prefix gsl_stats_int and are declared in the header files `gsl_stats_int.h'.

Mean, Standard Deviation and Variance

Statistics: double gsl_stats_mean (const double data[], size_t stride, size_t n)
This function returns the arithmetic mean of data, a dataset of length n with stride stride. The arithmetic mean, or sample mean, is denoted by \Hat\mu and defined as,

where x_i are the elements of the dataset data. For samples drawn from a gaussian distribution the variance of \Hat\mu is \sigma^2 / N.

Statistics: double gsl_stats_variance (const double data[], size_t stride, size_t n)
This function returns the estimated, or sample, variance of data, a dataset of length n with stride stride. The estimated variance is denoted by \Hat\sigma^2 and is defined by,

where x_i are the elements of the dataset data. Note that the normalization factor of 1/(N-1) results from the derivation of \Hat\sigma^2 as an unbiased estimator of the population variance \sigma^2. For samples drawn from a gaussian distribution the variance of \Hat\sigma^2 itself is 2 \sigma^4 / N.

This function computes the mean via a call to gsl_stats_mean. If you have already computed the mean then you can pass it directly to gsl_stats_variance_m.

Statistics: double gsl_stats_variance_m (const double data[], size_t stride, size_t n, double mean)
This function returns the sample variance of data relative to the given value of mean. The function is computed with \Hat\mu replaced by the value of mean that you supply,

Statistics: double gsl_stats_sd (const double data[], size_t stride, size_t n)
Statistics: double gsl_stats_sd_m (const double data[], size_t stride, size_t n, double mean)
The standard deviation is defined as the square root of the variance. These functions return the square root of the corresponding variance functions above.

Statistics: double gsl_stats_variance_with_fixed_mean (const double data[], size_t stride, size_t n, double mean)
This function computes an unbiased estimate of the variance of data when the population mean mean of the underlying distribution is known a priori. In this case the estimator for the variance uses the factor 1/N and the sample mean \Hat\mu is replaced by the known population mean \mu,

Statistics: double gsl_stats_sd_with_fixed_mean (const double data[], size_t stride, size_t n, double mean)
This function calculates the standard deviation of data for a a fixed population mean mean. The result is the square root of the corresponding variance function.

Absolute deviation

Statistics: double gsl_stats_absdev (const double data[], size_t stride, size_t n)
This function computes the absolute deviation from the mean of data, a dataset of length n with stride stride. The absolute deviation from the mean is defined as,

where x_i are the elements of the dataset data. The absolute deviation from the mean provides a more robust measure of the width of a distribution than the variance. This function computes the mean of data via a call to gsl_stats_mean.

Statistics: double gsl_stats_absdev_m (const double data[], size_t stride, size_t n, double mean)
This function computes the absolute deviation of the dataset data relative to the given value of mean,

This function is useful if you have already computed the mean of data (and want to avoid recomputing it), or wish to calculate the absolute deviation relative to another value (such as zero, or the median).

Higher moments (skewness and kurtosis)

Statistics: double gsl_stats_skew (const double data[], size_t stride, size_t n)
This function computes the skewness of data, a dataset of length n with stride stride. The skewness is defined as,

where x_i are the elements of the dataset data. The skewness measures the asymmetry of the tails of a distribution.

The function computes the mean and estimated standard deviation of data via calls to gsl_stats_mean and gsl_stats_sd.

Statistics: double gsl_stats_skew_m_sd (const double data[], size_t stride, size_t n, double mean, double sd)
This function computes the skewness of the dataset data using the given values of the mean mean and standard deviation sd,

These functions are useful if you have already computed the mean and standard deviation of data and want to avoid recomputing them.

Statistics: double gsl_stats_kurtosis (const double data[], size_t stride, size_t n)
This function computes the kurtosis of data, a dataset of length n with stride stride. The kurtosis is defined as,

The kurtosis measures how sharply peaked a distribution is, relative to its width. The kurtosis is normalized to zero for a gaussian distribution.

Statistics: double gsl_stats_kurtosis_m_sd (const double data[], size_t stride, size_t n, double mean, double sd)
This function computes the kurtosis of the dataset data using the given values of the mean mean and standard deviation sd,

This function is useful if you have already computed the mean and standard deviation of data and want to avoid recomputing them.

Autocorrelation

Function: double gsl_stats_lag1_autocorrelation (const double data[], const size_t stride, const size_t n)
This function computes the lag-1 autocorrelation of the dataset data.

Function: double gsl_stats_lag1_autocorrelation_m (const double data[], const size_t stride, const size_t n, const double mean)
This function computes the lag-1 autocorrelation of the dataset data using the given value of the mean mean.

Covariance

Function: double gsl_stats_covariance (const double data1[], const size_t stride1, const double data2[], const size_t stride2, const size_t n)
This function computes the covariance of the datasets data1 and data2 which must both be of the same length n.

Function: double gsl_stats_covariance_m (const double data1[], const size_t stride1, const double data2[], const size_t n, const double mean1, const double mean2)
This function computes the covariance of the datasets data1 and data2 using the given values of the means, mean1 and mean2.

Weighted Samples

The functions described in this section allow the computation of statistics for weighted samples. The functions accept an array of samples, x_i, with associated weights, w_i. Each sample x_i is considered as having been drawn from a Gaussian distribution with variance \sigma_i^2. The sample weight w_i is defined as the reciprocal of this variance, w_i = 1/\sigma_i^2. Setting a weight to zero corresponds to removing a sample from a dataset.

Statistics: double gsl_stats_wmean (const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function returns the weighted mean of the dataset data with stride stride and length n, using the set of weights w with stride wstride and length n. The weighted mean is defined as,

Statistics: double gsl_stats_wvariance (const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function returns the estimated variance of the dataset data with stride stride and length n, using the set of weights w with stride wstride and length n. The estimated variance of a weighted dataset is defined as,

Note that this expression reduces to an unweighted variance with the familiar 1/(N-1) factor when there are N equal non-zero weights.

Statistics: double gsl_stats_wvariance_m (const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean)
This function returns the estimated variance of the weighted dataset data using the given weighted mean wmean.

Statistics: double gsl_stats_wsd (const double w[], size_t wstride, const double data[], size_t stride, size_t n)
The standard deviation is defined as the square root of the variance. This function returns the square root of the corresponding variance function gsl_stats_wvariance above.

Statistics: double gsl_stats_wsd_m (const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean)
This function returns the square root of the corresponding variance function gsl_stats_wvariance_m above.

Statistics: double gsl_stats_wvariance_with_fixed_mean (const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function computes an unbiased estimate of the variance of weighted dataset data when the population mean mean of the underlying distribution is known a priori. In this case the estimator for the variance replaces the sample mean \Hat\mu by the known population mean \mu,

Statistics: double gsl_stats_wsd_with_fixed_mean (const double w[], size_t wstride, const double data[], size_t stride, size_t n)
The standard deviation is defined as the square root of the variance. This function returns the square root of the corresponding variance function above.

Statistics: double gsl_stats_wabsdev (const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function computes the weighted absolute deviation from the weighted mean of data. The absolute deviation from the mean is defined as,

Statistics: double gsl_stats_wabsdev_m (const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean)
This function computes the absolute deviation of the weighted dataset data about the given weighted mean wmean.

Statistics: double gsl_stats_wskew (const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function computes the weighted skewness of the dataset data.

Statistics: double gsl_stats_wskew_m_sd (const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean, double wsd)
This function computes the weighted skewness of the dataset data using the given values of the weighted mean and weighted standard deviation, wmean and wsd.

Statistics: double gsl_stats_wkurtosis (const double w[], size_t wstride, const double data[], size_t stride, size_t n)
This function computes the weighted kurtosis of the dataset data.

Statistics: double gsl_stats_wkurtosis_m_sd (const double w[], size_t wstride, const double data[], size_t stride, size_t n, double wmean, double wsd)
This function computes the weighted kurtosis of the dataset data using the given values of the weighted mean and weighted standard deviation, wmean and wsd.

Maximum and Minimum values

Statistics: double gsl_stats_max (const double data[], size_t stride, size_t n)
This function returns the maximum value in data, a dataset of length n with stride stride. The maximum value is defined as the value of the element x_i which satisfies @c{$x_i \ge x_j$} x_i >= x_j for all j.

If you want instead to find the element with the largest absolute magnitude you will need to apply fabs or abs to your data before calling this function.

Statistics: double gsl_stats_min (const double data[], size_t stride, size_t n)
This function returns the minimum value in data, a dataset of length n with stride stride. The minimum value is defined as the value of the element x_i which satisfies @c{$x_i \le x_j$} x_i <= x_j for all j.

If you want instead to find the element with the smallest absolute magnitude you will need to apply fabs or abs to your data before calling this function.

Statistics: void gsl_stats_minmax (double * min, double * max, const double data[], size_t stride, size_t n)
This function finds both the minimum and maximum values min, max in data in a single pass.

Statistics: size_t gsl_stats_max_index (const double data[], size_t stride, size_t n)
This function returns the index of the maximum value in data, a dataset of length n with stride stride. The maximum value is defined as the value of the element x_i which satisfies x_i >= x_j for all j. When there are several equal maximum elements then the first one is chosen.

Statistics: size_t gsl_stats_min_index (const double data[], size_t stride, size_t n)
This function returns the index of the minimum value in data, a dataset of length n with stride stride. The minimum value is defined as the value of the element x_i which satisfies x_i >= x_j for all j. When there are several equal minimum elements then the first one is chosen.

Statistics: void gsl_stats_minmax_index (size_t * min_index, size_t * max_index, const double data[], size_t stride, size_t n)
This function returns the indexes min_index, max_index of the minimum and maximum values in data in a single pass.

Median and Percentiles

The median and percentile functions described in this section operate on sorted data. For convenience we use quantiles, measured on a scale of 0 to 1, instead of percentiles (which use a scale of 0 to 100).

Statistics: double gsl_stats_median_from_sorted_data (const double sorted_data[], size_t stride, size_t n)
This function returns the median value of sorted_data, a dataset of length n with stride stride. The elements of the array must be in ascending numerical order. There are no checks to see whether the data are sorted, so the function gsl_sort should always be used first.

When the dataset has an odd number of elements the median is the value of element (n-1)/2. When the dataset has an even number of elements the median is the mean of the two nearest middle values, elements (n-1)/2 and n/2. Since the algorithm for computing the median involves interpolation this function always returns a floating-point number, even for integer data types.

Statistics: double gsl_stats_quantile_from_sorted_data (const double sorted_data[], size_t stride, size_t n, double f)
This function returns a quantile value of sorted_data, a double-precision array of length n with stride stride. The elements of the array must be in ascending numerical order. The quantile is determined by the f, a fraction between 0 and 1. For example, to compute the value of the 75th percentile f should have the value 0.75.

There are no checks to see whether the data are sorted, so the function gsl_sort should always be used first.

The quantile is found by interpolation, using the formula

where i is floor((n - 1)f) and \delta is (n-1)f - i.

Thus the minimum value of the array (data[0*stride]) is given by f equal to zero, the maximum value (data[(n-1)*stride]) is given by f equal to one and the median value is given by f equal to 0.5. Since the algorithm for computing quantiles involves interpolation this function always returns a floating-point number, even for integer data types.

Example statistical programs

Here is a basic example of how to use the statistical functions:

#include <stdio.h>
#include <gsl/gsl_statistics.h>

int
main(void)
{
  double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
  double mean, variance, largest, smallest;

  mean     = gsl_stats_mean(data, 1, 5);
  variance = gsl_stats_variance(data, 1, 5);
  largest  = gsl_stats_max(data, 1, 5);
  smallest = gsl_stats_min(data, 1, 5);

  printf("The dataset is %g, %g, %g, %g, %g\n",
        data[0], data[1], data[2], data[3], data[4]);

  printf("The sample mean is %g\n", mean);
  printf("The estimated variance is %g\n", variance);
  printf("The largest value is %g\n", largest);
  printf("The smallest value is %g\n", smallest);
  return 0;
}

The program should produce the following output,

The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
The sample mean is 16.54
The estimated variance is 4.2984
The largest value is 18.3
The smallest value is 12.6

Here is an example using sorted data,

#include <stdio.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_statistics.h>

int
main(void)
{
  double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
  double median, upperq, lowerq;

  printf("Original dataset:  %g, %g, %g, %g, %g\n",
        data[0], data[1], data[2], data[3], data[4]);

  gsl_sort (data, 1, 5);

  printf("Sorted dataset: %g, %g, %g, %g, %g\n",
        data[0], data[1], data[2], data[3], data[4]);

  median 
    = gsl_stats_median_from_sorted_data (data, 
                                         1, 5);

  upperq 
    = gsl_stats_quantile_from_sorted_data (data, 
                                           1, 5,
                                           0.75);
  lowerq 
    = gsl_stats_quantile_from_sorted_data (data, 
                                           1, 5,
                                           0.25);

  printf("The median is %g\n", median);
  printf("The upper quartile is %g\n", upperq);
  printf("The lower quartile is %g\n", lowerq);
  return 0;
}

This program should produce the following output,

Original dataset: 17.2, 18.1, 16.5, 18.3, 12.6
Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3
The median is 17.2
The upper quartile is 18.1
The lower quartile is 16.5

References and Further Reading

The standard reference for almost any topic in statistics is the multi-volume Advanced Theory of Statistics by Kendall and Stuart.

Many statistical concepts can be more easily understood by a Bayesian approach. The following book by Gelman, Carlin, Stern and Rubin gives a comprehensive coverage of the subject.

For physicists the Particle Data Group provides useful reviews of Probability and Statistics in the "Mathematical Tools" section of its Annual Review of Particle Physics.

The Review of Particle Physics is available online at http://pdg.lbl.gov/.

Histograms

This chapter describes functions for creating histograms. Histograms provide a convenient way of summarizing the distribution of a set of data. A histogram consists of a set of bins which count the number of events falling into a given range of a continuous variable x. In GSL the bins of a histogram contain floating-point numbers, so they can be used to record both integer and non-integer distributions. The bins can use arbitrary sets of ranges (uniformly spaced bins are the default). Both one and two-dimensional histograms are supported.

Once a histogram has been created it can also be converted into a probability distribution function. The library provides efficient routines for selecting random samples from probability distributions. This can be useful for generating simulations based real data.

The functions are declared in the header files `gsl_histogram.h' and `gsl_histogram2d.h'.

The histogram struct

A histogram is defined by the following struct,

Data Type: gsl_histogram
size_t n
This is the number of histogram bins
double * range
The ranges of the bins are stored in an array of n+1 elements pointed to by range.
double * bin
The counts for each bin are stored in an array of n elements pointed to by bin. The bins are floating-point numbers, so you can increment them by non-integer values if necessary.

The range for bin[i] is given by range[i] to range[i+1]. For n bins there are n+1 entries in the array range. Each bin is inclusive at the lower end and exclusive at the upper end. Mathematically this means that the bins are defined by the following inequality,

Here is a diagram of the correspondence between ranges and bins on the number-line for x,


     [ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[5] )
  ---|---------|---------|---------|---------|---------|---  x
   r[0]      r[1]      r[2]      r[3]      r[4]      r[5]

In this picture the values of the range array are denoted by r. On the left-hand side of each bin the square bracket "[" denotes an inclusive lower bound (@c{$r \le x$} r <= x), and the round parentheses ")" on the right-hand side denote an exclusive upper bound (x < r). Thus any samples which fall on the upper end of the histogram are excluded. If you want to include this value for the last bin you will need to add an extra bin to your histogram.

The gsl_histogram struct and its associated functions are defined in the header file `gsl_histogram.h'.

Histogram allocation

The functions for allocating memory to a histogram follow the style of malloc and free. In addition they also perform their own error checking. If there is insufficient memory available to allocate a histogram then the functions call the error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every histogram alloc.

Function: gsl_histogram * gsl_histogram_alloc (size_t n)
This function allocates memory for a histogram with n bins, and returns a pointer to a newly created gsl_histogram struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of GSL_ENOMEM. The bins and ranges are not initialized, and should be prepared using one of the range-setting functions below in order to make the histogram ready for use.

Function: int gsl_histogram_set_ranges (gsl_histogram * h, const double range[], size_t size)
This function sets the ranges of the existing histogram h using the array range of size size. The values of the histogram bins are reset to zero. The range array should contain the desired bin limits. The ranges can be arbitrary, subject to the restriction that they are monotonically increasing.

The following example shows how to create a histogram with logarithmic bins with ranges [1,10), [10,100) and [100,1000).

gsl_histogram * h = gsl_histogram_alloc (3);

/* bin[0] covers the range 1 <= x < 10 */
/* bin[1] covers the range 10 <= x < 100 */
/* bin[2] covers the range 100 <= x < 1000 */

double range[4] = { 1.0, 10.0, 100.0, 1000.0 };

gsl_histogram_set_ranges (h, range, 4);

Note that the size of the range array should be defined to be one element bigger than the number of bins. The additional element is required for the upper value of the final bin.

Function: int gsl_histogram_set_ranges_uniform (gsl_histogram * h, double xmin, double xmax)
This function sets the ranges of the existing histogram h to cover the range xmin to xmax uniformly. The values of the histogram bins are reset to zero. The bin ranges are shown in the table below,

where d is the bin spacing, d = (xmax-xmin)/n.

Function: void gsl_histogram_free (gsl_histogram * h)
This function frees the histogram h and all of the memory associated with it.

Copying Histograms

Function: int gsl_histogram_memcpy (gsl_histogram * dest, const gsl_histogram * src)
This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copy of src. The two histograms must be of the same size.

Function: gsl_histogram * gsl_histogram_clone (const gsl_histogram * src)
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.

Updating and accessing histogram elements

There are two ways to access histogram bins, either by specifying an x coordinate or by using the bin-index directly. The functions for accessing the histogram through x coordinates use a binary search to identify the bin which covers the appropriate range.

Function: int gsl_histogram_increment (gsl_histogram * h, double x)
This function updates the histogram h by adding one (1.0) to the bin whose range contains the coordinate x.

If x lies in the valid range of the histogram then the function returns zero to indicate success. If x is less than the lower limit of the histogram then the function returns GSL_EDOM, and none of bins are modified. Similarly, if the value of x is greater than or equal to the upper limit of the histogram then the function returns GSL_EDOM, and none of the bins are modified. The error handler is not called, however, since it is often necessary to compute histogram for a small range of a larger dataset, ignoring the values outside the range of interest.

Function: int gsl_histogram_accumulate (gsl_histogram * h, double x, double weight)
This function is similar to gsl_histogram_increment but increases the value of the appropriate bin in the histogram h by the floating-point number weight.

Function: double gsl_histogram_get (const gsl_histogram * h, size_t i)
This function returns the contents of the ith bin of the histogram h. If i lies outside the valid range of indices for the histogram then the error handler is called with an error code of GSL_EDOM and the function returns 0.

Function: int gsl_histogram_get_range (const gsl_histogram * h, size_t i, double * lower, double * upper)
This function finds the upper and lower range limits of the ith bin of the histogram h. If the index i is valid then the corresponding range limits are stored in lower and upper. The lower limit is inclusive (i.e. events with this coordinate are included in the bin) and the upper limit is exclusive (i.e. events with the coordinate of the upper limit are excluded and fall in the neighboring higher bin, if it exists). The function returns 0 to indicate success. If i lies outside the valid range of indices for the histogram then the error handler is called and the function returns an error code of GSL_EDOM.

Function: double gsl_histogram_max (const gsl_histogram * h)
Function: double gsl_histogram_min (const gsl_histogram * h)
Function: size_t gsl_histogram_bins (const gsl_histogram * h)
These functions return the maximum upper and minimum lower range limits and the number of bins of the histogram h. They provide a way of determining these values without accessing the gsl_histogram struct directly.

Function: void gsl_histogram_reset (gsl_histogram * h)
This function resets all the bins in the histogram h to zero.

Searching histogram ranges

The following functions are used by the access and update routines to locate the bin which corresponds to a given x coordinate.

Function: int gsl_histogram_find (const gsl_histogram * h, double x, size_t * i)
This function finds and sets the index i to the bin number which covers the coordinate x in the histogram h. The bin is located using a binary search. The search includes an optimization for histograms with uniform range, and will return the correct bin immediately in this case. If x is found in the range of the histogram then the function sets the index i and returns GSL_SUCCESS. If x lies outside the valid range of the histogram then the function returns GSL_EDOM and the error handler is invoked.

Histogram Statistics

Function: double gsl_histogram_max_val (const gsl_histogram * h)
This function returns the maximum value contained in the histogram bins.

Function: size_t gsl_histogram_max_bin (const gsl_histogram * h)
This function returns the index of the bin containing the maximum value. In the case where several bins contain the same maximum value the smallest index is returned.

Function: double gsl_histogram_min_val (const gsl_histogram * h)
This function returns the minimum value contained in the histogram bins.

Function: size_t gsl_histogram_min_bin (const gsl_histogram * h)
This function returns the index of the bin containing the minimum value. In the case where several bins contain the same maximum value the smallest index is returned.

Function: double gsl_histogram_mean (const gsl_histogram * h)
This function returns the mean of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.

Function: double gsl_histogram_sigma (const gsl_histogram * h)
This function returns the standard deviation of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.

Histogram Operations

Function: int gsl_histogram_equal_bins_p (const gsl_histogram *h1, const gsl_histogram *h2)
This function returns 1 if the all of the individual bin ranges of the two histograms are identical, and 0 otherwise.

Function: int gsl_histogram_add (gsl_histogram *h1, const gsl_histogram *h2)
This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e. h'_1(i) = h_1(i) + h_2(i). The two histograms must have identical bin ranges.

Function: int gsl_histogram_sub (gsl_histogram *h1, const gsl_histogram *h2)
This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1, i.e. h'_1(i) = h_1(i) - h_2(i). The two histograms must have identical bin ranges.

Function: int gsl_histogram_mul (gsl_histogram *h1, const gsl_histogram *h2)
This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i) = h_1(i) * h_2(i). The two histograms must have identical bin ranges.

Function: int gsl_histogram_div (gsl_histogram *h1, const gsl_histogram *h2)
This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i) = h_1(i) / h_2(i). The two histograms must have identical bin ranges.

Function: int gsl_histogram_scale (gsl_histogram *h, double scale)
This function multiplies the contents of the bins of histogram h by the constant scale, i.e. @c{$h'_1(i) = h_1(i) * \hbox{\it scale}$} h'_1(i) = h_1(i) * scale.

Function: int gsl_histogram_shift (gsl_histogram *h, double offset)
This function shifts the contents of the bins of histogram h by the constant offset, i.e. @c{$h'_1(i) = h_1(i) + \hbox{\it offset}$} h'_1(i) = h_1(i) + offset.

Reading and writing histograms

The library provides functions for reading and writing histograms to a file as binary data or formatted text.

Function: int gsl_histogram_fwrite (FILE * stream, const gsl_histogram * h)
This function writes the ranges and bins of the histogram h to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

Function: int gsl_histogram_fread (FILE * stream, gsl_histogram * h)
This function reads into the histogram h from the open stream stream in binary format. The histogram h must be preallocated with the correct size since the function uses the number of bins in h to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

Function: int gsl_histogram_fprintf (FILE * stream, const gsl_histogram * h, const char * range_format, const char * bin_format)
This function writes the ranges and bins of the histogram h line-by-line to the stream stream using the format specifiers range_format and bin_format. These should be one of the %g, %e or %f formats for floating point numbers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file. The histogram output is formatted in three columns, and the columns are separated by spaces, like this,
range[0] range[1] bin[0]
range[1] range[2] bin[1]
range[2] range[3] bin[2]
....
range[n-1] range[n] bin[n-1]

The values of the ranges are formatted using range_format and the value of the bins are formatted using bin_format. Each line contains the lower and upper limit of the range of the bins and the value of the bin itself. Since the upper limit of one bin is the lower limit of the next there is duplication of these values between lines but this allows the histogram to be manipulated with line-oriented tools.

Function: int gsl_histogram_fscanf (FILE * stream, gsl_histogram * h)
This function reads formatted data from the stream stream into the histogram h. The data is assumed to be in the three-column format used by gsl_histogram_fprintf. The histogram h must be preallocated with the correct length since the function uses the size of h to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

Resampling from histograms

A histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of an event where the value of x falls in the range of that bin. The probability distribution function has the one-dimensional form p(x)dx where,

In this equation n_i is the number of events in the bin which contains x, w_i is the width of the bin and N is the total number of events. The distribution of events within each bin is assumed to be uniform.

The histogram probability distribution struct

The probability distribution function for a histogram consists of a set of bins which measure the probability of an event falling into a given range of a continuous variable x. A probability distribution function is defined by the following struct, which actually stores the cumulative probability distribution function. This is the natural quantity for generating samples via the inverse transform method, because there is a one-to-one mapping between the cumulative probability distribution and the range [0,1]. It can be shown that by taking a uniform random number in this range and finding its corresponding coordinate in the cumulative probability distribution we obtain samples with the desired probability distribution.

Data Type: gsl_histogram_pdf
size_t n
This is the number of bins used to approximate the probability distribution function.
double * range
The ranges of the bins are stored in an array of n+1 elements pointed to by range.
double * sum
The cumulative probability for the bins is stored in an array of n elements pointed to by sum.

The following functions allow you to create a gsl_histogram_pdf struct which represents this probability distribution and generate random samples from it.

Function: gsl_histogram_pdf * gsl_histogram_pdf_alloc (size_t n)
This function allocates memory for a probability distribution with n bins and returns a pointer to a newly initialized gsl_histogram_pdf struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of GSL_ENOMEM.

Function: int gsl_histogram_pdf_init (gsl_histogram_pdf * p, const gsl_histogram * h)
This function initializes the probability distribution p with with the contents of the histogram h. If any of the bins of h are negative then the error handler is invoked with an error code of GSL_EDOM because a probability distribution cannot contain negative values.

Function: void gsl_histogram_pdf_free (gsl_histogram_pdf * p)
This function frees the probability distribution function p and all of the memory associated with it.

Function: double gsl_histogram_pdf_sample (const gsl_histogram_pdf * p, double r)
This function uses r, a uniform random number between zero and one, to compute a single random sample from the probability distribution p. The algorithm used to compute the sample s is given by the following formula,

where i is the index which satisfies sum[i] <= r < sum[i+1] and delta is (r - sum[i])/(sum[i+1] - sum[i]).

Example programs for histograms

The following program shows how to make a simple histogram of a column of numerical data supplied on stdin. The program takes three arguments, specifying the upper and lower bounds of the histogram and the number of bins. It then reads numbers from stdin, one line at a time, and adds them to the histogram. When there is no more data to read it prints out the accumulated histogram using gsl_histogram_fprintf.

#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_histogram.h>

int
main (int argc, char **argv)
{
  double a, b;
  size_t n;

  if (argc != 4)
    {
      printf ("Usage: gsl-histogram xmin xmax n\n"
              "Computes a histogram of the data "
              "on stdin using n bins from xmin "
              "to xmax\n");
      exit (0);
    }

  a = atof (argv[1]);
  b = atof (argv[2]);
  n = atoi (argv[3]);

  {
    int status;
    double x;

    gsl_histogram * h = gsl_histogram_alloc (n);
            
    gsl_histogram_set_uniform (h, a, b);

    while (fscanf(stdin, "%lg", &x) == 1)
      {
        gsl_histogram_increment(h, x);
      }

    gsl_histogram_fprintf (stdout, h, "%g", "%g");

    gsl_histogram_free (h);
  }
  
  exit (0);
}

Here is an example of the program in use. We generate 10000 random samples from a Cauchy distribution with a width of 30 and histogram them over the range -100 to 100, using 200 bins.

$ gsl-randist 0 10000 cauchy 30 
   | gsl-histogram -100 100 200 > histogram.dat

A plot of the resulting histogram shows the familiar shape of the Cauchy distribution and the fluctuations caused by the finite sample size.

$ awk '{print $1, $3 ; print $2, $3}' histogram.dat 
   | graph -T X

histogram

Two dimensional histograms

A two dimensional histogram consists of a set of bins which count the number of events falling in a given area of the (x,y) plane. The simplest way to use a two dimensional histogram is to record two-dimensional position information, n(x,y). Another possibility is to form a joint distribution by recording related variables. For example a detector might record both the position of an event (x) and the amount of energy it deposited E. These could be histogrammed as the joint distribution n(x,E).

The 2D histogram struct

Two dimensional histograms are defined by the following struct,

Data Type: gsl_histogram2d
size_t nx, ny
This is the number of histogram bins in the x and y directions.
double * xrange
The ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by xrange.
double * yrange
The ranges of the bins in the y-direction are stored in an array of ny + 1 pointed to by yrange.
double * bin
The counts for each bin are stored in an array pointed to by bin. The bins are floating-point numbers, so you can increment them by non-integer values if necessary. The array bin stores the two dimensional array of bins in a single block of memory according to the mapping bin(i,j) = bin[i * ny + j].

The range for bin(i,j) is given by xrange[i] to xrange[i+1] in the x-direction and yrange[j] to yrange[j+1] in the y-direction. Each bin is inclusive at the lower end and exclusive at the upper end. Mathematically this means that the bins are defined by the following inequality,

Note that any samples which fall on the upper sides of the histogram are excluded. If you want to include these values for the side bins you will need to add an extra row or column to your histogram.

The gsl_histogram2d struct and its associated functions are defined in the header file `gsl_histogram2d.h'.

2D Histogram allocation

The functions for allocating memory to a 2D histogram follow the style of malloc and free. In addition they also perform their own error checking. If there is insufficient memory available to allocate a histogram then the functions call the error handler (with an error number of GSL_ENOMEM) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every 2D histogram alloc.

Function: gsl_histogram2d * gsl_histogram2d_alloc (size_t nx, size_t ny)
This function allocates memory for a two-dimensional histogram with nx bins in the x direction and ny bins in the y direction. The function returns a pointer to a newly created gsl_histogram2d struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of GSL_ENOMEM. The bins and ranges must be initialized with one of the functions below before the histogram is ready for use.

Function: int gsl_histogram2d_set_ranges (gsl_histogram2d * h, const double xrange[], size_t xsize, const double yrange[], size_t ysize)
This function sets the ranges of the existing histogram h using the arrays xrange and yrange of size xsize and ysize respectively. The values of the histogram bins are reset to zero.

Function: int gsl_histogram2d_set_ranges_uniform (gsl_histogram2d * h, double xmin, double xmax, double ymin, double ymax)
This function sets the ranges of the existing histogram h to cover the ranges xmin to xmax and ymin to ymax uniformly. The values of the histogram bins are reset to zero.

Function: void gsl_histogram2d_free (gsl_histogram2d * h)
This function frees the 2D histogram h and all of the memory associated with it.

Copying 2D Histograms

Function: int gsl_histogram2d_memcpy (gsl_histogram2d * dest, const gsl_histogram2d * src)
This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copy of src. The two histograms must be of the same size.

Function: gsl_histogram2d * gsl_histogram2d_clone (const gsl_histogram2d * src)
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.

Updating and accessing 2D histogram elements

You can access the bins of a two-dimensional histogram either by specifying a pair of (x,y) coordinates or by using the bin indices (i,j) directly. The functions for accessing the histogram through (x,y) coordinates use binary searches in the x and y directions to identify the bin which covers the appropriate range.

Function: int gsl_histogram2d_increment (gsl_histogram2d * h, double x, double y)
This function updates the histogram h by adding one (1.0) to the bin whose x and y ranges contain the coordinates (x,y).

If the point (x,y) lies inside the valid ranges of the histogram then the function returns zero to indicate success. If (x,y) lies outside the limits of the histogram then the function returns GSL_EDOM, and none of bins are modified. The error handler is not called, since it is often necessary to compute histogram for a small range of a larger dataset, ignoring any coordinates outside the range of interest.

Function: int gsl_histogram2d_accumulate (gsl_histogram2d * h, double x, double y, double weight)
This function is similar to gsl_histogram2d_increment but increases the value of the appropriate bin in the histogram h by the floating-point number weight.

Function: double gsl_histogram2d_get (const gsl_histogram2d * h, size_t i, size_t j)
This function returns the contents of the (i,j)th bin of the histogram h. If (i,j) lies outside the valid range of indices for the histogram then the error handler is called with an error code of GSL_EDOM and the function returns 0.

Function: int gsl_histogram2d_get_xrange (const gsl_histogram2d * h, size_t i, double * xlower, double * xupper)
Function: int gsl_histogram2d_get_yrange (const gsl_histogram2d * h, size_t j, double * ylower, double * yupper)
These functions find the upper and lower range limits of the ith and jth bins in the x and y directions of the histogram h. The range limits are stored in xlower and xupper or ylower and yupper. The lower limits are inclusive (i.e. events with these coordinates are included in the bin) and the upper limits are exclusive (i.e. events with the value of the upper limit are not included and fall in the neighboring higher bin, if it exists). The functions return 0 to indicate success. If i or j lies outside the valid range of indices for the histogram then the error handler is called with an error code of GSL_EDOM.

Function: double gsl_histogram2d_xmax (const gsl_histogram2d * h)
Function: double gsl_histogram2d_xmin (const gsl_histogram2d * h)
Function: size_t gsl_histogram2d_nx (const gsl_histogram2d * h)
Function: double gsl_histogram2d_ymax (const gsl_histogram2d * h)
Function: double gsl_histogram2d_ymin (const gsl_histogram2d * h)
Function: size_t gsl_histogram2d_ny (const gsl_histogram2d * h)
These functions return the maximum upper and minimum lower range limits and the number of bins for the x and y directions of the histogram h. They provide a way of determining these values without accessing the gsl_histogram2d struct directly.

Function: void gsl_histogram2d_reset (gsl_histogram2d * h)
This function resets all the bins of the histogram h to zero.

Searching 2D histogram ranges

The following functions are used by the access and update routines to locate the bin which corresponds to a given (x\,y) coordinate.

Function: int gsl_histogram2d_find (const gsl_histogram2d * h, double x, double y, size_t * i, size_t * j)
This function finds and sets the indices i and j to the to the bin which covers the coordinates (x,y). The bin is located using a binary search. The search includes an optimization for histogram with uniform ranges, and will return the correct bin immediately in this case. If (x,y) is found then the function sets the indices (i,j) and returns GSL_SUCCESS. If (x,y) lies outside the valid range of the histogram then the function returns GSL_EDOM and the error handler is invoked.

2D Histogram Statistics

Function: double gsl_histogram2d_max_val (const gsl_histogram2d * h)
This function returns the maximum value contained in the histogram bins.

Function: void gsl_histogram2d_max_bin (const gsl_histogram2d * h, size_t * i, size_t * j)
This function returns the indices (i,j) of the bin containing the maximum value in the histogram h. In the case where several bins contain the same maximum value the first bin found is returned.

Function: double gsl_histogram2d_min_val (const gsl_histogram2d * h)
This function returns the minimum value contained in the histogram bins.

Function: void gsl_histogram2d_min_bin (const gsl_histogram2d * h, size_t * i, size_t * j)
This function returns the indices (i,j) of the bin containing the minimum value in the histogram h. In the case where several bins contain the same maximum value the first bin found is returned.

2D Histogram Operations

Function: int gsl_histogram2d_equal_bins_p (const gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function returns 1 if the all of the individual bin ranges of the two histograms are identical, and 0 otherwise.

Function: int gsl_histogram2d_add (gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e. h'_1(i,j) = h_1(i,j) + h_2(i,j). The two histograms must have identical bin ranges.

Function: int gsl_histogram2d_sub (gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1, i.e. h'_1(i,j) = h_1(i,j) - h_2(i,j). The two histograms must have identical bin ranges.

Function: int gsl_histogram2d_mul (gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i,j) = h_1(i,j) * h_2(i,j). The two histograms must have identical bin ranges.

Function: int gsl_histogram2d_div (gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i,j) = h_1(i,j) / h_2(i,j). The two histograms must have identical bin ranges.

Function: int gsl_histogram2d_scale (gsl_histogram2d *h, double scale)
This function multiplies the contents of the bins of histogram h by the constant scale, i.e. @c{$h'_1(i,j) = h_1(i,j) * \hbox{\it scale}$} h'_1(i,j) = h_1(i,j) scale.

Function: int gsl_histogram2d_shift (gsl_histogram2d *h, double offset)
This function shifts the contents of the bins of histogram h by the constant offset, i.e. @c{$h'_1(i,j) = h_1(i,j) + \hbox{\it offset}$} h'_1(i,j) = h_1(i,j) + offset.

Reading and writing 2D histograms

The library provides functions for reading and writing two dimensional histograms to a file as binary data or formatted text.

Function: int gsl_histogram2d_fwrite (FILE * stream, const gsl_histogram2d * h)
This function writes the ranges and bins of the histogram h to the stream stream in binary format. The return value is 0 for success and GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

Function: int gsl_histogram2d_fread (FILE * stream, gsl_histogram2d * h)
This function reads into the histogram h from the stream stream in binary format. The histogram h must be preallocated with the correct size since the function uses the number of x and y bins in h to determine how many bytes to read. The return value is 0 for success and GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

Function: int gsl_histogram2d_fprintf (FILE * stream, const gsl_histogram2d * h, const char * range_format, const char * bin_format)
This function writes the ranges and bins of the histogram h line-by-line to the stream stream using the format specifiers range_format and bin_format. These should be one of the %g, %e or %f formats for floating point numbers. The function returns 0 for success and GSL_EFAILED if there was a problem writing to the file. The histogram output is formatted in five columns, and the columns are separated by spaces, like this,
xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0)
xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1)
xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2)
....
xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1)

xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0)
xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1)
xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2)
....
xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1)

....

xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0)
xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1)
xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2)
....
xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)

Each line contains the lower and upper limits of the bin and the contents of the bin. Since the upper limits of the each bin are the lower limits of the neighboring bins there is duplication of these values but this allows the histogram to be manipulated with line-oriented tools.

Function: int gsl_histogram2d_fscanf (FILE * stream, gsl_histogram2d * h)
This function reads formatted data from the stream stream into the histogram h. The data is assumed to be in the five-column format used by gsl_histogram_fprintf. The histogram h must be preallocated with the correct lengths since the function uses the sizes of h to determine how many numbers to read. The function returns 0 for success and GSL_EFAILED if there was a problem reading from the file.

Resampling from 2D histograms

As in the one-dimensional case, a two-dimensional histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of an event where (x,y) falls in the range of that bin. For a two-dimensional histogram the probability distribution takes the form p(x,y) dx dy where,

In this equation n_{ij} is the number of events in the bin which contains (x,y), A_{ij} is the area of the bin and N is the total number of events. The distribution of events within each bin is assumed to be uniform.

Data Type: gsl_histogram2d_pdf
size_t nx, ny
This is the number of histogram bins used to approximate the probability distribution function in the x and y directions.
double * xrange
The ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by xrange.
double * yrange
The ranges of the bins in the y-direction are stored in an array of ny + 1 pointed to by yrange.
double * sum
The cumulative probability for the bins is stored in an array of nx*ny elements pointed to by sum.

The following functions allow you to create a gsl_histogram2d_pdf struct which represents a two dimensional probability distribution and generate random samples from it.

Function: gsl_histogram2d_pdf * gsl_histogram2d_pdf_alloc (size_t nx, size_t ny)
This function allocates memory for a two-dimensional probability distribution of size nx-by-ny and returns a pointer to a newly initialized gsl_histogram2d_pdf struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of GSL_ENOMEM.

Function: int gsl_histogram2d_pdf_init (gsl_histogram2d_pdf * p, const gsl_histogram2d * h)
This function initializes the two-dimensional probability distribution calculated p from the histogram h. If any of the bins of h are negative then the error handler is invoked with an error code of GSL_EDOM because a probability distribution cannot contain negative values.

Function: void gsl_histogram2d_pdf_free (gsl_histogram2d_pdf * p)
This function frees the two-dimensional probability distribution function p and all of the memory associated with it.

Function: int gsl_histogram2d_pdf_sample (const gsl_histogram2d_pdf * p, double r1, double r2, double * x, double * y)
This function uses two uniform random numbers between zero and one, r1 and r2, to compute a single random sample from the two-dimensional probability distribution p.

Example programs for 2D histograms

This program demonstrates two features of two-dimensional histograms. First a 10 by 10 2d-histogram is created with x and y running from 0 to 1. Then a few sample points are added to the histogram, at (0.3,0.3) with a height of 1, at (0.8,0.1) with a height of 5 and at (0.7,0.9) with a height of 0.5. This histogram with three events is used to generate a random sample of 1000 simulated events, which are printed out.

#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_histogram2d.h>

int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;

  gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10)

  gsl_histogram2d_set_ranges_uniform (h, 
                                      0.0, 1.0,
                                      0.0, 1.0);

  gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
  gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
  gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);

  gsl_rng_env_setup();
  
  T = gsl_rng_default;
  r = gsl_rng_alloc(T);

  {
    int i;
    gsl_histogram2d_pdf * p 
      = gsl_histogram2d_pdf_alloc (h->n);
    
    gsl_histogram2d_pdf_init (p, h);

    for (i = 0; i < 1000; i++) {
      double x, y;
      double u = gsl_rng_uniform (r);
      double v = gsl_rng_uniform (r);
       
      int status 
       = gsl_histogram2d_pdf_sample (p, u, v, &x, &y);
      
      printf("%g %g\n", x, y);
    }
  }
  
 return 0;
}

The following plot shows the distribution of the simulated events. Using a higher resolution grid we can see the original underlying histogram and also the statistical fluctuations caused by the events being uniformly distributed over the the area of the original bins.

histogram2d

N-tuples

This chapter describes functions for creating and manipulating ntuples, sets of values associated with events. The ntuples are stored in files. Their values can be extracted in any combination and booked in an histogram using a selection function.

The values to be stored are held in a user-defined data structure, and an ntuple is created associating this data structure with a file. The values are then written to the file (normally inside a loop) using the ntuple functions described below.

A histogram can be created from ntuple data by providing a selection function and a value function. The selection function specifies whether an event should be included in the subset to be analyzed or not. The value function computes the entry to be added to the histogram entry for each event.

All the ntuple functions are defined in the header file `gsl_ntuple.h'

The ntuple struct

Ntuples are manipulated using the gsl_ntuple struct. This struct contains information on the file where the ntuple data is stored, a pointer to the current ntuple data row and the size of the user-defined ntuple data struct.

typedef struct {
    FILE * file;
    void * ntuple_data;
    size_t size;
} gsl_ntuple;

Creating ntuples

Function: gsl_ntuple * gsl_ntuple_create (char * filename, void * ntuple_data, size_t size)
This function creates a new write-only ntuple file filename for ntuples of size size and returns a pointer to the newly created ntuple struct. Any existing file with the same name is truncated to zero length and overwritten. A pointer to memory for the current ntuple row ntuple_data must be supplied -- this is used to copy ntuples in and out of the file.

Opening an existing ntuple file

Function: gsl_ntuple * gsl_ntuple_open (char * filename, void * ntuple_data, size_t size)
This function opens an existing ntuple file filename for reading and returns a pointer to a corresponding ntuple struct. The ntuples in the file must have size size. A pointer to memory for the current ntuple row ntuple_data must be supplied -- this is used to copy ntuples in and out of the file.

Writing ntuples

Function: int gsl_ntuple_write (gsl_ntuple * ntuple)
This function writes the current ntuple ntuple->ntuple_data of size ntuple->size to the corresponding file.

Function: int gsl_ntuple_bookdata (gsl_ntuple * ntuple)
This function is a synonym for gsl_ntuple_write

Reading ntuples

Function: int gsl_ntuple_read (gsl_ntuple * ntuple)
This function reads the current row of the ntuple file for ntuple and stores the values in ntuple->data

Closing an ntuple file

Function: int gsl_ntuple_close (gsl_ntuple * ntuple)
This function closes the ntuple file ntuple and frees its associated allocated memory.

Histogramming ntuple values

Once an ntuple has been created its contents can be histogrammed in various ways using the function gsl_ntuple_project. Two user-defined functions must be provided, a function to select events and a function to compute scalar values. The selection function and the value function both accept the ntuple row as a first argument and other parameters as a second argument.

The selection function determines which ntuple rows are selected for histogramming. It is defined by the following struct,

typedef struct {
  int (* function) (void * ntuple_data, void * params);
  void * params;
} gsl_ntuple_select_fn;

The struct component function should return a non-zero value for each ntuple row that is to be included in the histogram.

The value function computes scalar values for those ntuple rows selected by the selection function,

typedef struct {
  double (* function) (void * ntuple_data, void * params);
  void * params;
} gsl_ntuple_value_fn;

In this case the struct component function should return the value to be added to the histogram for the ntuple row.

Function: int gsl_ntuple_project (gsl_histogram * h, gsl_ntuple * ntuple, gsl_ntuple_value_fn *value_func, gsl_ntuple_select_fn *select_func)
This function updates the histogram h from the ntuple ntuple using the functions value_func and select_func. For each ntuple row where the selection function select_func is non-zero the corresponding value of that row is computed using the function value_func and added to the histogram. Those ntuple rows where select_func returns zero are ignored. New entries are added to the histogram, so subsequent calls can be used to accumulate further data in the same histogram.

Example programs

The following example programs demonstrate the use of ntuples in managing a large dataset. The first program creates a set of 100,000 simulated "events", each with 3 associated values (x,y,z). These are generated from a gaussian distribution with unit variance, for demonstration purposes, and written to the ntuple file `test.dat'.

#include <config.h>
#include <gsl/gsl_ntuple.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>

struct data
{
  double x;
  double y;
  double z;
};

int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;

  struct data ntuple_row;
  int i;

  gsl_ntuple *ntuple 
    = gsl_ntuple_create ("test.dat", &ntuple_row, 
                         sizeof (ntuple_row));

  gsl_rng_env_setup();

  T = gsl_rng_default; 
  r = gsl_rng_alloc (T);

  for (i = 0; i < 10000; i++)
    {
      ntuple_row.x = gsl_ran_ugaussian (r);
      ntuple_row.y = gsl_ran_ugaussian (r);
      ntuple_row.z = gsl_ran_ugaussian (r);
      
      gsl_ntuple_write (ntuple);
    }
  
  gsl_ntuple_close(ntuple);
  return 0;
}

The next program analyses the ntuple data in the file `test.dat'. The analysis procedure is to compute the squared-magnitude of each event, E^2=x^2+y^2+z^2, and select only those which exceed a lower limit of 1.5. The selected events are then histogrammed using their E^2 values.

#include <config.h>
#include <math.h>
#include <gsl/gsl_ntuple.h>
#include <gsl/gsl_histogram.h>

struct data
{
  double x;
  double y;
  double z;
};

int sel_func (void *ntuple_data, void *params);
double val_func (void *ntuple_data, void *params);

int
main (void)
{
  struct data ntuple_row;
  int i;

  gsl_ntuple *ntuple 
    = gsl_ntuple_open ("test.dat", &ntuple_row,
                       sizeof (ntuple_row));
  double lower = 1.5;

  gsl_ntuple_select_fn S;
  gsl_ntuple_value_fn V;

  gsl_histogram *h = gsl_histogram_alloc (100);
  gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);

  S.function = &sel_func;
  S.params = &lower;

  V.function = &val_func;
  V.params = 0;

  gsl_ntuple_project (h, ntuple, &V, &S);

  gsl_histogram_fprintf (stdout, h, "%f", "%f");

  gsl_histogram_free (h);

  gsl_ntuple_close (ntuple);
  return 0;
}

int
sel_func (void *ntuple_data, void *params)
{
  double x, y, z, E, scale;
  scale = *(double *) params;

  x = ((struct data *) ntuple_data)->x;
  y = ((struct data *) ntuple_data)->y;
  z = ((struct data *) ntuple_data)->z;

  E2 = x * x + y * y + z * z;

  return E2 > scale;
}

double
val_func (void *ntuple_data, void *params)
{
  double x, y, z;

  x = ((struct data *) ntuple_data)->x;
  y = ((struct data *) ntuple_data)->y;
  z = ((struct data *) ntuple_data)->z;

  return x * x + y * y + z * z;
}

The following plot shows the distribution of the selected events. Note the cut-off at the lower bound.

ntuple

References and Further Reading

Further information on the use of ntuples can be found in the documentation for the CERN packages PAW and HBOOK (available online).

Monte Carlo Integration

This chapter describes routines for multidimensional Monte Carlo integration. These include the traditional Monte Carlo method and adaptive algorithms such as VEGAS and MISER which use importance sampling and stratified sampling techniques. Each algorithm computes an estimate of a multidimensional definite integral of the form,

over a hypercubic region ((x_l,x_u), (y_l,y_u), ...) using a fixed number of function calls. The routines also provide a statistical estimate of the error on the result. This error estimate should be taken as a guide rather than as a strict error bound --- random sampling of the region may not uncover all the important features of the function, resulting in an underestimate of the error.

The functions are defined in separate header files for each routine, gsl_monte_plain.h, `gsl_monte_miser.h' and `gsl_monte_vegas.h'.

Interface

All of the Monte Carlo integration routines use the same interface. There is an allocator to allocate memory for control variables and workspace, a routine to initialize those control variables, the integrator itself, and a function to free the space when done.

Each integration function requires a random number generator to be supplied, and returns an estimate of the integral and its standard deviation. The accuracy of the result is determined by the number of function calls specified by the user. If a known level of accuracy is required this can be achieved by calling the integrator several times and averaging the individual results until the desired accuracy is obtained.

Random sample points used within the Monte Carlo routines are always chosen strictly within the integration region, so that endpoint singularities are automatically avoided.

The function to be integrated has its own datatype, defined in the header file `gsl_monte.h'.

Data Type: gsl_monte_function

This data type defines a general function with parameters for Monte Carlo integration.

double (* function) (double * x, size_t dim, void * params)
this function should return the value f(x,params) for argument x and parameters params, where x is an array of size dim giving the coordinates of the point where the function is to be evaluated.
size_t dim
the number of dimensions for x
void * params
a pointer to the parameters of the function

Here is an example for a quadratic function in two dimensions,

with a = 3, b = 2, c = 1. The following code defines a gsl_monte_function F which you could pass to an integrator:

struct my_f_params { double a; double b; double c; };

double
my_f (double x, size_t dim, void * p) {
   struct my_f_params * fp = (struct my_f_params *)p;

   if (dim != 2)
      {
        fprintf(stderr, "error: dim != 2");
        abort();
      }

   return  fp->a * x[0] * x[0] 
             + fp->b * x[0] * x[1] 
               + fp->c * x[1] * x[1];
}

gsl_monte_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };

F.function = &my_f;
F.dim = 2;
F.params = &params;

The function f(x) can be evaluated using the following macro,

#define GSL_MONTE_FN_EVAL(F,x) 
    (*((F)->function))(x,(F)->dim,(F)->params)

PLAIN Monte Carlo

The plain Monte Carlo algorithm samples points randomly from the integration region to estimate the integral and its error. Using this algorithm the estimate of the integral E(f; N) for N randomly distributed points x_i is given by,

where V is the volume of the integration region. The error on this estimate \sigma(E;N) is calculated from the estimated variance of the mean,

For large N this variance decreases asymptotically as var(f)/N, where var(f) is the true variance of the function over the integration region. The error estimate itself should decrease as @c{$\sigma(f)/\sqrt{N}$} \sigma(f)/\sqrt{N}. The familiar law of errors decreasing as @c{$1/\sqrt{N}$} 1/\sqrt{N} applies -- to reduce the error by a factor of 10 requires a 100-fold increase in the number of sample points.

The functions described in this section are declared in the header file `gsl_monte_plain.h'.

Function: gsl_monte_plain_state * gsl_monte_plain_alloc (size_t dim)
This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions.

Function: int gsl_monte_plain_init (gsl_monte_plain_state* s)
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.

Function: int gsl_monte_plain_integrate (gsl_monte_function * f, double * xl, double * xu, size_t dim, size_t calls, gsl_rng * r, gsl_monte_plain_state * s, double * result, double * abserr)
This routines uses the plain Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr.

Function: void gsl_monte_plain_free (gsl_monte_plain_state* s),
This function frees the memory associated with the integrator state s.

MISER

The MISER algorithm of Press and Farrar is based on recursive stratified sampling. This technique aims to reduce the overall integration error by concentrating integration points in the regions of highest variance.

The idea of stratified sampling begins with the observation that for two disjoint regions a and b with Monte Carlo estimates of the integral E_a(f) and E_b(f) and variances \sigma_a^2(f) and \sigma_b^2(f), the variance Var(f) of the combined estimate E(f) = (1/2) (E_a(f) + E_b(f)) is given by,

It can be shown that this variance is minimized by distributing the points such that,

Hence the smallest error estimate is obtained by allocating sample points in proportion to the standard deviation of the function in each sub-region.

The MISER algorithm proceeds by bisecting the integration region along one coordinate axis to give two sub-regions at each step. The direction is chosen by examining all d possible bisections and selecting the one which will minimize the combined variance of the two sub-regions. The variance in the sub-regions is estimated by sampling with a fraction of the total number of points available to the current step. The same procedure is then repeated recursively for each of the two half-spaces from the best bisection. The remaining sample points are allocated to the sub-regions using the formula for N_a and N_b. This recursive allocation of integration points continues down to a user-specified depth where each sub-region is integrated using a plain Monte Carlo estimate. These individual values and their error estimates are then combined upwards to give an overall result and an estimate of its error.

The functions described in this section are declared in the header file `gsl_monte_miser.h'.

Function: gsl_monte_miser_state * gsl_monte_miser_alloc (size_t dim)
This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The workspace is used to maintain the state of the integration.

Function: int gsl_monte_miser_init (gsl_monte_miser_state* s)
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.

Function: int gsl_monte_miser_integrate (gsl_monte_function * f, double * xl, double * xu, size_t dim, size_t calls, gsl_rng * r, gsl_monte_miser_state * s, double * result, double * abserr)
This routines uses the MISER Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr.

Function: void gsl_monte_miser_free (gsl_monte_miser_state* s),
This function frees the memory associated with the integrator state s.

The MISER algorithm has several configurable parameters. The following variables can be accessed through the gsl_monte_miser_state struct,

Variable: double estimate_frac
This parameter specifies the fraction of the currently available number of function calls which are allocated to estimating the variance at each recursive step. The default value is 0.1.

Variable: size_t min_calls
This parameter specifies the minimum number of function calls required for each estimate of the variance. If the number of function calls allocated to the estimate using estimate_frac falls below min_calls then min_calls are used instead. This ensures that each estimate maintains a reasonable level of accuracy. The default value of min_calls is 16 * dim.

Variable: size_t min_calls_per_bisection
This parameter specifies the minimum number of function calls required to proceed with a bisection step. When a recursive step has fewer calls available than min_calls_per_bisection it performs a plain Monte Carlo estimate of the current sub-region and terminates its branch of the recursion. The default value of this parameter is 32 * min_calls.

Variable: double alpha
This parameter controls how the estimated variances for the two sub-regions of a bisection are combined when allocating points. With recursive sampling the overall variance should scale better than 1/N, since the values from the sub-regions will be obtained using a procedure which explicitly minimizes their variance. To accommodate this behavior the MISER algorithm allows the total variance to depend on a scaling parameter \alpha,

The authors of the original paper describing MISER recommend the value \alpha = 2 as a good choice, obtained from numerical experiments, and this is used as the default value in this implementation.

Variable: double dither
This parameter introduces a random fractional variation of size dither into each bisection, which can be used to break the symmetry of integrands which are concentrated near the exact center of the hypercubic integration region. The default value of dither is zero, so no variation is introduced. If needed, a typical value of dither is around 0.1.

VEGAS

The VEGAS algorithm of Lepage is based on importance sampling. It samples points from the probability distribution described by the function |f|, so that the points are concentrated in the regions that make the largest contribution to the integral.

In general, if the Monte Carlo integral of f is sampled with points distributed according to a probability distribution described by the function g, we obtain an estimate E_g(f; N),

with a corresponding variance,

If the probability distribution is chosen as g = |f|/I(|f|) then it can be shown that the variance V_g(f; N) vanishes, and the error in the estimate will be zero. In practice it is not possible to sample from the exact distribution g for an arbitrary function, so importance sampling algorithms aim to produce efficient approximations to the desired distribution.

The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region while histogramming the function f. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution. In order to avoid the number of histogram bins growing like K^d the probability distribution is approximated by a separable function: g(x_1, x_2, ...) = g_1(x_1) g_2(x_2) ... so that the number of bins required is only Kd. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes. The efficiency of VEGAS depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS.

VEGAS incorporates a number of additional features, and combines both stratified sampling and importance sampling. The integration region is divided into a number of "boxes", with each box getting in fixed number of points (the goal is 2). Each box can then have a fractional number of bins, but if bins/box is less than two, Vegas switches to a kind variance reduction (rather than importance sampling).

Function: gsl_monte_vegas_state * gsl_monte_vegas_alloc (size_t dim)
This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The workspace is used to maintain the state of the integration.

Function: int gsl_monte_vegas_init (gsl_monte_vegas_state* s)
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.

Function: int gsl_monte_vegas_integrate (gsl_monte_function * f, double * xl, double * xu, size_t dim, size_t calls, gsl_rng * r, gsl_monte_vegas_state * s, double * result, double * abserr)
This routines uses the VEGAS Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr. The result and its error estimate are based on a weighted average of independent samples. The chi-squared per degree of freedom for the weighted average is returned via the state struct component, s->chisq, and must be consistent with 1 for the weighted average to be reliable.

Function: void gsl_monte_vegas_free (gsl_monte_vegas_state* s),
This function frees the memory associated with the integrator state s.

The VEGAS algorithm computes a number of independent estimates of the integral internally, according to the iterations parameter described below, and returns their weighted average. Random sampling of the integrand can occasionally produce an estimate where the error is zero, particularly if the function is constant in some regions. An estimate with zero error causes the weighted average to break down and must be handled separately. In the original Fortran implementations of VEGAS the error estimate is made non-zero by substituting a small value (typically 1e-30). The implementation in GSL differs from this and avoids the use of an arbitrary constant -- it either assigns the value a weight which is the average weight of the preceding estimates or discards it according to the following procedure,

current estimate has zero error, weighted average has finite error
The current estimate is assigned a weight which is the average weight of the preceding estimates.
current estimate has finite error, previous estimates had zero error
The previous estimates are discarded and the weighted averaging procedure begins with the current estimate.
current estimate has zero error, previous estimates had zero error
The estimates are averaged using the arithmetic mean, but no error is computed.

The VEGAS algorithm is highly configurable. The following variables can be accessed through the gsl_monte_vegas_state struct,

Variable: double result
Variable: double sigma
These parameters contain the raw value of the integral result and its error sigma from the last iteration of the algorithm.

Variable: double chisq
This parameter gives the chi-squared per degree of freedom for the weighted estimate of the integral. The value of chisq should be close to 1. A value of chisq which differs significantly from 1 indicates that the values from different iterations are inconsistent. In this case the weighted error will be under-estimated, and further iterations of the algorithm are needed to obtain reliable results.

Variable: double alpha
The parameter alpha controls the stiffness of the rebinning algorithm. It is typically set between one and two. A value of zero prevents rebinning of the grid. The default value is 1.5.

Variable: size_t iterations
The number of iterations to perform for each call to the routine. The default value is 5 iterations.

Variable: int stage
Setting this determines the stage of the calculation. Normally, stage = 0 which begins with a new uniform grid and empty weighted average. Calling vegas with stage = 1 retains the grid from the previous run but discards the weighted average, so that one can "tune" the grid using a relatively small number of points and then do a large run with stage = 1 on the optimized grid. Setting stage = 2 keeps the grid and the weighted average from the previous run, but may increase (or decrease) the number of histogram bins in the grid depending on the number of calls available. Choosing stage = 3 enters at the main loop, so that nothing is changed, and is equivalent to performing additional iterations in a previous call.

Variable: int mode
The possible choices are GSL_VEGAS_MODE_IMPORTANCE, GSL_VEGAS_MODE_STRATIFIED, GSL_VEGAS_MODE_IMPORTANCE_ONLY. This determines whether VEGAS will use importance sampling or stratified sampling, or whether it can pick on its own. In low dimensions VEGAS uses strict stratified sampling (more precisely, stratified sampling is chosen if there are fewer than 2 bins per box).

Variable: int verbose
Variable: FILE * ostream
These parameters set the level of information printed by VEGAS. All information is written to the stream ostream. The default setting of verbose is -1, which turns off all output. A verbose value of 0 prints summary information about the weighted average and final result, while a value of 1 also displays the grid coordinates. A value of 2 prints information from the rebinning procedure for each iteration.

Examples

The example program below uses the Monte Carlo routines to estimate the value of the following 3-dimensional integral from the theory of random walks,

The analytic value of this integral can be shown to be I = \Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859.... The integral gives the mean time spent at the origin by a random walk on a body-centered cubic lattice in three dimensions.

For simplicity we will compute the integral over the region (0,0,0) to (\pi,\pi,\pi) and multiply by 8 to obtain the full result. The integral is slowly varying in the middle of the region but has integrable singularities at the corners (0,0,0), (0,\pi,\pi), (\pi,0,\pi) and (\pi,\pi,0). The Monte Carlo routines only select points which are strictly within the integration region and so no special measures are needed to avoid these singularities.

#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_monte.h>
#include <gsl/gsl_monte_plain.h>
#include <gsl/gsl_monte_miser.h>
#include <gsl/gsl_monte_vegas.h>

/* Computation of the integral,

      I = int (dx dy dz)/(2pi)^3  1/(1-cos(x)cos(y)cos(z))

   over (-pi,-pi,-pi) to (+pi, +pi, +pi).  The exact answer
   is Gamma(1/4)^4/(4 pi^3).  This example is taken from
   C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
   Volume 1", Section 1.1, p21, which cites the original
   paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
   1800 (1977) */

/* For simplicity we compute the integral over the region 
   (0,0,0) -> (pi,pi,pi) and multiply by 8 */

double exact = 1.3932039296856768591842462603255;

double
g (double *k, size_t dim, void *params)
{
  double A = 1.0 / (M_PI * M_PI * M_PI);
  return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
}

void
display_results (char *title, double result, double error)
{
  printf ("%s ==================\n", title);
  printf ("result = % .6f\n", result);
  printf ("sigma  = % .6f\n", error);
  printf ("exact  = % .6f\n", exact);
  printf ("error  = % .6f = %.1g sigma\n", result - exact,
          fabs (result - exact) / error);
}

int
main (void)
{
  double res, err;

  double xl[3] = { 0, 0, 0 };
  double xu[3] = { M_PI, M_PI, M_PI };

  const gsl_rng_type *T;
  gsl_rng *r;

  gsl_monte_function G = { &g, 3, 0 };

  size_t calls = 500000;

  gsl_rng_env_setup ();

  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

  {
    gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
    gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s, 
                               &res, &err);
    gsl_monte_plain_free (s);

    display_results ("plain", res, err);
  }

  {
    gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
    gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
                               &res, &err);
    gsl_monte_miser_free (s);

    display_results ("miser", res, err);
  }

  {
    gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);

    gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
                               &res, &err);
    display_results ("vegas warm-up", res, err);

    printf ("converging...\n");

    do
      {
        gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
                                   &res, &err);
        printf ("result = % .6f sigma = % .6f "
                "chisq/dof = %.1f\n", res, err, s->chisq);
      }
    while (fabs (s->chisq - 1.0) > 0.5);

    display_results ("vegas final", res, err);

    gsl_monte_vegas_free (s);
  }
  return 0;
}

With 500,000 function calls the plain Monte Carlo algorithm achieves a fractional error of 0.6%. The estimated error sigma is consistent with the actual error, and the computed result differs from the true result by about one standard deviation,

plain ==================
result =  1.385867
sigma  =  0.007938
exact  =  1.393204
error  = -0.007337 = 0.9 sigma

The MISER algorithm reduces the error by a factor of two, and also correctly estimates the error,

miser ==================
result =  1.390656
sigma  =  0.003743
exact  =  1.393204
error  = -0.002548 = 0.7 sigma

In the case of the VEGAS algorithm the program uses an initial warm-up run of 10,000 function calls to prepare, or "warm up", the grid. This is followed by a main run with five iterations of 100,000 function calls. The chi-squared per degree of freedom for the five iterations are checked for consistency with 1, and the run is repeated if the results have not converged. In this case the estimates are consistent on the first pass.

vegas warm-up ==================
result =  1.386925
sigma  =  0.002651
exact  =  1.393204
error  = -0.006278 = 2 sigma
converging...
result =  1.392957 sigma =  0.000452 chisq/dof = 1.1
vegas final ==================
result =  1.392957
sigma  =  0.000452
exact  =  1.393204
error  = -0.000247 = 0.5 sigma

If the value of chisq had differed significantly from 1 it would indicate inconsistent results, with a correspondingly underestimated error. The final estimate from VEGAS (using a similar number of function calls) is significantly more accurate than the other two algorithms.

References and Further Reading

The MISER algorithm is described in the following article,

The VEGAS algorithm is described in the following papers,

Simulated Annealing

Stochastic search techniques are used when the structure of a space is not well understood or is not smooth, so that techniques like Newton's method (which requires calculating Jacobian derivative matrices) cannot be used.

In particular, these techniques are frequently used to solve combinatorial optimization problems, such as the traveling salesman problem.

The basic problem layout is that we are looking for a point in the space at which a real valued energy function (or cost function) is minimized.

Simulated annealing is a technique which has given good results in avoiding local minima; it is based on the idea of taking a random walk through the space at successively lower temperatures, where the probability of taking a step is given by a Boltzmann distribution.

The functions described in this chapter are declared in the header file `gsl_siman.h'.

Simulated Annealing algorithm

We take random walks through the problem space, looking for points with low energies; in these random walks, the probability of taking a step is determined by the Boltzmann distribution

if E_{i+1} > E_i, and p = 1 when E_{i+1} <= E_i.

In other words, a step will occur if the new energy is lower. If the new energy is higher, the transition can still occur, and its likelihood is proportional to the temperature T and inversely proportional to the energy difference E_{i+1} - E_i.

The temperature T is initially set to a high value, and a random walk is carried out at that temperature. Then the temperature is lowered very slightly (according to a cooling schedule) and another random walk is taken.

This slight probability of taking a step that gives higher energy is what allows simulated annealing to frequently get out of local minima.

An initial guess is supplied. At each step, a point is chosen at a random distance from the current one, where the random distance r is distributed according to a Boltzmann distribution After a few search steps using this distribution, the temperature T is lowered according to some scheme, for example where \mu_T is slightly greater than 1.

Simulated Annealing functions

Simulated annealing: gsl_siman_Efunc_t
double (*gsl_siman_Efunc_t) (void *xp);

Simulated annealing: gsl_siman_step_t
void (*gsl_siman_step_t) (void *xp, double step_size);

Simulated annealing: gsl_siman_metric_t
double (*gsl_siman_metric_t) (void *xp, void *yp);

Simulated annealing: gsl_siman_print_t
void (*gsl_siman_print_t) (void *xp);

Simulated annealing: gsl_siman_params_t
These are the parameters that control a run of gsl_siman_solve.
/* this structure contains all the information 
   needed to structure the search, beyond the 
   energy function, the step function and the 
   initial guess. */

struct s_siman_params {
  /* how many points to try for each step */
  int n_tries;          

  /* how many iterations at each temperature? */
  int iters_fixed_T;    

  /* max step size in the random walk */
  double step_size;     

  /* the following parameters are for the 
     Boltzmann distribution */
  double k, t_initial, mu_t, t_min;
};

typedef struct s_siman_params gsl_siman_params_t;

Simulated annealing: void gsl_siman_solve (void *x0_p, gsl_siman_Efunc_t Ef, gsl_siman_metric_t distance, gsl_siman_print_t print_position, size_t element_size, gsl_siman_params_t params)
Does a simulated annealing search through a given space. The space is specified by providing the functions Ef, distance, print_position, element_size.

The params structure (described above) controls the run by providing the temperature schedule and other tunable parameters to the algorithm (see section Simulated Annealing algorithm). p The result (optimal point in the space) is placed in *x0_p.

If print_position is not null, a log will be printed to the screen with the following columns:

number_of_iterations temperature x x-(*x0_p) Ef(x)

If print_position is null, no information is printed to the screen.

Examples with Simulated Annealing

GSL's Simulated Annealing package is clumsy, and it has to be because it is written in C, for C callers, and tries to be polymorphic at the same time. But here we provide some examples which can be pasted into your application with little change and should make things easier.

Trivial example

The first example, in one dimensional cartesian space, sets up an energy function which is a damped sine wave; this has many local minima, but only one global minimum, somewhere between 1.0 and 1.5. The initial guess given is 15.5, which is several local minima away from the global minimum.

/* set up parameters for this simulated annealing run */

/* how many points do we try before stepping */
#define N_TRIES 200             

/* how many iterations for each T? */
#define ITERS_FIXED_T 10        

/* max step size in random walk */
#define STEP_SIZE 10            

/* Boltzmann constant */
#define K 1.0                   

/* initial temperature */
#define T_INITIAL 0.002         

/* damping factor for temperature */
#define MU_T 1.005              
#define T_MIN 2.0e-6

gsl_siman_params_t params 
  = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
     K, T_INITIAL, MU_T, T_MIN};

/* now some functions to test in one dimension */
double E1(void *xp)
{
  double x = * ((double *) xp);

  return exp(-square(x-1))*sin(8*x);
}

double M1(void *xp, void *yp)
{
  double x = *((double *) xp);
  double y = *((double *) yp);

  return fabs(x - y);
}

void S1(void *xp, double step_size)
{
  double r;
  double old_x = *((double *) xp);
  double new_x;

  r = gsl_ran_uniform();
  new_x = r*2*step_size - step_size + old_x;

  memcpy(xp, &new_x, sizeof(new_x));
}

void P1(void *xp)
{
  printf("%12g", *((double *) xp));
}

int
main(int argc, char *argv[])
{
  Element x0; /* initial guess for search */

  double x_initial = 15.5;

  gsl_siman_solve(&x_initial, E1, S1, M1, P1,
                  sizeof(double), params);
  return 0;
}

Here are a couple of plots that are generated by running siman_test in the following way:

./siman_test | grep -v "^#" 
  | xyplot -xyil -y -0.88 -0.83 -d "x...y" 
  | xyps -d > siman-test.eps
./siman_test | grep -v "^#" 
  | xyplot -xyil -xl "generation" -yl "energy" -d "x..y"
  | xyps -d > siman-energy.eps

siman-testsiman-energy

Example of a simulated annealing run: at higher temperatures (early in the plot) you see that the solution can fluctuate, but at lower temperatures it converges.

Traveling Salesman Problem

The TSP (Traveling Salesman Problem) is the classic combinatorial optimization problem. I have provided a very simple version of it, based on the coordinates of twelve cities in the southwestern United States. This should maybe be called the Flying Salesman Problem, since I am using the great-circle distance between cities, rather than the driving distance. Also: I assume the earth is a sphere, so I don't use geoid distances.

The gsl_siman_solve() routine finds a route which is 3490.62 Kilometers long; this is confirmed by an exhaustive search of all possible routes with the same initial city.

The full code can be found in `siman/siman_tsp.c', but I include here some plots generated with in the following way:

./siman_tsp > tsp.output
grep -v "^#" tsp.output  
  | xyplot -xyil -d "x................y" 
           -lx "generation" -ly "distance" 
           -lt "TSP -- 12 southwest cities" 
  | xyps -d > 12-cities.eps
grep initial_city_coord tsp.output 
  | awk '{print $2, $3, $4, $5}' 
  | xyplot -xyil -lb0 -cs 0.8 
           -lx "longitude (- means west)" 
           -ly "latitude" 
           -lt "TSP -- initial-order" 
  | xyps -d > initial-route.eps
grep final_city_coord tsp.output 
  | awk '{print $2, $3, $4, $5}' 
  | xyplot -xyil -lb0 -cs 0.8
           -lx "longitude (- means west)" 
           -ly "latitude" 
           -lt "TSP -- final-order" 
  | xyps -d > final-route.eps

This is the output showing the initial order of the cities; longitude is negative, since it is west and I want the plot to look like a map.

# initial coordinates of cities (longitude and latitude)
###initial_city_coord: -105.95 35.68 Santa Fe
###initial_city_coord: -112.07 33.54 Phoenix
###initial_city_coord: -106.62 35.12 Albuquerque
###initial_city_coord: -103.2 34.41 Clovis
###initial_city_coord: -107.87 37.29 Durango
###initial_city_coord: -96.77 32.79 Dallas
###initial_city_coord: -105.92 35.77 Tesuque
###initial_city_coord: -107.84 35.15 Grants
###initial_city_coord: -106.28 35.89 Los Alamos
###initial_city_coord: -106.76 32.34 Las Cruces
###initial_city_coord: -108.58 37.35 Cortez
###initial_city_coord: -108.74 35.52 Gallup
###initial_city_coord: -105.95 35.68 Santa Fe

The optimal route turns out to be:

# final coordinates of cities (longitude and latitude)
###final_city_coord: -105.95 35.68 Santa Fe
###final_city_coord: -106.28 35.89 Los Alamos
###final_city_coord: -106.62 35.12 Albuquerque
###final_city_coord: -107.84 35.15 Grants
###final_city_coord: -107.87 37.29 Durango
###final_city_coord: -108.58 37.35 Cortez
###final_city_coord: -108.74 35.52 Gallup
###final_city_coord: -112.07 33.54 Phoenix
###final_city_coord: -106.76 32.34 Las Cruces
###final_city_coord: -96.77 32.79 Dallas
###final_city_coord: -103.2 34.41 Clovis
###final_city_coord: -105.92 35.77 Tesuque
###final_city_coord: -105.95 35.68 Santa Fe

initial-routefinal-route

Initial and final (optimal) route for the 12 southwestern cities Flying Salesman Problem.

Here's a plot of the cost function (energy) versus generation (point in the calculation at which a new temperature is set) for this problem:

12-cities

Example of a simulated annealing run for the 12 southwestern cities Flying Salesman Problem.

Ordinary Differential Equations

This chapter describes functions for solving ordinary differential equation (ODE) initial value problems. The library provides a variety of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines, and higher-level components for adaptive step-size control. The components can be combined by the user to achieve the desired solution, with full access to any intermediate steps.

These functions are declared in the header file `gsl_odeiv.h'.

Defining the ODE System

The routines solve the general n-dimensional first-order system,

for i = 1, \dots, n. The stepping functions rely on the vector of derivatives f_i and the Jacobian matrix, J_{ij} = df_i(t,y(t)) / dy_j. A system of equations is defined using the gsl_odeiv_system datatype.

Data Type: gsl_odeiv_system
This data type defines a general ODE system with arbitrary parameters.
int (* function) (double t, const double y[], double dydt[], void * params)
This function should store the elements of f(t,y,params) in the array dydt, for arguments (t,y) and parameters params
int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);
This function should store the elements of f(t,y,params) in the array dfdt and the Jacobian matrix @c{$J_{ij}$} J_{ij} in the the array dfdy regarded as a row-ordered matrix J(i,j) = dfdy[i * dim + j] where dim is the dimension of the system.
size_t dimension;
This is the dimension of the system of equations
void * params
This is a pointer to the arbitrary parameters of the system.

Stepping Functions

The lowest level components are the stepping functions which advance a solution from time t to t+h for a fixed step-size h and estimate the resulting local error.

Function: gsl_odeiv_step * gsl_odeiv_step_alloc (const gsl_odeiv_step_type * T, size_t dim)
This function returns a pointer to a newly allocated instance of a stepping function of type T for a system of dim dimensions.

Function: int gsl_odeiv_step_reset (gsl_odeiv_step * s)
This function resets the stepping function s. It should be used whenever the next use of s will not be a continuation of a previous step.

Function: void gsl_odeiv_step_free (gsl_odeiv_step * s)
This function frees all the memory associated with the stepping function s.

Function: const char * gsl_odeiv_step_name (const gsl_odeiv_step * s)
This function returns a pointer to the name of the stepping function. For example,
printf("step method is '%s'\n",
        gsl_odeiv_step_name (s));

would print something like step method is 'rk4'.

Function: unsigned int gsl_odeiv_step_order (const gsl_odeiv_step * s)
This function returns the order of the stepping function on the previous step. This order can vary if the stepping function itself is adaptive.

Function: int gsl_odeiv_step_apply (gsl_odeiv_step * s, double t, double h, double y[], double yerr[], const double dydt_in[], double dydt_out[], const gsl_odeiv_system * dydt)
This function applies the stepping function s to the system of equations defined by dydt, using the step size h to advance the system from time t and state y to time t+h. The new state of the system is stored in y on output, with an estimate of the absolute error in each component stored in yerr. If the argument dydt_in is not null it should point an array containing the derivatives for the system at time t on input. This is optional as the derivatives will be computed internally if they are not provided, but allows the reuse of existing derivative information. On output the new derivatives of the system at time t+h will be stored in dydt_out if it is not null.

The following algorithms are available,

Step Type: gsl_odeiv_step_rk2
Embedded 2nd order Runge-Kutta with 3rd order error estimate.

Step Type: gsl_odeiv_step_rk4
4th order (classical) Runge-Kutta.

Step Type: gsl_odeiv_step_rkf45
Embedded 4th order Runge-Kutta-Fehlberg method with 5th order error estimate. This method is a good general-purpose integrator.

Step Type: gsl_odeiv_step_rkck
Embedded 4th order Runge-Kutta Cash-Karp method with 5th order error estimate.

Step Type: gsl_odeiv_step_rk8pd
Embedded 8th order Runge-Kutta Prince-Dormand method with 9th order error estimate.

Step Type: gsl_odeiv_step_rk2imp
Implicit 2nd order Runge-Kutta at Gaussian points

Step Type: gsl_odeiv_step_rk4imp
Implicit 4th order Runge-Kutta at Gaussian points

Step Type: gsl_odeiv_step_bsimp
Implicit Bulirsch-Stoer method of Bader and Deuflhard.

Step Type: gsl_odeiv_step_gear1
M=1 implicit Gear method

Step Type: gsl_odeiv_step_gear2
M=2 implicit Gear method

Adaptive Step-size Control

The control function examines the proposed change to the solution and its error estimate produced by a stepping function and attempts to determine the optimal step-size for a user-specified level of error.

Function: gsl_odeiv_control * gsl_odeiv_control_standard_new (double eps_abs, double eps_rel, double a_y, double a_dydt)
The standard control object is a four parameter heuristic based on absolute and relative errors eps_abs and eps_rel, and scaling factors a_y and a_dydt for the system state y(t) and derivatives y'(t) respectively.

The step-size adjustment procedure for this method begins by computing the desired error level D_i for each component,

and comparing it with the observed error E_i = |yerr_i|. If the observed error E exceeds the desired error level D by more than 10% for any component then the method reduces the step-size by an appropriate factor,

where q is the consistency order of method (e.g. q=4 for 4(5) embedded RK), and S is a safety factor of 0.9. The ratio D/E is taken to be the maximum of the ratios D_i/E_i.

If the observed error E is less than 50% of the desired error level D for the maximum ratio D_i/E_i then the algorithm takes the opportunity to increase the step-size to bring the error in line with the desired level,

This encompasses all the standard error scaling methods.

Function: gsl_odeiv_control * gsl_odeiv_control_y_new (double eps_abs, double eps_rel)
This function creates a new control object which will keep the local error on each step within an absolute error of eps_abs and relative error of eps_rel with respect to the solution y_i(t). This is equivalent to the standard control object with a_y=1 and a_dydt=0.

Function: gsl_odeiv_control * gsl_odeiv_control_yp_new (double eps_abs, double eps_rel)
This function creates a new control object which will keep the local error on each step within an absolute error of eps_abs and relative error of eps_rel with respect to the derivatives of the solution y'_i(t) . This is equivalent to the standard control object with a_y=0 and a_dydt=1.

Function: gsl_odeiv_control * gsl_odeiv_control_alloc (const gsl_odeiv_control_type * T)
This function returns a pointer to a newly allocated instance of a control function of type T. This function is only needed for defining new types of control functions. For most purposes the standard control functions described above should be sufficient.

Function: int gsl_odeiv_control_init (gsl_odeiv_control * c, double eps_abs, double eps_rel, double a_y, double a_dydt)
This function initializes the control function c with the parameters eps_abs (absolute error), eps_rel (relative error), a_y (scaling factor for y) and a_dydt (scaling factor for derivatives).

Function: void gsl_odeiv_control_free (gsl_odeiv_control * c)
This function frees all the memory associated with the control function c.

Function: int gsl_odeiv_control_hadjust (gsl_odeiv_control * c, gsl_odeiv_step * s, const double y0[], const double yerr[], const double dydt[], double * h)
This function adjusts the step-size h using the control function c, and the current values of y, yerr and dydt. The stepping function step is also needed to determine the order of the method. If the error in the y-values yerr is found to be too large then the step-size h is reduced and the function returns GSL_ODEIV_HADJ_DEC. If the error is sufficiently small then h may be increased and GSL_ODEIV_HADJ_INC is returned. The function returns GSL_ODEIV_HADJ_NIL if the step-size is unchanged. The goal of the function is to estimate the largest step-size which satisfies the user-specified accuracy requirements for the current point.

Function: const char * gsl_odeiv_control_name (const gsl_odeiv_control * c)
This function returns a pointer to the name of the control function. For example,
printf("control method is '%s'\n", 
       gsl_odeiv_control_name (c));

would print something like control method is 'standard'

Evolution

The highest level of the system is the evolution function which combines the results of a stepping function and control function to reliably advance the solution forward over an interval (t_0, t_1). If the control function signals that the step-size should be decreased the evolution function backs out of the current step and tries the proposed smaller step-size. This is process is continued until an acceptable step-size is found.

Function: gsl_odeiv_evolve * gsl_odeiv_evolve_alloc (size_t dim)
This function returns a pointer to a newly allocated instance of an evolution function for a system of dim dimensions.

Function: int gsl_odeiv_evolve_apply (gsl_odeiv_evolve * e, gsl_odeiv_control * con, gsl_odeiv_step * step, const gsl_odeiv_system * dydt, double * t, double t1, double * h, double y[])
This function advances the system (e, dydt) from time t and position y using the stepping function step. The new time and position are stored in t and y on output. The initial step-size is taken as h, but this will be modified using the control function c to achieve the appropriate error bound if necessary. The routine may make several calls to step in order to determine the optimum step-size. If the step-size has been changed the value of h will be modified on output. The maximum time t1 is guaranteed not to be exceeded by the time-step. On the final time-step the value of t will be set to t1 exactly.

Function: int gsl_odeiv_evolve_reset (gsl_odeiv_evolve * e)
This function resets the evolution function e. It should be used whenever the next use of e will not be a continuation of a previous step.

Function: void gsl_odeiv_evolve_free (gsl_odeiv_evolve * e)
This function frees all the memory associated with the evolution function e.

Examples

The following program solves the second-order nonlinear Van der Pol oscillator equation,

This can be converted into a first order system suitable for use with the library by introducing a separate variable for the velocity, y = x'(t),

The program begins by defining functions for these derivatives and their Jacobian,

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv.h>

int
func (double t, const double y[], double f[],
      void *params)
{
  double mu = *(double *)params;
  f[0] = y[1];
  f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
  return GSL_SUCCESS;
}

int
jac (double t, const double y[], double *dfdy, 
     double dfdt[], void *params)
{
  double mu = *(double *)params;
  gsl_matrix_view dfdy_mat 
    = gsl_matrix_view_array (dfdy, 2, 2);
  gsl_matrix * m = &dfdy_mat.matrix; 
  gsl_matrix_set (m, 0, 0, 0.0);
  gsl_matrix_set (m, 0, 1, 1.0);
  gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
  gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
  dfdt[0] = 0.0;
  dfdt[1] = 0.0;
  return GSL_SUCCESS;
}

int
main (void)
{
  const gsl_odeiv_step_type * T 
    = gsl_odeiv_step_rk8pd;

  gsl_odeiv_step * s 
    = gsl_odeiv_step_alloc (T, 2);
  gsl_odeiv_control * c 
    = gsl_odeiv_control_y_new (1e-6, 0.0);
  gsl_odeiv_evolve * e 
    = gsl_odeiv_evolve_alloc (2);

  double mu = 10;
  gsl_odeiv_system sys = {func, jac, 2, &mu};

  double t = 0.0, t1 = 100.0;
  double h = 1e-6;
  double y[2] = { 1.0, 0.0 };

  gsl_ieee_env_setup();

  while (t < t1)
    {
      int status = gsl_odeiv_evolve_apply (e, c, s,
                                           &sys, 
                                           &t, t1,
                                           &h, y);

      if (status != GSL_SUCCESS)
          break;

      printf("%.5e %.5e %.5e\n", t, y[0], y[1]);
    }

  gsl_odeiv_evolve_free(e);
  gsl_odeiv_control_free(c);
  gsl_odeiv_step_free(s);
  return 0;
}

The main loop of the program evolves the solution from (y, y') = (1, 0) at t=0 to t=100. The step-size h is automatically adjusted by the controller to maintain an absolute accuracy of @c{$10^{-6}$} 10^{-6} in the function values y.

vdp}
Numerical solution of the Van der Pol oscillator equation using Prince-Dormand 8th order Runge-Kutta.

To obtain the values at regular intervals, rather than the variable spacings chosen by the control function, the main loop can be modified to advance the solution from one point to the next. For example, the following main loop prints the solution at the fixed points t = 0, 1, 2, \dots, 100,

  for (i = 1; i <= 100; i++)
    {
      double ti = i * t1 / 100.0;

      while (t < ti)
        {
          gsl_odeiv_evolve_apply (e, c, s, 
                                  &sys, 
                                  &t, ti, &h,
                                  y);
        }
 
      printf("%.5e %.5e %.5e\n", t, y[0], y[1]);
    }

References and Further Reading

Many of the the basic Runge-Kutta formulas can be found in the Handbook of Mathematical Functions,

The implicit Bulirsch-Stoer algorithm bsimp is described in the following paper,

Interpolation

This chapter describes functions for performing interpolation. The library provides a variety of interpolation methods, including Cubic splines and Akima splines. The interpolation types are interchangeable, allowing different methods to be used without recompiling. Interpolations can be defined for both normal and periodic boundary conditions. Additional functions are available for computing derivatives and integrals of interpolating functions.

The functions described in this section are declared in the header files `gsl_interp.h' and `gsl_spline.h'.

Introduction

Given a set of data points (x_1, y_1) \dots (x_n, y_n) the routines described in this section compute a continuous interpolating function y(x) such that y_i = y(x_i). The interpolation is piecewise smooth, and its behavior at the points is determined by the type of interpolation used.

Interpolation Functions

The interpolation function for a given dataset is stored in a gsl_interp object. These are created by the following functions.

Function: gsl_interp * gsl_interp_alloc (const gsl_interp_type * T, size_t size)
This function returns a pointer to a newly allocated interpolation object of type T for size data-points.

Function: int gsl_interp_init (gsl_interp * interp, const double xa[], const double ya[], size_t size)
This function initializes the interpolation object interp for the data (xa,ya) where xa and ya are arrays of size size. The interpolation object (gsl_interp) does not save the data arrays xa and ya and only stores the static state computed from the data. The xa data array is always assumed to be strictly ordered; the behavior for other arrangements is not defined.

Function: void gsl_interp_free (gsl_interp * interp)
This function frees the interpolation object interp.

Interpolation Types

The interpolation library provides five interpolation types:

Interpolation Type: gsl_interp_linear
Linear interpolation. This interpolation method does not require any additional memory.

Interpolation Type: gsl_interp_cspline
Cubic spline with natural boundary conditions

Interpolation Type: gsl_interp_cspline_periodic
Cubic spline with periodic boundary conditions

Interpolation Type: gsl_interp_akima
Akima spline with natural boundary conditions

Interpolation Type: gsl_interp_akima_periodic
Akima spline with periodic boundary conditions

The following related functions are available,

Function: const char * gsl_interp_name (const gsl_interp * interp)
This function returns the name of the interpolation type used by interp. For example,
printf("interp uses '%s' interpolation\n", 
       gsl_interp_name (interp));

would print something like,

interp uses 'cspline' interpolation.

Function: unsigned int gsl_interp_min_size (const gsl_interp * interp)
This function returns the minimum number of points required by the interpolation type of interp. For example, cubic interpolation requires a minimum of 3 points.

Index Look-up and Acceleration

The state of searches can be stored in a gsl_interp_accel object, which is a kind of iterator for interpolation lookups. It caches the previous value of an index lookup. When the subsequent interpolation point falls in the same interval its index value can be returned immediately.

Function: size_t gsl_interp_bsearch (const double x_array[], double x, size_t index_lo, size_t index_hi)
This function returns the index i of the array x_array such that x_array[i] <= x < x_array[i+1]. The index is searched for in the range [index_lo,index_hi].

Function: gsl_interp_accel * gsl_interp_accel_alloc (void)
This function returns a pointer to an accelerator object, which is a kind of iterator for interpolation lookups. It tracks the state of lookups, thus allowing for application of various acceleration strategies.

Function: size_t gsl_interp_accel_find (gsl_interp_accel * a, const double x_array[], size_t size, double x)
This function performs a lookup action on the data array x_array of size size, using the given accelerator a. This is how lookups are performed during evaluation of an interpolation. The function returns an index i such that xarray[i] <= x < xarray[i+1].

Function: void gsl_interp_accel_free (gsl_interp_accel* a)
This function frees the accelerator object a.

Evaluation of interpolating functions

Function: double gsl_interp_eval (const gsl_interp * interp, const double xa[], const double ya[], double x, gsl_interp_accel * a)
Function: int gsl_interp_eval_e (const gsl_interp * interp, const double xa[], const double ya[], double x, gsl_interp_accel * a, double * y)
These functions return the interpolated value of y for a given point x, using the interpolation object interp, data arrays xa and ya and the accelerator a.

Function: double gsl_interp_eval_deriv (const gsl_interp * interp, const double xa[], const double ya[], double x, gsl_interp_accel * a)
Function: int gsl_interp_eval_deriv_e (const gsl_interp * interp, const double xa[], const double ya[], double x, gsl_interp_accel * a, double * d)
These functions return the derivative d of an interpolated function for a given point x, using the interpolation object interp, data arrays xa and ya and the accelerator a.

Function: double gsl_interp_eval_deriv2 (const gsl_interp * interp, const double xa[], const double ya[], double x, gsl_interp_accel * a)
Function: int gsl_interp_eval_deriv2_e (const gsl_interp * interp, const double xa[], const double ya[], double x, gsl_interp_accel * a, double * d2)
These functions return the second derivative d2 of an interpolated function for a given point x, using the interpolation object interp, data arrays xa and ya and the accelerator a.

Function: double gsl_interp_eval_integ (const gsl_interp * interp, const double xa[], const double ya[], double a, double bdouble x, gsl_interp_accel * a)
Function: int gsl_interp_eval_integ_e (const gsl_interp * interp, const double xa[], const double ya[], , double a, double b, gsl_interp_accel * a, double * result)
These functions return the numerical integral result of an interpolated function over the range [a, b], using the interpolation object interp, data arrays xa and ya and the accelerator a.

Higher-level interface

The functions described in the previous sections required the user to supply pointers to the x and y arrays on each call. The following functions are equivalent to the corresponding gsl_interp functions but maintain a copy of this data in the gsl_spline object. This removes the need to pass both xa and ya as arguments on each evaluation. These functions are defined in the header file `gsl_spline.h'.

Function: gsl_spline * gsl_spline_alloc (const gsl_interp_type * T, size_t n)

Function: int gsl_spline_init (gsl_spline * spline, const double xa[], const double ya[], size_t size)

Function: void gsl_spline_free (gsl_spline * spline)

Function: double gsl_spline_eval (const gsl_spline * spline, double x, gsl_interp_accel * a)
Function: int gsl_spline_eval_e (const gsl_spline * spline, double x, gsl_interp_accel * a, double * y)

Function: double gsl_spline_eval_deriv (const gsl_spline * spline, double x, gsl_interp_accel * a)
Function: int gsl_spline_eval_deriv_e (const gsl_spline * spline, double x, gsl_interp_accel * a, double * d)

Function: double gsl_spline_eval_deriv2 (const gsl_spline * spline, double x, gsl_interp_accel * a)
Function: int gsl_spline_eval_deriv2_e (const gsl_spline * spline, double x, gsl_interp_accel * a, double * d2)

Function: double gsl_spline_eval_integ (const gsl_spline * spline, double a, double b, gsl_interp_accel * acc)
Function: int gsl_spline_eval_integ_e (const gsl_spline * spline, double a, double b, gsl_interp_accel * acc, double * result)

Examples

The following program demonstrates the use of the interpolation and spline functions. It computes a cubic spline interpolation of the 10-point dataset (x_i, y_i) where x_i = i + \sin(i)/2 and y_i = i + \cos(i^2) for i = 0 \dots 9.

#include <config.h>
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_spline.h>

int
main (void)
{
  int i;
  double xi, yi, x[10], y[10];

  printf ("#m=0,S=2\n");

  for (i = 0; i < 10; i++)
    {
      x[i] = i + 0.5 * sin (i);
      y[i] = i + cos (i * i);
      printf ("%g %g\n", x[i], y[i]);
    }

  printf ("#m=1,S=0\n");

  {
    gsl_interp_accel *acc 
      = gsl_interp_accel_alloc ();
    gsl_spline *spline 
      = gsl_spline_alloc (gsl_interp_cspline, 10);

    gsl_spline_init (spline, x, y, 10);

    for (xi = x[0]; xi < x[9]; xi += 0.01)
      {
        double yi = gsl_spline_eval (spline, xi, acc);
        printf ("%g %g\n", xi, yi);
      }
    gsl_spline_free (spline);
    gsl_interp_accel_free(acc);
  }
  return 0;
}

The output is designed to be used with the GNU plotutils graph program,

$ ./a.out > interp.dat
$ graph -T ps < interp.dat > interp.ps

interpThe result shows a smooth interpolation of the original points. The interpolation method can changed simply by varying the first argument of gsl_spline_alloc.

References and Further Reading

Descriptions of the interpolation algorithms and further references can be found in the following book,

Numerical Differentiation

The functions described in this chapter compute numerical derivatives by finite differencing. An adaptive algorithm is used to find the best choice of finite difference and to estimate the error in the derivative. These functions are declared in the header file `gsl_diff.h'

Functions

Function: int gsl_diff_central (const gsl_function *f, double x, double *result, double *abserr)
This function computes the numerical derivative of the function f at the point x using an adaptive central difference algorithm. The derivative is returned in result and an estimate of its absolute error is returned in abserr.

Function: int gsl_diff_forward (const gsl_function *f, double x, double *result, double *abserr)
This function computes the numerical derivative of the function f at the point x using an adaptive forward difference algorithm. The function is evaluated only at points greater than x and at x itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. This function should be used if f(x) has a singularity or is undefined for values less than x.

Function: int gsl_diff_backward (const gsl_function *f, double x, double *result, double *abserr)
This function computes the numerical derivative of the function f at the point x using an adaptive backward difference algorithm. The function is evaluated only at points less than x and at x itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. This function should be used if f(x) has a singularity or is undefined for values greater than x.

Example

The following code estimates the derivative of the function f(x) = x^{3/2} at x=2 and at x=0. The function f(x) is undefined for x<0 so the derivative at x=0 is computed using gsl_diff_forward.

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_diff.h>

double f (double x, void * params)
{
  return pow (x, 1.5);
}

int
main (void)
{
  gsl_function F;
  double result, abserr;

  F.function = &f;
  F.params = 0;

  printf("f(x) = x^(3/2)\n");

  gsl_diff_central (&F, 2.0, &result, &abserr);
  printf("x = 2.0\n");
  printf("f'(x) = %.10f +/- %.5f\n", result, abserr);
  printf("exact = %.10f\n\n", 1.5 * sqrt(2.0));

  gsl_diff_forward (&F, 0.0, &result, &abserr);
  printf("x = 0.0\n");
  printf("f'(x) = %.10f +/- %.5f\n", result, abserr);
  printf("exact = %.10f\n", 0.0);

  return 0;
}

Here is the output of the program,

$ ./demo 
f(x) = x^(3/2)

x = 2.0
f'(x) = 2.1213203435 +/- 0.01490
exact = 2.1213203436

x = 0.0
f'(x) = 0.0012172897 +/- 0.05028
exact = 0.0000000000

References and Further Reading

The algorithms used by these functions are described in the following book,

Chebyshev Approximations

This chapter describes routines for computing Chebyshev approximations to univariate functions. A Chebyshev approximation is a truncation of the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials on the interval [-1,1] with the weight function @c{$1 / \sqrt{1-x^2}$} 1 / \sqrt{1-x^2}. The first few Chebyshev polynomials are, T_0(x) = 1, T_1(x) = x, T_2(x) = 2 x^2 - 1.

The functions described in this chapter are declared in the header file `gsl_chebyshev.h'.

The gsl_cheb_series struct

A Chebyshev series is stored using the following structure,

typedef struct
{
  double * c;   /* coefficients  c[0] .. c[order] */
  int order;    /* order of expansion             */
  double a;     /* lower interval point           */
  double b;     /* upper interval point           */
} gsl_cheb_struct

The approximation is made over the range [a,b] using order+1 terms, including the coefficient c[0].

Creation and Calculation of Chebyshev Series

Function: gsl_cheb_series * gsl_cheb_alloc (const size_t n)
This function allocates space for a Chebyshev series of order n and returns a pointer to a new gsl_cheb_series struct.

Function: void gsl_cheb_free (gsl_cheb_series * cs)
This function frees a previously allocated Chebyshev series cs.

Function: int gsl_cheb_init (gsl_cheb_series * cs, const gsl_function * f, const double a, const double b)
This function computes the Chebyshev approximation cs for the function f over the range (a,b) to the previously specified order. The computation of the Chebyshev approximation is an O(n^2) process, and requires n function evaluations.

Chebyshev Series Evaluation

Function: double gsl_cheb_eval (const gsl_cheb_series * cs, double x)
This function evaluates the Chebyshev series cs at a given point x.

Function: int gsl_cheb_eval_err (const gsl_cheb_series * cs, const double x, double * result, double * abserr)
This function computes the Chebyshev series cs at a given point x, estimating both the series result and its absolute error abserr. The error estimate is made from the first neglected term in the series.

Function: double gsl_cheb_eval_n (const gsl_cheb_series * cs, size_t order, double x)
This function evaluates the Chebyshev series cs at a given point n, to (at most) the given order order.

Function: int gsl_cheb_eval_n_err (const gsl_cheb_series * cs, const size_t order, const double x, double * result, double * abserr)
This function evaluates a Chebyshev series cs at a given point x, estimating both the series result and its absolute error abserr, to (at most) the given order order. The error estimate is made from the first neglected term in the series.

Derivatives and Integrals

The following functions allow a Chebyshev series to be differentiated or integrated, producing a new Chebyshev series. Note that the error estimate produced by evaluating the derivative series will be underestimated due to the contribution of higher order terms being neglected.

Function: int gsl_cheb_calc_deriv (gsl_cheb_series * deriv, const gsl_cheb_series * cs)
This function computes the derivative of the series cs, storing the derivative coefficients in the previously allocated deriv. The two series cs and deriv must have been allocated with the same order.

Function: int gsl_cheb_calc_integ (gsl_cheb_series * integ, const gsl_cheb_series * cs)
This function computes the integral of the series cs, storing the integral coefficients in the previously allocated integ. The two series cs and integ must have been allocated with the same order. The lower limit of the integration is taken to be the left hand end of the range a.

Examples

The following example program computes Chebyshev approximations to a step function. This is an extremely difficult approximation to make, due to the discontinuity, and was chosen as an example where approximation error is visible. For smooth functions the Chebyshev approximation converges extremely rapidly and errors would not be visible.

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_chebyshev.h>

double
f (double x, void *p)
{
  if (x < 0.5)
    return 0.25;
  else
    return 0.75;
}

int
main (void)
{
  int i, n = 10000; 

  gsl_cheb_series *cs = gsl_cheb_alloc (40);

  gsl_function F;

  F.function = f;
  F.params = 0;

  gsl_cheb_init (cs, &F, 0.0, 1.0);

  for (i = 0; i < n; i++)
    {
      double x = i / (double)n;
      double r10 = gsl_cheb_eval_n (cs, 10, x);
      double r40 = gsl_cheb_eval (cs, x);
      printf ("%g %g %g %g\n", 
              x, GSL_FN_EVAL (&F, x), r10, r40);
    }

  gsl_cheb_free (cs);

  return 0;
}

The output from the program gives the original function, 10-th order approximation and 40-th order approximation, all sampled at intervals of 0.001 in x.

cheb

References and Further Reading

The following paper describes the use of Chebyshev series,

Series Acceleration

The functions described in this chapter accelerate the convergence of a series using the Levin u-transform. This method takes a small number of terms from the start of a series and uses a systematic approximation to compute an extrapolated value and an estimate of its error. The u-transform works for both convergent and divergent series, including asymptotic series.

These functions are declared in the header file `gsl_sum.h'.

Acceleration functions

The following functions compute the full Levin u-transform of a series with its error estimate. The error estimate is computed by propagating rounding errors from each term through to the final extrapolation.

These functions are intended for summing analytic series where each term is known to high accuracy, and the rounding errors are assumed to originate from finite precision. They are taken to be relative errors of order GSL_DBL_EPSILON for each term.

The calculation of the error in the extrapolated value is an O(N^2) process, which is expensive in time and memory. A faster but less reliable method which estimates the error from the convergence of the extrapolated value is described in the next section For the method described here a full table of intermediate values and derivatives through to O(N) must be computed and stored, but this does give a reliable error estimate. .

Function: gsl_sum_levin_u_workspace * gsl_sum_levin_u_alloc (size_t n)
This function allocates a workspace for a Levin u-transform of n terms. The size of the workspace is O(2n^2 + 3n).

Function: int gsl_sum_levin_u_free (gsl_sum_levin_u_workspace * w)
This function frees the memory associated with the workspace w.

Function: int gsl_sum_levin_u_accel (const double * array, size_t array_size, gsl_sum_levin_u_workspace * w, double * sum_accel, double * abserr)
This function takes the terms of a series in array of size array_size and computes the extrapolated limit of the series using a Levin u-transform. Additional working space must be provided in w. The extrapolated sum is stored in sum_accel, with an estimate of the absolute error stored in abserr. The actual term-by-term sum is returned in w->sum_plain. The algorithm calculates the truncation error (the difference between two successive extrapolations) and round-off error (propagated from the individual terms) to choose an optimal number of terms for the extrapolation.

Acceleration functions without error estimation

The functions described in this section compute the Levin u-transform of series and attempt to estimate the error from the "truncation error" in the extrapolation, the difference between the final two approximations. Using this method avoids the need to compute an intermediate table of derivatives because the error is estimated from the behavior of the extrapolated value itself. Consequently this algorithm is an O(N) process and only requires O(N) terms of storage. If the series converges sufficiently fast then this procedure can be acceptable. It is appropriate to use this method when there is a need to compute many extrapolations of series with similar converge properties at high-speed. For example, when numerically integrating a function defined by a parameterized series where the parameter varies only slightly. A reliable error estimate should be computed first using the full algorithm described above in order to verify the consistency of the results.

Function: gsl_sum_levin_utrunc_workspace * gsl_sum_levin_utrunc_alloc (size_t n)
This function allocates a workspace for a Levin u-transform of n terms, without error estimation. The size of the workspace is O(3n).

Function: int gsl_sum_levin_utrunc_free (gsl_sum_levin_utrunc_workspace * w)
This function frees the memory associated with the workspace w.

Function: int gsl_sum_levin_utrunc_accel (const double * array, size_t array_size, gsl_sum_levin_utrunc_workspace * w, double * sum_accel, double * abserr_trunc)
This function takes the terms of a series in array of size array_size and computes the extrapolated limit of the series using a Levin u-transform. Additional working space must be provided in w. The extrapolated sum is stored in sum_accel. The actual term-by-term sum is returned in w->sum_plain. The algorithm terminates when the difference between two successive extrapolations reaches a minimum or is sufficiently small. The difference between these two values is used as estimate of the error and is stored in abserr_trunc. To improve the reliability of the algorithm the extrapolated values are replaced by moving averages when calculating the truncation error, smoothing out any fluctuations.

Example of accelerating a series

The following code calculates an estimate of \zeta(2) = \pi^2 / 6 using the series,

After N terms the error in the sum is O(1/N), making direct summation of the series converge slowly.

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_sum.h>

#define N 20

int
main (void)
{
  double t[N];
  double sum_accel, err;
  double sum = 0;
  int n;
  
  gsl_sum_levin_u_workspace * w 
    = gsl_sum_levin_u_alloc (N);

  const double zeta_2 = M_PI * M_PI / 6.0;
  
  /* terms for zeta(2) = \sum_{n=0}^{\infty} 1/n^2 */

  for (n = 0; n < N; n++)
    {
      double np1 = n + 1.0;
      t[n] = 1.0 / (np1 * np1);
      sum += t[n];
    }
  
  gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);

  printf("term-by-term sum = % .16f using %d terms\n", 
         sum, N);

  printf("term-by-term sum = % .16f using %d terms\n", 
         w->sum_plain, w->terms_used);

  printf("exact value      = % .16f\n", zeta_2);
  printf("accelerated sum  = % .16f using %d terms\n", 
         sum_accel, w->terms_used);

  printf("estimated error  = % .16f\n", err);
  printf("actual error     = % .16f\n", 
         sum_accel - zeta_2);

  gsl_sum_levin_u_free (w);
  return 0;
}

The output below shows that the Levin u-transform is able to obtain an estimate of the sum to 1 part in 10^10 using the first eleven terms of the series. The error estimate returned by the function is also accurate, giving the correct number of significant digits.

bash$ ./a.out 
term-by-term sum =  1.5961632439130233 using 20 terms
term-by-term sum =  1.5759958390005426 using 13 terms
exact value      =  1.6449340668482264
accelerated sum  =  1.6449340668166479 using 13 terms
estimated error  =  0.0000000000508580
actual error     = -0.0000000000315785

Note that a direct summation of this series would require 10^10 terms to achieve the same precision as the accelerated sum does in 13 terms.

References and Further Reading

The algorithms used by these functions are described in the following papers,

The theory of the u-transform was presented by Levin,

A review paper on the Levin Transform is available online,

Discrete Hankel Transforms

This chapter describes functions for performing Discrete Hankel Transforms (DHTs). The functions are declared in the header file `gsl_dht.h'.

Definitions

The discrete Hankel transform acts on a vector of sampled data, where the samples are assumed to have been taken at points related to the zeroes of a Bessel function of fixed order; compare this to the case of the discrete Fourier transform, where samples are taken at points related to the zeroes of the sine or cosine function.

Specifically, let f(t) be a function on the unit interval. Then the finite \nu-Hankel transform of f(t) is defined to be the set of numbers g_m given by

so that

Suppose that f is band-limited in the sense that g_m=0 for m > M. Then we have the following fundamental sampling theorem.

It is this discrete expression which defines the discrete Hankel transform. The kernel in the summation above defines the matrix of the \nu-Hankel transform of size M-1. The coefficients of this matrix, being dependent on \nu and M, must be precomputed and stored; the gsl_dht object encapsulates this data. The allocation function gsl_dht_alloc returns a gsl_dht object which must be properly initialized with gsl_dht_init before it can be used to perform transforms on data sample vectors, for fixed \nu and M, using the gsl_dht_apply function. The implementation allows a scaling of the fundamental interval, for convenience, so that one take assume the function is defined on the interval [0,X], rather than the unit interval.

Notice that by assumption f(t) vanishes at the endpoints of the interval, consistent with the inversion formula and the sampling formula given above. Therefore, this transform corresponds to an orthogonal expansion in eigenfunctions of the Dirichlet problem for the Bessel differential equation.

Functions

Function: gsl_dht * gsl_dht_alloc (size_t size)
This function allocates a Discrete Hankel transform object of size size.

Function: int gsl_dht_init (gsl_dht * t, double nu, double xmax)
This function initializes the transform t for the given values of nu and x.

Function: gsl_dht * gsl_dht_new (size_t size, double nu, double xmax)
This function allocates a Discrete Hankel transform object of size size and initializes it for the given values of nu and x.

Function: void gsl_dht_free (gsl_dht * t)
This function frees the transform t.

Function: int gsl_dht_apply (const gsl_dht * t, double * f_in, double * f_out)
This function applies the transform t to the array f_in whose size is equal to the size of the transform. The result is stored in the array f_out which must be of the same length.

Function: double gsl_dht_x_sample (const gsl_dht * t, int n)
This function returns the value of the n'th sample point in the unit interval, (j_{\nu,n+1}/j_{\nu,M}) X. These are the points where the function f(t) is assumed to be sampled.

Function: double gsl_dht_k_sample (const gsl_dht * t, int n)
This function returns the value of the n'th sample point in "k-space", j_{\nu,n+1}/X.

References and Further Reading

The algorithms used by these functions are described in the following papers,

One dimensional Root-Finding

この章では任意の1次元関数の解を探索するルーチンについて述べる. このライ ブラリは様々な反復的解探査と収束テストの低レベルのコンポーネントを実装し ている. これらをユーザーが組みあわせて, 反復の途中段階すべてを見て望まし い解を得ることができる. メソッドの各クラスで同じ枠組が使われているので, プログラムを再コンパイルせずに実行時に解法を切り替えることができる. 解法 の各インスタンスで状態を保持しているので, マルチスレッドプログラムでも使 うことができる.

ヘッダファイル`gsl_roots.h'に解探査関数のプロトタイプと関連する定義 がある.

Overview

1次元解探査アルゴリズムは2つのクラスに分けることができる. 解の囲い 込み解の洗練である. 解を囲い込むアルゴリズムは収束を保証できる. 囲い込みアルゴリズムは解のあることを知っている区間が出発点となる. 区間の 幅は反復的に縮小され, 望む許容誤差以下に区間幅が縮小するまで探査が続く. これは解の位置の誤差評価が厳密に行える利点がある.

解の洗練の技法は, 解の初期推測値を改良しようとする. このアルゴリズ ムは解に「十分近い」出発点を与えたときのみ収束する. またスピードのため厳 密な誤差評価を犠牲にする. 解近傍で関数の挙動を近似することで, 初期値の高 次の改良を試みる. 関数の挙動がアルゴリズムのものに近く, 初期値がよく推定 できる時には洗練アルゴリズムは素早く収束する.

GSL ではどちらのタイプのアルゴリズムも同じ枠組みに収めた. ユーザーはアル ゴリズムの高レベルドライバを実装し, ライブラリは各ステップに必要な個々の 関数を実装する. 反復には3つのメインフェーズが存在する:

囲い込み法の状態はgsl_root_fsolver構造体に保持される. 更新プロシー ジャは関数値のみ用いる(導関数値は用いない). 洗練法の状態は gsl_root_fdfsolver構造体に保持される. 更新には関数とその導関数が 必要となる(このためfdfという名前になっている).

Caveats

解探査関数は1度に1つの解しか探査しない. 探査領域内にいくつかの解があると き, 始めに見付かった解を返す. しかしどの解が見付かったのかを調べるのは難 しい. 多くの場合, 複数の解がある領域で解を探している場合にも何も警 告されない.

重解を持つ関数の取り扱いには注意を要する(たとえば のような). 偶数次の重解に対しては囲い込み法は使えない. これらの方法では, 初期範囲は0をよぎらなければならない. つまり両端点で片方は正, もう片方は 負の値をとらなければならない. 偶数次の重解は0をよぎらず, 接するだけであ る. 囲い込み法は奇数次の重解(3次, 5次...)では機能する. 解の洗練法は 一般的に高次の重解を探査することができるが, 収束が悪くなる. この場合収束 の加速にSteffensonアルゴリズムを用いるとよい.

探査領域に が解を持つことは絶対必要というわけではないが, 解の存在を調べるの に数値解探査関数を使うべきではない. もっとよい方法がある. 数値解探査が間 違う可能性があるのだから, よく知らない関数を丸投げするのはよくない. 一般 的に解を探す前にグラフを描いてチェックするのがよいだろう.

Initializing the Solver

Function: gsl_root_fsolver * gsl_root_fsolver_alloc (const gsl_root_fsolver_type * T)

この関数はタイプTのソルバのインスタンスを新しく割りあててポインタ を返す. 例えば, 次のコードはbisectionソルバのインスタンスを作成する:

const gsl_root_fsolver_type * T 
  = gsl_root_fsolver_bisection;
gsl_root_fsolver * s 
  = gsl_root_fsolver_alloc (T);

ソルバを作るのに十分なメモリがない場合, 関数はヌルポインタを返し, エラー コードGSL_ENOMEMによりエラーハンドラが呼びだされる.

Function: gsl_root_fdfsolver * gsl_root_fdfsolver_alloc (const gsl_root_fdfsolver_type * T)

この関数はタイプTの導関数ベースのソルバのインスタンスを新しく割り あててポインタを返す. 例えば, 次のコードはNewton-Raphsonソルバのインスタ ンスを作成する:

const gsl_root_fdfsolver_type * T 
  = gsl_root_fdfsolver_newton;
gsl_root_fdfsolver * s 
  = gsl_root_fdfsolver_alloc (T);

ソルバを作るのに十分なメモリがない場合, 関数はヌルポインタを返し, エラー コードGSL_ENOMEMによりエラーハンドラが呼びだされる.

Function: int gsl_root_fsolver_set (gsl_root_fsolver * s, gsl_function * f, double x_lower, double x_upper)

この関数は存在するソルバsが関数fの解を初期探査範囲 [x_lower, x_upper]で探査するよう(再)初期化する.

Function: int gsl_root_fdfsolver_set (gsl_root_fdfsolver * s, gsl_function_fdf * fdf, double root)

この関数は存在するソルバsが関数とその微分fdfおよび初期値 rootで探査するように(再)初期化する.

Function: void gsl_root_fsolver_free (gsl_root_fsolver * s)
Function: void gsl_root_fdfsolver_free (gsl_root_fdfsolver * s)

この関数はソルバsに割りあてられたメモリを解放する.

Function: const char * gsl_root_fsolver_name (const gsl_root_fsolver * s)
Function: const char * gsl_root_fdfsolver_name (const gsl_root_fdfsolver * s)

これらの関数はソルバの名前のポインタを返す. 例えば

printf("s is a '%s' solver\n",
       gsl_root_fsolver_name (s)) ;

は, s is a 'bisection' solverのような出力を返す.

Providing the function to solve

解探査をさせるため, 1変数の連続関数, および場合によりその導関数を与える 必要がある. 一般のパラメータを許すため, 関数は次のデータ型で定義されてい る必要がある:

Data Type: gsl_function
このデータ型はパラメータをもつ一般の関数を定義する.
double (* function) (double x, void * params)
この関数は, 引数x, パラメータparamsの値 を返す.
void * params
関数のパラメータのポインタ.

一般の2次関数を例にあげる:

ここで だとしよう. 次のコードは解探査が可能な関数gsl_function Fを 定義する:

struct my_f_params { double a; double b; double c; } ;

double
my_f (double x, void * p) {
   struct my_f_params * params
     = (struct my_f_params *)p;
   double a = (params->a);
   double b = (params->b);
   double c = (params->c);

   return  (a * x + b) * x + c;
}

gsl_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };

F.function = &my_f;
F.params = &params;

関数 は次のマクロにより評価できる:

#define GSL_FN_EVAL(F,x)
    (*((F)->function))(x,(F)->params)

Data Type: gsl_function_fdf
このデータ型でパラメータをもつ一般の関数およびその導関数を定義する.
double (* f) (double x, void * params)
この関数は引数x, パラメータparamsの値 を返す.
double (* df) (double x, void * params)
この関数は引数x, パラメータparamsで, 関数fxに 関する微分値 を返す.
void (* fdf) (double x, void * params, double * f, double * df)
この関数は, 引数x, パラメータparamsでの関数fの値を に, そして導関数dfの値を で求めることを宣言する. この関数は別々の関数 , を最適化する--関数値とその導関数値を同時に求めることにより高速化を図って いる.
void * params
関数のパラメータへのポインタ.

次の関数を例に挙げる:

double
my_f (double x, void * params)
{
   return exp (2 * x);
}

double
my_df (double x, void * params)
{
   return 2 * exp (2 * x);
}

void
my_fdf (double x, void * params,
        double * f, double * df)
{
   double t = exp (2 * x);

   *f = t;
   *df = 2 * t;   /* computing using existing values */
}

gsl_function_fdf FDF;

FDF.f = &my_f;
FDF.df = &my_df;
FDF.fdf = &my_fdf;
FDF.params = 0;

関数 は次のマクロで評価できる:

#define GSL_FN_FDF_EVAL_F(FDF,x) 
     (*((FDF)->f))(x,(FDF)->params)

導関数 は次のマクロで評価できる:

#define GSL_FN_FDF_EVAL_DF(FDF,x) 
     (*((FDF)->df))(x,(FDF)->params)

そして関数値 および導関数値 を同時に求めるには次のマクロを使う:

#define GSL_FN_FDF_EVAL_F_DF(FDF,x,y,dy) 
     (*((FDF)->fdf))(x,(FDF)->params,(y),(dy))

このマクロは yに, そして dyに格納する---これらはともにdouble型のポインタである.

Search Bounds and Guesses

ユーザーは初期範囲か初期値を与えることになる. この章では探査範囲や推定値 がどのように機能し, 関数の引数がどのようにそれらを制御するのかを解説する.

推定値は単にxの値である. 解に対して許容範囲内に近づくまで反復され る. double型をもつ.

探査範囲は区間の両端点である. 区間幅が要求精度内になるまで反復される. 区 間は上限と下限を示す2つの値で指定される. 端点を区間に含むかどうかは区間 の使われる状況による.

Iteration

以下の関数は各アルゴリズムでの反復を制御する. 各関数は相当する型のソルバ の状態を更新する反復を1回行う. 同じ関数であらゆるソルバに対応するため, コードを変更することなく実行時にメソッドを変更することができる.

Function: int gsl_root_fsolver_iterate (gsl_root_fsolver * s)
Function: int gsl_root_fdfsolver_iterate (gsl_root_fdfsolver * s)
これらの関数はソルバsの反復を1回行う. 予期しない問題が生じた場合は エラーコードが返される.
GSL_EBADFUNC
関数値や導関数値がInfまたはNaNとなる特異点が発生した.
GSL_EZERODIV
導関数値が反復点で消滅し, 0で割る操作が発生しそうになったため強制終了し た.

ソルバはその時点での解の最も良い推定値を保持している. 囲い込み法ではその 時点で最も良い解区間を保持している. この情報は次の関数で参照することがで きる:

Function: double gsl_root_fsolver_root (const gsl_root_fsolver * s)
Function: double gsl_root_fdfsolver_root (const gsl_root_fdfsolver * s)
これらの関数はソルバsのもつ現時点での解の推定値を返す.

Function: double gsl_root_fsolver_x_lower (const gsl_root_fsolver * s)
Function: double gsl_root_fsolver_x_upper (const gsl_root_fsolver * s)
これらの関数はソルバsのもつ現時点での解区間を返す.

Search Stopping Parameters

解探査プロシージャは以下の条件が真になったときに停止する:

これらの条件の処理はユーザーに任せられている. 以下の関数はユーザーが標準 的な方法により現在の結果の精度を調べるために用意されている.

Function: int gsl_root_test_interval (double x_lower, double x_upper, double epsrel, double epsabs)
この関数は区間[x_lower, x_upper]の収束を絶対誤差epsabs および相対誤差epsrelで調べるものである. 以下の条件が満たされた場合 にはGSL_SUCCESSを返す.

ただし, 区間 が原点を含んでいないことが必要である. 原点を含んでいる場合には は(区間の最小値である)0に置きかえられる. これは原点近くの解での相対誤差 の正確な評価に必要な操作である.

区間に対するこの条件により, 区間内の推定値rが真の解 r* に対して同様の条件を満たすことを要請する:

ただし区間内に真の解が存在すると仮定している.

Function: int gsl_root_test_delta (double x1, double x0, double epsrel, double epsabs)
この関数は数列..., x0, x1 が絶対誤差epsabsおよび相 対誤差epsrelで収束しているかどうかを調べる. 以下の条件が満たされて いる場合にGSL_SUCCESSを返す:

満たされていない場合にはGSL_CONTINUEを返す.

Function: int gsl_root_test_residual (double f, double epsabs)
この関数は絶対誤差epsabsに対する残差fを調べる. 以下の条件が 満たされたときはGSL_SUCCESSを返す:

満たされない場合はGSL_CONTINUEを返す. この条件は残差|f(x)| が十分小さいため, 真の解xの正確な場所が重要でなくなる場合には適し ている.

Root Bracketing Algorithms

この章で説明する解の囲い込み法は解が含まれていることを知っている初期範囲 が必要となる--abが範囲の端点であれば, f(a)f(b)は符号が異なる必要がある. つまり与えられた区間で少なくとも1回 は0をよぎる必要がある. 妥当な初期範囲が与えられれば, 関数の振舞いがよけ ればこれらのアルゴリズムは失敗することはない.

囲い込み法では偶数次の重解は見つけることができない. x軸に接するか らである.

Solver: gsl_root_fsolver_bisection

2分法は囲い込み法のうち最も簡単な方法である. 線型収束を示し, ライ ブラリ中で最も遅く収束するアルゴリズムである.

各段で, 区間は2等分され, 中点での関数値が求められる. この値の符号により どちらの区間に解が含まれるかを決定する. 区間の半分が捨てられ残る半分が新 しい推定区間となる. このプロシージャが, 区間幅が十分小さくなるまで繰り返 される.

どの時点でも, 解の推定値は中点の値である.

roots-bisection

2分法を4回繰り返した. a_nは区間始点のn番目の位置, b_nは区間終点のn番目の位置. 各段での中点の位置も示してある.

Solver: gsl_root_fsolver_falsepos

false position アルゴリズムは線型補完による解探査法である. 収束は 線型だが, 一般に2分法より早い.

各段で, 端点(a,f(a)), (b,f(b))を結ぶ直線が引かれ, x 軸との交点を「中点」とする. この点での関数値が計算され, その符号でどちら の領域を採用するかを決定し, 片方を捨て残りを次の領域とする. このプロシー ジャが区間が十分に小さくなるまで繰り返される.

現在の推定値は線型補完により求められる.

Solver: gsl_root_fsolver_brent

Brent-Dekker法(Brent法と略すことにする)は2分法アルゴリズムと 内挿を組みあわせたものである. この方法は荒っぽいが早いアルゴリズムである.

各段でBrent法は内挿曲線で関数を近似する. 初段では2端点の直線近似だが, 段 が進むにつれ 最新の3点について逆2乗フィットを行う. 内挿曲線とx軸 の交点を解の推定値とする. 推定値が現在の推定区間にある場合は内挿点を用い, 区間を縮小する. 内挿点が求められない時は通常の2分法に切りかえる.

最新の推定値は最新の内挿もしくは2分法の値である.

Root Finding Algorithms using Derivatives

この章で説明する解の洗練法は解の位置をあらかじめ推定しておく必要がある. 収束するという絶対的な保証はない--関数はこの手法に適した形をしていなくて はならず, 初期値は真の解に十分近くなければならない. 条件が満たされたとき には収束は2次である.

これらのアルゴリズムは関数とその導関数が必要となる.

Derivative Solver: gsl_root_fdfsolver_newton

Newton法は標準的な解洗練アルゴリズムである. このアルゴリズムは解の位置の 初期推定値から始まる. 各段で, 関数fの接線がひかれる. 接線と x軸の交点が新しい推定値となる. 繰り返しは次の数列で定義される:

Newton法は単一の解に対しては2次で, 複数の解には線型で収束する.

roots-newtons-method

Newton法での数回の繰り返し. g_nn番目の推定値である.

Derivative Solver: gsl_root_fdfsolver_secant

割線法はNewton法を単純化したもので, 導関数を毎回求める必要がない.

初段ではアルゴリズムはNewton法から始め, 導関数を用いる.

続く繰り返しでは, 微分値を求める代わりに数値的な推定値で代用する. 前の2 点の傾きを使うことになる:

導関数が解の近傍でさほど変化しないときには, 割線法はかなりの省力化を図る ことができる. 導関数の評価が関数自身の評価よりコストが0.44倍以上かかるの であれば, 割線法はニュートン法より早い. 数値微分の計算と同様, 2点の間隔 が小さくなりすぎると桁落ちが生じる.

単一の解であれば, 収束は (約1.62)次である. 複数の解では線型で収束する.

Derivative Solver: gsl_root_fdfsolver_steffenson

Steffenson法はこれらルーチンの中で最も早い収束を示す. これは基本的 なNewton法にAitkenの「デルタ2乗」加速を組みあわせたものである. Newton法 の各段をx_iとしたとき, 加速により新しい数列R_iを生成する:

これは合理的な条件のもとで元の数列よりも早く収束する. 加速数列を生成する には最低3項必要である. 初段では元のNewton法の結果を返す. 加速項の分母が0 になる場合もNewton法の結果を返す.

加速プロシージャを用いているため, 関数の振舞いが悪い場合にはこの方法は不 安定になる.

Examples

どの解探査法であっても, 解くべき方程式を用意する必要がある. この例では, 前に示した一般の2次方程式を解くことにしよう. まず関数のパラメータを定義 するためにヘッダファイル(`demo_fn.h')が必要である.

struct quadratic_params
  {
    double a, b, c;
  };

double quadratic (double x, void *params);
double quadratic_deriv (double x, void *params);
void quadratic_fdf (double x, void *params, 
                    double *y, double *dy);

関数の定義は別のファイル(`demo_fn.c')で行うことにしよう.

double
quadratic (double x, void *params)
{
  struct quadratic_params *p 
    = (struct quadratic_params *) params;

  double a = p->a;
  double b = p->b;
  double c = p->c;

  return (a * x + b) * x + c;
}

double
quadratic_deriv (double x, void *params)
{
  struct quadratic_params *p 
    = (struct quadratic_params *) params;

  double a = p->a;
  double b = p->b;
  double c = p->c;

  return 2.0 * a * x + b;
}

void
quadratic_fdf (double x, void *params,
               double *y, double *dy)
{
  struct quadratic_params *p 
    = (struct quadratic_params *) params;

  double a = p->a;
  double b = p->b;
  double c = p->c;

  *y = (a * x + b) * x + c;
  *dy = 2.0 * a * x + b;
}

最初のプログラムはBrent法のソルバgsl_root_fsolver_brentを用い, 次 の方程式を解くものである.

この解は である.

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_roots.h>

#include "demo_fn.h"
#include "demo_fn.c"

int
main (void)
{
  int status;
  int iter = 0, max_iter = 100;
  const gsl_root_fsolver_type *T;
  gsl_root_fsolver *s;
  double r = 0, r_expected = sqrt (5.0);
  double x_lo = 0.0, x_hi = 5.0;
  gsl_function F;
  struct quadratic_params params = {1.0, 0.0, -5.0};

  F.function = &quadratic;
  F.params = &params;

  T = gsl_root_fsolver_brent;
  s = gsl_root_fsolver_alloc (T);
  gsl_root_fsolver_set (s, &F, x_lo, x_hi);

  printf ("using %s method\n",
          gsl_root_fsolver_name (s));

  printf ("%5s [%9s, %9s] %9s %10s %9s\n",
          "iter", "lower", "upper", "root",
          "err", "err(est)");

  do
    {
      iter++;
      status = gsl_root_fsolver_iterate (s);
      r = gsl_root_fsolver_root (s);
      x_lo = gsl_root_fsolver_x_lower (s);
      x_hi = gsl_root_fsolver_x_upper (s);
      status = gsl_root_test_interval (x_lo, x_hi,
                                       0, 0.001);

      if (status == GSL_SUCCESS)
        printf ("Converged:\n");

      printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
              iter, x_lo, x_hi,
              r, r - r_expected, 
              x_hi - x_lo);
    }
  while (status == GSL_CONTINUE && iter < max_iter);
  return status;
}

以下に結果を挙げる:

bash$ ./a.out 
using brent method
 iter [    lower,     upper]      root        err  err(est)
    1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
    2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
    3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
    4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
    5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
Converged:                            
    6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666

Brent法ではなく2分法を使うようにする場合は, gsl_root_fsolver_brentgsl_root_fsolver_bisectionに変更す ればよい. この場合, 収束が遅くなる:

bash$ ./a.out 
using bisection method
 iter [    lower,     upper]      root        err  err(est)
    1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
    2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
    3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
    4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
    5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
    6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
    7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
    8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
    9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
   10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
   11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
Converged:                            
   12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207

次のプログラムは同じ関数を導関数も用いて解を探査するものである.

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_roots.h>

#include "demo_fn.h"
#include "demo_fn.c"

int
main (void)
{
  int status;
  int iter = 0, max_iter = 100;
  const gsl_root_fdfsolver_type *T;
  gsl_root_fdfsolver *s;
  double x0, x = 5.0, r_expected = sqrt (5.0);
  gsl_function_fdf FDF;
  struct quadratic_params params = {1.0, 0.0, -5.0};

  FDF.f = &quadratic;
  FDF.df = &quadratic_deriv;
  FDF.fdf = &quadratic_fdf;
  FDF.params = &params;

  T = gsl_root_fdfsolver_newton;
  s = gsl_root_fdfsolver_alloc (T);
  gsl_root_fdfsolver_set (s, &FDF, x);

  printf ("using %s method\n",
          gsl_root_fdfsolver_name (s));

  printf ("%-5s %10s %10s %10s\n",
          "iter", "root", "err", "err(est)");
  do
    {
      iter++;
      status = gsl_root_fdfsolver_iterate (s);
      x0 = x;
      x = gsl_root_fdfsolver_root (s);
      status = gsl_root_test_delta (x, x0, 0, 1e-3);

      if (status == GSL_SUCCESS)
        printf ("Converged:\n");

      printf ("%5d %10.7f %+10.7f %10.7f\n",
              iter, x, x - r_expected, x - x0);
    }
  while (status == GSL_CONTINUE && iter < max_iter);
  return status;
}

Newton法の結果を以下に挙げる:

bash$ ./a.out 
using newton method
iter        root        err   err(est)
    1  3.0000000 +0.7639320 -2.0000000
    2  2.3333333 +0.0972654 -0.6666667
    3  2.2380952 +0.0020273 -0.0952381
Converged:      
    4  2.2360689 +0.0000009 -0.0020263

誤差は前の段との差を取るより次の段との差をとる方がより正確となる. gsl_root_fdfsolver_newtongsl_root_fdfsolver_secantgsl_root_fdfsolver_steffensonに変えることで他の方法に切りかえるこ とができる.

References and Further Reading

Brent-Dekkerアルゴリズムについては以下の2つの論文を参照すること.

One dimensional Minimization

This chapter describes routines for finding minima of arbitrary one-dimensional functions. The library provides low level components for a variety of iterative minimizers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the algorithms. Each class of methods uses the same framework, so that you can switch between minimizers at runtime without needing to recompile your program. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multi-threaded programs.

The header file `gsl_min.h' contains prototypes for the minimization functions and related declarations. To use the minimization algorithms to find the maximum of a function simply invert its sign.

Overview

The minimization algorithms begin with a bounded region known to contain a minimum. The region is described by an lower bound a and an upper bound b, with an estimate of the minimum x.

min-interval}

The value of the function at x must be less than the value of the function at the ends of the interval,

This condition guarantees that a minimum is contained somewhere within the interval. On each iteration a new point x' is selected using one of the available algorithms. If the new point is a better estimate of the minimum, f(x') < f(x), then the current estimate of the minimum x is updated. The new point also allows the size of the bounded interval to be reduced, by choosing the most compact set of points which satisfies the constraint f(a) > f(x) < f(b). The interval is reduced until it encloses the true minimum to a desired tolerance. This provides a best estimate of the location of the minimum and a rigorous error estimate.

Several bracketing algorithms are available within a single framework. The user provides a high-level driver for the algorithm, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

The state for the minimizers is held in a gsl_min_fminimizer struct. The updating procedure uses only function evaluations (not derivatives).

Caveats

Note that minimization functions can only search for one minimum at a time. When there are several minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima this will be. In most cases, no error will be reported if you try to find a minimum in an area where there is more than one.

With all minimization algorithms it can be difficult to determine the location of the minimum to full numerical precision. The behavior of the function in the region of the minimum x^* can be approximated by a Taylor expansion,

and the second term of this expansion can be lost when added to the first term at finite precision. This magnifies the error in locating x^*, making it proportional to \sqrt \epsilon (where \epsilon is the relative accuracy of the floating point numbers). For functions with higher order minima, such as x^4, the magnification of the error is correspondingly worse. The best that can be achieved is to converge to the limit of numerical accuracy in the function values, rather than the location of the minimum itself.

Initializing the Minimizer

Function: gsl_min_fminimizer * gsl_min_fminimizer_alloc (const gsl_min_fminimizer_type * T)
This function returns a pointer to a a newly allocated instance of a minimizer of type T. For example, the following code creates an instance of a golden section minimizer,
const gsl_min_fminimizer_type * T 
  = gsl_min_fminimizer_goldensection;
gsl_min_fminimizer * s 
  = gsl_min_fminimizer_alloc (T);

If there is insufficient memory to create the minimizer then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

Function: int gsl_min_fminimizer_set (gsl_min_fminimizer * s, gsl_function * f, double minimum, double x_lower, double x_upper)
This function sets, or resets, an existing minimizer s to use the function f and the initial search interval [x_lower, x_upper], with a guess for the location of the minimum minimum.

If the interval given does not contain a minimum, then the function returns an error code of GSL_FAILURE.

Function: int gsl_min_fminimizer_set_with_values (gsl_min_fminimizer * s, gsl_function * f, double minimum, double f_minimum, double x_lower, double f_lower, double x_upper, double f_upper)
This function is equivalent to gsl_min_fminimizer_set but uses the values f_minimum, f_lower and f_upper instead of computing f(minimum), f(x_lower) and f(x_upper).

Function: void gsl_min_fminimizer_free (gsl_min_fminimizer * s)
This function frees all the memory associated with the minimizer s.

Function: const char * gsl_min_fminimizer_name (const gsl_min_fminimizer * s)
This function returns a pointer to the name of the minimizer. For example,
printf("s is a '%s' minimizer\n",
       gsl_min_fminimizer_name (s));

would print something like s is a 'brent' minimizer.

Providing the function to minimize

You must provide a continuous function of one variable for the minimizers to operate on. In order to allow for general parameters the functions are defined by a gsl_function data type (see section Providing the function to solve).

Iteration

The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any minimizer of the corresponding type. The same functions work for all minimizers so that different methods can be substituted at runtime without modifications to the code.

Function: int gsl_min_fminimizer_iterate (gsl_min_fminimizer * s)
This function performs a single iteration of the minimizer s. If the iteration encounters an unexpected problem then an error code will be returned,
GSL_EBADFUNC
the iteration encountered a singular point where the function evaluated to Inf or NaN.
GSL_FAILURE
the algorithm could not improve the current best approximation or bounding interval.

The minimizer maintains a current best estimate of the minimum at all times, and the current interval bounding the minimum. This information can be accessed with the following auxiliary functions,

Function: double gsl_min_fminimizer_minimum (const gsl_min_fminimizer * s)
This function returns the current estimate of the minimum for the minimizer s.

Function: double gsl_interval gsl_min_fminimizer_x_upper (const gsl_min_fminimizer * s)
Function: double gsl_interval gsl_min_fminimizer_x_lower (const gsl_min_fminimizer * s)
These functions return the current upper and lower bound of the interval for the minimizer s.

Stopping Parameters

A minimization procedure should stop when one of the following conditions is true:

The handling of these conditions is under user control. The function below allows the user to test the precision of the current result.

Function: int gsl_min_test_interval (double x_lower, double x_upper, double epsrel, double epsabs)
This function tests for the convergence of the interval [x_lower, x_upper] with absolute error epsabs and relative error epsrel. The test returns GSL_SUCCESS if the following condition is achieved,

when the interval x = [a,b] does not include the origin. If the interval includes the origin then \min(|a|,|b|) is replaced by zero (which is the minimum value of |x| over the interval). This ensures that the relative error is accurately estimated for minima close to the origin.

This condition on the interval also implies that any estimate of the minimum x_m in the interval satisfies the same condition with respect to the true minimum x_m^*,

assuming that the true minimum x_m^* is contained within the interval.

Minimization Algorithms

The minimization algorithms described in this section require an initial interval which is guaranteed to contain a minimum -- if a and b are the endpoints of the interval and x is an estimate of the minimum then f(a) > f(x) < f(b). This ensures that the function has at least one minimum somewhere in the interval. If a valid initial interval is used then these algorithm cannot fail, provided the function is well-behaved.

Minimizer: gsl_min_fminimizer_goldensection

The golden section algorithm is the simplest method of bracketing the minimum of a function. It is the slowest algorithm provided by the library, with linear convergence.

On each iteration, the algorithm first compares the subintervals from the endpoints to the current minimum. The larger subinterval is divided in a golden section (using the famous ratio (3-\sqrt 5)/2 = 0.3189660...) and the value of the function at this new point is calculated. The new value is used with the constraint f(a') > f(x') < f(b') to a select new interval containing the minimum, by discarding the least useful point. This procedure can be continued indefinitely until the interval is sufficiently small. Choosing the golden section as the bisection ratio can be shown to provide the fastest convergence for this type of algorithm.

Minimizer: gsl_min_fminimizer_brent

The Brent minimization algorithm combines a parabolic interpolation with the golden section algorithm. This produces a fast algorithm which is still robust.

The outline of the algorithm can be summarized as follows: on each iteration Brent's method approximates the function using an interpolating parabola through three existing points. The minimum of the parabola is taken as a guess for the minimum. If it lies within the bounds of the current interval then the interpolating point is accepted, and used to generate a smaller interval. If the interpolating point is not accepted then the algorithm falls back to an ordinary golden section step. The full details of Brent's method include some additional checks to improve convergence.

Examples

The following program uses the Brent algorithm to find the minimum of the function f(x) = \cos(x) + 1, which occurs at x = \pi. The starting interval is (0,6), with an initial guess for the minimum of 2.

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_min.h>

double fn1 (double x, void * params)
{
  return cos(x) + 1.0;
}

int
main (void)
{
  int status;
  int iter = 0, max_iter = 100;
  const gsl_min_fminimizer_type *T;
  gsl_min_fminimizer *s;
  double m = 2.0, m_expected = M_PI;
  double a = 0.0, b = 6.0;
  gsl_function F;

  F.function = &fn1;
  F.params = 0;

  T = gsl_min_fminimizer_brent;
  s = gsl_min_fminimizer_alloc (T);
  gsl_min_fminimizer_set (s, &F, m, a, b);

  printf ("using %s method\n",
          gsl_min_fminimizer_name (s));

  printf ("%5s [%9s, %9s] %9s %10s %9s\n",
          "iter", "lower", "upper", "min",
          "err", "err(est)");

  printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
          iter, a, b,
          m, m - m_expected, b - a);

  do
    {
      iter++;
      status = gsl_min_fminimizer_iterate (s);

      m = gsl_min_fminimizer_minimum (s);
      a = gsl_min_fminimizer_x_lower (s);
      b = gsl_min_fminimizer_x_upper (s);

      status 
        = gsl_min_test_interval (a, b, 0.001, 0.0);

      if (status == GSL_SUCCESS)
        printf ("Converged:\n");

      printf ("%5d [%.7f, %.7f] "
              "%.7f %.7f %+.7f %.7f\n",
              iter, a, b,
              m, m_expected, m - m_expected, b - a);
    }
  while (status == GSL_CONTINUE && iter < max_iter);

  return status;
}

Here are the results of the minimization procedure.

bash$ ./a.out 
    0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
    1 [2.0000000, 6.0000000] 3.2758640 +0.1342713 4.0000000
    2 [2.0000000, 3.2831929] 3.2758640 +0.1342713 1.2831929
    3 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
    4 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
    5 [2.8689068, 3.2758640] 3.1460585 +0.0044658 0.4069572
    6 [3.1346075, 3.2758640] 3.1460585 +0.0044658 0.1412565
    7 [3.1346075, 3.1874620] 3.1460585 +0.0044658 0.0528545
    8 [3.1346075, 3.1460585] 3.1460585 +0.0044658 0.0114510
    9 [3.1346075, 3.1460585] 3.1424060 +0.0008133 0.0114510
   10 [3.1346075, 3.1424060] 3.1415885 -0.0000041 0.0077985
Converged:                            
   11 [3.1415885, 3.1424060] 3.1415927 -0.0000000 0.0008175

References and Further Reading

Further information on Brent's algorithm is available in the following book,

Multidimensional Root-Finding

This chapter describes functions for multidimensional root-finding (solving nonlinear systems with n equations in n unknowns). The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs. The solvers are based on the original Fortran library MINPACK.

The header file `gsl_multiroots.h' contains prototypes for the multidimensional root finding functions and related declarations.

Overview

The problem of multidimensional root finding requires the simultaneous solution of n equations, f_i, in n variables, x_i,

In general there are no bracketing methods available for n dimensional systems, and no way of knowing whether any solutions exist. All algorithms proceed from an initial guess using a variant of the Newton iteration,

where x, f are vector quantities and J is the Jacobian matrix @c{$J_{ij} = \partial f_i / \partial x_j$} J_{ij} = d f_i / d x_j. Additional strategies can be used to enlarge the region of convergence. These include requiring a decrease in the norm |f| on each step proposed by Newton's method, or taking steepest-descent steps in the direction of the negative gradient of |f|.

Several root-finding algorithms are available within a single framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

The evaluation of the Jacobian matrix can be problematic, either because programming the derivatives is intractable or because computation of the n^2 terms of the matrix becomes too expensive. For these reasons the algorithms provided by the library are divided into two classes according to whether the derivatives are available or not.

The state for solvers with an analytic Jacobian matrix is held in a gsl_multiroot_fdfsolver struct. The updating procedure requires both the function and its derivatives to be supplied by the user.

The state for solvers which do not use an analytic Jacobian matrix is held in a gsl_multiroot_fsolver struct. The updating procedure uses only function evaluations (not derivatives). The algorithms estimate the matrix J or @c{$J^{-1}$} J^{-1} by approximate methods.

Initializing the Solver

The following functions initialize a multidimensional solver, either with or without derivatives. The solver itself depends only on the dimension of the problem and the algorithm and can be reused for different problems.

Function: gsl_multiroot_fsolver * gsl_multiroot_fsolver_alloc (const gsl_multiroot_fsolver_type * T, size_t n)
This function returns a pointer to a a newly allocated instance of a solver of type T for a system of n dimensions. For example, the following code creates an instance of a hybrid solver, to solve a 3-dimensional system of equations.
const gsl_multiroot_fsolver_type * T 
    = gsl_multiroot_fsolver_hybrid;
gsl_multiroot_fsolver * s 
    = gsl_multiroot_fsolver_alloc (T, 3);

If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

Function: gsl_multiroot_fdfsolver * gsl_multiroot_fdfsolver_alloc (const gsl_multiroot_fdfsolver_type * T, size_t n)
This function returns a pointer to a a newly allocated instance of a derivative solver of type T for a system of n dimensions. For example, the following code creates an instance of a Newton-Raphson solver, for a 2-dimensional system of equations.
const gsl_multiroot_fdfsolver_type * T 
    = gsl_multiroot_fdfsolver_newton;
gsl_multiroot_fdfsolver * s = 
    gsl_multiroot_fdfsolver_alloc (T, 2);

If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

Function: int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * s, gsl_multiroot_function * f, gsl_vector * x)
This function sets, or resets, an existing solver s to use the function f and the initial guess x.

Function: int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver * s, gsl_function_fdf * fdf, gsl_vector * x)
This function sets, or resets, an existing solver s to use the function and derivative fdf and the initial guess x.

Function: void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver * s)
Function: void gsl_multiroot_fdfsolver_free (gsl_multiroot_fdfsolver * s)
These functions free all the memory associated with the solver s.

Function: const char * gsl_multiroot_fsolver_name (const gsl_multiroot_fsolver * s)
Function: const char * gsl_multiroot_fdfsolver_name (const gsl_multiroot_fdfsolver * s)
These functions return a pointer to the name of the solver. For example,
printf("s is a '%s' solver\n", 
       gsl_multiroot_fdfsolver_name (s));

would print something like s is a 'newton' solver.

Providing the function to solve

You must provide n functions of n variables for the root finders to operate on. In order to allow for general parameters the functions are defined by the following data types:

Data Type: gsl_multiroot_function
This data type defines a general system of functions with parameters.
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)
this function should store the vector result f(x,params) in f for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
size_t n
the dimension of the system, i.e. the number of components of the vectors x and f.
void * params
a pointer to the parameters of the function.

Here is an example using Powell's test function,

with A = 10^4. The following code defines a gsl_multiroot_function system F which you could pass to a solver:

struct powell_params { double A; };

int
powell (gsl_vector * x, void * p, gsl_vector * f) {
   struct powell_params * params 
     = *(struct powell_params *)p;
   double A = (params->A);
   double x0 = gsl_vector_get(x,0);
   double x1 = gsl_vector_get(x,1);

   gsl_vector_set (f, 0, A * x0 * x1 - 1)
   gsl_vector_set (f, 1, (exp(-x0) + exp(-x1) 
                          - (1.0 + 1.0/A)))
   return GSL_SUCCESS
}

gsl_multiroot_function F;
struct powell_params params = { 10000.0 };

F.function = &powell;
F.n = 2;
F.params = &params;

Data Type: gsl_multiroot_function_fdf
This data type defines a general system of functions with parameters and the corresponding Jacobian matrix of derivatives,
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)
this function should store the vector result f(x,params) in f for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)
this function should store the n-by-n matrix result J_ij = d f_i(x,params) / d x_j in J for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
int (* fdf) (const gsl_vector * x, void * params, gsl_vector * f, gsl_matrix * J)
This function should set the values of the f and J as above, for arguments x and parameters params. This function provides an optimization of the separate functions for f(x) and J(x) -- it is always faster to compute the function and its derivative at the same time.
size_t n
the dimension of the system, i.e. the number of components of the vectors x and f.
void * params
a pointer to the parameters of the function.

The example of Powell's test function defined above can be extended to include analytic derivatives using the following code,

int
powell_df (gsl_vector * x, void * p, gsl_matrix * J) 
{
   struct powell_params * params 
     = *(struct powell_params *)p;
   double A = (params->A);
   double x0 = gsl_vector_get(x,0);
   double x1 = gsl_vector_get(x,1);
   gsl_vector_set (J, 0, 0, A * x1)
   gsl_vector_set (J, 0, 1, A * x0)
   gsl_vector_set (J, 1, 0, -exp(-x0))
   gsl_vector_set (J, 1, 1, -exp(-x1))
   return GSL_SUCCESS
}

int
powell_fdf (gsl_vector * x, void * p, 
            gsl_matrix * f, gsl_matrix * J) {
   struct powell_params * params 
     = *(struct powell_params *)p;
   double A = (params->A);
   double x0 = gsl_vector_get(x,0);
   double x1 = gsl_vector_get(x,1);

   double u0 = exp(-x0);
   double u1 = exp(-x1);

   gsl_vector_set (f, 0, A * x0 * x1 - 1)
   gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A))

   gsl_vector_set (J, 0, 0, A * x1)
   gsl_vector_set (J, 0, 1, A * x0)
   gsl_vector_set (J, 1, 0, -u0)
   gsl_vector_set (J, 1, 1, -u1)
   return GSL_SUCCESS
}

gsl_multiroot_function_fdf FDF;

FDF.f = &powell_f;
FDF.df = &powell_df;
FDF.fdf = &powell_fdf;
FDF.n = 2;
FDF.params = 0;

Note that the function powell_fdf is able to reuse existing terms from the function when calculating the Jacobian, thus saving time.

Iteration

The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any solver of the corresponding type. The same functions work for all solvers so that different methods can be substituted at runtime without modifications to the code.

Function: int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver * s)
Function: int gsl_multiroot_fdfsolver_iterate (gsl_multiroot_fdfsolver * s)
These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem then an error code will be returned,
GSL_EBADFUNC
the iteration encountered a singular point where the function or its derivative evaluated to Inf or NaN.
GSL_ENOPROG
the iteration is not making any progress, preventing the algorithm from continuing.

The solver maintains a current best estimate of the root at all times. This information can be accessed with the following auxiliary functions,

Function: gsl_vector * gsl_multiroot_fsolver_root (const gsl_multiroot_fsolver * s)
Function: gsl_vector * gsl_multiroot_fdfsolver_root (const gsl_multiroot_fdfsolver * s)
These functions return the current estimate of the root for the solver s.

Search Stopping Parameters

A root finding procedure should stop when one of the following conditions is true:

The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result in several standard ways.

Function: int gsl_multiroot_test_delta (const gsl_vector * dx, const gsl_vector * x, double epsabs, double epsrel)

This function tests for the convergence of the sequence by comparing the last step dx with the absolute error epsabs and relative error epsrel to the current position x. The test returns GSL_SUCCESS if the following condition is achieved,

for each component of x and returns GSL_CONTINUE otherwise.

Function: int gsl_multiroot_test_residual (const gsl_vector * f, double epsabs)
This function tests the residual value f against the absolute error bound epsabs. The test returns GSL_SUCCESS if the following condition is achieved,

and returns GSL_CONTINUE otherwise. This criterion is suitable for situations where the the precise location of the root, x, is unimportant provided a value can be found where the residual is small enough.

Algorithms using Derivatives

The root finding algorithms described in this section make use of both the function and its derivative. They require an initial guess for the location of the root, but there is no absolute guarantee of convergence -- the function must be suitable for this technique and the initial guess must be sufficiently close to the root for it to work. When the conditions are satisfied then convergence is quadratic.

Derivative Solver: gsl_multiroot_fdfsolver_hybridsj
This is a modified version of Powell's Hybrid method as implemented in the HYBRJ algorithm in MINPACK. Minpack was written by Jorge J. Mor'e, Burton S. Garbow and Kenneth E. Hillstrom. The Hybrid algorithm retains the fast convergence of Newton's method but will also reduce the residual when Newton's method is unreliable.

The algorithm uses a generalized trust region to keep each step under control. In order to be accepted a proposed new position x' must satisfy the condition |D (x' - x)| < \delta, where D is a diagonal scaling matrix and \delta is the size of the trust region. The components of D are computed internally, using the column norms of the Jacobian to estimate the sensitivity of the residual to each component of x. This improves the behavior of the algorithm for badly scaled functions.

On each iteration the algorithm first determines the standard Newton step by solving the system J dx = - f. If this step falls inside the trust region it is used as a trial step in the next stage. If not, the algorithm uses the linear combination of the Newton and gradient directions which is predicted to minimize the norm of the function while staying inside the trust region.

This combination of Newton and gradient directions is referred to as a dogleg step.

The proposed step is now tested by evaluating the function at the resulting point, x'. If the step reduces the norm of the function sufficiently then it is accepted and size of the trust region is increased. If the proposed step fails to improve the solution then the size of the trust region is decreased and another trial step is computed.

The speed of the algorithm is increased by computing the changes to the Jacobian approximately, using a rank-1 update. If two successive attempts fail to reduce the residual then the full Jacobian is recomputed. The algorithm also monitors the progress of the solution and returns an error if several steps fail to make any improvement,

GSL_ENOPROG
the iteration is not making any progress, preventing the algorithm from continuing.
GSL_ENOPROGJ
re-evaluations of the Jacobian indicate that the iteration is not making any progress, preventing the algorithm from continuing.

Derivative Solver: gsl_multiroot_fdfsolver_hybridj
This algorithm is an unscaled version of hybridsj. The steps are controlled by a spherical trust region |x' - x| < \delta, instead of a generalized region. This can be useful if the generalized region estimated by hybridsj is inappropriate.

Derivative Solver: gsl_multiroot_fdfsolver_newton

Newton's Method is the standard root-polishing algorithm. The algorithm begins with an initial guess for the location of the solution. On each iteration a linear approximation to the function F is used to estimate the step which will zero all the components of the residual. The iteration is defined by the following sequence,

where the Jacobian matrix J is computed from the derivative functions provided by f. The step dx is obtained by solving the linear system,

using LU decomposition.

Derivative Solver: gsl_multiroot_fdfsolver_gnewton
This is a modified version of Newton's method which attempts to improve global convergence by requiring every step to reduce the Euclidean norm of the residual, |f(x)|. If the Newton step leads to an increase in the norm then a reduced step of relative size,

is proposed, with r being the ratio of norms |f(x')|^2/|f(x)|^2. This procedure is repeated until a suitable step size is found.

Algorithms without Derivatives

The algorithms described in this section do not require any derivative information to be supplied by the user. Any derivatives needed are approximated from by finite difference.

Solver: gsl_multiroot_fsolver_hybrids
This is a version of the Hybrid algorithm which replaces calls to the Jacobian function by its finite difference approximation. The finite difference approximation is computed using gsl_multiroots_fdjac with a relative step size of GSL_SQRT_DBL_EPSILON.

Solver: gsl_multiroot_fsolver_hybrid
This is a finite difference version of the Hybrid algorithm without internal scaling.

Solver: gsl_multiroot_fsolver_dnewton

The discrete Newton algorithm is the simplest method of solving a multidimensional system. It uses the Newton iteration

where the Jacobian matrix J is approximated by taking finite differences of the function f. The approximation scheme used by this implementation is,

where \delta_j is a step of size \sqrt\epsilon |x_j| with \epsilon being the machine precision (@c{$\epsilon \approx 2.22 \times 10^{-16}$} \epsilon \approx 2.22 \times 10^-16). The order of convergence of Newton's algorithm is quadratic, but the finite differences require n^2 function evaluations on each iteration. The algorithm may become unstable if the finite differences are not a good approximation to the true derivatives.

Solver: gsl_multiroot_fsolver_broyden

The Broyden algorithm is a version of the discrete Newton algorithm which attempts to avoids the expensive update of the Jacobian matrix on each iteration. The changes to the Jacobian are also approximated, using a rank-1 update,

where the vectors dx and df are the changes in x and f. On the first iteration the inverse Jacobian is estimated using finite differences, as in the discrete Newton algorithm. This approximation gives a fast update but is unreliable if the changes are not small, and the estimate of the inverse Jacobian becomes worse as time passes. The algorithm has a tendency to become unstable unless it starts close to the root. The Jacobian is refreshed if this instability is detected (consult the source for details).

This algorithm is not recommended and is included only for demonstration purposes.

Examples

The multidimensional solvers are used in a similar way to the one-dimensional root finding algorithms. This first example demonstrates the hybrids scaled-hybrid algorithm, which does not require derivatives. The program solves the Rosenbrock system of equations,

with a = 1, b = 10. The solution of this system lies at (x,y) = (1,1) in a narrow valley.

The first stage of the program is to define the system of equations,

#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_multiroots.h>

struct rparams
  {
    double a;
    double b;
  };

int
rosenbrock_f (const gsl_vector * x, void *params, 
              gsl_vector * f)
{
  double a = ((struct rparams *) params)->a;
  double b = ((struct rparams *) params)->b;

  double x0 = gsl_vector_get (x, 0);
  double x1 = gsl_vector_get (x, 1);

  double y0 = a * (1 - x0);
  double y1 = b * (x1 - x0 * x0);

  gsl_vector_set (f, 0, y0);
  gsl_vector_set (f, 1, y1);

  return GSL_SUCCESS;
}

The main program begins by creating the function object f, with the arguments (x,y) and parameters (a,b). The solver s is initialized to use this function, with the hybrids method.

int
main (void)
{
  const gsl_multiroot_fsolver_type *T;
  gsl_multiroot_fsolver *s;

  int status;
  size_t i, iter = 0;

  const size_t n = 2;
  struct rparams p = {1.0, 10.0};
  gsl_multiroot_function f = {&rosenbrock_f, n, &p};

  double x_init[2] = {-10.0, -5.0};
  gsl_vector *x = gsl_vector_alloc (n);

  gsl_vector_set (x, 0, x_init[0]);
  gsl_vector_set (x, 1, x_init[1]);

  T = gsl_multiroot_fsolver_hybrids;
  s = gsl_multiroot_fsolver_alloc (T, 2);
  gsl_multiroot_fsolver_set (s, &f, x);

  print_state (iter, s);

  do
    {
      iter++;
      status = gsl_multiroot_fsolver_iterate (s);

      print_state (iter, s);

      if (status)   /* check if solver is stuck */
        break;

      status = 
        gsl_multiroot_test_residual (s->f, 1e-7);
    }
  while (status == GSL_CONTINUE && iter < 1000);

  printf ("status = %s\n", gsl_strerror (status));

  gsl_multiroot_fsolver_free (s);
  gsl_vector_free (x);
  return 0;
}

Note that it is important to check the return status of each solver step, in case the algorithm becomes stuck. If an error condition is detected, indicating that the algorithm cannot proceed, then the error can be reported to the user, a new starting point chosen or a different algorithm used.

The intermediate state of the solution is displayed by the following function. The solver state contains the vector s->x which is the current position, and the vector s->f with corresponding function values.

int
print_state (size_t iter, gsl_multiroot_fsolver * s)
{
  printf ("iter = %3u x = % .3f % .3f "
          "f(x) = % .3e % .3e\n",
          iter,
          gsl_vector_get (s->x, 0), 
          gsl_vector_get (s->x, 1),
          gsl_vector_get (s->f, 0), 
          gsl_vector_get (s->f, 1));
}

Here are the results of running the program. The algorithm is started at (-10,-5) far from the solution. Since the solution is hidden in a narrow valley the earliest steps follow the gradient of the function downhill, in an attempt to reduce the large value of the residual. Once the root has been approximately located, on iteration 8, the Newton behavior takes over and convergence is very rapid.

iter =  0 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
iter =  1 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
iter =  2 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
iter =  3 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
iter =  4 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
iter =  5 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
iter =  6 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
iter =  7 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
iter =  8 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
iter =  9 x =   1.000   0.878  f(x) = 1.268e-10 -1.218e+00
iter = 10 x =   1.000   0.989  f(x) = 1.124e-11 -1.080e-01
iter = 11 x =   1.000   1.000  f(x) = 0.000e+00  0.000e+00
status = success

Note that the algorithm does not update the location on every iteration. Some iterations are used to adjust the trust-region parameter, after trying a step which was found to be divergent, or to recompute the Jacobian, when poor convergence behavior is detected.

The next example program adds derivative information, in order to accelerate the solution. There are two derivative functions rosenbrock_df and rosenbrock_fdf. The latter computes both the function and its derivative simultaneously. This allows the optimization of any common terms. For simplicity we substitute calls to the separate f and df functions at this point in the code below.

int
rosenbrock_df (const gsl_vector * x, void *params, 
               gsl_matrix * J)
{
  double a = ((struct rparams *) params)->a;
  double b = ((struct rparams *) params)->b;

  double x0 = gsl_vector_get (x, 0);

  double df00 = -a;
  double df01 = 0;
  double df10 = -2 * b  * x0;
  double df11 = b;

  gsl_matrix_set (J, 0, 0, df00);
  gsl_matrix_set (J, 0, 1, df01);
  gsl_matrix_set (J, 1, 0, df10);
  gsl_matrix_set (J, 1, 1, df11);

  return GSL_SUCCESS;
}

int
rosenbrock_fdf (const gsl_vector * x, void *params,
                gsl_vector * f, gsl_matrix * J)
{
  rosenbrock_f (x, params, f);
  rosenbrock_df (x, params, J);

  return GSL_SUCCESS;
}

The main program now makes calls to the corresponding fdfsolver versions of the functions,

int
main (void)
{
  const gsl_multiroot_fdfsolver_type *T;
  gsl_multiroot_fdfsolver *s;

  int status;
  size_t i, iter = 0;

  const size_t n = 2;
  struct rparams p = {1.0, 10.0};
  gsl_multiroot_function_fdf f = {&rosenbrock_f, 
                                  &rosenbrock_df, 
                                  &rosenbrock_fdf, 
                                  n, &p};

  double x_init[2] = {-10.0, -5.0};
  gsl_vector *x = gsl_vector_alloc (n);

  gsl_vector_set (x, 0, x_init[0]);
  gsl_vector_set (x, 1, x_init[1]);

  T = gsl_multiroot_fdfsolver_gnewton;
  s = gsl_multiroot_fdfsolver_alloc (T, n);
  gsl_multiroot_fdfsolver_set (s, &f, x);

  print_state (iter, s);

  do
    {
      iter++;

      status = gsl_multiroot_fdfsolver_iterate (s);

      print_state (iter, s);

      if (status)
        break;

      status = gsl_multiroot_test_residual (s->f, 1e-7);
    }
  while (status == GSL_CONTINUE && iter < 1000);

  printf ("status = %s\n", gsl_strerror (status));

  gsl_multiroot_fdfsolver_free (s);
  gsl_vector_free (x);
  return 0;
}

The addition of derivative information to the hybrids solver does not make any significant difference to its behavior, since it able to approximate the Jacobian numerically with sufficient accuracy. To illustrate the behavior of a different derivative solver we switch to gnewton. This is a traditional newton solver with the constraint that it scales back its step if the full step would lead "uphill". Here is the output for the gnewton algorithm,

iter = 0 x = -10.000  -5.000 f(x) =  1.100e+01 -1.050e+03
iter = 1 x =  -4.231 -65.317 f(x) =  5.231e+00 -8.321e+02
iter = 2 x =   1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
iter = 3 x =   1.000   1.000 f(x) = -2.220e-16 -4.441e-15
status = success

The convergence is much more rapid, but takes a wide excursion out to the point (-4.23,-65.3). This could cause the algorithm to go astray in a realistic application. The hybrid algorithm follows the downhill path to the solution more reliably.

References and Further Reading

The original version of the Hybrid method is described in the following articles by Powell,

The following papers are also relevant to the algorithms described in this section,

Multidimensional Minimization

This chapter describes routines for finding minima of arbitrary multidimensional functions. The library provides low level components for a variety of iterative minimizers and convergence tests. These can be combined by the user to achieve the desired solution, while providing full access to the intermediate steps of the algorithms. Each class of methods uses the same framework, so that you can switch between minimizers at runtime without needing to recompile your program. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multi-threaded programs. The minimization algorithms can be used to maximize a function by inverting its sign.

The header file `gsl_multimin.h' contains prototypes for the minimization functions and related declarations.

Overview

The problem of multidimensional minimization requires finding a point x such that the scalar function,

takes a value which is lower than at any neighboring point. For smooth functions the gradient g = \nabla f vanishes at the minimum. In general there are no bracketing methods available for the minimization of n-dimensional functions. All algorithms proceed from an initial guess using a search algorithm which attempts to move in a downhill direction. A one-dimensional line minimisation is performed along this direction until the lowest point is found to a suitable tolerance. The search direction is then updated with local information from the function and its derivatives, and the whole process repeated until the true n-dimensional minimum is found.

Several minimization algorithms are available within a single framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,

Each iteration step consists either of an improvement to the line-mimisation in the current direction or an update to the search direction itself. The state for the minimizers is held in a gsl_multimin_fdfminimizer struct.

Caveats

Note that the minimization algorithms can only search for one local minimum at a time. When there are several local minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima this will be. In most cases, no error will be reported if you try to find a local minimum in an area where there is more than one.

It is also important to note that the minimization algorithms find local minima; there is no way to determine whether a minimum is a global minimum of the function in question.

Initializing the Multidimensional Minimizer

The following function initializes a multidimensional minimizer. The minimizer itself depends only on the dimension of the problem and the algorithm and can be reused for different problems.

Function: gsl_multimin_fdfminimizer * gsl_multimin_fdfminimizer_alloc (const gsl_multimin_fdfminimizer_type *T, size_t n)
This function returns a pointer to a a newly allocated instance of a minimizer of type T for an n-dimension function. If there is insufficient memory to create the minimizer then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

Function: int gsl_multimin_fdfminimizer_set (gsl_multimin_fdfminimizer * s, gsl_multimin_function_fdf *fdf, const gsl_vector * x, double step_size, double tol)
This function initializes the minimizer s to minimize the function fdf starting from the initial point x. The size of the first trial step is given by step_size. The accuracy of the line minimization is specified by tol. The precise meaning of this parameter depends on the method used. Typically the line minimization is considered successful if the gradient of the function g is orthogonal to the current search direction p to a relative accuracy of tol, where @c{$p\cdot g < tol |p| |g|$} dot(p,g) < tol |p| |g|.

Function: void gsl_multimin_fdfminimizer_free (gsl_multimin_fdfminimizer *s)
This function frees all the memory associated with the minimizer s.

Function: const char * gsl_multimin_fdfminimizer_name (const gsl_multimin_fdfminimizer * s)
This function returns a pointer to the name of the minimizer. For example,
printf("s is a '%s' minimizer\n", 
       gsl_multimin_fdfminimizer_name (s));

would print something like s is a 'conjugate_pr' minimizer.

Providing a function to minimize

You must provide a parametric function of n variables for the minimizers to operate on. You also need to provide a routine which calculates the gradient of the function and a third routine which calculates both the function value and the gradient together. In order to allow for general parameters the functions are defined by the following data type:

Data Type: gsl_multimin_function_fdf
This data type defines a general function of n variables with parameters and the corresponding gradient vector of derivatives,
double (* f) (const gsl_vector * x, void * params)
this function should return the result f(x,params) for argument x and parameters params.
int (* df) (const gsl_vector * x, void * params, gsl_vector * g)
this function should store the n-dimensional gradient g_i = d f(x,params) / d x_i in the vector g for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
int (* fdf) (const gsl_vector * x, void * params, double * f, gsl_vector * g)
This function should set the values of the f and g as above, for arguments x and parameters params. This function provides an optimization of the separate functions for f(x) and g(x) -- it is always faster to compute the function and its derivative at the same time.
size_t n
the dimension of the system, i.e. the number of components of the vectors x.
void * params
a pointer to the parameters of the function.

The following example function defines a simple paraboloid with two parameters,

/* Paraboloid centered on (dp[0],dp[1]) */

double
my_f (const gsl_vector *v, void *params)
{
  double x, y;
  double *dp = (double *)params;
  
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
 
  return 10.0 * (x - dp[0]) * (x - dp[0]) +
           20.0 * (y - dp[1]) * (y - dp[1]) + 30.0; 
}

/* The gradient of f, df = (df/dx, df/dy). */
void 
my_df (const gsl_vector *v, void *params, 
       gsl_vector *df)
{
  double x, y;
  double *dp = (double *)params;
  
  x = gsl_vector_get(v, 0);
  y = gsl_vector_get(v, 1);
 
  gsl_vector_set(df, 0, 20.0 * (x - dp[0]));
  gsl_vector_set(df, 1, 40.0 * (y - dp[1]));
}

/* Compute both f and df together. */
void 
my_fdf (const gsl_vector *x, void *params, 
        double *f, gsl_vector *df) 
{
  *f = my_f(x, params); 
  my_df(x, params, df);
}

The function can be initialized using the following code,

gsl_multimin_function_fdf my_func;

double p[2] = { 1.0, 2.0 }; /* center at (1,2) */

my_func.f = &my_f;
my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.n = 2;
my_func.params = (void *)p;

Iteration

The following function drives the iteration of each algorithm. The function performs one iteration to update the state of the minimizer. The same function works for all minimizers so that different methods can be substituted at runtime without modifications to the code.

Function: int gsl_multimin_fdfminimizer_iterate (gsl_multimin_fdfminimizer *s)
These functions perform a single iteration of the minimizer s. If the iteration encounters an unexpected problem then an error code will be returned.
The minimizer maintains a current best estimate of the minimum at all times. This information can be accessed with the following auxiliary functions,

Function: gsl_vector * gsl_multiroot_fdfsolver_x (const gsl_multiroot_fdfsolver * s)
Function: double gsl_multiroot_fdfsolver_minimum (const gsl_multiroot_fdfsolver * s)
Function: gsl_vector * gsl_multiroot_fdfsolver_gradient (const gsl_multiroot_fdfsolver * s)
These functions return the current best estimate of the location of the minimum, the value of the function at that point and its gradient, for the minimizer s.

Function: int gsl_multimin_fdfminimizer_restart (gsl_multimin_fdfminimizer *s)
This function resets the minimizer s to use the current point as a new starting point.

Stopping Criteria

A minimization procedure should stop when one of the following conditions is true:

The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result.

Function: int gsl_multimin_test_gradient (const gsl_vector * g,double epsabs)
This function tests the norm of the gradient g against the absolute tolerance epsabs. The gradient of a multidimensional function goes to zero at a minimum. The test returns GSL_SUCCESS if the following condition is achieved,

and returns GSL_CONTINUE otherwise. A suitable choice of epsabs can be made from the desired accuracy in the function for small variations in x. The relationship between these quantities is given by @c{$\delta f = g\,\delta x$} \delta f = g \delta x.

Algorithms

There are several minimization methods available. The best choice of algorithm depends on the problem. Each of the algorithms uses the value of the function and its gradient at each evaluation point.

Minimizer: gsl_multimin_fdfminimizer_conjugate_fr
This is the Fletcher-Reeves conjugate gradient algorithm. The conjugate gradient algorithm proceeds as a succession of line minimizations. The sequence of search directions to build up an approximation to the curvature of the function in the neighborhood of the minimum. An initial search direction p is chosen using the gradient and line minimization is carried out in that direction. The accuracy of the line minimization is specified by the parameter tol. At the minimum along this line the function gradient g and the search direction p are orthogonal. The line minimization terminates when dot(p,g) < tol |p| |g|. The search direction is updated using the Fletcher-Reeves formula p' = g' - \beta g where \beta=-|g'|^2/|g|^2, and the line minimization is then repeated for the new search direction.

Minimizer: gsl_multimin_fdfminimizer_conjugate_pr
This is the Polak-Ribiere conjugate gradient algorithm. It is similar to the Fletcher-Reeves method, differing only in the choice of the coefficient \beta. Both methods work well when the evaluation point is close enough to the minimum of the objective function that it is well approximated by a quadratic hypersurface.

Minimizer: gsl_multimin_fdfminimizer_vector_bfgs
This is the vector Broyden-Fletcher-Goldfarb-Shanno conjugate gradient algorithm. It is a quasi-Newton method which builds up an approximation to the second derivatives of the function f using the difference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior in that region.

Minimizer: gsl_multimin_fdfminimizer_steepest_descent
The steepest descent algorithm follows the downhill gradient of the function at each step. When a downhill step is successful the step-size is increased by factor of two. If the downhill step leads to a higher function value then the algorithm backtracks and the step size is decreased using the parameter tol. A suitable value of tol for most applications is 0.1. The steepest descent method is inefficient and is included only for demonstration purposes.

Examples

This example program finds the minimum of the paraboloid function defined earlier. The location of the minimum is offset from the origin in x and y, and the function value at the minimum is non-zero. The main program is given below, it requires the example function given earlier in this chapter.

int
main (void)
{
  size_t iter = 0;
  int status;

  const gsl_multimin_fdfminimizer_type *T;
  gsl_multimin_fdfminimizer *s;

  /* Position of the minimum (1,2). */
  double par[2] = { 1.0, 2.0 };

  gsl_vector *x;
  gsl_multimin_function_fdf my_func;

  my_func.f = &my_f;
  my_func.df = &my_df;
  my_func.fdf = &my_fdf;
  my_func.n = 2;
  my_func.params = &par;

  /* Starting point, x = (5,7) */

  x = gsl_vector_alloc (2);
  gsl_vector_set (x, 0, 5.0);
  gsl_vector_set (x, 1, 7.0);

  T = gsl_multimin_fdfminimizer_conjugate_fr;
  s = gsl_multimin_fdfminimizer_alloc (T, 2);

  gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);

  do
    {
      iter++;
      status = gsl_multimin_fdfminimizer_iterate (s);

      if (status)
        break;

      status = gsl_multimin_test_gradient (s->gradient, 1e-3);

      if (status == GSL_SUCCESS)
        printf ("Minimum found at:\n");

      printf ("%5d %.5f %.5f %10.5f\n", iter,
              gsl_vector_get (s->x, 0), 
              gsl_vector_get (s->x, 1), 
              s->f);

    }
  while (status == GSL_CONTINUE && iter < 100);

  gsl_multimin_fdfminimizer_free (s);
  gsl_vector_free (x);

  return 0;
}

The initial step-size is chosen as 0.01, a conservative estimate in this case, and the line minimization parameter is set at 0.0001. The program terminates when the norm of the gradient has been reduced below 0.001. The output of the program is shown below,

         x       y         f
    1 4.99629 6.99072  687.84780
    2 4.98886 6.97215  683.55456
    3 4.97400 6.93501  675.01278
    4 4.94429 6.86073  658.10798
    5 4.88487 6.71217  625.01340
    6 4.76602 6.41506  561.68440
    7 4.52833 5.82083  446.46694
    8 4.05295 4.63238  261.79422
    9 3.10219 2.25548   75.49762
   10 2.85185 1.62963   67.03704
   11 2.19088 1.76182   45.31640
   12 0.86892 2.02622   30.18555
Minimum found at:
   13 1.00000 2.00000   30.00000

Note that the algorithm gradually increases the step size as it successfully moves downhill, as can be seen by plotting the successive points.

multiminThe conjugate gradient algorithm finds the minimum on its second direction because the function is purely quadratic. Additional iterations would be needed for a more complicated function.

References and Further Reading

A brief description of multidimensional minimization algorithms and further references can be found in the following book,

Least-Squares Fitting

This chapter describes routines for performing least squares fits to experimental data using linear combinations of functions. The data may be weighted or unweighted. For weighted data the functions compute the best fit parameters and their associated covariance matrix. For unweighted data the covariance matrix is estimated from the scatter of the points, giving a variance-covariance matrix. The functions are divided into separate versions for simple one- or two-parameter regression and multiple-parameter fits. The functions are declared in the header file `gsl_fit.h'

Linear regression

The functions described in this section can be used to perform least-squares fits to a straight line model, Y = c_0 + c_1 X. For weighted data the best-fit is found by minimizing the weighted sum of squared residuals, \chi^2,

for the parameters c_0, c_1. For unweighted data the sum is computed with w_i = 1.

Function: int gsl_fit_linear (const double * x, const size_t xstride, const double * y, const size_t ystride, size_t n, double * c0, double * c1, double * cov00, double * cov01, double * cov11, double * sumsq)
This function computes the best-fit linear regression coefficients (c0,c1) of the model Y = c_0 + c_1 X for the datasets (x, y), two vectors of length n with strides xstride and ystride. The variance-covariance matrix for the parameters (c0, c1) is estimated from the scatter of the points around the best-fit line and returned via the parameters (cov00, cov01, cov11). The sum of squares of the residuals from the best-fit line is returned in sumsq.

Function: int gsl_fit_wlinear (const double * x, const size_t xstride, const double * w, const size_t wstride, const double * y, const size_t ystride, size_t n, double * c0, double * c1, double * cov00, double * cov01, double * cov11, double * chisq)
This function computes the best-fit linear regression coefficients (c0,c1) of the model Y = c_0 + c_1 X for the weighted datasets (x, y), two vectors of length n with strides xstride and ystride. The vector w, of length n and stride wstride, specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapoint in y.

The covariance matrix for the parameters (c0, c1) is estimated from weighted data and returned via the parameters (cov00, cov01, cov11). The weighted sum of squares of the residuals from the best-fit line, \chi^2, is returned in chisq.

Function: int gsl_fit_linear_est (double x, double c0, double c1, double c00, double c01, double c11, double *y, double *y_err)
This function uses the best-fit linear regression coefficients c0,c1 and their estimated covariance cov00,cov01,cov11 to compute the fitted function y and its standard deviation y_err for the model Y = c_0 + c_1 X at the point x.

Linear fitting without a constant term

The functions described in this section can be used to perform least-squares fits to a straight line model without a constant term, Y = c_1 X. For weighted data the best-fit is found by minimizing the weighted sum of squared residuals, \chi^2,

for the parameter c_1. For unweighted data the sum is computed with w_i = 1.

Function: int gsl_fit_mul (const double * x, const size_t xstride, const double * y, const size_t ystride, size_t n, double * c1, double * cov11, double * sumsq)
This function computes the best-fit linear regression coefficient c1 of the model Y = c_1 X for the datasets (x, y), two vectors of length n with strides xstride and ystride. The variance of the parameter c1 is estimated from the scatter of the points around the best-fit line and returned via the parameter cov11. The sum of squares of the residuals from the best-fit line is returned in sumsq.

Function: int gsl_fit_wmul (const double * x, const size_t xstride, const double * w, const size_t wstride, const double * y, const size_t ystride, size_t n, double * c1, double * cov11, double * sumsq)
This function computes the best-fit linear regression coefficient c1 of the model Y = c_1 X for the weighted datasets (x, y), two vectors of length n with strides xstride and ystride. The vector w, of length n and stride wstride, specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapoint in y.

The variance of the parameter c1 is estimated from the weighted data and returned via the parameters cov11. The weighted sum of squares of the residuals from the best-fit line, \chi^2, is returned in chisq.

Function: int gsl_fit_mul_est (double x, double c1, double c11, double *y, double *y_err)
This function uses the best-fit linear regression coefficient c1 and its estimated covariance cov11 to compute the fitted function y and its standard deviation y_err for the model Y = c_1 X at the point x.

Multi-parameter fitting

The functions described in this section perform least-squares fits to a general linear model, y = X c where y is a vector of n observations, X is an n by p matrix of predictor variables, and c are the p unknown best-fit parameters, which are to be estimated.

The best-fit is found by minimizing the weighted sums of squared residuals, \chi^2,

with respect to the parameters c. The weights are specified by the diagonal elements of the n by n matrix W. For unweighted data W is replaced by the identity matrix.

This formulation can be used for fits to any number of functions and/or variables by preparing the n-by-p matrix X appropriately. For example, to fit to a p-th order polynomial in x, use the following matrix,

where the index i runs over the observations and the index j runs from 0 to p-1.

To fit to a set of p sinusoidal functions with fixed frequencies \omega_1, \omega_2, ..., \omega_p, use,

To fit to p independent variables x_1, x_2, ..., x_p, use,

where x_j(i) is the i-th value of the predictor variable x_j.

The functions described in this section are declared in the header file `gsl_multifit.h'.

The solution of the general linear least-squares system requires an additional working space for intermediate results, such as the singular value decomposition of the matrix X.

Function: gsl_multifit_linear_workspace * gsl_multifit_linear_alloc (size_t n, size_t p)
This function allocates a workspace for fitting a model to n observations using p parameters.

Function: void gsl_multifit_linear_free (gsl_multifit_linear_workspace * work)
This function frees the memory associated with the workspace w.

Function: int gsl_multifit_linear (const gsl_matrix * X, const gsl_vector * y, gsl_vector * c, gsl_matrix * cov, double * chisq, gsl_multifit_linear_workspace * work)
This function computes the best-fit parameters c of the model y = X c for the observations y and the matrix of predictor variables X. The variance-covariance matrix of the model parameters cov is estimated from the scatter of the observations about the best-fit. The sum of squares of the residuals from the best-fit, \chi^2, is returned in chisq.

The best-fit is found by singular value decomposition of the matrix X using the preallocated workspace provided in work. The modified Golub-Reinsch SVD algorithm is used, with column scaling to improve the accuracy of the singular values. Any components which have zero singular value (to machine precision) are discarded from the fit.

Function: int gsl_multifit_wlinear (const gsl_matrix * X, const gsl_vector * w, const gsl_vector * y, gsl_vector * c, gsl_matrix * cov, double * chisq, gsl_multifit_linear_workspace * work)

This function computes the best-fit parameters c of the model y = X c for the observations y and the matrix of predictor variables X. The covariance matrix of the model parameters cov is estimated from the weighted data. The weighted sum of squares of the residuals from the best-fit, \chi^2, is returned in chisq.

The best-fit is found by singular value decomposition of the matrix X using the preallocated workspace provided in work. Any components which have zero singular value (to machine precision) are discarded from the fit.

Examples

The following program computes a least squares straight-line fit to a simple (fictitious) dataset, and outputs the best-fit line and its associated one standard-deviation error bars.

#include <stdio.h>
#include <gsl/gsl_fit.h>

int
main (void)
{
  int i, n = 4;
  double x[4] = { 1970, 1980, 1990, 2000 };
  double y[4] = {   12,   11,   14,   13 };
  double w[4] = {  0.1,  0.2,  0.3,  0.4 };

  double c0, c1, cov00, cov01, cov11, chisq;

  gsl_fit_wlinear (x, 1, w, 1, y, 1, n, 
                   &c0, &c1, &cov00, &cov01, &cov11, 
                   &chisq);

  printf("# best fit: Y = %g + %g X\n", c0, c1);
  printf("# covariance matrix:\n");
  printf("# [ %g, %g\n#   %g, %g]\n", 
         cov00, cov01, cov01, cov11);
  printf("# chisq = %g\n", chisq);

  for (i = 0; i < n; i++)
    printf("data: %g %g %g\n", 
                  x[i], y[i], 1/sqrt(w[i]));

  printf("\n");

  for (i = -30; i < 130; i++)
    {
      double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);
      double yf, yf_err;

      gsl_fit_linear_est (xf, 
                          c0, c1, 
                          cov00, cov01, cov11, 
                          &yf, &yf_err);

      printf("fit: %g %g\n", xf, yf);
      printf("hi : %g %g\n", xf, yf + yf_err);
      printf("lo : %g %g\n", xf, yf - yf_err);
    }
  return 0;
}

The following commands extract the data from the output of the program and display it using the GNU plotutils graph utility,

$ ./demo > tmp
$ more tmp
# best fit: Y = -106.6 + 0.06 X
# covariance matrix:
# [ 39602, -19.9
#   -19.9, 0.01]
# chisq = 0.8

$ for n in data fit hi lo ; 
   do 
     grep "^$n" tmp | cut -d: -f2 > $n ; 
   done
$ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data 
     -S 0 -I a -m 1 fit -m 2 hi -m 2 lo

fit-wlinear

The next program performs a quadratic fit y = c_0 + c_1 x + c_2 x^2 to a weighted dataset using the generalised linear fitting function gsl_multifit_wlinear. The model matrix X for a quadratic fit is given by,

where the column of ones corresponds to the constant term c_0. The two remaining columns corresponds to the terms c_1 x and and c_2 x^2.

The program reads n lines of data in the format (x, y, err) where err is the error (standard deviation) in the value y.

#include <stdio.h>
#include <gsl/gsl_multifit.h>

int
main (int argc, char **argv)
{
  int i, n;
  double xi, yi, ei, chisq;
  gsl_matrix *X, *cov;
  gsl_vector *y, *w, *c;

  if (argc != 2)
    {
      fprintf(stderr,"usage: fit n < data\n");
      exit (-1);
    }

  n = atoi(argv[1]);

  X = gsl_matrix_alloc (n, 3);
  y = gsl_vector_alloc (n);
  w = gsl_vector_alloc (n);

  c = gsl_vector_alloc (3);
  cov = gsl_matrix_alloc (3, 3);

  for (i = 0; i < n; i++)
    {
      int count = fscanf(stdin, "%lg %lg %lg",
                         &xi, &yi, &ei);

      if (count != 3)
        {
          fprintf(stderr, "error reading file\n");
          exit(-1);
        }

      printf("%g %g +/- %g\n", xi, yi, ei);
      
      gsl_matrix_set (X, i, 0, 1.0);
      gsl_matrix_set (X, i, 1, xi);
      gsl_matrix_set (X, i, 2, xi*xi);
      
      gsl_vector_set (y, i, yi);
      gsl_vector_set (w, i, 1.0/(ei*ei));
    }

  {
    gsl_multifit_linear_workspace * work 
      = gsl_multifit_linear_alloc (n, 3);
    gsl_multifit_wlinear (X, w, y, c, cov,
                          &chisq, work);
    gsl_multifit_linear_free (work);
  }

#define C(i) (gsl_vector_get(c,(i)))
#define COV(i,j) (gsl_matrix_get(cov,(i),(j)))

  {
    printf("# best fit: Y = %g + %g X + %g X^2\n", 
           C(0), C(1), C(2));

    printf("# covariance matrix:\n");
    printf("[ %+.5e, %+.5e, %+.5e  \n",
              COV(0,0), COV(0,1), COV(0,2));
    printf("  %+.5e, %+.5e, %+.5e  \n", 
              COV(1,0), COV(1,1), COV(1,2));
    printf("  %+.5e, %+.5e, %+.5e ]\n", 
              COV(2,0), COV(2,1), COV(2,2));
    printf("# chisq = %g\n", chisq);
  }
  return 0;
}

A suitable set of data for fitting can be generated using the following program. It outputs a set of points with gaussian errors from the curve y = e^x in the region 0 < x < 2.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_randist.h>

int
main (void)
{
  double x;
  const gsl_rng_type * T;
  gsl_rng * r;
  
  gsl_rng_env_setup();
  
  T = gsl_rng_default;
  r = gsl_rng_alloc(T);

  for (x = 0.1; x < 2; x+= 0.1)
    {
      double y0 = exp(x);
      double sigma = 0.1*y0;
      double dy = gsl_ran_gaussian(r, sigma)

      printf("%g %g %g\n", x, y0 + dy, sigma);
    }
  return 0;
}

The data can be prepared by running the resulting executable program,

$ ./generate > exp.dat
$ more exp.dat
0.1 0.97935 0.110517
0.2 1.3359 0.12214
0.3 1.52573 0.134986
0.4 1.60318 0.149182
0.5 1.81731 0.164872
0.6 1.92475 0.182212
....

To fit the data use the previous program, with the number of data points given as the first argument. In this case there are 19 data points.

$ ./fit 19 < exp.dat
0.1 0.97935 +/- 0.110517
0.2 1.3359 +/- 0.12214
...
# best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
# covariance matrix:
[ +1.25612e-02, -3.64387e-02, +1.94389e-02  
  -3.64387e-02, +1.42339e-01, -8.48761e-02  
  +1.94389e-02, -8.48761e-02, +5.60243e-02 ]
# chisq = 23.0987

The parameters of the quadratic fit match the coefficients of the expansion of e^x, taking into account the errors on the parameters and the O(x^3) difference between the exponential and quadratic functions for the larger values of x. The errors on the parameters are given by the square-root of the corresponding diagonal elements of the covariance matrix. The chi-squared per degree of freedom is 1.4, indicating a reasonable fit to the data.

fit-wlinear2

References and Further Reading

A summary of formulas and techniques for least squares fitting can be found in the "Statistics" chapter of the Annual Review of Particle Physics prepared by the Particle Data Group.

The Review of Particle Physics is available online at the website given above.

The tests used to prepare these routines are based on the NIST Statistical Reference Datasets. The datasets and their documentation are available from NIST at the following website,

http://www.nist.gov/itl/div898/strd/index.html.

Nonlinear Least-Squares Fitting

This chapter describes functions for multidimensional nonlinear least-squares fitting. The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs.

The header file `gsl_multifit_nlin.h' contains prototypes for the multidimensional nonlinear fitting functions and related declarations.

Overview

The problem of multidimensional nonlinear least-squares fitting requires the minimization of the squared residuals of n functions, f_i, in p parameters, x_i,

All algorithms proceed from an initial guess using the linearization,

where x is the initial point, p is the proposed step and J is the Jacobian matrix @c{$J_{ij} = \partial f_i / \partial x_j$} J_{ij} = d f_i / d x_j. Additional strategies are used to enlarge the region of convergence. These include requiring a decrease in the norm ||F|| on each step or using a trust region to avoid steps which fall outside the linear regime.

Initializing the Solver

Function: gsl_multifit_fsolver * gsl_multifit_fsolver_alloc (const gsl_multifit_fsolver_type * T, size_t n, size_t p)
This function returns a pointer to a a newly allocated instance of a solver of type T for n observations and p parameters.

If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

Function: gsl_multifit_fdfsolver * gsl_multifit_fdfsolver_alloc (const gsl_multifit_fdfsolver_type * T, size_t n, size_t p)
This function returns a pointer to a a newly allocated instance of a derivative solver of type T for n observations and p parameters. For example, the following code creates an instance of a Levenberg-Marquardt solver for 100 data points and 3 parameters,
const gsl_multifit_fdfsolver_type * T 
    = gsl_multifit_fdfsolver_lmder;
gsl_multifit_fdfsolver * s 
    = gsl_multifit_fdfsolver_alloc (T, 100, 3);

If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

Function: int gsl_multifit_fsolver_set (gsl_multifit_fsolver * s, gsl_multifit_function * f, gsl_vector * x)
This function initializes, or reinitializes, an existing solver s to use the function f and the initial guess x.

Function: int gsl_multifit_fdfsolver_set (gsl_multifit_fdfsolver * s, gsl_function_fdf * fdf, gsl_vector * x)
This function initializes, or reinitializes, an existing solver s to use the function and derivative fdf and the initial guess x.

Function: void gsl_multifit_fsolver_free (gsl_multifit_fsolver * s)
Function: void gsl_multifit_fdfsolver_free (gsl_multifit_fdfsolver * s)
These functions free all the memory associated with the solver s.

Function: const char * gsl_multifit_fsolver_name (const gsl_multifit_fdfsolver * s)
Function: const char * gsl_multifit_fdfsolver_name (const gsl_multifit_fdfsolver * s)
These functions return a pointer to the name of the solver. For example,
printf("s is a '%s' solver\n", 
       gsl_multifit_fdfsolver_name (s));

would print something like s is a 'lmder' solver.

Providing the Function to be Minimized

You must provide n functions of p variables for the minimization algorithms to operate on. In order to allow for general parameters the functions are defined by the following data types:

Data Type: gsl_multifit_function
This data type defines a general system of functions with parameters.
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)
this function should store the vector result f(x,params) in f for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
size_t n
the number of functions, i.e. the number of components of the vector f
size_t p
the number of independent variables, i.e. the number of components of the vectors x
void * params
a pointer to the parameters of the function

Data Type: gsl_multifit_function_fdf
This data type defines a general system of functions with parameters and the corresponding Jacobian matrix of derivatives,
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)
this function should store the vector result f(x,params) in f for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)
this function should store the n-by-p matrix result J_ij = d f_i(x,params) / d x_j in J for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
int (* fdf) (const gsl_vector * x, void * params, gsl_vector * f, gsl_matrix * J)
This function should set the values of the f and J as above, for arguments x and parameters params. This function provides an optimization of the separate functions for f(x) and J(x) -- it is always faster to compute the function and its derivative at the same time.
size_t n
the number of functions, i.e. the number of components of the vector f
size_t p
the number of independent variables, i.e. the number of components of the vectors x
void * params
a pointer to the parameters of the function

Iteration

The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any solver of the corresponding type. The same functions work for all solvers so that different methods can be substituted at runtime without modifications to the code.

Function: int gsl_multifit_fsolver_iterate (gsl_multifit_fsolver * s)
Function: int gsl_multifit_fdfsolver_iterate (gsl_multifit_fdfsolver * s)
These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem then an error code will be returned. The solver maintains a current estimate of the best-fit parameters at all times. This information can be accessed with the following auxiliary functions,

Function: gsl_vector * gsl_multifit_fsolver_position (const gsl_multifit_fsolver * s)
Function: gsl_vector * gsl_multifit_fdfsolver_position (const gsl_multifit_fdfsolver * s)
These functions return the current position (i.e. best-fit parameters) of the solver s.

Search Stopping Parameters

A minimization procedure should stop when one of the following conditions is true:

The handling of these conditions is under user control. The functions below allow the user to test the current estimate of the best-fit parameters in several standard ways.

Function: int gsl_multifit_test_delta (const gsl_vector * dx, const gsl_vector * x, double epsabs, double epsrel)

This function tests for the convergence of the sequence by comparing the last step dx with the absolute error epsabs and relative error epsrel to the current position x. The test returns GSL_SUCCESS if the following condition is achieved,

for each component of x and returns GSL_CONTINUE otherwise.

Function: int gsl_multifit_test_gradient (const gsl_vector * g, double epsabs)
This function tests the residual gradient g against the absolute error bound epsabs. Mathematically, the gradient should be exactly zero at the minimum. The test returns GSL_SUCCESS if the following condition is achieved,

and returns GSL_CONTINUE otherwise. This criterion is suitable for situations where the the precise location of the minimum, x, is unimportant provided a value can be found where the gradient is small enough.

Function: int gsl_multifit_gradient (const gsl_matrix * J, const gsl_vector * f, gsl_vector * g)
This function computes the gradient g of \Phi(x) = (1/2) ||F(x)||^2 from the Jacobian matrix J and the function values f, using the formula g = J^T f.

Minimization Algorithms using Derivatives

The minimization algorithms described in this section make use of both the function and its derivative. They require an initial guess for the location of the minimum. There is no absolute guarantee of convergence -- the function must be suitable for this technique and the initial guess must be sufficiently close to the minimum for it to work.

Derivative Solver: gsl_multifit_fdfsolver_lmsder
This is a robust and efficient version of the Levenberg-Marquardt algorithm as implemented in the scaled LMDER routine in MINPACK. Minpack was written by Jorge J. Mor'e, Burton S. Garbow and Kenneth E. Hillstrom.

The algorithm uses a generalized trust region to keep each step under control. In order to be accepted a proposed new position x' must satisfy the condition |D (x' - x)| < \delta, where D is a diagonal scaling matrix and \delta is the size of the trust region. The components of D are computed internally, using the column norms of the Jacobian to estimate the sensitivity of the residual to each component of x. This improves the behavior of the algorithm for badly scaled functions.

On each iteration the algorithm attempts to minimize the linear system |F + J p| subject to the constraint |D p| < \Delta. The solution to this constrained linear system is found using the Levenberg-Marquardt method.

The proposed step is now tested by evaluating the function at the resulting point, x'. If the step reduces the norm of the function sufficiently, and follows the predicted behavior of the function within the trust region. then it is accepted and size of the trust region is increased. If the proposed step fails to improve the solution, or differs significantly from the expected behavior within the trust region, then the size of the trust region is decreased and another trial step is computed.

The algorithm also monitors the progress of the solution and returns an error if the changes in the solution are smaller than the machine precision. The possible error codes are,

GSL_ETOLF
the decrease in the function falls below machine precision
GSL_ETOLX
the change in the position vector falls below machine precision
GSL_ETOLG
the norm of the gradient, relative to the norm of the function, falls below machine precision

These error codes indicate that further iterations will be unlikely to change the solution from its current value.

Derivative Solver: gsl_multifit_fdfsolver_lmder
This is an unscaled version of the LMDER algorithm. The elements of the diagonal scaling matrix D are set to 1. This algorithm may be useful in circumstances where the scaled version of LMDER converges too slowly, or the function is already scaled appropriately.

Minimization Algorithms without Derivatives

There are no algorithms implemented in this section at the moment.

Computing the covariance matrix of best fit parameters

Function: int gsl_multifit_covar (const gsl_matrix * J, double epsrel, gsl_matrix * covar)
This function uses the Jacobian matrix J to compute the covariance matrix of the best-fit parameters, covar. The parameter epsrel is used to remove linear-dependent columns when J is rank deficient.

The covariance matrix is given by,

and is computed by QR decomposition of J with column-pivoting. Any columns of R which satisfy

are considered linearly-dependent and are excluded from the covariance matrix (the corresponding rows and columns of the covariance matrix are set to zero).

Examples

The following example program fits a weighted exponential model with background to experimental data, Y = A \exp(-\lambda t) + b. The first part of the program sets up the functions expb_f and expb_df to calculate the model and its Jacobian. The appropriate fitting function is given by,

where we have chosen t_i = i. The Jacobian matrix J is the derivative of these functions with respect to the three parameters (A, \lambda, b). It is given by,

where x_0 = A, x_1 = \lambda and x_2 = b.

#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlin.h>

struct data {
  size_t n;
  double * y;
  double * sigma;
};

int
expb_f (const gsl_vector * x, void *params, 
        gsl_vector * f)
{
  size_t n = ((struct data *)params)->n;
  double *y = ((struct data *)params)->y;
  double *sigma = ((struct data *) params)->sigma;

  double A = gsl_vector_get (x, 0);
  double lambda = gsl_vector_get (x, 1);
  double b = gsl_vector_get (x, 2);

  size_t i;

  for (i = 0; i < n; i++)
    {
      /* Model Yi = A * exp(-lambda * i) + b */
      double t = i;
      double Yi = A * exp (-lambda * t) + b;
      gsl_vector_set (f, i, (Yi - y[i])/sigma[i]);
    }

  return GSL_SUCCESS;
}

int
expb_df (const gsl_vector * x, void *params, 
         gsl_matrix * J)
{
  size_t n = ((struct data *)params)->n;
  double *sigma = ((struct data *) params)->sigma;

  double A = gsl_vector_get (x, 0);
  double lambda = gsl_vector_get (x, 1);

  size_t i;

  for (i = 0; i < n; i++)
    {
      /* Jacobian matrix J(i,j) = dfi / dxj, */
      /* where fi = (Yi - yi)/sigma[i],      */
      /*       Yi = A * exp(-lambda * i) + b  */
      /* and the xj are the parameters (A,lambda,b) */
      double t = i;
      double s = sigma[i];
      double e = exp(-lambda * t);
      gsl_matrix_set (J, i, 0, e/s); 
      gsl_matrix_set (J, i, 1, -t * A * e/s);
      gsl_matrix_set (J, i, 2, 1/s);

    }
  return GSL_SUCCESS;
}

int
expb_fdf (const gsl_vector * x, void *params,
          gsl_vector * f, gsl_matrix * J)
{
  expb_f (x, params, f);
  expb_df (x, params, J);

  return GSL_SUCCESS;
}

The main part of the program sets up a Levenberg-Marquardt solver and some simulated random data. The data uses the known parameters (1.0,5.0,0.1) combined with gaussian noise (standard deviation = 0.1) over a range of 40 timesteps. The initial guess for the parameters is chosen as (0.0, 1.0, 0.0).

int
main (void)
{
  const gsl_multifit_fdfsolver_type *T;
  gsl_multifit_fdfsolver *s;

  int status;
  size_t i, iter = 0;

  const size_t n = 40;
  const size_t p = 3;

  gsl_matrix *covar = gsl_matrix_alloc (p, p);

  double y[n], sigma[n];

  struct data d = { n, y, sigma};
  
  gsl_multifit_function_fdf f;

  double x_init[3] = { 1.0, 0.0, 0.0 };

  gsl_vector_view x = gsl_vector_view_array (x_init, p);

  const gsl_rng_type * type;
  gsl_rng * r;

  gsl_rng_env_setup();

  type = gsl_rng_default;
  r = gsl_rng_alloc (type);

  f.f = &expb_f;
  f.df = &expb_df;
  f.fdf = &expb_fdf;
  f.n = n;
  f.p = p;
  f.params = &d;

  /* This is the data to be fitted */

  for (i = 0; i < n; i++)
    {
      double t = i;
      y[i] = 1.0 + 5 * exp (-0.1 * t) 
                 + gsl_ran_gaussian(r, 0.1);
      sigma[i] = 0.1;
      printf("data: %d %g %g\n", i, y[i], sigma[i]);
    };

  T = gsl_multifit_fdfsolver_lmsder;
  s = gsl_multifit_fdfsolver_alloc (T, n, p);
  gsl_multifit_fdfsolver_set (s, &f, &x.vector);

  print_state (iter, s);

  do
    {
      iter++;
      status = gsl_multifit_fdfsolver_iterate (s);

      printf ("status = %s\n", gsl_strerror (status));

      print_state (iter, s);

      if (status)
        break;

      status = gsl_multifit_test_delta (s->dx, s->x,
                                        1e-4, 1e-4);
    }
  while (status == GSL_CONTINUE && iter < 500);

  gsl_multifit_covar (s->J, 0.0, covar);

  gsl_matrix_fprintf (stdout, covar, "%g");

#define FIT(i) gsl_vector_get(s->x, i)
#define ERR(i) sqrt(gsl_matrix_get(covar,i,i))

  printf("A      = %.5f +/- %.5f\n", FIT(0), ERR(0));
  printf("lambda = %.5f +/- %.5f\n", FIT(1), ERR(1));
  printf("b      = %.5f +/- %.5f\n", FIT(2), ERR(2));

  printf ("status = %s\n", gsl_strerror (status));

  gsl_multifit_fdfsolver_free (s);
  return 0;
}

int
print_state (size_t iter, gsl_multifit_fdfsolver * s)
{
  printf ("iter: %3u x = % 15.8f % 15.8f % 15.8f "
          "|f(x)| = %g\n",
          iter,
          gsl_vector_get (s->x, 0), 
          gsl_vector_get (s->x, 1),
          gsl_vector_get (s->x, 2), 
          gsl_blas_dnrm2 (s->f));
}

The iteration terminates when the change in x is smaller than 0.0001, as both an absolute and relative change. Here are the results of running the program,

iter: 0 x = 1.00000000 0.00000000 0.00000000 |f(x)| = 118.574
iter: 1 x = 1.64919392 0.01780040 0.64919392 |f(x)| = 77.2068
iter: 2 x = 2.86269020 0.08032198 1.45913464 |f(x)| = 38.0579
iter: 3 x = 4.97908864 0.11510525 1.06649948 |f(x)| = 10.1548
iter: 4 x = 5.03295496 0.09912462 1.00939075 |f(x)| = 6.4982
iter: 5 x = 5.05811477 0.10055914 0.99819876 |f(x)| = 6.33121
iter: 6 x = 5.05827645 0.10051697 0.99756444 |f(x)| = 6.33119
iter: 7 x = 5.05828006 0.10051819 0.99757710 |f(x)| = 6.33119

A      = 5.05828 +/- 0.05983
lambda = 0.10052 +/- 0.00309
b      = 0.99758 +/- 0.03944
status = success

The approximate values of the parameters are found correctly. The errors on the parameters are given by the square roots of the diagonal elements of the covariance matrix.

fit-exp

References and Further Reading

The MINPACK algorithm is described in the following article,

The following paper is also relevant to the algorithms described in this section,

Physical Constants

This chapter describes macros for the values of physical constants, such as the speed of light, c, and gravitational constant, G. The values are available in different unit systems, including the standard MKS system (meters, kilograms, seconds) and the CGS system (centimeters, grams, seconds), which is commonly used in Astronomy.

The definitions of constants in the MKS system are available in the file `gsl_const_mks.h'. The constants in the CGS system are defined in `gsl_const_cgs.h'. Dimensionless constants, such as the fine structure constant, which are pure numbers are defined in `gsl_const_num.h'.

The full list of constants is described briefly below. Consult the header files themselves for the values of the constants used in the library.

Fundamental Constants

GSL_CONST_MKS_SPEED_OF_LIGHT
The speed of light in vacuum, c.
GSL_CONST_NUM_AVOGADRO
Avogadro's number, N_a.
GSL_CONST_MKS_FARADAY
The molar charge of 1 Faraday.
GSL_CONST_MKS_BOLTZMANN
The Boltzmann constant, k.
GSL_CONST_MKS_MOLAR_GAS
The molar gas constant, R_0.
GSL_CONST_MKS_STANDARD_GAS_VOLUME
The standard gas volume, V_0.
GSL_CONST_MKS_GAUSS
The magnetic field of 1 Gauss.
GSL_CONST_MKS_MICRON
The length of 1 micron.
GSL_CONST_MKS_HECTARE
The area of 1 hectare.
GSL_CONST_MKS_MILES_PER_HOUR
The speed of 1 mile per hour.
GSL_CONST_MKS_KILOMETERS_PER_HOUR
The speed of 1 kilometer per hour.

Astronomy and Astrophysics

GSL_CONST_MKS_ASTRONOMICAL_UNIT
The length of 1 astronomical unit (mean earth-sun distance), au.
GSL_CONST_MKS_GRAVITATIONAL_CONSTANT
The gravitational constant, G.
GSL_CONST_MKS_LIGHT_YEAR
The distance of 1 light-year, ly.
GSL_CONST_MKS_PARSEC
The distance of 1 parsec, pc.
GSL_CONST_MKS_GRAV_ACCEL
The standard gravitational acceleration on Earth, g.
GSL_CONST_MKS_SOLAR_MASS
The mass of the Sun.

Atomic and Nuclear Physics

GSL_CONST_MKS_ELECTRON_CHARGE
The charge of the electron, e.
GSL_CONST_MKS_ELECTRON_VOLT
The energy of 1 electron volt, eV.
GSL_CONST_MKS_UNIFIED_ATOMIC_MASS
The unified atomic mass, amu.
GSL_CONST_MKS_MASS_ELECTRON
The mass of the electron, m_e.
GSL_CONST_MKS_MASS_MUON
The mass of the muon, m_\mu.
GSL_CONST_MKS_MASS_PROTON
The mass of the proton, m_p.
GSL_CONST_MKS_MASS_NEUTRON
The mass of the neutron, m_n.
GSL_CONST_NUM_FINE_STRUCTURE
The electromagnetic fine structure constant \alpha.
GSL_CONST_MKS_RYDBERG
The Rydberg constant, Ry.
GSL_CONST_MKS_ANGSTROM
The length of 1 angstrom.
GSL_CONST_MKS_BARN
The area of 1 barn.
GSL_CONST_MKS_BOHR_MAGNETON
The Bohr Magneton, \mu_B.
GSL_CONST_MKS_NUCLEAR_MAGNETON
The Nuclear Magneton, \mu_N.
GSL_CONST_MKS_ELECTRON_MAGNETIC_MOMENT
The magnetic moment of the electron, \mu_e.
GSL_CONST_MKS_PROTON_MAGNETIC_MOMENT
The magnetic moment of the proton, \mu_p.

Measurement of Time

GSL_CONST_MKS_MINUTE
The number of seconds in 1 minute.
GSL_CONST_MKS_HOUR
The number of seconds in 1 hour.
GSL_CONST_MKS_DAY
The number of seconds in 1 day.
GSL_CONST_MKS_WEEK
The number of seconds in 1 week.

Imperial Units

GSL_CONST_MKS_INCH
The length of 1 inch.
GSL_CONST_MKS_FOOT
The length of 1 foot.
GSL_CONST_MKS_YARD
The length of 1 yard.
GSL_CONST_MKS_MILE
The length of 1 mile.
GSL_CONST_MKS_MIL
The length of 1 mil (1/1000th of an inch).

Nautical Units

GSL_CONST_MKS_NAUTICAL_MILE
The length of 1 nautical mile.
GSL_CONST_MKS_FATHOM
The length of 1 fathom.
GSL_CONST_MKS_KNOT
The speed of 1 knot.

Printers Units

GSL_CONST_MKS_POINT
The length of 1 printer's point (1/72 inch).
GSL_CONST_MKS_TEXPOINT
The length of 1 TeX point (1/72.27 inch).

Volume

GSL_CONST_MKS_ACRE
The area of 1 acre.
GSL_CONST_MKS_LITER
The volume of 1 liter.
GSL_CONST_MKS_US_GALLON
The volume of 1 US gallon.
GSL_CONST_MKS_CANADIAN_GALLON
The volume of 1 Canadian gallon.
GSL_CONST_MKS_UK_GALLON
The volume of 1 UK gallon.
GSL_CONST_MKS_QUART
The volume of 1 quart.
GSL_CONST_MKS_PINT
The volume of 1 pint.

Mass and Weight

GSL_CONST_MKS_POUND_MASS
The mass of 1 pound.
GSL_CONST_MKS_OUNCE_MASS
The mass of 1 ounce.
GSL_CONST_MKS_TON
The mass of 1 ton.
GSL_CONST_MKS_METRIC_TON
The mass of 1 metric ton (1000 kg).
GSL_CONST_MKS_UK_TON
The mass of 1 UK ton.
GSL_CONST_MKS_TROY_OUNCE
The mass of 1 troy ounce.
GSL_CONST_MKS_CARAT
The mass of 1 carat.
GSL_CONST_MKS_GRAM_FORCE
The force of 1 gram weight.
GSL_CONST_MKS_POUND_FORCE
The force of 1 pound weight.
GSL_CONST_MKS_KILOPOUND_FORCE
The force of 1 kilopound weight.
GSL_CONST_MKS_POUNDAL
The force of 1 poundal.

Thermal Energy and Power

GSL_CONST_MKS_CALORIE
The energy of 1 calorie.
GSL_CONST_MKS_BTU
The energy of 1 British Thermal Unit, btu.
GSL_CONST_MKS_THERM
The energy of 1 Therm.
GSL_CONST_MKS_HORSEPOWER
The power of 1 horsepower.

Pressure

GSL_CONST_MKS_BAR
The pressure of 1 bar.
GSL_CONST_MKS_STD_ATMOSPHERE
The pressure of 1 standard atmosphere.
GSL_CONST_MKS_TORR
The pressure of 1 torr.
GSL_CONST_MKS_METER_OF_MERCURY
The pressure of 1 meter of mercury.
GSL_CONST_MKS_INCH_OF_MERCURY
The pressure of 1 inch of mercury.
GSL_CONST_MKS_INCH_OF_WATER
The pressure of 1 inch of water.
GSL_CONST_MKS_PSI
The pressure of 1 pound per square inch.

Viscosity

GSL_CONST_MKS_POISE
The dynamic viscosity of 1 poise.
GSL_CONST_MKS_STOKES
The kinematic viscosity of 1 stokes.

Light and Illumination

GSL_CONST_MKS_STILB
The luminance of 1 stilb.
GSL_CONST_MKS_LUMEN
The luminous flux of 1 lumen.
GSL_CONST_MKS_LUX
The illuminance of 1 lux.
GSL_CONST_MKS_PHOT
The illuminance of 1 phot.
GSL_CONST_MKS_FOOTCANDLE
The illuminance of 1 footcandle.
GSL_CONST_MKS_LAMBERT
The luminance of 1 lambert.
GSL_CONST_MKS_FOOTLAMBERT
The luminance of 1 footlambert.

Radioactivity

GSL_CONST_MKS_CURIE
The activity of 1 curie.
GSL_CONST_MKS_ROENTGEN
The exposure of 1 roentgen.
GSL_CONST_MKS_RAD
The absorbed dose of 1 rad.

Examples

The following program demonstrates the use of the physical constants in a calculation. In this case, the goal is to calculate the range of light-travel times from Earth to Mars.

The required data is the average distance of each planet from the Sun in astronomical units (the eccentricities of the orbits will be neglected for the purposes of this calculation). The average radius of the orbit of Mars is 1.52 astronomical units, and for the orbit of Earth it is 1 astronomical unit (by definition). These values are combined with the MKS values of the constants for the speed of light and the length of an astronomical unit to produce a result for the shortest and longest light-travel times in seconds. The figures are converted into minutes before being displayed.

#include <stdio.h>
#include <gsl/gsl_const_mks.h>

int
main (void)
{
  double c  = GSL_CONST_MKS_SPEED_OF_LIGHT;
  double au = GSL_CONST_MKS_ASTRONOMICAL_UNIT;
  double minutes = GSL_CONST_MKS_MINUTE;

  /* distance stored in meters */
  double r_earth = 1.00 * au;  
  double r_mars  = 1.52 * au;

  double t_min, t_max;

  t_min = (r_mars - r_earth) / c;
  t_max = (r_mars + r_earth) / c;

  printf("light travel time from Earth to Mars:\n");
  printf("minimum = %.1f minutes\n", t_min / minutes);
  printf("maximum = %.1f minutes\n", t_max / minutes);

  return 0;
}

Here is the output from the program,

light travel time from Earth to Mars:
minimum = 4.3 minutes
maximum = 21.0 minutes

References and Further Reading

Further information on the values of physical constants is available from the NIST website,

IEEE floating-point arithmetic

This chapter describes functions for examining the representation of floating point numbers and controlling the floating point environment of your program. The functions described in this chapter are declared in the header file `gsl_ieee_utils.h'.

Representation of floating point numbers

The IEEE Standard for Binary Floating-Point Arithmetic defines binary formats for single and double precision numbers. Each number is composed of three parts: a sign bit (s), an exponent (E) and a fraction (f). The numerical value of the combination (s,E,f) is given by the following formula,

The sign bit is either zero or one. The exponent ranges from a minimum value E_min to a maximum value E_max depending on the precision. The exponent is converted to an unsigned number e, known as the biased exponent, for storage by adding a bias parameter, e = E + bias. The sequence fffff... represents the digits of the binary fraction f. The binary digits are stored in normalized form, by adjusting the exponent to give a leading digit of 1. Since the leading digit is always 1 for normalized numbers it is assumed implicitly and does not have to be stored. Numbers smaller than 2^(E_min) are be stored in denormalized form with a leading zero,

This allows gradual underflow down to 2^(E_min - p) for p bits of precision. A zero is encoded with the special exponent of 2^(E_min - 1) and infinities with the exponent of 2^(E_max + 1).

The format for single precision numbers uses 32 bits divided in the following way,

seeeeeeeefffffffffffffffffffffff
    
s = sign bit, 1 bit
e = exponent, 8 bits  (E_min=-126, E_max=127, bias=127)
f = fraction, 23 bits

The format for double precision numbers uses 64 bits divided in the following way,

seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff

s = sign bit, 1 bit
e = exponent, 11 bits  (E_min=-1022, E_max=1023, bias=1023)
f = fraction, 52 bits

It is often useful to be able to investigate the behavior of a calculation at the bit-level and the library provides functions for printing the IEEE representations in a human-readable form.

Function: void gsl_ieee_fprintf_float (FILE * stream, const float * x)
Function: void gsl_ieee_fprintf_double (FILE * stream, const double * x)
These functions output a formatted version of the IEEE floating-point number pointed to by x to the stream stream. A pointer is used to pass the number indirectly, to avoid any undesired promotion from float to double. The output takes one of the following forms,
NaN
the Not-a-Number symbol
Inf, -Inf
positive or negative infinity
1.fffff...*2^E, -1.fffff...*2^E
a normalized floating point number
0.fffff...*2^E, -0.fffff...*2^E
a denormalized floating point number
0, -0
positive or negative zero

The output can be used directly in GNU Emacs Calc mode by preceding it with 2# to indicate binary.

Function: void gsl_ieee_printf_float (const float * x)
Function: void gsl_ieee_printf_double (const double * x)
These functions output a formatted version of the IEEE floating-point number pointed to by x to the stream stdout.
The following program demonstrates the use of the functions by printing the single and double precision representations of the fraction 1/3. For comparison the representation of the value promoted from single to double precision is also printed.
#include <stdio.h>
#include <gsl/gsl_ieee_utils.h>

int
main (void) 
{
  float f = 1.0/3.0;
  double d = 1.0/3.0;

  double fd = f; /* promote from float to double */
  
  printf(" f="); gsl_ieee_printf_float(&f); 
  printf("\n");

  printf("fd="); gsl_ieee_printf_double(&fd); 
  printf("\n");

  printf(" d="); gsl_ieee_printf_double(&d); 
  printf("\n");

  return 0;
}

The binary representation of 1/3 is 0.01010101... . The output below shows that the IEEE format normalizes this fraction to give a leading digit of 1,

 f= 1.01010101010101010101011*2^-2
fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
 d= 1.0101010101010101010101010101010101010101010101010101*2^-2

The output also shows that a single-precision number is promoted to double-precision by adding zeros in the binary representation.

Setting up your IEEE environment

The IEEE standard defines several modes for controlling the behavior of floating point operations. These modes specify the important properties of computer arithmetic: the direction used for rounding (e.g. whether numbers should be rounded up, down or to the nearest number), the rounding precision and how the program should handle arithmetic exceptions, such as division by zero.

Many of these features can now be controlled via standard functions such as fpsetround, which should be used whenever they are available. Unfortunately in the past there has been no universal API for controlling their behavior -- each system has had its own way of accessing them. For example, the Linux kernel provides the function __setfpucw (set-fpu-control-word) to set IEEE modes, while HP-UX and Solaris use the functions fpsetround and fpsetmask. To help you write portable programs GSL allows you to specify modes in a platform-independent way using the environment variable GSL_IEEE_MODE. The library then takes care of all the necessary machine-specific initializations for you when you call the function gsl_ieee_env_setup.

Function: void gsl_ieee_env_setup ()
This function reads the environment variable GSL_IEEE_MODE and attempts to set up the corresponding specified IEEE modes. The environment variable should be a list of keywords, separated by commas, like this,
GSL_IEEE_MODE = "keyword,keyword,..."

where keyword is one of the following mode-names,

If GSL_IEEE_MODE is empty or undefined then the function returns immediately and no attempt is made to change the system's IEEE mode. When the modes from GSL_IEEE_MODE are turned on the function prints a short message showing the new settings to remind you that the results of the program will be affected.

If the requested modes are not supported by the platform being used then the function calls the error handler and returns an error code of GSL_EUNSUP.

The following combination of modes is convenient for many purposes,

GSL_IEEE_MODE="double-precision,"\
                "mask-underflow,"\
                  "mask-denormalized"

This choice ignores any errors relating to small numbers (either denormalized, or underflowing to zero) but traps overflows, division by zero and invalid operations.

To demonstrate the effects of different rounding modes consider the following program which computes e, the base of natural logarithms, by summing a rapidly-decreasing series,

#include <math.h>
#include <stdio.h>
#include <gsl/gsl_ieee_utils.h>

int
main (void)
{
  double x = 1, oldsum = 0, sum = 0; 
  int i = 0;

  gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */

  do 
    {
      i++;
      
      oldsum = sum;
      sum += x;
      x = x / i;
      
      printf("i=%2d sum=%.18f error=%g\n",
             i, sum, sum - M_E);

      if (i > 30)
         break;
    }  
  while (sum != oldsum);

  return 0;
}

Here are the results of running the program in round-to-nearest mode. This is the IEEE default so it isn't really necessary to specify it here,

GSL_IEEE_MODE="round-to-nearest" ./a.out 
i= 1 sum=1.000000000000000000 error=-1.71828
i= 2 sum=2.000000000000000000 error=-0.718282
....
i=18 sum=2.718281828459045535 error=4.44089e-16
i=19 sum=2.718281828459045535 error=4.44089e-16

After nineteen terms the sum converges to within @c{$4 \times 10^{-16}$} 4 \times 10^-16 of the correct value. If we now change the rounding mode to round-down the final result is less accurate,

GSL_IEEE_MODE="round-down" ./a.out 
i= 1 sum=1.000000000000000000 error=-1.71828
....
i=19 sum=2.718281828459041094 error=-3.9968e-15

The result is about 4 \times 10^-15 below the correct value, an order of magnitude worse than the result obtained in the round-to-nearest mode.

If we change to rounding mode to round-up then the series no longer converges (the reason is that when we add each term to the sum the final result is always rounded up. This is guaranteed to increase the sum by at least one tick on each iteration). To avoid this problem we would need to use a safer converge criterion, such as while (fabs(sum - oldsum) > epsilon), with a suitably chosen value of epsilon.

Finally we can see the effect of computing the sum using single-precision rounding, in the default round-to-nearest mode. In this case the program thinks it is still using double precision numbers but the CPU rounds the result of each floating point operation to single-precision accuracy. This simulates the effect of writing the program using single-precision float variables instead of double variables. The iteration stops after about half the number of iterations and the final result is much less accurate,

GSL_IEEE_MODE="single-precision" ./a.out 
....
i=12 sum=2.718281984329223633 error=1.5587e-07

with an error of O(10^-7), which corresponds to single precision accuracy (about 1 part in 10^7). Continuing the iterations further does not decrease the error because all the subsequent results are rounded to the same value.

References and Further Reading

The reference for the IEEE standard is,

A more pedagogical introduction to the standard can be found in the paper "What Every Computer Scientist Should Know About Floating-Point Arithmetic".

Debugging Numerical Programs

This chapter describes some tips and tricks for debugging numerical programs which use GSL.

Using gdb

Any errors reported by the library are passed to the function gsl_error. By running your programs under gdb and setting a breakpoint in this function you can automatically catch any library errors. You can add a breakpoint for every session by putting

break gsl_error

into your `.gdbinit' file in the directory where your program is started.

If the breakpoint catches an error then you can use a backtrace (bt) to see the call-tree, and the arguments which possibly caused the error. By moving up into the calling function you can investigate the values of variable at that point. Here is an example from the program fft/test_trap, which contains the following line,

status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);

The function gsl_fft_complex_wavetable_alloc takes the length of an FFT as its first argument. When this line is executed an error will be generated because the length of an FFT is not allowed to be zero.

To debug this problem we start gdb, using the file `.gdbinit' to define a breakpoint in gsl_error,

bash$ gdb test_trap

GDB is free software and you are welcome to distribute copies
of it under certain conditions; type "show copying" to see
the conditions.  There is absolutely no warranty for GDB;
type "show warranty" for details.  GDB 4.16 (i586-debian-linux), 
Copyright 1996 Free Software Foundation, Inc.

Breakpoint 1 at 0x8050b1e: file error.c, line 14.

When we run the program this breakpoint catches the error and shows the reason for it.

(gdb) run
Starting program: test_trap 

Breakpoint 1, gsl_error (reason=0x8052b0d 
    "length n must be positive integer", 
    file=0x8052b04 "c_init.c", line=108, gsl_errno=1) 
    at error.c:14
14        if (gsl_error_handler) 

The first argument of gsl_error is always a string describing the error. Now we can look at the backtrace to see what caused the problem,

(gdb) bt
#0  gsl_error (reason=0x8052b0d 
    "length n must be positive integer", 
    file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
    at error.c:14
#1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
    wavetable=0xbffff778) at c_init.c:108
#2  0x8048a00 in main (argc=1, argv=0xbffff9bc) 
    at test_trap.c:94
#3  0x80488be in ___crt_dummy__ ()

We can see that the error was generated in the function gsl_fft_complex_wavetable_alloc when it was called with an argument of n=0. The original call came from line 94 in the file `test_trap.c'.

By moving up to the level of the original call we can find the line that caused the error,

(gdb) up
#1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
    wavetable=0xbffff778) at c_init.c:108
108   GSL_ERROR ("length n must be positive integer", GSL_EDOM);
(gdb) up
#2  0x8048a00 in main (argc=1, argv=0xbffff9bc) 
    at test_trap.c:94
94    status = gsl_fft_complex_wavetable_alloc (0,
        &complex_wavetable);

Thus we have found the line that caused the problem. From this point we could also print out the values of other variables such as complex_wavetable.

Examining floating point registers

The contents of floating point registers can be examined using the command info float (not available on all platforms).

(gdb) info float
     st0: 0xc4018b895aa17a945000  Valid Normal -7.838871e+308
     st1: 0x3ff9ea3f50e4d7275000  Valid Normal 0.0285946
     st2: 0x3fe790c64ce27dad4800  Valid Normal 6.7415931e-08
     st3: 0x3ffaa3ef0df6607d7800  Spec  Normal 0.0400229
     st4: 0x3c028000000000000000  Valid Normal 4.4501477e-308
     st5: 0x3ffef5412c22219d9000  Zero  Normal 0.9580257
     st6: 0x3fff8000000000000000  Valid Normal 1
     st7: 0xc4028b65a1f6d243c800  Valid Normal -1.566206e+309
   fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
   fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS
    ftag: 0x3fff
     fip: 0x08048b5c
     fcs: 0x051a0023
  fopoff: 0x08086820
  fopsel: 0x002b

Individual registers can be examined using the variables $reg, where reg is the register name.

(gdb) p $st1 
$1 = 0.02859464454261210347719

Handling floating point exceptions

It is possible to stop the program whenever a SIGFPE floating point exception occurs. This can be useful for finding the cause of an unexpected infinity or NaN. The current handler settings can be shown with the command info signal SIGFPE.

(gdb) info signal SIGFPE
Signal  Stop  Print  Pass to program Description
SIGFPE  Yes   Yes    Yes             Arithmetic exception

Unless the program uses a signal handler the default setting should be changed so that SIGFPE is not passed to the program, as this would cause it to exit. The command handle SIGFPE stop nopass prevents this.

(gdb) handle SIGFPE stop nopass
Signal  Stop  Print  Pass to program Description
SIGFPE  Yes   Yes    No              Arithmetic exception

Depending on the platform it may be necessary to instruct the kernel to generate signals for floating point exceptions. For programs using GSL this can be achieved using the GSL_IEEE_MODE environment variable in conjunction with the function gsl_ieee_env_setup() as described in see section IEEE floating-point arithmetic.

(gdb) set env GSL_IEEE_MODE=double-precision

GCC warning options for numerical programs

Writing reliable numerical programs in C requires great care. The following GCC warning options are recommended when compiling numerical programs:

gcc -ansi -pedantic -Werror -Wall -W 
  -Wmissing-prototypes -Wstrict-prototypes 
  -Wtraditional -Wconversion -Wshadow
  -Wpointer-arith -Wcast-qual -Wcast-align 
  -Wwrite-strings -Wnested-externs 
  -fshort-enums -fno-common -Dinline= -g -O4

For details of each option consult the manual Using and Porting GCC. The following table gives a brief explanation of what types of errors these options catch.

-ansi -pedantic
Use ANSI C, and reject any non-ANSI extensions. These flags help in writing portable programs that will compile on other systems.
-Werror
Consider warnings to be errors, so that compilation stops. This prevents warnings from scrolling off the top of the screen and being lost. You won't be able to compile the program until it is completely warning-free.
-Wall
This turns on a set of warnings for common programming problems. You need -Wall, but it is not enough on its own.
-O4
Turn on optimization. The warnings for uninitialized variables in -Wall rely on the optimizer to analyze the code. If there is no optimization then the warnings aren't generated.
-W
This turns on some extra warnings not included in -Wall, such as missing return values and comparisons between signed and unsigned integers.
-Wmissing-prototypes -Wstrict-prototypes
Warn if there are any missing or inconsistent prototypes. Without prototypes it is harder to detect problems with incorrect arguments.
-Wtraditional
This warns about certain constructs that behave differently in traditional and ANSI C. Whether the traditional or ANSI interpretation is used might be unpredictable on other compilers.
-Wconversion
The main use of this option is to warn about conversions from signed to unsigned integers. For example, unsigned int x = -1. If you need to perform such a conversion you can use an explicit cast.
-Wshadow
This warns whenever a local variable shadows another local variable. If two variables have the same name then it is a potential source of confusion.
-Wpointer-arith -Wcast-qual -Wcast-align
These options warn if you try to do pointer arithmetic for types which don't have a size, such as void, if you remove a const cast from a pointer, or if you cast a pointer to a type which has a different size, causing an invalid alignment.
-Wwrite-strings
This option gives string constants a const qualifier so that it will be a compile-time error to attempt to overwrite them.
-fshort-enums
This option makes the type of enum as short as possible. Normally this makes an enum different from an int. Consequently any attempts to assign a pointer-to-int to a pointer-to-enum will generate a cast-alignment warning.
-fno-common
This option prevents global variables being simultaneously defined in different object files (you get an error at link time). Such a variable should be defined in one file and referred to in other files with an extern declaration.
-Wnested-externs
This warns if an extern declaration is encountered within an function.
-Dinline=
The inline keyword is not part of ANSI C. Thus if you want to use -ansi with a program which uses inline functions you can use this preprocessor definition to remove the inline keywords.
-g
It always makes sense to put debugging symbols in the executable so that you can debug it using gdb. The only effect of debugging symbols is to increase the size of the file, and you can use the strip command to remove them later if necessary.

References and Further Reading

The following books are essential reading for anyone writing and debugging numerical programs with GCC and GDB.

Contributors to GSL

(See the AUTHORS file in the distribution for up-to-date information.)

Mark Galassi
Conceived GSL (with James Theiler) and wrote the design document. Wrote the simulated annealing package and the relevant chapter in the manual.
James Theiler
Conceived GSL (with Mark Galassi). Wrote the random number generators and the relevant chapter in this manual.
Jim Davies
Wrote the statistical routines and the relevant chapter in this manual.
Brian Gough
FFTs, numerical integration, random number generators and distributions, root finding, minimization and fitting, polynomial solvers, complex numbers, physical constants, permutations, vector and matrix functions, histograms, statistics, ieee-utils, revised CBLAS Level 2 & 3, matrix decompositions and eigensystems.
Reid Priedhorsky
Wrote and documented the initial version of the root finding routines while at Los Alamos National Laboratory, Mathematical Modeling and Analysis Group.
Gerard Jungman
Series acceleration, ODEs, BLAS, Linear Algebra, Eigensystems, Hankel Transforms.
Mike Booth
Wrote the Monte Carlo library.
Jorma Olavi T@"ahtinen
Wrote the initial complex arithmetic functions.
Thomas Walter
Wrote the initial heapsort routines and cholesky decomposition.
Fabrice Rossi
Multidimensional minimization.

Autoconf Macros

The following autoconf test will check for extern inline,

dnl Check for "extern inline", using a modified version
dnl of the test for AC_C_INLINE from acspecific.mt
dnl
AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,
[ac_cv_c_extern_inline=no
AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
extern $ac_cv_c_inline double foo(double x) { return x+1.0; };
double foo (double x) { return x + 1.0; };], 
[  foo(1.0)  ],
[ac_cv_c_extern_inline="yes"])
])

if test "$ac_cv_c_extern_inline" != no ; then
  AC_DEFINE(HAVE_INLINE,1)
  AC_SUBST(HAVE_INLINE)
fi

GSL CBLAS Library

The prototypes for the low-level CBLAS functions are declared in the file gsl_cblas.h. For the definition of the functions consult the documentation available from Netlib (see section References and Further Reading).

Level 1

Function: float cblas_sdsdot (const int N, const float alpha, const float *x, const int incx, const float *y, const int incy)

Function: double cblas_dsdot (const int N, const float *x, const int incx, const float *y, const int incy)

Function: float cblas_sdot (const int N, const float *x, const int incx, const float *y, const int incy)

Function: double cblas_ddot (const int N, const double *x, const int incx, const double *y, const int incy)

Function: void cblas_cdotu_sub (const int N, const void *x, const int incx, const void *y, const int incy, void *dotu)

Function: void cblas_cdotc_sub (const int N, const void *x, const int incx, const void *y, const int incy, void *dotc)

Function: void cblas_zdotu_sub (const int N, const void *x, const int incx, const void *y, const int incy, void *dotu)

Function: void cblas_zdotc_sub (const int N, const void *x, const int incx, const void *y, const int incy, void *dotc)

Function: float cblas_snrm2 (const int N, const float *x, const int incx)

Function: float cblas_sasum (const int N, const float *x, const int incx)

Function: double cblas_dnrm2 (const int N, const double *x, const int incx)

Function: double cblas_dasum (const int N, const double *x, const int incx)

Function: float cblas_scnrm2 (const int N, const void *x, const int incx)

Function: float cblas_scasum (const int N, const void *x, const int incx)

Function: double cblas_dznrm2 (const int N, const void *x, const int incx)

Function: double cblas_dzasum (const int N, const void *x, const int incx)

Function: CBLAS_INDEX cblas_isamax (const int N, const float *x, const int incx)

Function: CBLAS_INDEX cblas_idamax (const int N, const double *x, const int incx)

Function: CBLAS_INDEX cblas_icamax (const int N, const void *x, const int incx)

Function: CBLAS_INDEX cblas_izamax (const int N, const void *x, const int incx)

Function: void cblas_sswap (const int N, float *x, const int incx, float *y, const int incy)

Function: void cblas_scopy (const int N, const float *x, const int incx, float *y, const int incy)

Function: void cblas_saxpy (const int N, const float alpha, const float *x, const int incx, float *y, const int incy)

Function: void cblas_dswap (const int N, double *x, const int incx, double *y, const int incy)

Function: void cblas_dcopy (const int N, const double *x, const int incx, double *y, const int incy)

Function: void cblas_daxpy (const int N, const double alpha, const double *x, const int incx, double *y, const int incy)

Function: void cblas_cswap (const int N, void *x, const int incx, void *y, const int incy)

Function: void cblas_ccopy (const int N, const void *x, const int incx, void *y, const int incy)

Function: void cblas_caxpy (const int N, const void *alpha, const void *x, const int incx, void *y, const int incy)

Function: void cblas_zswap (const int N, void *x, const int incx, void *y, const int incy)

Function: void cblas_zcopy (const int N, const void *x, const int incx, void *y, const int incy)

Function: void cblas_zaxpy (const int N, const void *alpha, const void *x, const int incx, void *y, const int incy)

Function: void cblas_srotg (float *a, float *b, float *c, float *s)

Function: void cblas_srotmg (float *d1, float *d2, float *b1, const float b2, float *P)

Function: void cblas_srot (const int N, float *x, const int incx, float *y, const int incy, const float c, const float s)

Function: void cblas_srotm (const int N, float *x, const int incx, float *y, const int incy, const float *P)

Function: void cblas_drotg (double *a, double *b, double *c, double *s)

Function: void cblas_drotmg (double *d1, double *d2, double *b1, const double b2, double *P)

Function: void cblas_drot (const int N, double *x, const int incx, double *y, const int incy, const double c, const double s)

Function: void cblas_drotm (const int N, double *x, const int incx, double *y, const int incy, const double *P)

Function: void cblas_sscal (const int N, const float alpha, float *x, const int incx)

Function: void cblas_dscal (const int N, const double alpha, double *x, const int incx)

Function: void cblas_cscal (const int N, const void *alpha, void *x, const int incx)

Function: void cblas_zscal (const int N, const void *alpha, void *x, const int incx)

Function: void cblas_csscal (const int N, const float alpha, void *x, const int incx)

Function: void cblas_zdscal (const int N, const double alpha, void *x, const int incx)

Level 2

Function: void cblas_sgemv (const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M, const int N, const float alpha, const float *A, const int lda, const float *x, const int incx, const float beta, float *y, const int incy)

Function: void cblas_sgbmv (const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M, const int N, const int KL, const int KU, const float alpha, const float *A, const int lda, const float *x, const int incx, const float beta, float *y, const int incy)

Function: void cblas_strmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const float *A, const int lda, float *x, const int incx)

Function: void cblas_stbmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const float *A, const int lda, float *x, const int incx)

Function: void cblas_stpmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const float *Ap, float *x, const int incx)

Function: void cblas_strsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const float *A, const int lda, float *x, const int incx)

Function: void cblas_stbsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const float *A, const int lda, float *x, const int incx)

Function: void cblas_stpsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const float *Ap, float *x, const int incx)

Function: void cblas_dgemv (const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M, const int N, const double alpha, const double *A, const int lda, const double *x, const int incx, const double beta, double *y, const int incy)

Function: void cblas_dgbmv (const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M, const int N, const int KL, const int KU, const double alpha, const double *A, const int lda, const double *x, const int incx, const double beta, double *y, const int incy)

Function: void cblas_dtrmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const double *A, const int lda, double *x, const int incx)

Function: void cblas_dtbmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const double *A, const int lda, double *x, const int incx)

Function: void cblas_dtpmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const double *Ap, double *x, const int incx)

Function: void cblas_dtrsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const double *A, const int lda, double *x, const int incx)

Function: void cblas_dtbsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const double *A, const int lda, double *x, const int incx)

Function: void cblas_dtpsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const double *Ap, double *x, const int incx)

Function: void cblas_cgemv (const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M, const int N, const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_cgbmv (const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M, const int N, const int KL, const int KU, const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_ctrmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *A, const int lda, void *x, const int incx)

Function: void cblas_ctbmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const void *A, const int lda, void *x, const int incx)

Function: void cblas_ctpmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *Ap, void *x, const int incx)

Function: void cblas_ctrsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *A, const int lda, void *x, const int incx)

Function: void cblas_ctbsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const void *A, const int lda, void *x, const int incx)

Function: void cblas_ctpsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *Ap, void *x, const int incx)

Function: void cblas_zgemv (const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M, const int N, const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_zgbmv (const enum CBLAS_ORDER order, const enum CBLAS_TRANSPOSE TransA, const int M, const int N, const int KL, const int KU, const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_ztrmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *A, const int lda, void *x, const int incx)

Function: void cblas_ztbmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const void *A, const int lda, void *x, const int incx)

Function: void cblas_ztpmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *Ap, void *x, const int incx)

Function: void cblas_ztrsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *A, const int lda, void *x, const int incx)

Function: void cblas_ztbsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const int K, const void *A, const int lda, void *x, const int incx)

Function: void cblas_ztpsv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int N, const void *Ap, void *x, const int incx)

Function: void cblas_ssymv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const float alpha, const float *A, const int lda, const float *x, const int incx, const float beta, float *y, const int incy)

Function: void cblas_ssbmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const int K, const float alpha, const float *A, const int lda, const float *x, const int incx, const float beta, float *y, const int incy)

Function: void cblas_sspmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const float alpha, const float *Ap, const float *x, const int incx, const float beta, float *y, const int incy)

Function: void cblas_sger (const enum CBLAS_ORDER order, const int M, const int N, const float alpha, const float *x, const int incx, const float *y, const int incy, float *A, const int lda)

Function: void cblas_ssyr (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const float alpha, const float *x, const int incx, float *A, const int lda)

Function: void cblas_sspr (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const float alpha, const float *x, const int incx, float *Ap)

Function: void cblas_ssyr2 (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const float alpha, const float *x, const int incx, const float *y, const int incy, float *A, const int lda)

Function: void cblas_sspr2 (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const float alpha, const float *x, const int incx, const float *y, const int incy, float *A)

Function: void cblas_dsymv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const double alpha, const double *A, const int lda, const double *x, const int incx, const double beta, double *y, const int incy)

Function: void cblas_dsbmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const int K, const double alpha, const double *A, const int lda, const double *x, const int incx, const double beta, double *y, const int incy)

Function: void cblas_dspmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const double alpha, const double *Ap, const double *x, const int incx, const double beta, double *y, const int incy)

Function: void cblas_dger (const enum CBLAS_ORDER order, const int M, const int N, const double alpha, const double *x, const int incx, const double *y, const int incy, double *A, const int lda)

Function: void cblas_dsyr (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const double alpha, const double *x, const int incx, double *A, const int lda)

Function: void cblas_dspr (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const double alpha, const double *x, const int incx, double *Ap)

Function: void cblas_dsyr2 (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const double alpha, const double *x, const int incx, const double *y, const int incy, double *A, const int lda)

Function: void cblas_dspr2 (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const double alpha, const double *x, const int incx, const double *y, const int incy, double *A)

Function: void cblas_chemv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_chbmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const int K, const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_chpmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void *alpha, const void *Ap, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_cgeru (const enum CBLAS_ORDER order, const int M, const int N, const void *alpha, const void *x, const int incx, const void *y, const int incy, void *A, const int lda)

Function: void cblas_cgerc (const enum CBLAS_ORDER order, const int M, const int N, const void *alpha, const void *x, const int incx, const void *y, const int incy, void *A, const int lda)

Function: void cblas_cher (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const float alpha, const void *x, const int incx, void *A, const int lda)

Function: void cblas_chpr (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const float alpha, const void *x, const int incx, void *A)

Function: void cblas_cher2 (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void *alpha, const void *x, const int incx, const void *y, const int incy, void *A, const int lda)

Function: void cblas_chpr2 (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void *alpha, const void *x, const int incx, const void *y, const int incy, void *Ap)

Function: void cblas_zhemv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_zhbmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const int K, const void *alpha, const void *A, const int lda, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_zhpmv (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void *alpha, const void *Ap, const void *x, const int incx, const void *beta, void *y, const int incy)

Function: void cblas_zgeru (const enum CBLAS_ORDER order, const int M, const int N, const void *alpha, const void *x, const int incx, const void *y, const int incy, void *A, const int lda)

Function: void cblas_zgerc (const enum CBLAS_ORDER order, const int M, const int N, const void *alpha, const void *x, const int incx, const void *y, const int incy, void *A, const int lda)

Function: void cblas_zher (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const double alpha, const void *x, const int incx, void *A, const int lda)

Function: void cblas_zhpr (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const double alpha, const void *x, const int incx, void *A)

Function: void cblas_zher2 (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void *alpha, const void *x, const int incx, const void *y, const int incy, void *A, const int lda)

Function: void cblas_zhpr2 (const enum CBLAS_ORDER order, const enum CBLAS_UPLO Uplo, const int N, const void *alpha, const void *x, const int incx, const void *y, const int incy, void *Ap)

Level 3

Function: void cblas_sgemm (const enum CBLAS_ORDER Order, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const float alpha, const float *A, const int lda, const float *B, const int ldb, const float beta, float *C, const int ldc)

Function: void cblas_ssymm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M, const int N, const float alpha, const float *A, const int lda, const float *B, const int ldb, const float beta, float *C, const int ldc)

Function: void cblas_ssyrk (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const float alpha, const float *A, const int lda, const float beta, float *C, const int ldc)

Function: void cblas_ssyr2k (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const float alpha, const float *A, const int lda, const float *B, const int ldb, const float beta, float *C, const int ldc)

Function: void cblas_strmm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int M, const int N, const float alpha, const float *A, const int lda, float *B, const int ldb)

Function: void cblas_strsm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int M, const int N, const float alpha, const float *A, const int lda, float *B, const int ldb)

Function: void cblas_dgemm (const enum CBLAS_ORDER Order, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const double alpha, const double *A, const int lda, const double *B, const int ldb, const double beta, double *C, const int ldc)

Function: void cblas_dsymm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M, const int N, const double alpha, const double *A, const int lda, const double *B, const int ldb, const double beta, double *C, const int ldc)

Function: void cblas_dsyrk (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const double alpha, const double *A, const int lda, const double beta, double *C, const int ldc)

Function: void cblas_dsyr2k (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const double alpha, const double *A, const int lda, const double *B, const int ldb, const double beta, double *C, const int ldc)

Function: void cblas_dtrmm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int M, const int N, const double alpha, const double *A, const int lda, double *B, const int ldb)

Function: void cblas_dtrsm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int M, const int N, const double alpha, const double *A, const int lda, double *B, const int ldb)

Function: void cblas_cgemm (const enum CBLAS_ORDER Order, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)

Function: void cblas_csymm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M, const int N, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)

Function: void cblas_csyrk (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const int lda, const void *beta, void *C, const int ldc)

Function: void cblas_csyr2k (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)

Function: void cblas_ctrmm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int M, const int N, const void *alpha, const void *A, const int lda, void *B, const int ldb)

Function: void cblas_ctrsm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int M, const int N, const void *alpha, const void *A, const int lda, void *B, const int ldb)

Function: void cblas_zgemm (const enum CBLAS_ORDER Order, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)

Function: void cblas_zsymm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M, const int N, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)

Function: void cblas_zsyrk (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const int lda, const void *beta, void *C, const int ldc)

Function: void cblas_zsyr2k (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)

Function: void cblas_ztrmm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int M, const int N, const void *alpha, const void *A, const int lda, void *B, const int ldb)

Function: void cblas_ztrsm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int M, const int N, const void *alpha, const void *A, const int lda, void *B, const int ldb)

Function: void cblas_chemm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M, const int N, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)

Function: void cblas_cherk (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const float alpha, const void *A, const int lda, const float beta, void *C, const int ldc)

Function: void cblas_cher2k (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const float beta, void *C, const int ldc)

Function: void cblas_zhemm (const enum CBLAS_ORDER Order, const enum CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M, const int N, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const void *beta, void *C, const int ldc)

Function: void cblas_zherk (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const double alpha, const void *A, const int lda, const double beta, void *C, const int ldc)

Function: void cblas_zher2k (const enum CBLAS_ORDER Order, const enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int N, const int K, const void *alpha, const void *A, const int lda, const void *B, const int ldb, const double beta, void *C, const int ldc)

Function: void cblas_xerbla (int p, const char *rout, const char *form, ...)

Examples

The following program computes the product of two matrices using the Level-3 BLAS function SGEMM,

The matrices are stored in row major order but could be stored in column major order if the first argument of the call to cblas_sgemm was changed to CblasColMajor.

#include <stdio.h>
#include <gsl/gsl_cblas.h>

int
main (void)
{
  int lda = 3;

  float A[] = { 0.11, 0.12, 0.13,
                0.21, 0.22, 0.23 };

  int ldb = 2;
  
  float B[] = { 1011, 1012,
                1021, 1022,
                1031, 1032 };

  int ldc = 2;

  float C[] = { 0.00, 0.00,
                0.00, 0.00 };

  /* Compute C = A B */

  cblas_sgemm (CblasRowMajor, 
               CblasNoTrans, CblasNoTrans, 2, 2, 3,
               1.0, A, lda, B, ldb, 0.0, C, ldc);

  printf("[ %g, %g\n", C[0], C[1]);
  printf("  %g, %g ]\n", C[2], C[3]);

  return 0;  
}

To compile the program use the following command line,

gcc demo.c -lgslcblas

There is no need to link with the main library -lgsl in this case as the CBLAS library is an independent unit. Here is the output from the program,

$ ./a.out
[ 367.76, 368.12
  674.06, 674.72 ]

GNU General Public License

Version 2, June 1991

Copyright (C) 1989, 1991 Free Software Foundation, Inc.
59 Temple Place - Suite 330, Boston, MA  02111-1307, USA

Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.

Preamble

The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software--to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things.

To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.

We protect your rights with two steps: (1) copyright the software, and (2) offer you this license which gives you legal permission to copy, distribute and/or modify the software.

Also, for each author's protection and ours, we want to make certain that everyone understands that there is no warranty for this free software. If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors' reputations.

Finally, any free program is threatened constantly by software patents. We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses, in effect making the program proprietary. To prevent this, we have made it clear that any patent must be licensed for everyone's free use or not licensed at all.

The precise terms and conditions for copying, distribution and modification follow.

TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION

  1. This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License. The "Program", below, refers to any such program or work, and a "work based on the Program" means either the Program or any derivative work under copyright law: that is to say, a work containing the Program or a portion of it, either verbatim or with modifications and/or translated into another language. (Hereinafter, translation is included without limitation in the term "modification".) Each licensee is addressed as "you". Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program). Whether that is true depends on what the Program does.
  2. You may copy and distribute verbatim copies of the Program's source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and give any other recipients of the Program a copy of this License along with the Program. You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee.
  3. You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions:
    1. You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change.
    2. You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.
    3. If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License. (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.)
    These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it. Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Program. In addition, mere aggregation of another work not based on the Program with the Program (or with a work based on the Program) on a volume of a storage or distribution medium does not bring the other work under the scope of this License.
  4. You may copy and distribute the Program (or a work based on it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following:
    1. Accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,
    2. Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,
    3. Accompany it with the information you received as to the offer to distribute corresponding source code. (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.)
    The source code for a work means the preferred form of the work for making modifications to it. For an executable work, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the executable. However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable. If distribution of executable or object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place counts as distribution of the source code, even though third parties are not compelled to copy the source along with the object code.
  5. You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.
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  8. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program. If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances. It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice. This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License.
  9. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License.
  10. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and "any later version", you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation.
  11. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.

    NO WARRANTY

  12. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
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END OF TERMS AND CONDITIONS

Appendix: How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found.

one line to give the program's name and a brief idea of what it does.
Copyright (C) yyyy  name of author

This program is free software; you can redistribute it
and/or modify it under the terms of the GNU General Public
License as published by the Free Software Foundation; either
version 2 of the License, or (at your option) any later
version.

This program is distributed in the hope that it will be
useful, but WITHOUT ANY WARRANTY; without even the implied
warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
PURPOSE.  See the GNU General Public License for more
details.

You should have received a copy of the GNU General Public
License along with this program; if not, write to the Free
Software Foundation, Inc., 59 Temple Place - Suite 330,
Boston, MA 02111-1307, USA.

Also add information on how to contact you by electronic and paper mail.

If the program is interactive, make it output a short notice like this when it starts in an interactive mode:

Gnomovision version 69, Copyright (C) 19yy name of author
Gnomovision comes with ABSOLUTELY NO WARRANTY; for details
type `show w'.  This is free software, and you are welcome
to redistribute it under certain conditions; type `show c'
for details.

The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w' and `show c'; they could even be mouse-clicks or menu items--whatever suits your program.

You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the program, if necessary. Here is a sample; alter the names:

Yoyodyne, Inc., hereby disclaims all copyright interest in
the program `Gnomovision' (which makes passes at compilers)
written by James Hacker.

signature of Ty Coon, 1 April 1989
Ty Coon, President of Vice

This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License.

GNU Free Documentation License

Version 1.1, March 2000

Copyright (C) 2000 Free Software Foundation, Inc.
59 Temple Place, Suite 330, Boston, MA  02111-1307, USA

Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
  1. PREAMBLE The purpose of this License is to make a manual, textbook, or other written document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. This License is a kind of "copyleft", which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software. We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.
  2. APPLICABILITY AND DEFINITIONS This License applies to any manual or other work that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. The "Document", below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as "you". A "Modified Version" of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language. A "Secondary Section" is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (For example, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them. The "Invariant Sections" are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. The "Cover Texts" are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A "Transparent" copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, whose contents can be viewed and edited directly and straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup has been designed to thwart or discourage subsequent modification by readers is not Transparent. A copy that is not "Transparent" is called "Opaque". Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, @acronym{SGML} or @acronym{XML} using a publicly available @acronym{DTD}, and standard-conforming simple @acronym{HTML} designed for human modification. Opaque formats include PostScript, @acronym{PDF}, proprietary formats that can be read and edited only by proprietary word processors, @acronym{SGML} or @acronym{XML} for which the @acronym{DTD} and/or processing tools are not generally available, and the machine-generated @acronym{HTML} produced by some word processors for output purposes only. The "Title Page" means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, "Title Page" means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text.
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    5. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.
    6. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.
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    9. Preserve the section entitled "History", and its title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section entitled "History" in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.
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  6. COMBINING DOCUMENTS You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice. The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work. In the combination, you must combine any sections entitled "History" in the various original documents, forming one section entitled "History"; likewise combine any sections entitled "Acknowledgments", and any sections entitled "Dedications". You must delete all sections entitled "Endorsements."
  7. COLLECTIONS OF DOCUMENTS You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects. You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.
  8. AGGREGATION WITH INDEPENDENT WORKS A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, does not as a whole count as a Modified Version of the Document, provided no compilation copyright is claimed for the compilation. Such a compilation is called an "aggregate", and this License does not apply to the other self-contained works thus compiled with the Document, on account of their being thus compiled, if they are not themselves derivative works of the Document. If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one quarter of the entire aggregate, the Document's Cover Texts may be placed on covers that surround only the Document within the aggregate. Otherwise they must appear on covers around the whole aggregate.
  9. TRANSLATION Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License provided that you also include the original English version of this License. In case of a disagreement between the translation and the original English version of this License, the original English version will prevail.
  10. TERMINATION You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.
  11. FUTURE REVISIONS OF THIS LICENSE The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/. Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

ADDENDUM: How to use this License for your documents

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

  Copyright (C) year your name.  Permission is
  granted to copy, distribute and/or modify this document
  under the terms of the GNU Free Documentation License,
  Version 1.1 or any later version published by the Free
  Software Foundation; with the Invariant Sections being
  list their titles, with the Front-Cover Texts being
  list, and with the Back-Cover Texts being
  list.  A copy of the license is included in the
  section entitled ``GNU Free Documentation License''.

If you have no Invariant Sections, write "with no Invariant Sections" instead of saying which ones are invariant. If you have no Front-Cover Texts, write "no Front-Cover Texts" instead of "Front-Cover Texts being list"; likewise for Back-Cover Texts.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

@normalbottom

Function Index

Jump to: c - g

c

  • cblas_caxpy
  • cblas_ccopy
  • cblas_cdotc_sub
  • cblas_cdotu_sub
  • cblas_cgbmv
  • cblas_cgemm
  • cblas_cgemv
  • cblas_cgerc
  • cblas_cgeru
  • cblas_chbmv
  • cblas_chemm
  • cblas_chemv
  • cblas_cher
  • cblas_cher2
  • cblas_cher2k
  • cblas_cherk
  • cblas_chpmv
  • cblas_chpr
  • cblas_chpr2
  • cblas_cscal
  • cblas_csscal
  • cblas_cswap
  • cblas_csymm
  • cblas_csyr2k
  • cblas_csyrk
  • cblas_ctbmv
  • cblas_ctbsv
  • cblas_ctpmv
  • cblas_ctpsv
  • cblas_ctrmm
  • cblas_ctrmv
  • cblas_ctrsm
  • cblas_ctrsv
  • cblas_dasum
  • cblas_daxpy
  • cblas_dcopy
  • cblas_ddot
  • cblas_dgbmv
  • cblas_dgemm
  • cblas_dgemv
  • cblas_dger
  • cblas_dnrm2
  • cblas_drot
  • cblas_drotg
  • cblas_drotm
  • cblas_drotmg
  • cblas_dsbmv
  • cblas_dscal
  • cblas_dsdot
  • cblas_dspmv
  • cblas_dspr
  • cblas_dspr2
  • cblas_dswap
  • cblas_dsymm
  • cblas_dsymv
  • cblas_dsyr
  • cblas_dsyr2
  • cblas_dsyr2k
  • cblas_dsyrk
  • cblas_dtbmv
  • cblas_dtbsv
  • cblas_dtpmv
  • cblas_dtpsv
  • cblas_dtrmm
  • cblas_dtrmv
  • cblas_dtrsm
  • cblas_dtrsv
  • cblas_dzasum
  • cblas_dznrm2
  • cblas_icamax
  • cblas_idamax
  • cblas_isamax
  • cblas_izamax
  • cblas_sasum
  • cblas_saxpy
  • cblas_scasum
  • cblas_scnrm2
  • cblas_scopy
  • cblas_sdot
  • cblas_sdsdot
  • cblas_sgbmv
  • cblas_sgemm
  • cblas_sgemv
  • cblas_sger
  • cblas_snrm2
  • cblas_srot
  • cblas_srotg
  • cblas_srotm
  • cblas_srotmg
  • cblas_ssbmv
  • cblas_sscal
  • cblas_sspmv
  • cblas_sspr
  • cblas_sspr2
  • cblas_sswap
  • cblas_ssymm
  • cblas_ssymv
  • cblas_ssyr
  • cblas_ssyr2
  • cblas_ssyr2k
  • cblas_ssyrk
  • cblas_stbmv
  • cblas_stbsv
  • cblas_stpmv
  • cblas_stpsv
  • cblas_strmm
  • cblas_strmv
  • cblas_strsm
  • cblas_strsv
  • cblas_xerbla
  • cblas_zaxpy
  • cblas_zcopy
  • cblas_zdotc_sub
  • cblas_zdotu_sub
  • cblas_zdscal
  • cblas_zgbmv
  • cblas_zgemm
  • cblas_zgemv
  • cblas_zgerc
  • cblas_zgeru
  • cblas_zhbmv
  • cblas_zhemm
  • cblas_zhemv
  • cblas_zher
  • cblas_zher2
  • cblas_zher2k
  • cblas_zherk
  • cblas_zhpmv
  • cblas_zhpr
  • cblas_zhpr2
  • cblas_zscal
  • cblas_zswap
  • cblas_zsymm
  • cblas_zsyr2k
  • cblas_zsyrk
  • cblas_ztbmv
  • cblas_ztbsv
  • cblas_ztpmv
  • cblas_ztpsv
  • cblas_ztrmm
  • cblas_ztrmv
  • cblas_ztrsm
  • cblas_ztrsv
  • g

  • gsl_acosh
  • gsl_asinh
  • gsl_atanh
  • gsl_blas_caxpy
  • gsl_blas_ccopy
  • gsl_blas_cdotc
  • gsl_blas_cdotu
  • gsl_blas_cgemm
  • gsl_blas_cgemv
  • gsl_blas_cgerc
  • gsl_blas_cgeru
  • gsl_blas_chemm
  • gsl_blas_chemv
  • gsl_blas_cher
  • gsl_blas_cher2
  • gsl_blas_cher2k
  • gsl_blas_cherk
  • gsl_blas_cscal
  • gsl_blas_csscal
  • gsl_blas_cswap
  • gsl_blas_csymm
  • gsl_blas_csyr2k
  • gsl_blas_csyrk
  • gsl_blas_ctrmm
  • gsl_blas_ctrmv
  • gsl_blas_ctrsm
  • gsl_blas_ctrsv
  • gsl_blas_dasum
  • gsl_blas_daxpy
  • gsl_blas_dcopy
  • gsl_blas_ddot
  • gsl_blas_dgemm
  • gsl_blas_dgemv
  • gsl_blas_dger
  • gsl_blas_dnrm2
  • gsl_blas_drot
  • gsl_blas_drotg
  • gsl_blas_drotm
  • gsl_blas_drotmg
  • gsl_blas_dscal
  • gsl_blas_dsdot
  • gsl_blas_dswap
  • gsl_blas_dsymm
  • gsl_blas_dsymv
  • gsl_blas_dsyr
  • gsl_blas_dsyr2
  • gsl_blas_dsyr2k
  • gsl_blas_dsyrk
  • gsl_blas_dtrmm
  • gsl_blas_dtrmv
  • gsl_blas_dtrsm
  • gsl_blas_dtrsv
  • gsl_blas_dzasum
  • gsl_blas_dznrm2
  • gsl_blas_icamax
  • gsl_blas_idamax
  • gsl_blas_isamax
  • gsl_blas_izamax
  • gsl_blas_sasum
  • gsl_blas_saxpy
  • gsl_blas_scasum
  • gsl_blas_scnrm2
  • gsl_blas_scopy
  • gsl_blas_sdot
  • gsl_blas_sdsdot
  • gsl_blas_sgemm
  • gsl_blas_sgemv
  • gsl_blas_sger
  • gsl_blas_snrm2
  • gsl_blas_srot
  • gsl_blas_srotg
  • gsl_blas_srotm
  • gsl_blas_srotmg
  • gsl_blas_sscal
  • gsl_blas_sswap
  • gsl_blas_ssymm
  • gsl_blas_ssymv
  • gsl_blas_ssyr
  • gsl_blas_ssyr2
  • gsl_blas_ssyr2k
  • gsl_blas_ssyrk
  • gsl_blas_strmm
  • gsl_blas_strmv
  • gsl_blas_strsm
  • gsl_blas_strsv
  • gsl_blas_zaxpy
  • gsl_blas_zcopy
  • gsl_blas_zdotc
  • gsl_blas_zdotu
  • gsl_blas_zdscal
  • gsl_blas_zgemm
  • gsl_blas_zgemv
  • gsl_blas_zgerc