GNU Scientific Library – Reference Manual

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GSL

This file documents the GNU Scientific Library (GSL), a collection of numerical routines for scientific computing. It corresponds to release 1.16+ of the library. Please report any errors in this manual to bug-gsl@gnu.org.

More information about GSL can be found at the project homepage, http://www.gnu.org/software/gsl/.

Printed copies of this manual can be purchased from Network Theory Ltd at http://www.network-theory.co.uk/gsl/manual/. The money raised from sales of the manual helps support the development of GSL.

A Japanese translation of this manual is available from the GSL project homepage thanks to Daisuke Tominaga.

Copyright © 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 The GSL Team.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections and no cover texts. A copy of the license is included in the section entitled “GNU Free Documentation License”.



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1 Introduction

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


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1.1 Routines available in GSL

The library covers a wide range of topics in numerical computing. Routines are available for the following areas,

Complex NumbersRoots of Polynomials
Special FunctionsVectors and Matrices
PermutationsCombinations
SortingBLAS Support
Linear AlgebraCBLAS Library
Fast Fourier TransformsEigensystems
Random NumbersQuadrature
Random DistributionsQuasi-Random Sequences
HistogramsStatistics
Monte Carlo IntegrationN-Tuples
Differential EquationsSimulated Annealing
Numerical DifferentiationInterpolation
Series AccelerationChebyshev Approximations
Root-FindingDiscrete Hankel Transforms
Least-Squares FittingMinimization
IEEE Floating-PointPhysical Constants
Basis SplinesWavelets

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.

Where possible the routines have been based on reliable public-domain packages such as FFTPACK and QUADPACK, which the developers of GSL have reimplemented in C with modern coding conventions.


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1.2 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 share copies of programs that you are given which use 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.

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 code which uses the GNU Scientific Library, you must give the recipients all the rights that you have received. You must make sure that they, too, receive or can get the source code, both to the library and the code which uses it. And you must tell them their rights. This means that the library should not be redistributed in proprietary programs.

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 GNU General Public License). Further information about this license is available from the GNU Project webpage Frequently Asked Questions about the GNU GPL,

The Free Software Foundation also operates a license consulting service for commercial users (contact details available from http://www.fsf.org/).


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1.3 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 GNU website,

The preferred platform for the library is a GNU system, which allows it to take advantage of additional features in the GNU C compiler and GNU C library. However, the library is fully portable and should compile on most systems with a C compiler.

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

To: info-gsl-request@gnu.org 
Subject: subscribe

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


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1.4 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, or to purchase support and warranties from commercial redistributors. Consult the GNU General Public license for further details (see GNU General Public License).


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1.5 Reporting Bugs

A list of known bugs can be found in the BUGS file included in the GSL distribution or online in the GSL bug tracker.1 Details of compilation problems can be found in the INSTALL file.

If you find a bug which is not listed in these files, please report it to bug-gsl@gnu.org.

All bug reports should include:

It is useful if you can check whether the same problem occurs when the library is compiled without optimization. Thank you.

Any errors or omissions in this manual can also be reported to the same address.


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1.6 Further Information

Additional information, including online copies of this manual, links to related projects, and mailing list archives are available from the website mentioned above.

Any questions about the use and installation of the library can be asked on the mailing list help-gsl@gnu.org. To subscribe to this list, send an email of the following form:

To: help-gsl-request@gnu.org
Subject: subscribe

This mailing list can be used to ask questions not covered by this manual, and to contact the developers of the library.

If you would like to refer to the GNU Scientific Library in a journal article, the recommended way is to cite this reference manual, e.g. M. Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078.

If you want to give a url, use “http://www.gnu.org/software/gsl/”.


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1.7 Conventions used in this manual

This manual contains many examples which can be typed at the keyboard. A command entered at the terminal is shown like this,

$ command

The first character on the line is the terminal prompt, and should not be typed. The dollar sign ‘$’ is used as the standard prompt in this manual, although some systems may use a different character.

The examples assume the use of the GNU operating system. There may be minor differences in the output on other systems. The commands for setting environment variables use the Bourne shell syntax of the standard GNU shell (bash).


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2 Using the library

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


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2.1 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,2

J0(5) = -1.775967713143382920e-01

The steps needed to compile this program are described in the following sections.


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2.2 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. A typical compilation command for a source file example.c with the GNU C compiler gcc is,

$ gcc -Wall -I/usr/local/include -c example.c

This results in an object file example.o. The default include path for gcc searches /usr/local/include automatically so the -I option can actually be omitted when GSL is installed in its default location.


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2.2.1 Linking programs with the library

The library is installed as a single file, libgsl.a. A shared version of the library libgsl.so is also installed on systems that support shared libraries. The default location of these files is /usr/local/lib. If this directory is not on the standard search path of your linker you will also need to provide its location as a command line flag.

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 -L/usr/local/lib example.o -lgsl -lgslcblas -lm

The default library path for gcc searches /usr/local/lib automatically so the -L option can be omitted when GSL is installed in its default location.

The option -lm links with the system math library. On some systems it is not needed.3

For a tutorial introduction to the GNU C Compiler and related programs, see An Introduction to GCC (ISBN 0954161793).4


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2.2.2 Linking with an alternative BLAS library

The following command line shows how you would link the same application with an alternative CBLAS library libcblas.a,

$ gcc example.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 example.o -lgsl -lcblas -latlas -lm

If the ATLAS library is installed in a non-standard directory use the -L option to add it to the search path, as described above.

For more information about BLAS functions see BLAS Support.


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2.3 Shared Libraries

To run a program linked with the shared version of the library the operating system must be able to locate the corresponding .so file at runtime. If the library cannot be found, the following error will occur:

$ ./a.out 
./a.out: error while loading shared libraries: 
libgsl.so.0: cannot open shared object file: No such 
file or directory

To avoid this error, either modify the system dynamic linker configuration5 or define the shell variable LD_LIBRARY_PATH to include the directory where the library is installed.

For example, in the Bourne shell (/bin/sh or /bin/bash), the library search path can be set with the following commands:

$ LD_LIBRARY_PATH=/usr/local/lib
$ export LD_LIBRARY_PATH
$ ./example

In the C-shell (/bin/csh or /bin/tcsh) the equivalent command is,

% setenv LD_LIBRARY_PATH /usr/local/lib

The standard prompt for the C-shell in the example above is the percent character ‘%’, and should not be typed as part of the command.

To save retyping these commands each session they can be placed in an individual or system-wide login file.

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

$ gcc -static example.o -lgsl -lgslcblas -lm

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2.4 ANSI C Compliance

The library is written in ANSI C and is intended to conform to the ANSI C standard (C89). 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_.


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2.5 Inline functions

The inline keyword is not part of the original ANSI C standard (C89) so the library does not export any inline function definitions by default. Inline functions were introduced officially in the newer C99 standard but most C89 compilers have also included inline as an extension for a long time.

To allow the use of inline functions, the library provides optional inline versions of performance-critical routines by conditional compilation in the exported header files. The inline versions of these functions can be included by defining the macro HAVE_INLINE when compiling an application,

$ gcc -Wall -c -DHAVE_INLINE example.c

If you use autoconf this macro can be defined automatically. If you do not define the macro HAVE_INLINE then the slower non-inlined versions of the functions will be used instead.

By default, the actual form of the inline keyword is extern inline, which is a gcc extension that eliminates unnecessary function definitions. If the form extern inline causes problems with other compilers a stricter autoconf test can be used, see Autoconf Macros.

When compiling with gcc in C99 mode (gcc -std=c99) the header files automatically switch to C99-compatible inline function declarations instead of extern inline. With other C99 compilers, define the macro GSL_C99_INLINE to use these declarations.


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2.6 Long double

In general, the algorithms in the library are written for double precision only. The long double type is not supported for actual computation.

One reason for this choice is that the precision of long double is platform dependent. The IEEE standard only specifies the minimum precision of extended precision numbers, while the precision of double is the same on all platforms.

However, it is sometimes necessary to interact with external data in long-double format, so the vector and matrix datatypes include long-double versions.

It should be noted that 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.


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2.7 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.

For example, after determining whether the BSD function hypot is available you can include the following macro definitions in a file config.h with your application,

/* 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 replace each occurrence of hypot by gsl_hypot when hypot is not available. This substitution can be made automatically if you use autoconf, see Autoconf Macros.

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.


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2.8 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 alternative 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.


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2.9 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 file, or any other specific type, 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 

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2.10 Compatibility with C++

The library header files automatically define functions to have extern "C" linkage when included in C++ programs. This allows the functions to be called directly from C++.

To use C++ exception handling within user-defined functions passed to the library as parameters, the library must be built with the additional CFLAGS compilation option -fexceptions.


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2.11 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.


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2.12 Thread-safety

The library can be used in multi-threaded programs. All the functions are thread-safe, in the sense that they do not use static variables. Memory is always associated with objects and not with functions. For functions which use workspace objects as temporary storage the workspaces should be allocated on a per-thread basis. For functions which use table objects as read-only memory the tables can be used by multiple threads simultaneously. Table arguments are always declared const in function prototypes, to indicate that they may be safely accessed by different threads.

There are a small number of static global variables which are used to control the overall behavior of the library (e.g. whether to use range-checking, the function to call on fatal error, etc). These variables are set directly by the user, so they should be initialized once at program startup and not modified by different threads.


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2.13 Deprecated Functions

From time to time, it may be necessary for the definitions of some functions to be altered or removed from the library. In these circumstances the functions will first be declared deprecated and then removed from subsequent versions of the library. Functions that are deprecated can be disabled in the current release by setting the preprocessor definition GSL_DISABLE_DEPRECATED. This allows existing code to be tested for forwards compatibility.


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2.14 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.


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3 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.


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3.1 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. The caller can examine the return code and decide what action to take, including ignoring the error if it is not considered serious.

In addition to reporting errors by return codes the library also has an error handler function gsl_error. This function is called by other library functions when they report an error, just before they return to the caller. The default behavior of the error handler is to print a message and abort the program,

gsl: file.c:67: ERROR: invalid argument supplied by user
Default GSL error handler invoked.
Aborted

The purpose of the gsl_error 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 return codes.


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3.2 Error Codes

The error code numbers returned by library functions are defined in the file gsl_errno.h. They all have the prefix GSL_ and expand to non-zero constant integer values. Error codes above 1024 are reserved for applications, and are not used by the library. Many of the error codes use the same base name as the corresponding error code in the 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).

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.


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3.3 Error Handlers

The default behavior of the GSL error handler is to print a short message and call abort. When this default is in use programs 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 routines and handle them yourself. You can also 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 function 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 be only 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.


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3.4 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 value of value instead of an error code. It can be used for mathematical functions that return a floating point value.

The following example shows how to return a NaN at a mathematical singularity using the GSL_ERROR_VAL macro,

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

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3.5 Examples

Here is an example of some code which checks the return value of a function where an error might be reported,

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>

...
  int status;
  size_t n = 37;

  gsl_set_error_handler_off();

  status = gsl_fft_complex_radix2_forward (data, stride, 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 function call to gsl_set_error_handler_off stops the default error handler from aborting the program. The else clause catches any other possible errors.


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4 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.


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4.1 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, \log_10 (e)

M_SQRT2

The square root of two, \sqrt 2

M_SQRT1_2

The square root of one-half, \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


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4.2 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.6

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.


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4.3 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 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_hypot3 (const double x, const double y, const double z)

This function computes the value of \sqrt{x^2 + y^2 + z^2} in a way that avoids overflow.

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).

Function: double gsl_ldexp (double x, int e)

This function computes the value of x * 2^e. It provides an alternative to the standard math function ldexp(x,e).

Function: double gsl_frexp (double x, int * e)

This function splits the number x into its normalized fraction f and exponent e, such that x = f * 2^e and 0.5 <= f < 1. The function returns f and stores the exponent in e. If x is zero, both f and e are set to zero. This function provides an alternative to the standard math function frexp(x, e).


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4.4 Small integer powers

A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL provides some 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)
Function: double gsl_pow_uint (double x, unsigned int n)

These routines computes the power x^n for integer n. The power is computed efficiently—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 HAVE_INLINE is defined, 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 */

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4.5 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).


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4.6 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.


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4.7 Maximum and Minimum functions

Note that the following macros perform multiple evaluations of their arguments, so they should not be used with arguments that have side effects (such as a call to a random number generator).

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.


Previous: Maximum and Minimum functions, Up: Mathematical Functions   [Index]

4.8 Approximate Comparison of Floating Point Numbers

It is sometimes useful to be able to compare two floating point numbers approximately, to allow for rounding and truncation errors. The following function implements the approximate floating-point comparison algorithm proposed by D.E. Knuth in Section 4.2.2 of Seminumerical Algorithms (3rd edition).

Function: int gsl_fcmp (double x, double y, double epsilon)

This function determines whether x and y are approximately equal to a relative accuracy epsilon.

The relative accuracy is measured using an interval of size 2 \delta, where \delta = 2^k \epsilon and k is the maximum base-2 exponent of x and y as computed by the function frexp.

If x and y lie within this interval, they are considered approximately equal and the function returns 0. Otherwise if x < y, the function returns -1, or if x > y, the function returns +1.

Note that x and y are compared to relative accuracy, so this function is not suitable for testing whether a value is approximately zero.

The implementation is based on the package fcmp by T.C. Belding.


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5 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 be evaluated over 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)7 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.


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5.1 Representation of 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. An inline version of this function is used when HAVE_INLINE is defined.

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.


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5.2 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 -\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.


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5.3 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).


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5.4 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, \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).


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5.5 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).


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5.6 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/2,\pi/2]. 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 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).


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5.7 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).


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5.8 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. Note that in this case we use the negative square root in formula 4.6.21 of Abramowitz & Stegun giving \arccosh(z)=\log(z-\sqrt{z^2-1}).

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).


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5.9 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,


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6 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.


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6.1 Polynomial Evaluation

The functions described here evaluate the polynomial P(x) = c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^{len-1} using Horner’s method for stability. Inline versions of these functions are used when HAVE_INLINE is defined.

Function: double gsl_poly_eval (const double c[], const int len, const double x)

This function evaluates a polynomial with real coefficients for the real variable x.

Function: gsl_complex gsl_poly_complex_eval (const double c[], const int len, const gsl_complex z)

This function evaluates a polynomial with real coefficients for the complex variable z.

Function: gsl_complex gsl_complex_poly_complex_eval (const gsl_complex c[], const int len, const gsl_complex z)

This function evaluates a polynomial with complex coefficients for the complex variable z.

Function: int gsl_poly_eval_derivs (const double c[], const size_t lenc, const double x, double res[], const size_t lenres)

This function evaluates a polynomial and its derivatives storing the results in the array res of size lenres. The output array contains the values of d^k P/d x^k for the specified value of x starting with k = 0.


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6.2 Divided Difference Representation of Polynomials

The functions described here manipulate polynomials stored in Newton’s divided-difference representation. The use of divided-differences is described in Abramowitz & Stegun sections 25.1.4 and 25.2.26, and Burden and Faires, chapter 3, and discussed briefly below.

Given a function f(x), an nth degree interpolating polynomial P_{n}(x) can be constructed which agrees with f at n+1 distinct points x_0,x_1,...,x_{n}. This polynomial can be written in a form known as Newton’s divided-difference representation:

P_n(x) = f(x_0) + \sum_(k=1)^n [x_0,x_1,...,x_k] (x-x_0)(x-x_1)...(x-x_(k-1))

where the divided differences [x_0,x_1,...,x_k] are defined in section 25.1.4 of Abramowitz and Stegun. Additionally, it is possible to construct an interpolating polynomial of degree 2n+1 which also matches the first derivatives of f at the points x_0,x_1,...,x_n. This is called the Hermite interpolating polynomial and is defined as

H_(2n+1)(x) = f(z_0) + \sum_(k=1)^(2n+1) [z_0,z_1,...,z_k] (x-z_0)(x-z_1)...(x-z_(k-1))

where the elements of z = \{x_0,x_0,x_1,x_1,...,x_n,x_n\} are defined by z_{2k} = z_{2k+1} = x_k. The divided-differences [z_0,z_1,...,z_k] are discussed in Burden and Faires, section 3.4.

Function: int gsl_poly_dd_init (double dd[], const double xa[], const double ya[], size_t size)

This function computes a divided-difference representation of the interpolating polynomial for the points (x, y) stored in the arrays xa and ya of length size. On output the divided-differences of (xa,ya) are stored in the array dd, also of length size. Using the notation above, dd[k] = [x_0,x_1,...,x_k].

Function: double gsl_poly_dd_eval (const double dd[], const double xa[], const size_t size, const double x)

This function evaluates the polynomial stored in divided-difference form in the arrays dd and xa of length size at the point x. An inline version of this function is used when HAVE_INLINE is defined.

Function: int gsl_poly_dd_taylor (double c[], double xp, const double dd[], const double xa[], size_t size, double w[])

This function converts the divided-difference representation of a polynomial to a Taylor expansion. The divided-difference representation is supplied in the arrays dd and xa of length size. On output the Taylor coefficients of the polynomial expanded about the point xp are stored in the array c also of length size. A workspace of length size must be provided in the array w.

Function: int gsl_poly_dd_hermite_init (double dd[], double za[], const double xa[], const double ya[], const double dya[], const size_t size)

This function computes a divided-difference representation of the interpolating Hermite polynomial for the points (x, y) stored in the arrays xa and ya of length size. Hermite interpolation constructs polynomials which also match first derivatives dy/dx which are provided in the array dya also of length size. The first derivatives can be incorported into the usual divided-difference algorithm by forming a new dataset z = \{x_0,x_0,x_1,x_1,...\}, which is stored in the array za of length 2*size on output. On output the divided-differences of the Hermite representation are stored in the array dd, also of length 2*size. Using the notation above, dd[k] = [z_0,z_1,...,z_k]. The resulting Hermite polynomial can be evaluated by calling gsl_poly_dd_eval and using za for the input argument xa.


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6.3 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,

a x^2 + b x + c = 0

The number of real roots (either zero, one 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. If one real root is found (i.e. if a=0) then it is stored in x0. 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,

a z^2 + b z + c = 0

The number of complex roots is returned (either one or 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. If only one real root is found (i.e. if a=0) then it is stored in z0.


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6.4 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,

x^3 + a x^2 + b x + c = 0

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. As in the quadratic case, finite precision may cause equal or closely-spaced real roots to move off the real axis into the complex plane, leading to a discrete change in the number of real roots.

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,

z^3 + a z^2 + b z + c = 0

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.


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6.5 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. If the QR reduction does not converge, the error handler is invoked with an error code of GSL_EFAILED. Note that due to finite precision, roots of higher multiplicity are returned as a cluster of simple roots with reduced accuracy. The solution of polynomials with higher-order roots requires specialized algorithms that take the multiplicity structure into account (see e.g. Z. Zeng, Algorithm 835, ACM Transactions on Mathematical Software, Volume 30, Issue 2 (2004), pp 218–236).


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6.6 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,

1, e^{2\pi i /5}, e^{4\pi i /5}, e^{6\pi i /5}, e^{8\pi i /5}

The following program will find these roots.

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

int
main (void)
{
  int i;
  /* coefficients 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,

$ ./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).


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6.7 References and Further Reading

The balanced-QR method and its error analysis are described in the following papers,

The formulas for divided differences are given in the following texts,


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7 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 in this chapter 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.


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7.1 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.


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7.2 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;

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7.3 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 2 * 10^-16.

GSL_PREC_SINGLE

Single-precision, a relative accuracy of approximately 10^-7.

GSL_PREC_APPROX

Approximate values, a relative accuracy of approximately 5 * 10^-4.

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


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7.4 Airy Functions and Derivatives

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

Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt
Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3 + xt) + \sin((1/3) t^3 + xt)) dt

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


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7.4.1 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 \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 exp(-(2/3) x^(3/2)), and is 1 for x<0.


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7.4.2 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 scaled Airy function derivative S_A(x) Ai'(x). For x>0 the scaling factor S_A(x) is \exp(+(2/3) x^(3/2)), and is 1 for x<0.

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 scaled Airy function derivative S_B(x) Bi'(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)), and is 1 for x<0.


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7.4.3 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).


Previous: Zeros of Airy Functions, Up: Airy Functions and Derivatives   [Index]

7.4.4 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).


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7.5 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.


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7.5.1 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.


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7.5.2 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.


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7.5.3 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.


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7.5.4 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.


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7.5.5 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 l >= 0 and 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 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.

Function: int gsl_sf_bessel_jl_steed_array (int lmax, double x, double * result_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 lmax >= 0 and 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.


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7.5.6 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^3 + 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 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 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.


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7.5.7 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 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.


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7.5.8 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 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.


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7.5.9 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).


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7.5.10 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).


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7.5.11 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.


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7.5.12 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.


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7.5.13 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). The current implementation does not support negative values of nu.


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7.6 Clausen Functions

The Clausen function is defined by the following integral,

Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))

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).


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7.7 Coulomb Functions

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


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7.7.1 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 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,

R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Zr/n)^l
          L^{2l+1}_{n-l-1}(2Zr/n).  

where L^a_b(x) is the generalized Laguerre polynomial (see Laguerre Functions). The normalization is chosen such that the wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.


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7.7.2 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,

F_L(eta,x)  =  fc[k_L] * exp(exp_F)
G_L(eta,x)  =  gc[k_L] * exp(exp_G)

F_L'(eta,x) = fcp[k_L] * exp(exp_F)
G_L'(eta,x) = gcp[k_L] * exp(exp_G)
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 F'_L(\eta,x), G'_{L-k}(\eta,x) with respect to 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 Coulomb wave 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.


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7.7.3 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.


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7.8 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.


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7.8.1 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,

(ja jb jc
 ma mb mc)

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


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7.8.2 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,

{ja jb jc
 jd je jf}

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


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7.8.3 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,

{ja jb jc
 jd je jf
 jg jh ji}

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


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7.9 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.


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7.10 Debye Functions

The Debye functions D_n(x) are defined by the following 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)).

Function: double gsl_sf_debye_5 (double x)
Function: int gsl_sf_debye_5_e (double x, gsl_sf_result * result)

These routines compute the fifth-order Debye function D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1)).

Function: double gsl_sf_debye_6 (double x)
Function: int gsl_sf_debye_6_e (double x, gsl_sf_result * result)

These routines compute the sixth-order Debye function D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1)).


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7.11 Dilogarithm

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


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7.11.1 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 x <= 1, and -\pi\log(x) for x > 1.

Note that Abramowitz & Stegun refer to the Spence integral S(x)=Li_2(1-x) as the dilogarithm rather than Li_2(x).


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7.11.2 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.


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7.12 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.


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7.13 Elliptic Integrals

The functions described in this section are declared in the header file gsl_sf_ellint.h. Further information about the elliptic integrals can be found in Abramowitz & Stegun, Chapter 17.


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7.13.1 Definition of Legendre Forms

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

  F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))

  E(\phi,k) = \int_0^\phi dt   \sqrt((1 - k^2 \sin^2(t)))

Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))

The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k) = E(\pi/2, k).

The notation used here is based on Carlson, Numerische Mathematik 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun, where the functions are given in terms of the parameter m = k^2 and n is replaced by -n.


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7.13.2 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,

    RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)

  RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)

  RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)

RJ(x,y,z,p) = 3/2 \int_0^\infty dt 
                 (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)

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7.13.3 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. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.

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. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.

Function: double gsl_sf_ellint_Pcomp (double k, double n, gsl_mode_t mode)
Function: int gsl_sf_ellint_Pcomp_e (double k, double n, gsl_mode_t mode, gsl_sf_result * result)

These routines compute the complete elliptic integral \Pi(k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.


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7.13.4 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. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.

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. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.

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 \Pi(\phi,k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.

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) which is defined through the Carlson form RD(x,y,z) by the following relation,

D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).

The argument n is not used and will be removed in a future release.


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7.13.5 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.


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7.14 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.


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7.15 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.


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7.15.1 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), where erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).


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7.15.2 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).


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7.15.3 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)).


Previous: Log Complementary Error Function, Up: Error Functions   [Index]

7.15.4 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 density function Z(x) = (1/\sqrt{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/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2).

The hazard function for the normal distribution, also known as the inverse Mills’ ratio, is defined as,

h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)

It decreases rapidly as x approaches -\infty and asymptotes to h(x) \sim x as x approaches +\infty.

Function: double gsl_sf_hazard (double x)
Function: int gsl_sf_hazard_e (double x, gsl_sf_result * result)

These routines compute the hazard function for the normal distribution.


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7.16 Exponential Functions

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


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7.16.1 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.


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7.16.2 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_exprel_2. The N-relative exponential is given by,

exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
            = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ...
            = 1F1 (1,1+N,x)

Previous: Relative Exponential Functions, Up: Exponential Functions   [Index]

7.16.3 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 function exponentiates 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.


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7.17 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.


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7.17.1 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),

E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.
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),

E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
Function: double gsl_sf_expint_En (int n, double x)
Function: int gsl_sf_expint_En_e (int n, double x, gsl_sf_result * result)

These routines compute the exponential integral E_n(x) of order n,

E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.

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7.17.2 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 Ei(x),

Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t)

where PV denotes the principal value of the integral.


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7.17.3 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).


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7.17.4 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 third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0.


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7.17.5 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.


Previous: Trigonometric Integrals, Up: Exponential Integrals   [Index]

7.17.6 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, which is defined as AtanInt(x) = \int_0^x dt \arctan(t)/t.


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7.18 Fermi-Dirac Function

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


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7.18.1 Complete Fermi-Dirac Integrals

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

F_j(x)   := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

Note that the Fermi-Dirac integral is sometimes defined without the normalisation factor in other texts.

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).


Previous: Complete Fermi-Dirac Integrals, Up: Fermi-Dirac Function   [Index]

7.18.2 Incomplete Fermi-Dirac Integrals

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

F_j(x,b)   := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1))
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).


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7.19 Gamma and Beta Functions

This following routines compute the gamma and beta functions in their full and incomplete forms, as well as various kinds of factorials. The functions described in this section are declared in the header file gsl_sf_gamma.h.


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7.19.1 Gamma Functions

The Gamma function is defined by the following integral,

\Gamma(x) = \int_0^\infty dt  t^{x-1} \exp(-t)

It is related to the factorial function by \Gamma(n)=(n-1)! for positive integer n. Further information on the Gamma function can be found in Abramowitz & Stegun, Chapter 6.

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 or zero. 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 being a negative integer or zero. 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 of its magnitude, subject to x not being a negative integer or zero. The function is computed using the real Lanczos method. The value of the gamma function and its error can be reconstructed using the relation \Gamma(x) = sgn * \exp(result\_lg), taking into account the two components of result_lg.

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,

\Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))
            = (1 + (1/12x) + ...)  for x \to \infty

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 or zero, 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.


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7.19.2 Factorials

Although factorials can be computed from the Gamma function, using the relation n! = \Gamma(n+1) for non-negative integer n, it is usually more efficient to call the functions in this section, particularly for small values of n, whose factorial values are maintained in hardcoded tables.

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). The maximum value of n such that n! is not considered an overflow is given by the macro GSL_SF_FACT_NMAX and is 170.

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. The maximum value of n such that n!! is not considered an overflow is given by the macro GSL_SF_DOUBLEFACT_NMAX and is 297.

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_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.


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7.19.3 Pochhammer Symbol

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(a). The Pochhammer symbol is also known as the Apell symbol and sometimes written as (a,x). When a and a+x are negative integers or zero, the limiting value of the ratio is returned.

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)).

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|) with a corresponding error term, and sgn = \sgn((a)_x) where (a)_x = \Gamma(a + x)/\Gamma(a).

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 = \Gamma(a + x)/\Gamma(a).


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7.19.4 Incomplete Gamma Functions

Function: double gsl_sf_gamma_inc (double a, double x)
Function: int gsl_sf_gamma_inc_e (double a, double x, gsl_sf_result * result)

These functions compute the unnormalized incomplete Gamma Function \Gamma(a,x) = \int_x^\infty dt t^{a-1} \exp(-t) for a real and x >= 0.

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 Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^{a-1} \exp(-t) for a > 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 - Q(a,x) = 1/\Gamma(a) \int_0^x dt t^{a-1} \exp(-t) for a > 0, x >= 0.

Note that Abramowitz & Stegun call P(a,x) the incomplete gamma function (section 6.5).


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7.19.5 Beta Functions

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) subject to a and b not being negative integers.

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)) subject to a and b not being negative integers.


Previous: Beta Functions, Up: Gamma and Beta Functions   [Index]

7.19.6 Incomplete Beta Function

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 normalized incomplete Beta function I_x(a,b)=B_x(a,b)/B(a,b) where B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt for 0 <= x <= 1. For a > 0, b > 0 the value is computed using a continued fraction expansion. For all other values it is computed using the relation I_x(a,b,x) = (1/a) x^a 2F1(a,1-b,a+1,x)/B(a,b).


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7.20 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^{(\lambda)}_n(x) for a specific value of n, lambda, x subject to \lambda > -1/2, 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, nmax >= 0.


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7.21 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 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, gsl_sf_result * result)

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) = F(a,b,c,x) for |x| < 1.

If the arguments (a,b,c,x) are too close to a singularity then the function can return the error code GSL_EMAXITER when the series approximation converges too slowly. This occurs in the region of x=1, c - a - b = m for integer m.

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.

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 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)


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7.22 Laguerre Functions

The generalized Laguerre polynomials are defined in terms of confluent hypergeometric functions as L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x), and are sometimes referred to as the associated Laguerre polynomials. They are related to the plain Laguerre polynomials L_n(x) by L^0_n(x) = L_n(x) and L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x). For more information see Abramowitz & Stegun, Chapter 22.

The functions described in this section 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)

These routines evaluate the generalized Laguerre polynomials L^a_n(x) for a > -1, n >= 0.


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7.23 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).


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7.24 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.


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7.24.1 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 P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1

Function: int gsl_sf_legendre_Pl_array (int lmax, double x, double result_array[])
Function: int gsl_sf_legendre_Pl_deriv_array (int lmax, double x, double result_array[], double result_deriv_array[])

These functions compute arrays of Legendre polynomials P_l(x) and derivatives dP_l(x)/dx, 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, 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, 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, x != 1 and l >= 0.


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7.24.2 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 m >= 0, l >= m, |x| <= 1.

Function: int gsl_sf_legendre_Plm_array (int lmax, int m, double x, double result_array[])
Function: int gsl_sf_legendre_Plm_deriv_array (int lmax, int m, double x, double result_array[], double result_deriv_array[])

These functions compute arrays of Legendre polynomials P_l^m(x) and derivatives dP_l^m(x)/dx, for m >= 0, l = |m|, ..., lmax, |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 m >= 0, l >= m, |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[])
Function: int gsl_sf_legendre_sphPlm_deriv_array (int lmax, int m, double x, double result_array[], double result_deriv_array[])

These functions compute arrays of normalized associated Legendre functions \sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x), and derivatives, for m >= 0, l = |m|, ..., lmax, |x| <= 1.0

Function: int gsl_sf_legendre_array_size (const int lmax, const int m)

This function returns the size of result_array[] needed for the array versions of P_l^m(x), lmax - m + 1. An inline version of this function is used when HAVE_INLINE is defined.


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7.24.3 Conical Functions

The Conical Functions P^\mu_{-(1/2)+i\lambda}(x) and 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, 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, m >= -1.


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7.24.4 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 \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 \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 \eta >= 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 0 <= l <= lmax.


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7.25 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.


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7.26 Mathieu Functions

The routines described in this section compute the angular and radial Mathieu functions, and their characteristic values. Mathieu functions are the solutions of the following two differential equations:

d^2y/dv^2 + (a - 2q\cos 2v)y = 0
d^2f/du^2 - (a - 2q\cosh 2u)f = 0

The angular Mathieu functions ce_r(x,q), se_r(x,q) are the even and odd periodic solutions of the first equation, which is known as Mathieu’s equation. These exist only for the discrete sequence of characteristic values a=a_r(q) (even-periodic) and a=b_r(q) (odd-periodic).

The radial Mathieu functions Mc^{(j)}_{r}(z,q), Ms^{(j)}_{r}(z,q) are the solutions of the second equation, which is referred to as Mathieu’s modified equation. The radial Mathieu functions of the first, second, third and fourth kind are denoted by the parameter j, which takes the value 1, 2, 3 or 4.

For more information on the Mathieu functions, see Abramowitz and Stegun, Chapter 20. These functions are defined in the header file gsl_sf_mathieu.h.


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7.26.1 Mathieu Function Workspace

The Mathieu functions can be computed for a single order or for multiple orders, using array-based routines. The array-based routines require a preallocated workspace.

Function: gsl_sf_mathieu_workspace * gsl_sf_mathieu_alloc (size_t n, double qmax)

This function returns a workspace for the array versions of the Mathieu routines. The arguments n and qmax specify the maximum order and q-value of Mathieu functions which can be computed with this workspace.

Function: void gsl_sf_mathieu_free (gsl_sf_mathieu_workspace * work)

This function frees the workspace work.


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7.26.2 Mathieu Function Characteristic Values

Function: int gsl_sf_mathieu_a (int n, double q, gsl_sf_result * result)
Function: int gsl_sf_mathieu_b (int n, double q, gsl_sf_result * result)

These routines compute the characteristic values a_n(q), b_n(q) of the Mathieu functions ce_n(q,x) and se_n(q,x), respectively.

Function: int gsl_sf_mathieu_a_array (int order_min, int order_max, double q, gsl_sf_mathieu_workspace * work, double result_array[])
Function: int gsl_sf_mathieu_b_array (int order_min, int order_max, double q, gsl_sf_mathieu_workspace * work, double result_array[])

These routines compute a series of Mathieu characteristic values a_n(q), b_n(q) for n from order_min to order_max inclusive, storing the results in the array result_array.


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7.26.3 Angular Mathieu Functions

Function: int gsl_sf_mathieu_ce (int n, double q, double x, gsl_sf_result * result)
Function: int gsl_sf_mathieu_se (int n, double q, double x, gsl_sf_result * result)

These routines compute the angular Mathieu functions ce_n(q,x) and se_n(q,x), respectively.

Function: int gsl_sf_mathieu_ce_array (int nmin, int nmax, double q, double x, gsl_sf_mathieu_workspace * work, double result_array[])
Function: int gsl_sf_mathieu_se_array (int nmin, int nmax, double q, double x, gsl_sf_mathieu_workspace * work, double result_array[])

These routines compute a series of the angular Mathieu functions ce_n(q,x) and se_n(q,x) of order n from nmin to nmax inclusive, storing the results in the array result_array.


Previous: Angular Mathieu Functions, Up: Mathieu Functions   [Index]

7.26.4 Radial Mathieu Functions

Function: int gsl_sf_mathieu_Mc (int j, int n, double q, double x, gsl_sf_result * result)
Function: int gsl_sf_mathieu_Ms (int j, int n, double q, double x, gsl_sf_result * result)

These routines compute the radial j-th kind Mathieu functions Mc_n^{(j)}(q,x) and Ms_n^{(j)}(q,x) of order n.

The allowed values of j are 1 and 2. The functions for j = 3,4 can be computed as M_n^{(3)} = M_n^{(1)} + iM_n^{(2)} and M_n^{(4)} = M_n^{(1)} - iM_n^{(2)}, where M_n^{(j)} = Mc_n^{(j)} or Ms_n^{(j)}.

Function: int gsl_sf_mathieu_Mc_array (int j, int nmin, int nmax, double q, double x, gsl_sf_mathieu_workspace * work, double result_array[])
Function: int gsl_sf_mathieu_Ms_array (int j, int nmin, int nmax, double q, double x, gsl_sf_mathieu_workspace * work, double result_array[])

These routines compute a series of the radial Mathieu functions of kind j, with order from nmin to nmax inclusive, storing the results in the array result_array.


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7.27 Power Function

The following functions are equivalent to the function gsl_pow_int (see 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); 

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7.28 Psi (Digamma) Function

The polygamma functions of order n are defined by

\psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+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.


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7.28.1 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)].


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7.28.2 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.

Function: double gsl_sf_psi_1 (double x)
Function: int gsl_sf_psi_1_e (double x, gsl_sf_result * result)

These routines compute the Trigamma function \psi'(x) for general x.


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7.28.3 Polygamma Function

Function: double gsl_sf_psi_n (int n, double x)
Function: int gsl_sf_psi_n_e (int n, double x, gsl_sf_result * result)

These routines compute the polygamma function \psi^{(n)}(x) for n >= 0, x > 0.


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7.29 Synchrotron Functions

The functions described in this section are declared in the header file gsl_sf_synchrotron.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 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 x >= 0.


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7.30 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).


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7.31 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.


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7.31.1 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 \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.


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7.31.2 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 czr, czi.

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 lszr, lszi.


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7.31.3 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.


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7.31.4 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].


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7.31.5 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].

Note that the mathematical value of \pi is slightly greater than M_PI, so the machine numbers M_PI and -M_PI are included in the range.

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).

Note that the mathematical value of 2\pi is slightly greater than 2*M_PI, so the machine number 2*M_PI is included in the range.


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7.31.6 Trigonometric Functions With Error Estimates

Function: int gsl_sf_sin_err_e (double x, double dx, gsl_sf_result * result)

This routine computes the sine of an angle x with an associated absolute error dx, \sin(x \pm dx). Note that this function is provided in the error-handling form only since its purpose is to compute the propagated error.

Function: int gsl_sf_cos_err_e (double x, double dx, gsl_sf_result * result)

This routine computes the cosine of an angle x with an associated absolute error dx, \cos(x \pm dx). Note that this function is provided in the error-handling form only since its purpose is to compute the propagated error.


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7.32 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.


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7.32.1 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.


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7.32.2 Riemann Zeta Function Minus One

For large positive argument, the Riemann zeta function approaches one. In this region the fractional part is interesting, and therefore we need a function to evaluate it explicitly.

Function: double gsl_sf_zetam1_int (int n)
Function: int gsl_sf_zetam1_int_e (int n, gsl_sf_result * result)

These routines compute \zeta(n) - 1 for integer n, n \ne 1.

Function: double gsl_sf_zetam1 (double s)
Function: int gsl_sf_zetam1_e (double s, gsl_sf_result * result)

These routines compute \zeta(s) - 1 for arbitrary s, s \ne 1.


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7.32.3 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.


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7.32.4 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.


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7.33 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_errno.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

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7.34 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,


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8 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.


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8.1 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.


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8.2 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


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8.2.1 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.


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8.2.2 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.


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8.2.3 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

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8.3 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 with the vector.

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


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8.3.1 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.


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8.3.2 Accessing vector elements

Unlike FORTRAN compilers, C compilers do not usually provide support for range checking of vectors and matrices.8 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. You must compile your program with the preprocessor macro HAVE_INLINE defined to use these functions.

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 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.

If you use a C99 compiler which requires inline functions in header files to be declared inline instead of extern inline, define the macro GSL_C99_INLINE (see Inline functions). With GCC this is selected automatically when compiling in C99 mode (-std=c99).

If inline functions are not used, calls to the functions gsl_vector_get and gsl_vector_set will link to the compiled versions of these functions in the library itself. The range checking in these functions is controlled by the global integer variable gsl_check_range. It is enabled by default—to disable it, set gsl_check_range to zero. Due to function-call overhead, there is less benefit in disabling range checking here than for inline functions.

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. An inline version of this function is used when HAVE_INLINE is defined.

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. An inline version of this function is used when HAVE_INLINE is defined.

Function: double * gsl_vector_ptr (gsl_vector * v, size_t i)
Function: const double * gsl_vector_const_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. Inline versions of these functions are used when HAVE_INLINE is defined.


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8.3.3 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.


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8.3.4 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.


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8.3.5 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.

When using this pointer it is important to ensure that the view itself remains in scope—the simplest way to do so is by always writing the pointer as &view.vector, and never storing this value in another variable.

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_view 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 the 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.


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8.3.6 Copying vectors

Common operations on vectors such as addition and multiplication are available in the BLAS part of the library (see 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.


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8.3.7 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.


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8.3.8 Vector operations

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. The result a_i \leftarrow a_i + b_i is stored in a and b remains unchanged. 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. The result a_i \leftarrow a_i - b_i is stored in a and b remains unchanged. 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. The result a_i \leftarrow a_i * b_i is stored in a and b remains unchanged. 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. The result a_i \leftarrow a_i / b_i is stored in a and b remains unchanged. 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. The result a_i \leftarrow x a_i is stored in a.

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. The result a_i \leftarrow a_i + x is stored in a.


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8.3.9 Finding maximum and minimum elements of vectors

The following operations are only defined for real 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.


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8.3.10 Vector properties

The following functions are defined for real and complex vectors. For complex vectors both the real and imaginary parts must satisfy the conditions.

Function: int gsl_vector_isnull (const gsl_vector * v)
Function: int gsl_vector_ispos (const gsl_vector * v)
Function: int gsl_vector_isneg (const gsl_vector * v)
Function: int gsl_vector_isnonneg (const gsl_vector * v)

These functions return 1 if all the elements of the vector v are zero, strictly positive, strictly negative, or non-negative respectively, and 0 otherwise.

Function: int gsl_vector_equal (const gsl_vector * u, const gsl_vector * v)

This function returns 1 if the vectors u and v are equal (by comparison of element values) and 0 otherwise.


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8.3.11 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++) /* OUT OF RANGE ERROR */
    {
      printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
    }

  gsl_vector_free (v);
  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.

$ ./a.out
v_0 = 1.23
v_1 = 2.23
v_2 = 3.23
gsl: vector_source.c:12: ERROR: index out of range
Default GSL error handler invoked.
Aborted (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);
  }

  gsl_vector_free (v);
  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) as follows:

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

int
main (void)
{
  int i; 
  gsl_vector * v = gsl_vector_alloc (10);

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

  for (i = 0; i < 10; i++)
    {
      printf ("%g\n", gsl_vector_get(v, i));
    }

  gsl_vector_free (v);
  return 0;
}

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8.4 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


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8.4.1 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 matrix 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.


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8.4.2 Accessing matrix elements

The functions for accessing the elements of a matrix use the same range checking system as vectors. You can 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. An inline version of this function is used when HAVE_INLINE is defined.

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. An inline version of this function is used when HAVE_INLINE is defined.

Function: double * gsl_matrix_ptr (gsl_matrix * m, size_t i, size_t j)
Function: const double * gsl_matrix_const_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. Inline versions of these functions are used when HAVE_INLINE is defined.


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8.4.3 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.


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8.4.4 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.


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8.4.5 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 k1, size_t k2, size_t n1, size_t n2)
Function: gsl_matrix_const_view gsl_matrix_const_submatrix (const gsl_matrix * m, size_t k1, size_t k2, 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 (i,j,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 the corresponding 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.


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8.4.6 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 is equivalent to gsl_matrix_column but can be used for matrices which are declared const.

Function: gsl_vector_view gsl_matrix_subrow (gsl_matrix * m, size_t i, size_t offset, size_t n)
Function: gsl_vector_const_view gsl_matrix_const_subrow (const gsl_matrix * m, size_t i, size_t offset, size_t n)

These functions return a vector view of the i-th row of the matrix m beginning at offset elements past the first column and containing n elements. The data pointer of the new vector is set to null if i, offset, or n are out of range.

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

Function: gsl_vector_view gsl_matrix_subcolumn (gsl_matrix * m, size_t j, size_t offset, size_t n)
Function: gsl_vector_const_view gsl_matrix_const_subcolumn (const gsl_matrix * m, size_t j, size_t offset, size_t n)

These functions return a vector view of the j-th column of the matrix m beginning at offset elements past the first row and containing n elements. The data pointer of the new vector is set to null if j, offset, or n are out of range.

The function gsl_vector_const_subcolumn is equivalent to gsl_matrix_subcolumn 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 return 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.


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8.4.7 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, gsl_matrix * m2)

This function exchanges the elements of the matrices m1 and m2 by copying. The two matrices must have the same size.


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8.4.8 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 j-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 j-th column of the matrix m. The length of the vector must be the same as the length of the column.


Next: , Previous: Copying rows and columns, Up: Matrices   [Index]

8.4.9 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, const 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.


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8.4.10 Matrix operations

The following operations are defined for real and complex 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. The result a(i,j) \leftarrow a(i,j) + b(i,j) is stored in a and b remains unchanged. 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. The result a(i,j) \leftarrow a(i,j) - b(i,j) is stored in a and b remains unchanged. 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. The result a(i,j) \leftarrow a(i,j) * b(i,j) is stored in a and b remains unchanged. 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. The result a(i,j) \leftarrow a(i,j) / b(i,j) is stored in a and b remains unchanged. 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. The result a(i,j) \leftarrow x a(i,j) is stored in a.

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. The result a(i,j) \leftarrow a(i,j) + x is stored in a.


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8.4.11 Finding maximum and minimum elements of matrices

The following operations are only defined for real 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, searching in row-major order.

Function: void gsl_matrix_min_index (const gsl_matrix * m, size_t * imin, size_t * jmin)

This function returns the indices of the minimum value in the matrix m, storing them in imin and jmin. When there are several equal minimum elements then the first element found is returned, searching in row-major order.

Function: void gsl_matrix_minmax_index (const gsl_matrix * m, size_t * imin, size_t * jmin, size_t * imax, size_t * jmax)

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, searching in row-major order.


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8.4.12 Matrix properties

The following functions are defined for real and complex matrices. For complex matrices both the real and imaginary parts must satisfy the conditions.

Function: int gsl_matrix_isnull (const gsl_matrix * m)
Function: int gsl_matrix_ispos (const gsl_matrix * m)
Function: int gsl_matrix_isneg (const gsl_matrix * m)
Function: int gsl_matrix_isnonneg (const gsl_matrix * m)

These functions return 1 if all the elements of the matrix m are zero, strictly positive, strictly negative, or non-negative respectively, and 0 otherwise. To test whether a matrix is positive-definite, use the Cholesky decomposition (see Cholesky Decomposition).

Function: int gsl_matrix_equal (const gsl_matrix * a, const gsl_matrix * b)

This function returns 1 if the matrices a and b are equal (by comparison of element values) and 0 otherwise.


Previous: Matrix properties, Up: Matrices   [Index]

8.4.13 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++)  /* OUT OF RANGE ERROR */
    for (j = 0; j < 3; j++)
      printf ("m(%d,%d) = %g\n", i, j, 
              gsl_matrix_get (m, i, j));

  gsl_matrix_free (m);

  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.

$ ./a.out
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
Default GSL error handler invoked.
Aborted (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", "wb");
     gsl_matrix_fwrite (f, m);
     fclose (f);
  }

  {  
     FILE * f = fopen ("test.dat", "rb");
     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++;
      }

  gsl_matrix_free (m);
  gsl_matrix_free (a);

  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);

  return 0;
}

Here is the output of the program,

$ ./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

The results can be confirmed using GNU OCTAVE,

$ octave
GNU Octave, version 2.0.16.92
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

Previous: Matrices, Up: Vectors and Matrices   [Index]

8.5 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,


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9 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 to 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.


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9.1 The Permutation struct

A permutation is defined 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;

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9.2 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.

Function: int gsl_permutation_memcpy (gsl_permutation * dest, const gsl_permutation * src)

This function copies the elements of the permutation src into the permutation dest. The two permutations must have the same size.


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9.3 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. An inline version of this function is used when HAVE_INLINE is defined.

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.


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9.4 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 (const gsl_permutation * p)

This function checks that the permutation p is valid. The n elements should contain each of the numbers 0 to n-1 once and only once.


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9.5 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.


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9.6 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.

Function: int gsl_permutation_mul (gsl_permutation * p, const gsl_permutation * pa, const gsl_permutation * pb)

This function combines the two permutations pa and pb into a single permutation p, where p = pa * pb. The permutation p is equivalent to applying pb first and then pa.


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9.7 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. In ISO C99 the type modifier z represents size_t, so "%zu\n" is a suitable format.9 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.


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9.8 Permutations in cyclic form

A permutation can be represented in both linear and cyclic notations. The functions described in this section convert between the two forms. The linear notation is an index mapping, and has already been described above. The cyclic notation expresses a permutation as a series of circular rearrangements of groups of elements, or cycles.

For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced by 3 and 3 is replaced by 1 in a circular fashion. Cycles of different sets of elements can be combined independently, for example (1 2 3) (4 5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange of elements 4 and 5. A cycle of length one represents an element which is unchanged by the permutation and is referred to as a singleton.

It can be shown that every permutation can be decomposed into combinations of cycles. The decomposition is not unique, but can always be rearranged into a standard canonical form by a reordering of elements. The library uses the canonical form defined in Knuth’s Art of Computer Programming (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.

The procedure for obtaining the canonical form given by Knuth is,

  1. Write all singleton cycles explicitly
  2. Within each cycle, put the smallest number first
  3. Order the cycles in decreasing order of the first number in the cycle.

For example, the linear representation (2 4 3 0 1) is represented as (1 4) (0 2 3) in canonical form. The permutation corresponds to an exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.

The important property of the canonical form is that it can be reconstructed from the contents of each cycle without the brackets. In addition, by removing the brackets it can be considered as a linear representation of a different permutation. In the example given above the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping has many applications in the theory of permutations.

Function: int gsl_permutation_linear_to_canonical (gsl_permutation * q, const gsl_permutation * p)

This function computes the canonical form of the permutation p and stores it in the output argument q.

Function: int gsl_permutation_canonical_to_linear (gsl_permutation * p, const gsl_permutation * q)

This function converts a permutation q in canonical form back into linear form storing it in the output argument p.

Function: size_t gsl_permutation_inversions (const gsl_permutation * p)

This function counts the number of inversions in the permutation p. An inversion is any pair of elements that are not in order. For example, the permutation 2031 has three inversions, corresponding to the pairs (2,0) (2,1) and (3,1). The identity permutation has no inversions.

Function: size_t gsl_permutation_linear_cycles (const gsl_permutation * p)

This function counts the number of cycles in the permutation p, given in linear form.

Function: size_t gsl_permutation_canonical_cycles (const gsl_permutation * q)

This function counts the number of cycles in the permutation q, given in canonical form.


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9.9 Examples

The example program below creates a random permutation (by shuffling the elements of the identity) 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_inverse (q, p);
  gsl_permutation_fprintf (stdout, q, " %u");
  printf ("\n");

  gsl_permutation_free (p);
  gsl_permutation_free (q);
  gsl_rng_free (r);

  return 0;
}

Here is the output from the program,

$ ./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 third 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);

  gsl_permutation_free (p);

  return 0;
}

Here is the output from the program,

$ ./a.out 
 0 1 2
 0 2 1
 1 0 2
 1 2 0
 2 0 1
 2 1 0

The 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.


Previous: Permutation Examples, Up: Permutations   [Index]

9.10 References and Further Reading

The subject of permutations is covered extensively in Knuth’s Sorting and Searching,

For the definition of the canonical form see,


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10 Combinations

This chapter describes functions for creating and manipulating combinations. A combination c is represented by an array of k integers in the range 0 to n-1, where each value c_i occurs at most once. The combination c corresponds to indices of k elements chosen from an n element vector. Combinations are useful for iterating over all k-element subsets of a set.

The functions described in this chapter are defined in the header file gsl_combination.h.


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10.1 The Combination struct

A combination is defined by a structure containing three components, the values of n and k, and a pointer to the combination array. The elements of the combination array are all of type size_t, and are stored in increasing order. The gsl_combination structure looks like this,

typedef struct
{
  size_t n;
  size_t k;
  size_t *data;
} gsl_combination;

Next: , Previous: The Combination struct, Up: Combinations   [Index]

10.2 Combination allocation

Function: gsl_combination * gsl_combination_alloc (size_t n, size_t k)

This function allocates memory for a new combination with parameters n, k. The combination is not initialized and its elements are undefined. Use the function gsl_combination_calloc if you want to create a combination which is initialized to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination.

Function: gsl_combination * gsl_combination_calloc (size_t n, size_t k)

This function allocates memory for a new combination with parameters n, k and initializes it to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination.

Function: void gsl_combination_init_first (gsl_combination * c)

This function initializes the combination c to the lexicographically first combination, i.e. (0,1,2,…,k-1).

Function: void gsl_combination_init_last (gsl_combination * c)

This function initializes the combination c to the lexicographically last combination, i.e. (n-k,n-k+1,…,n-1).

Function: void gsl_combination_free (gsl_combination * c)

This function frees all the memory used by the combination c.

Function: int gsl_combination_memcpy (gsl_combination * dest, const gsl_combination * src)

This function copies the elements of the combination src into the combination dest. The two combinations must have the same size.


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10.3 Accessing combination elements

The following function can be used to access the elements of a combination.

Function: size_t gsl_combination_get (const gsl_combination * c, const size_t i)

This function returns the value of the i-th element of the combination c. If i lies outside the allowed range of 0 to k-1 then the error handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE is defined.


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10.4 Combination properties

Function: size_t gsl_combination_n (const gsl_combination * c)

This function returns the range (n) of the combination c.

Function: size_t gsl_combination_k (const gsl_combination * c)

This function returns the number of elements (k) in the combination c.

Function: size_t * gsl_combination_data (const gsl_combination * c)

This function returns a pointer to the array of elements in the combination c.

Function: int gsl_combination_valid (gsl_combination * c)

This function checks that the combination c is valid. The k elements should lie in the range 0 to n-1, with each value occurring once at most and in increasing order.


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10.5 Combination functions

Function: int gsl_combination_next (gsl_combination * c)

This function advances the combination c to the next combination in lexicographic order and returns GSL_SUCCESS. If no further combinations are available it returns GSL_FAILURE and leaves c unmodified. Starting with the first combination and repeatedly applying this function will iterate through all possible combinations of a given order.

Function: int gsl_combination_prev (gsl_combination * c)

This function steps backwards from the combination c to the previous combination in lexicographic order, returning GSL_SUCCESS. If no previous combination is available it returns GSL_FAILURE and leaves c unmodified.


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10.6 Reading and writing combinations

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

Function: int gsl_combination_fwrite (FILE * stream, const gsl_combination * c)

This function writes the elements of the combination c 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_combination_fread (FILE * stream, gsl_combination * c)

This function reads elements from the open stream stream into the combination c in binary format. The combination c must be preallocated with correct values of n and k since the function uses the size of c 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_combination_fprintf (FILE * stream, const gsl_combination * c, const char * format)

This function writes the elements of the combination c line-by-line to the stream stream using the format specifier format, which should be suitable for a type of size_t. In ISO C99 the type modifier z represents size_t, so "%zu\n" is a suitable format.10 The function returns GSL_EFAILED if there was a problem writing to the file.

Function: int gsl_combination_fscanf (FILE * stream, gsl_combination * c)

This function reads formatted data from the stream stream into the combination c. The combination c must be preallocated with correct values of n and k since the function uses the size of c to determine how many numbers to read. The function returns GSL_EFAILED if there was a problem reading from the file.


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10.7 Examples

The example program below prints all subsets of the set {0,1,2,3} ordered by size. Subsets of the same size are ordered lexicographically.

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

int 
main (void) 
{
  gsl_combination * c;
  size_t i;

  printf ("All subsets of {0,1,2,3} by size:\n") ;
  for (i = 0; i <= 4; i++)
    {
      c = gsl_combination_calloc (4, i);
      do
        {
          printf ("{");
          gsl_combination_fprintf (stdout, c, " %u");
          printf (" }\n");
        }
      while (gsl_combination_next (c) == GSL_SUCCESS);
      gsl_combination_free (c);
    }

  return 0;
}

Here is the output from the program,

$ ./a.out 
All subsets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 2 }
{ 1 3 }
{ 2 3 }
{ 0 1 2 }
{ 0 1 3 }
{ 0 2 3 }
{ 1 2 3 }
{ 0 1 2 3 }

All 16 subsets are generated, and the subsets of each size are sorted lexicographically.


Previous: Combination Examples, Up: Combinations   [Index]

10.8 References and Further Reading

Further information on combinations can be found in,


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11 Multisets

This chapter describes functions for creating and manipulating multisets. A multiset c is represented by an array of k integers in the range 0 to n-1, where each value c_i may occur more than once. The multiset c corresponds to indices of k elements chosen from an n element vector with replacement. In mathematical terms, n is the cardinality of the multiset while k is the maximum multiplicity of any value. Multisets are useful, for example, when iterating over the indices of a k-th order symmetric tensor in n-space.

The functions described in this chapter are defined in the header file gsl_multiset.h.


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11.1 The Multiset struct

A multiset is defined by a structure containing three components, the values of n and k, and a pointer to the multiset array. The elements of the multiset array are all of type size_t, and are stored in increasing order. The gsl_multiset structure looks like this,

typedef struct
{
  size_t n;
  size_t k;
  size_t *data;
} gsl_multiset;

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11.2 Multiset allocation

Function: gsl_multiset * gsl_multiset_alloc (size_t n, size_t k)

This function allocates memory for a new multiset with parameters n, k. The multiset is not initialized and its elements are undefined. Use the function gsl_multiset_calloc if you want to create a multiset which is initialized to the lexicographically first multiset element. A null pointer is returned if insufficient memory is available to create the multiset.

Function: gsl_multiset * gsl_multiset_calloc (size_t n, size_t k)

This function allocates memory for a new multiset with parameters n, k and initializes it to the lexicographically first multiset element. A null pointer is returned if insufficient memory is available to create the multiset.

Function: void gsl_multiset_init_first (gsl_multiset * c)

This function initializes the multiset c to the lexicographically first multiset element, i.e. 0 repeated k times.

Function: void gsl_multiset_init_last (gsl_multiset * c)

This function initializes the multiset c to the lexicographically last multiset element, i.e. n-1 repeated k times.

Function: void gsl_multiset_free (gsl_multiset * c)

This function frees all the memory used by the multiset c.

Function: int gsl_multiset_memcpy (gsl_multiset * dest, const gsl_multiset * src)

This function copies the elements of the multiset src into the multiset dest. The two multisets must have the same size.


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11.3 Accessing multiset elements

The following function can be used to access the elements of a multiset.

Function: size_t gsl_multiset_get (const gsl_multiset * c, const size_t i)

This function returns the value of the i-th element of the multiset c. If i lies outside the allowed range of 0 to k-1 then the error handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE is defined.


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11.4 Multiset properties

Function: size_t gsl_multiset_n (const gsl_multiset * c)

This function returns the range (n) of the multiset c.

Function: size_t gsl_multiset_k (const gsl_multiset * c)

This function returns the number of elements (k) in the multiset c.

Function: size_t * gsl_multiset_data (const gsl_multiset * c)

This function returns a pointer to the array of elements in the multiset c.

Function: int gsl_multiset_valid (gsl_multiset * c)

This function checks that the multiset c is valid. The k elements should lie in the range 0 to n-1, with each value occurring in nondecreasing order.


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11.5 Multiset functions

Function: int gsl_multiset_next (gsl_multiset * c)

This function advances the multiset c to the next multiset element in lexicographic order and returns GSL_SUCCESS. If no further multisets elements are available it returns GSL_FAILURE and leaves c unmodified. Starting with the first multiset and repeatedly applying this function will iterate through all possible multisets of a given order.

Function: int gsl_multiset_prev (gsl_multiset * c)

This function steps backwards from the multiset c to the previous multiset element in lexicographic order, returning GSL_SUCCESS. If no previous multiset is available it returns GSL_FAILURE and leaves c unmodified.


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11.6 Reading and writing multisets

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

Function: int gsl_multiset_fwrite (FILE * stream, const gsl_multiset * c)

This function writes the elements of the multiset c 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_multiset_fread (FILE * stream, gsl_multiset * c)

This function reads elements from the open stream stream into the multiset c in binary format. The multiset c must be preallocated with correct values of n and k since the function uses the size of c 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_multiset_fprintf (FILE * stream, const gsl_multiset * c, const char * format)

This function writes the elements of the multiset c line-by-line to the stream stream using the format specifier format, which should be suitable for a type of size_t. In ISO C99 the type modifier z represents size_t, so "%zu\n" is a suitable format.11 The function returns GSL_EFAILED if there was a problem writing to the file.

Function: int gsl_multiset_fscanf (FILE * stream, gsl_multiset * c)

This function reads formatted data from the stream stream into the multiset c. The multiset c must be preallocated with correct values of n and k since the function uses the size of c to determine how many numbers to read. The function returns GSL_EFAILED if there was a problem reading from the file.


Previous: Reading and writing multisets, Up: Multisets   [Index]

11.7 Examples

The example program below prints all multisets elements containing the values {0,1,2,3} ordered by size. Multiset elements of the same size are ordered lexicographically.

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

int
main (void)
{
  gsl_multiset * c;
  size_t i;

  printf ("All multisets of {0,1,2,3} by size:\n") ;
  for (i = 0; i <= 4; i++)
    {
      c = gsl_multiset_calloc (4, i);
      do
        {
          printf ("{");
          gsl_multiset_fprintf (stdout, c, " %u");
          printf (" }\n");
        }
      while (gsl_multiset_next (c) == GSL_SUCCESS);
      gsl_multiset_free (c);
    }

  return 0;
}

Here is the output from the program,

$ ./a.out
all multisets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 0 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 1 }
{ 1 2 }
{ 1 3 }
{ 2 2 }
{ 2 3 }
{ 3 3 }
{ 0 0 0 }
{ 0 0 1 }
{ 0 0 2 }
{ 0 0 3 }
{ 0 1 1 }
{ 0 1 2 }
{ 0 1 3 }
{ 0 2 2 }
{ 0 2 3 }
{ 0 3 3 }
{ 1 1 1 }
{ 1 1 2 }
{ 1 1 3 }
{ 1 2 2 }
{ 1 2 3 }
{ 1 3 3 }
{ 2 2 2 }
{ 2 2 3 }
{ 2 3 3 }
{ 3 3 3 }
{ 0 0 0 0 }
{ 0 0 0 1 }
{ 0 0 0 2 }
{ 0 0 0 3 }
{ 0 0 1 1 }
{ 0 0 1 2 }
{ 0 0 1 3 }
{ 0 0 2 2 }
{ 0 0 2 3 }
{ 0 0 3 3 }
{ 0 1 1 1 }
{ 0 1 1 2 }
{ 0 1 1 3 }
{ 0 1 2 2 }
{ 0 1 2 3 }
{ 0 1 3 3 }
{ 0 2 2 2 }
{ 0 2 2 3 }
{ 0 2 3 3 }
{ 0 3 3 3 }
{ 1 1 1 1 }
{ 1 1 1 2 }
{ 1 1 1 3 }
{ 1 1 2 2 }
{ 1 1 2 3 }
{ 1 1 3 3 }
{ 1 2 2 2 }
{ 1 2 2 3 }
{ 1 2 3 3 }
{ 1 3 3 3 }
{ 2 2 2 2 }
{ 2 2 2 3 }
{ 2 2 3 3 }
{ 2 3 3 3 }
{ 3 3 3 3 }

All 70 multisets are generated and sorted lexicographically.


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12 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 and does not require any additional storage. It also provides consistent performance, the running time for its worst-case (ordered data) being 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.


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12.1 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)
{
    if (*a > *b)
       return 1;
    else if (*a < *b)
       return -1;
    else
       return 0;
}

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.


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12.2 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 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, const 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_sort2 (double * data1, const size_t stride1, double * data2, const size_t stride2, size_t n)

This function sorts the n elements of the array data1 with stride stride1 into ascending numerical order, while making the same rearrangement of the array data2 with stride stride2, also of size n.

Function: void gsl_sort_vector (gsl_vector * v)

This function sorts the elements of the vector v into ascending numerical order.

Function: void gsl_sort_vector2 (gsl_vector * v1, gsl_vector * v2)

This function sorts the elements of the vector v1 into ascending numerical order, while making the same rearrangement of the vector v2.

Function: void 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 with 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.


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12.3 Selecting the k smallest or largest elements

The functions described in this section select the k 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: int gsl_sort_smallest (double * dest, size_t k, const double * src, size_t stride, size_t n)

This function copies the k smallest elements of the array src, of size n and stride stride, in ascending numerical order into the array dest. The size k of the subset must be less than or equal to n. The data src is not modified by this operation.

Function: int gsl_sort_largest (double * dest, size_t k, const double * src, size_t stride, size_t n)

This function copies the k largest elements of the array src, of size n and stride stride, in descending numerical order into the array dest. k must be less than or equal to n. The data src is not modified by this operation.

Function: int gsl_sort_vector_smallest (double * dest, size_t k, const gsl_vector * v)
Function: int gsl_sort_vector_largest (double * dest, size_t k, const gsl_vector * v)

These functions copy the k smallest or largest elements of the vector v into the array dest. k must be less than or equal to the length of the vector v.

The following functions find the indices of the k smallest or largest elements of a dataset,

Function: int 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 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. k must be less than or equal to n. The data src is not modified by this operation.

Function: int 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 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. k must be less than or equal to n. The data src is not modified by this operation.

Function: int gsl_sort_vector_smallest_index (size_t * p, size_t k, const gsl_vector * v)
Function: int gsl_sort_vector_largest_index (size_t * p, size_t k, const gsl_vector * v)

These functions store the indices of the k smallest or largest elements of the vector v in the array p. k must be less than or equal to the length of the vector v.


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12.4 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_inverse(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_inverse (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);
}

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12.5 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;

  size_t 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]);
    }

  free (x);
  free (small);
  gsl_rng_free (r);
  return 0;
}

The output lists the 5 smallest values, in ascending order,

$ ./a.out 
5 smallest values from 100000
0: 0.000003489200025797
1: 0.000008199829608202
2: 0.000008953968062997
3: 0.000010712770745158
4: 0.000033531803637743

Previous: Sorting Examples, Up: Sorting   [Index]

12.6 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,


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13 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 described in this chapter. The functions are declared in the file gsl_blas.h and 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. Similarly, GSL vectors are restricted to positive strides, whereas the low-level CBLAS interface supports negative strides as specified in the BLAS standard.12

The interface for the gsl_cblas layer is specified in the file gsl_cblas.h. This interface corresponds to the BLAS Technical Forum’s 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 CBLAS standard and can be obtained from Netlib. The complete set of CBLAS functions is listed in an appendix (see 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 types 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”.

Note that the vector and matrix arguments to BLAS functions must not be aliased, as the results are undefined when the underlying arrays overlap (see Aliasing of arrays).


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13.1 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.


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13.1.1 Level 1

Function: int gsl_blas_sdsdot (float alpha, const gsl_vector_float * x, const gsl_vector_float * y, float * result)

This function computes 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_dsdot (const gsl_vector_float * x, const gsl_vector_float * y, double * 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 dotu

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 dotc

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,

||x||_2 = \sqrt {\sum (\Re(x_i)^2 + \Im(x_i)^2)}.
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 sum of the magnitudes of the real and imaginary parts of the complex vector x, \sum |\Re(x_i)| + |\Im(x_i)|.

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 vectors 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),

[  c  s ] [ a ] = [ r ]
[ -s  c ] [ b ]   [ 0 ]

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 Givens transformation. The modified Givens transformation is defined in the original Level-1 BLAS specification, given in the references.

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 Givens transformation.


Next: , Previous: Level 1 GSL BLAS Interface, Up: GSL BLAS Interface   [Index]

13.1.2 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 x = op(A) x 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.


Previous: Level 2 GSL BLAS Interface, Up: GSL BLAS Interface   [Index]

13.1.3 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 inverse-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 CblasConjTrans. 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 CblasConjTrans. 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.


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13.2 Examples

The following program computes the product of two matrices using the Level-3 BLAS function DGEMM,

[ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
[ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
                    [ 1031 1032 ]

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 ]

Previous: BLAS Examples, Up: BLAS Support   [Index]

13.3 References and Further Reading

Information on the BLAS standards, including both the legacy and updated 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.


Next: , Previous: BLAS Support, Up: Top   [Index]

14 Linear Algebra

This chapter describes functions for solving linear systems. The library provides linear algebra operations which operate directly on the gsl_vector and gsl_matrix objects. These routines use the standard algorithms from Golub & Van Loan’s Matrix Computations with Level-1 and Level-2 BLAS calls for efficiency.

The functions described in this chapter are declared in the header file gsl_linalg.h.


Next: , Up: Linear Algebra   [Index]

14.1 LU Decomposition

A general square matrix A has an LU decomposition into upper and lower triangular matrices,

P A = L U

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. Note that the LU decomposition is valid for singular matrices.

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 square 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 as input.

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 square system A x = b in-place using the precomputed 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, from the precomputed 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_linalg_complex_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. It is preferable to avoid direct use of the inverse whenever possible, as the linear solver functions can obtain the same result more efficiently and reliably (consult any introductory textbook on numerical linear algebra for details).

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.


Next: , Previous: LU Decomposition, Up: Linear Algebra   [Index]

14.2 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,

A = Q 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 square system A x = b using the QR decomposition of A held in (QR, tau) which must have been computed previously with gsl_linalg_QR_decomp. The least-squares solution for rectangular systems can be found using gsl_linalg_QR_lssolve.

Function: int gsl_linalg_QR_svx (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * x)

This function solves the square system A x = b in-place using the QR decomposition of A held in (QR,tau) which must have been computed previously 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 requires as input 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_QTmat (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * A)

This function applies the matrix Q^T encoded in the decomposition (QR,tau) to the matrix A, storing the result Q^T A in A. 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_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.


Next: , Previous: QR Decomposition, Up: Linear Algebra   [Index]

14.3 QR Decomposition with Column Pivoting

The QR decomposition can be extended to the rank deficient case by introducing a column permutation P,

A P = Q R

The first r columns of 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. The vector norm is a workspace of length N used for column pivoting.

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 square system A x = b using the QRP^T decomposition of A held in (QR, tau, p) which must have been computed previously 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 square system A x = b in-place using the QRP^T decomposition of A held in (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 square 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 * w, 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 P) 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.


Next: , Previous: QR Decomposition with Column Pivoting, Up: Linear Algebra   [Index]

14.4 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 N-by-N orthogonal square matrix V,

A = U S V^T

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.

For a rank-deficient matrix, the null space of A is given by the columns of V corresponding to the zero singular values. Similarly, the range of A is given by columns of U corresponding to the non-zero singular values.

Note that the routines here compute the “thin” version of the SVD with U as M-by-N orthogonal matrix. This allows in-place computation and is the most commonly-used form in practice. Mathematically, the “full” SVD is defined with U as an M-by-M orthogonal matrix and S as an M-by-N diagonal matrix (with additional rows of zeros).

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 for M >= N. 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 of length N 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 of the M-by-N matrix A using one-sided Jacobi orthogonalization for M >= N. The Jacobi method can compute singular values to higher relative accuracy than Golub-Reinsch algorithms (see references for details).

Function: int gsl_linalg_SV_solve (const gsl_matrix * U, const gsl_matrix * V, const 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 which must have been computed previously with 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.

Function: int gsl_linalg_SV_leverage (const gsl_matrix * U, gsl_vector * h)

This function computes the statistical leverage values h_i of a matrix A using its singular value decomposition (U, S, V) previously computed with gsl_linalg_SV_decomp. h_i are the diagonal values of the matrix A (A^T A)^{-1} A^T and depend only on the matrix U which is the input to this function.


Next: , Previous: Singular Value Decomposition, Up: Linear Algebra   [Index]

14.5 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,

A = L 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)
Function: int gsl_linalg_complex_cholesky_decomp (gsl_matrix_complex * A)

These functions factorize the symmetric, positive-definite square matrix A into the Cholesky decomposition A = L L^T (or A = L L^H for the complex case). On input, the values from the diagonal and lower-triangular part of the matrix A are used (the upper triangular part is ignored). On output the diagonal and lower triangular part of the input matrix A contain the matrix L, while the upper triangular part of the input matrix is overwritten with 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.

When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.

Function: int gsl_linalg_cholesky_solve (const gsl_matrix * cholesky, const gsl_vector * b, gsl_vector * x)
Function: int gsl_linalg_complex_cholesky_solve (const gsl_matrix_complex * cholesky, const gsl_vector_complex * b, gsl_vector_complex * x)

These functions solve the system A x = b using the Cholesky decomposition of A held in the matrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp or gsl_linalg_complex_cholesky_decomp.

Function: int gsl_linalg_cholesky_svx (const gsl_matrix * cholesky, gsl_vector * x)
Function: int gsl_linalg_complex_cholesky_svx (const gsl_matrix_complex * cholesky, gsl_vector_complex * x)

These functions solve the system A x = b in-place using the Cholesky decomposition of A held in the matrix cholesky which must have been previously computed by gsl_linalg_cholesky_decomp or gsl_linalg_complex_cholesky_decomp. On input x should contain the right-hand side b, which is replaced by the solution on output.

Function: int gsl_linalg_cholesky_invert (gsl_matrix * cholesky)
Function: int gsl_linalg_complex_cholesky_invert (gsl_matrix_complex * cholesky)

These functions compute the inverse of a matrix from its Cholesky decomposition cholesky, which must have been previously computed by gsl_linalg_cholesky_decomp or gsl_linalg_complex_cholesky_decomp. On output, the inverse is stored in-place in cholesky.


Next: , Previous: Cholesky Decomposition, Up: Linear Algebra   [Index]

14.6 Tridiagonal Decomposition of Real Symmetric Matrices

A symmetric matrix A can be factorized by similarity transformations into the form,

A = Q T Q^T

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 * diag, gsl_vector * subdiag)

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 diag and the vector of subdiagonal elements subdiag.

Function: int gsl_linalg_symmtd_unpack_T (const gsl_matrix * A, gsl_vector * diag, gsl_vector * subdiag)

This function unpacks the diagonal and subdiagonal of the encoded symmetric tridiagonal decomposition (A, tau) obtained from gsl_linalg_symmtd_decomp into the vectors diag and subdiag.


Next: , Previous: Tridiagonal Decomposition of Real Symmetric Matrices, Up: Linear Algebra   [Index]

14.7 Tridiagonal Decomposition of Hermitian Matrices

A hermitian matrix A can be factorized by similarity transformations into the form,

A = U T U^T

where U is a 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 unitary matrix U. 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 * U, gsl_vector * diag, gsl_vector * subdiag)

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 diag and the real vector of subdiagonal elements subdiag.

Function: int gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex * A, gsl_vector * diag, gsl_vector * subdiag)

This function unpacks the diagonal and subdiagonal of the encoded tridiagonal decomposition (A, tau) obtained from the gsl_linalg_hermtd_decomp into the real vectors diag and subdiag.


Next: , Previous: Tridiagonal Decomposition of Hermitian Matrices, Up: Linear Algebra   [Index]

14.8 Hessenberg Decomposition of Real Matrices

A general real matrix A can be decomposed by orthogonal similarity transformations into the form

A = U H U^T

where U is orthogonal and H is an upper Hessenberg matrix, meaning that it has zeros below the first subdiagonal. The Hessenberg reduction is the first step in the Schur decomposition for the nonsymmetric eigenvalue problem, but has applications in other areas as well.

Function: int gsl_linalg_hessenberg_decomp (gsl_matrix * A, gsl_vector * tau)

This function computes the Hessenberg decomposition of the matrix A by applying the similarity transformation H = U^T A U. On output, H is stored in the upper portion of A. The information required to construct the matrix U is stored in the lower triangular portion of A. U is a product of N - 2 Householder matrices. The Householder vectors are stored in the lower portion of A (below the subdiagonal) and the Householder coefficients are stored in the vector tau. tau must be of length N.

Function: int gsl_linalg_hessenberg_unpack (gsl_matrix * H, gsl_vector * tau, gsl_matrix * U)

This function constructs the orthogonal matrix U from the information stored in the Hessenberg matrix H along with the vector tau. H and tau are outputs from gsl_linalg_hessenberg_decomp.

Function: int gsl_linalg_hessenberg_unpack_accum (gsl_matrix * H, gsl_vector * tau, gsl_matrix * V)

This function is similar to gsl_linalg_hessenberg_unpack, except it accumulates the matrix U into V, so that V' = VU. The matrix V must be initialized prior to calling this function. Setting V to the identity matrix provides the same result as gsl_linalg_hessenberg_unpack. If H is order N, then V must have N columns but may have any number of rows.

Function: int gsl_linalg_hessenberg_set_zero (gsl_matrix * H)

This function sets the lower triangular portion of H, below the subdiagonal, to zero. It is useful for clearing out the Householder vectors after calling gsl_linalg_hessenberg_decomp.


Next: , Previous: Hessenberg Decomposition of Real Matrices, Up: Linear Algebra   [Index]

14.9 Hessenberg-Triangular Decomposition of Real Matrices

A general real matrix pair (A, B) can be decomposed by orthogonal similarity transformations into the form

A = U H V^T
B = U R V^T

where U and V are orthogonal, H is an upper Hessenberg matrix, and R is upper triangular. The Hessenberg-Triangular reduction is the first step in the generalized Schur decomposition for the generalized eigenvalue problem.

Function: int gsl_linalg_hesstri_decomp (gsl_matrix * A, gsl_matrix * B, gsl_matrix * U, gsl_matrix * V, gsl_vector * work)

This function computes the Hessenberg-Triangular decomposition of the matrix pair (A, B). On output, H is stored in A, and R is stored in B. If U and V are provided (they may be null), the similarity transformations are stored in them. Additional workspace of length N is needed in work.


Next: , Previous: Hessenberg-Triangular Decomposition of Real Matrices, Up: Linear Algebra   [Index]

14.10 Bidiagonalization

A general matrix A can be factorized by similarity transformations into the form,

A = U B V^T

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 produced 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. Note that U is stored as a compact M-by-N orthogonal matrix satisfying U^T U = I for efficiency.

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 produced 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 from gsl_linalg_bidiag_decomp, into the diagonal vector diag and superdiagonal vector superdiag.


Next: , Previous: Bidiagonalization, Up: Linear Algebra   [Index]

14.11 Householder Transformations

A Householder transformation is a rank-1 modification of the identity matrix which can be used to zero out selected elements of a vector. A Householder matrix P takes the form,

P = I - \tau v v^T

where v is a vector (called the Householder vector) and \tau = 2/(v^T v). The functions described in this section use the rank-1 structure of the Householder matrix to create and apply Householder transformations efficiently.

Function: double gsl_linalg_householder_transform (gsl_vector * v)
Function: gsl_complex gsl_linalg_complex_householder_transform (gsl_vector_complex * v)

This function prepares a Householder transformation P = I - \tau v v^T which can be used to zero all the elements of the input vector except the first. On output the transformation is stored in the vector v and the scalar \tau is returned.

Function: int gsl_linalg_householder_hm (double tau, const gsl_vector * v, gsl_matrix * A)
Function: int gsl_linalg_complex_householder_hm (gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A)

This function applies the Householder matrix P defined by the scalar tau and the vector v to the left-hand side of the matrix A. On output the result P A is stored in A.

Function: int gsl_linalg_householder_mh (double tau, const gsl_vector * v, gsl_matrix * A)
Function: int gsl_linalg_complex_householder_mh (gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A)

This function applies the Householder matrix P defined by the scalar tau and the vector v to the right-hand side of the matrix A. On output the result A P is stored in A.

Function: int gsl_linalg_householder_hv (double tau, const gsl_vector * v, gsl_vector * w)
Function: int gsl_linalg_complex_householder_hv (gsl_complex tau, const gsl_vector_complex * v, gsl_vector_complex * w)

This function applies the Householder transformation P defined by the scalar tau and the vector v to the vector w. On output the result P w is stored in w.


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14.12 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.


Next: , Previous: Householder solver for linear systems, Up: Linear Algebra   [Index]

14.13 Tridiagonal Systems

The functions described in this section efficiently solve symmetric, non-symmetric and cyclic tridiagonal systems with minimal storage. Note that the current implementations of these functions use a variant of Cholesky decomposition, so the tridiagonal matrix must be positive definite. For non-positive definite matrices, the functions return the error code GSL_ESING.

Function: int gsl_linalg_solve_tridiag (const gsl_vector * diag, const gsl_vector * e, const gsl_vector * f, const gsl_vector * b, gsl_vector * x)

This function solves the general N-by-N system A x = b where A is tridiagonal (N >= 2). The super-diagonal and sub-diagonal vectors e and f must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

A = ( d_0 e_0  0   0  )
    ( f_0 d_1 e_1  0  )
    (  0  f_1 d_2 e_2 )
    (  0   0  f_2 d_3 )
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 (N >= 2). The off-diagonal vector e must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

A = ( d_0 e_0  0   0  )
    ( e_0 d_1 e_1  0  )
    (  0  e_1 d_2 e_2 )
    (  0   0  e_2 d_3 )
Function: int gsl_linalg_solve_cyc_tridiag (const gsl_vector * diag, const gsl_vector * e, const gsl_vector * f, const gsl_vector * b, gsl_vector * x)

This function solves the general N-by-N system A x = b where A is cyclic tridiagonal (N >= 3). The cyclic super-diagonal and sub-diagonal vectors e and f must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

A = ( d_0 e_0  0  f_3 )
    ( f_0 d_1 e_1  0  )
    (  0  f_1 d_2 e_2 )
    ( e_3  0  f_2 d_3 )
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 (N >= 3). The cyclic off-diagonal vector e must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

A = ( d_0 e_0  0  e_3 )
    ( e_0 d_1 e_1  0  )
    (  0  e_1 d_2 e_2 )
    ( e_3  0  e_2 d_3 )

Next: , Previous: Tridiagonal Systems, Up: Linear Algebra   [Index]

14.14 Balancing

The process of balancing a matrix applies similarity transformations to make the rows and columns have comparable norms. This is useful, for example, to reduce roundoff errors in the solution of eigenvalue problems. Balancing a matrix A consists of replacing A with a similar matrix

A' = D^(-1) A D

where D is a diagonal matrix whose entries are powers of the floating point radix.

Function: int gsl_linalg_balance_matrix (gsl_matrix * A, gsl_vector * D)

This function replaces the matrix A with its balanced counterpart and stores the diagonal elements of the similarity transformation into the vector D.


Next: , Previous: Balancing, Up: Linear Algebra   [Index]

14.15 Examples

The following program solves the linear system A x = b. The system to be solved is,

[ 0.18 0.60 0.57 0.96 ] [x0]   [1.0]
[ 0.41 0.24 0.99 0.58 ] [x1] = [2.0]
[ 0.14 0.30 0.97 0.66 ] [x2]   [3.0]
[ 0.51 0.13 0.19 0.85 ] [x3]   [4.0]

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);
  gsl_vector_free (x);
  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.


Previous: Linear Algebra Examples, Up: Linear Algebra   [Index]

14.16 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 following manual,

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,


Next: , Previous: Linear Algebra, Up: Top   [Index]

15 Eigensystems

This chapter describes functions for computing eigenvalues and eigenvectors of matrices. There are routines for real symmetric, real nonsymmetric, complex hermitian, real generalized symmetric-definite, complex generalized hermitian-definite, and real generalized nonsymmetric eigensystems. Eigenvalues can be computed with or without eigenvectors. The hermitian and real symmetric matrix algorithms are symmetric bidiagonalization followed by QR reduction. The nonsymmetric algorithm is the Francis QR double-shift. The generalized nonsymmetric algorithm is the QZ method due to Moler and Stewart.

The functions described in this chapter are declared in the header file gsl_eigen.h.


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15.1 Real Symmetric Matrices

For real symmetric matrices, the library uses the symmetric bidiagonalization and QR reduction method. This is described in Golub & van Loan, section 8.3. The computed eigenvalues are accurate to an absolute accuracy of \epsilon ||A||_2, where \epsilon is the machine precision.

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.


Next: , Previous: Real Symmetric Matrices, Up: Eigensystems   [Index]

15.2 Complex Hermitian Matrices

For hermitian matrices, the library uses the complex form of the symmetric bidiagonalization and QR reduction method.

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.


Next: , Previous: Complex Hermitian Matrices, Up: Eigensystems   [Index]

15.3 Real Nonsymmetric Matrices

The solution of the real nonsymmetric eigensystem problem for a matrix A involves computing the Schur decomposition

A = Z T Z^T

where Z is an orthogonal matrix of Schur vectors and T, the Schur form, is quasi upper triangular with diagonal 1-by-1 blocks which are real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are complex conjugate eigenvalues of A. The algorithm used is the double-shift Francis method.

Function: gsl_eigen_nonsymm_workspace * gsl_eigen_nonsymm_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues of n-by-n real nonsymmetric matrices. The size of the workspace is O(2n).

Function: void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace * w)

This function frees the memory associated with the workspace w.

Function: void gsl_eigen_nonsymm_params (const int compute_t, const int balance, gsl_eigen_nonsymm_workspace * w)

This function sets some parameters which determine how the eigenvalue problem is solved in subsequent calls to gsl_eigen_nonsymm.

If compute_t is set to 1, the full Schur form T will be computed by gsl_eigen_nonsymm. If it is set to 0, T will not be computed (this is the default setting). Computing the full Schur form T requires approximately 1.5–2 times the number of flops.

If balance is set to 1, a balancing transformation is applied to the matrix prior to computing eigenvalues. This transformation is designed to make the rows and columns of the matrix have comparable norms, and can result in more accurate eigenvalues for matrices whose entries vary widely in magnitude. See Balancing for more information. Note that the balancing transformation does not preserve the orthogonality of the Schur vectors, so if you wish to compute the Schur vectors with gsl_eigen_nonsymm_Z you will obtain the Schur vectors of the balanced matrix instead of the original matrix. The relationship will be

T = Q^t D^(-1) A D Q

where Q is the matrix of Schur vectors for the balanced matrix, and D is the balancing transformation. Then gsl_eigen_nonsymm_Z will compute a matrix Z which satisfies

T = Z^(-1) A Z

with Z = D Q. Note that Z will not be orthogonal. For this reason, balancing is not performed by default.

Function: int gsl_eigen_nonsymm (gsl_matrix * A, gsl_vector_complex * eval, gsl_eigen_nonsymm_workspace * w)

This function computes the eigenvalues of the real nonsymmetric matrix A and stores them in the vector eval. If T is desired, it is stored in the upper portion of A on output. Otherwise, on output, the diagonal of A will contain the 1-by-1 real eigenvalues and 2-by-2 complex conjugate eigenvalue systems, and the rest of A is destroyed. In rare cases, this function may fail to find all eigenvalues. If this happens, an error code is returned and the number of converged eigenvalues is stored in w->n_evals. The converged eigenvalues are stored in the beginning of eval.

Function: int gsl_eigen_nonsymm_Z (gsl_matrix * A, gsl_vector_complex * eval, gsl_matrix * Z, gsl_eigen_nonsymm_workspace * w)

This function is identical to gsl_eigen_nonsymm except that it also computes the Schur vectors and stores them into Z.

Function: gsl_eigen_nonsymmv_workspace * gsl_eigen_nonsymmv_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real nonsymmetric matrices. The size of the workspace is O(5n).

Function: void gsl_eigen_nonsymmv_free (gsl_eigen_nonsymmv_workspace * w)

This function frees the memory associated with the workspace w.

Function: void gsl_eigen_nonsymmv_params (const int balance, gsl_eigen_nonsymm_workspace * w)

This function sets parameters which determine how the eigenvalue problem is solved in subsequent calls to gsl_eigen_nonsymmv. If balance is set to 1, a balancing transformation is applied to the matrix. See gsl_eigen_nonsymm_params for more information. Balancing is turned off by default since it does not preserve the orthogonality of the Schur vectors.

Function: int gsl_eigen_nonsymmv (gsl_matrix * A, gsl_vector_complex * eval, gsl_matrix_complex * evec, gsl_eigen_nonsymmv_workspace * w)

This function computes eigenvalues and right eigenvectors of the n-by-n real nonsymmetric matrix A. It first calls gsl_eigen_nonsymm to compute the eigenvalues, Schur form T, and Schur vectors. Then it finds eigenvectors of T and backtransforms them using the Schur vectors. The Schur vectors are destroyed in the process, but can be saved by using gsl_eigen_nonsymmv_Z. The computed eigenvectors are normalized to have unit magnitude. On output, the upper portion of A contains the Schur form T. If gsl_eigen_nonsymm fails, no eigenvectors are computed, and an error code is returned.

Function: int gsl_eigen_nonsymmv_Z (gsl_matrix * A, gsl_vector_complex * eval, gsl_matrix_complex * evec, gsl_matrix * Z, gsl_eigen_nonsymmv_workspace * w)

This function is identical to gsl_eigen_nonsymmv except that it also saves the Schur vectors into Z.


Next: , Previous: Real Nonsymmetric Matrices, Up: Eigensystems   [Index]

15.4 Real Generalized Symmetric-Definite Eigensystems

The real generalized symmetric-definite eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that

A x = \lambda B x

where A and B are symmetric matrices, and B is positive-definite. This problem reduces to the standard symmetric eigenvalue problem by applying the Cholesky decomposition to B:

                      A x = \lambda B x
                      A x = \lambda L L^t x
( L^{-1} A L^{-t} ) L^t x = \lambda L^t x

Therefore, the problem becomes C y = \lambda y where C = L^{-1} A L^{-t} is symmetric, and y = L^t x. The standard symmetric eigensolver can be applied to the matrix C. The resulting eigenvectors are backtransformed to find the vectors of the original problem. The eigenvalues and eigenvectors of the generalized symmetric-definite eigenproblem are always real.

Function: gsl_eigen_gensymm_workspace * gsl_eigen_gensymm_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues of n-by-n real generalized symmetric-definite eigensystems. The size of the workspace is O(2n).

Function: void gsl_eigen_gensymm_free (gsl_eigen_gensymm_workspace * w)

This function frees the memory associated with the workspace w.

Function: int gsl_eigen_gensymm (gsl_matrix * A, gsl_matrix * B, gsl_vector * eval, gsl_eigen_gensymm_workspace * w)

This function computes the eigenvalues of the real generalized symmetric-definite matrix pair (A, B), and stores them in eval, using the method outlined above. On output, B contains its Cholesky decomposition and A is destroyed.

Function: gsl_eigen_gensymmv_workspace * gsl_eigen_gensymmv_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real generalized symmetric-definite eigensystems. The size of the workspace is O(4n).

Function: void gsl_eigen_gensymmv_free (gsl_eigen_gensymmv_workspace * w)

This function frees the memory associated with the workspace w.

Function: int gsl_eigen_gensymmv (gsl_matrix * A, gsl_matrix * B, gsl_vector * eval, gsl_matrix * evec, gsl_eigen_gensymmv_workspace * w)

This function computes the eigenvalues and eigenvectors of the real generalized symmetric-definite matrix pair (A, B), and stores them in eval and evec respectively. The computed eigenvectors are normalized to have unit magnitude. On output, B contains its Cholesky decomposition and A is destroyed.


Next: , Previous: Real Generalized Symmetric-Definite Eigensystems, Up: Eigensystems   [Index]

15.5 Complex Generalized Hermitian-Definite Eigensystems

The complex generalized hermitian-definite eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that

A x = \lambda B x

where A and B are hermitian matrices, and B is positive-definite. Similarly to the real case, this can be reduced to C y = \lambda y where C = L^{-1} A L^{-H} is hermitian, and y = L^H x. The standard hermitian eigensolver can be applied to the matrix C. The resulting eigenvectors are backtransformed to find the vectors of the original problem. The eigenvalues of the generalized hermitian-definite eigenproblem are always real.

Function: gsl_eigen_genherm_workspace * gsl_eigen_genherm_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues of n-by-n complex generalized hermitian-definite eigensystems. The size of the workspace is O(3n).

Function: void gsl_eigen_genherm_free (gsl_eigen_genherm_workspace * w)

This function frees the memory associated with the workspace w.

Function: int gsl_eigen_genherm (gsl_matrix_complex * A, gsl_matrix_complex * B, gsl_vector * eval, gsl_eigen_genherm_workspace * w)

This function computes the eigenvalues of the complex generalized hermitian-definite matrix pair (A, B), and stores them in eval, using the method outlined above. On output, B contains its Cholesky decomposition and A is destroyed.

Function: gsl_eigen_genhermv_workspace * gsl_eigen_genhermv_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n complex generalized hermitian-definite eigensystems. The size of the workspace is O(5n).

Function: void gsl_eigen_genhermv_free (gsl_eigen_genhermv_workspace * w)

This function frees the memory associated with the workspace w.

Function: int gsl_eigen_genhermv (gsl_matrix_complex * A, gsl_matrix_complex * B, gsl_vector * eval, gsl_matrix_complex * evec, gsl_eigen_genhermv_workspace * w)

This function computes the eigenvalues and eigenvectors of the complex generalized hermitian-definite matrix pair (A, B), and stores them in eval and evec respectively. The computed eigenvectors are normalized to have unit magnitude. On output, B contains its Cholesky decomposition and A is destroyed.


Next: , Previous: Complex Generalized Hermitian-Definite Eigensystems, Up: Eigensystems   [Index]

15.6 Real Generalized Nonsymmetric Eigensystems

Given two square matrices (A, B), the generalized nonsymmetric eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that

A x = \lambda B x

We may also define the problem as finding eigenvalues \mu and eigenvectors y such that

\mu A y = B y

Note that these two problems are equivalent (with \lambda = 1/\mu) if neither \lambda nor \mu is zero. If say, \lambda is zero, then it is still a well defined eigenproblem, but its alternate problem involving \mu is not. Therefore, to allow for zero (and infinite) eigenvalues, the problem which is actually solved is

\beta A x = \alpha B x

The eigensolver routines below will return two values \alpha and \beta and leave it to the user to perform the divisions \lambda = \alpha / \beta and \mu = \beta / \alpha.

If the determinant of the matrix pencil A - \lambda B is zero for all \lambda, the problem is said to be singular; otherwise it is called regular. Singularity normally leads to some \alpha = \beta = 0 which means the eigenproblem is ill-conditioned and generally does not have well defined eigenvalue solutions. The routines below are intended for regular matrix pencils and could yield unpredictable results when applied to singular pencils.

The solution of the real generalized nonsymmetric eigensystem problem for a matrix pair (A, B) involves computing the generalized Schur decomposition

A = Q S Z^T
B = Q T Z^T

where Q and Z are orthogonal matrices of left and right Schur vectors respectively, and (S, T) is the generalized Schur form whose diagonal elements give the \alpha and \beta values. The algorithm used is the QZ method due to Moler and Stewart (see references).

Function: gsl_eigen_gen_workspace * gsl_eigen_gen_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(n).

Function: void gsl_eigen_gen_free (gsl_eigen_gen_workspace * w)

This function frees the memory associated with the workspace w.

Function: void gsl_eigen_gen_params (const int compute_s, const int compute_t, const int balance, gsl_eigen_gen_workspace * w)

This function sets some parameters which determine how the eigenvalue problem is solved in subsequent calls to gsl_eigen_gen.

If compute_s is set to 1, the full Schur form S will be computed by gsl_eigen_gen. If it is set to 0, S will not be computed (this is the default setting). S is a quasi upper triangular matrix with 1-by-1 and 2-by-2 blocks on its diagonal. 1-by-1 blocks correspond to real eigenvalues, and 2-by-2 blocks correspond to complex eigenvalues.

If compute_t is set to 1, the full Schur form T will be computed by gsl_eigen_gen. If it is set to 0, T will not be computed (this is the default setting). T is an upper triangular matrix with non-negative elements on its diagonal. Any 2-by-2 blocks in S will correspond to a 2-by-2 diagonal block in T.

The balance parameter is currently ignored, since generalized balancing is not yet implemented.

Function: int gsl_eigen_gen (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_eigen_gen_workspace * w)

This function computes the eigenvalues of the real generalized nonsymmetric matrix pair (A, B), and stores them as pairs in (alpha, beta), where alpha is complex and beta is real. If \beta_i is non-zero, then \lambda = \alpha_i / \beta_i is an eigenvalue. Likewise, if \alpha_i is non-zero, then \mu = \beta_i / \alpha_i is an eigenvalue of the alternate problem \mu A y = B y. The elements of beta are normalized to be non-negative.

If S is desired, it is stored in A on output. If T is desired, it is stored in B on output. The ordering of eigenvalues in (alpha, beta) follows the ordering of the diagonal blocks in the Schur forms S and T. In rare cases, this function may fail to find all eigenvalues. If this occurs, an error code is returned.

Function: int gsl_eigen_gen_QZ (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix * Q, gsl_matrix * Z, gsl_eigen_gen_workspace * w)

This function is identical to gsl_eigen_gen except that it also computes the left and right Schur vectors and stores them into Q and Z respectively.

Function: gsl_eigen_genv_workspace * gsl_eigen_genv_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(7n).

Function: void gsl_eigen_genv_free (gsl_eigen_genv_workspace * w)

This function frees the memory associated with the workspace w.

Function: int gsl_eigen_genv (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix_complex * evec, gsl_eigen_genv_workspace * w)

This function computes eigenvalues and right eigenvectors of the n-by-n real generalized nonsymmetric matrix pair (A, B). The eigenvalues are stored in (alpha, beta) and the eigenvectors are stored in evec. It first calls gsl_eigen_gen to compute the eigenvalues, Schur forms, and Schur vectors. Then it finds eigenvectors of the Schur forms and backtransforms them using the Schur vectors. The Schur vectors are destroyed in the process, but can be saved by using gsl_eigen_genv_QZ. The computed eigenvectors are normalized to have unit magnitude. On output, (A, B) contains the generalized Schur form (S, T). If gsl_eigen_gen fails, no eigenvectors are computed, and an error code is returned.

Function: int gsl_eigen_genv_QZ (gsl_matrix * A, gsl_matrix * B, gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix_complex * evec, gsl_matrix * Q, gsl_matrix * Z, gsl_eigen_genv_workspace * w)

This function is identical to gsl_eigen_genv except that it also computes the left and right Schur vectors and stores them into Q and Z respectively.


Next: , Previous: Real Generalized Nonsymmetric Eigensystems, Up: Eigensystems   [Index]

15.7 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.

Function: int gsl_eigen_nonsymmv_sort (gsl_vector_complex * 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. Only GSL_EIGEN_SORT_ABS_ASC and GSL_EIGEN_SORT_ABS_DESC are supported due to the eigenvalues being complex.

Function: int gsl_eigen_gensymmv_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 as shown above.

Function: int gsl_eigen_genhermv_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.

Function: int gsl_eigen_genv_sort (gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix_complex * evec, gsl_eigen_sort_t sort_type)

This function simultaneously sorts the eigenvalues stored in the vectors (alpha, beta) 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. Only GSL_EIGEN_SORT_ABS_ASC and GSL_EIGEN_SORT_ABS_DESC are supported due to the eigenvalues being complex.


Next: , Previous: Sorting Eigenvalues and Eigenvectors, Up: Eigensystems   [Index]

15.8 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");
      }
  }

  gsl_vector_free (eval);
  gsl_matrix_free (evec);

  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.

The following program illustrates the use of the nonsymmetric eigensolver, by computing the eigenvalues and eigenvectors of the Vandermonde matrix V(x;i,j) = x_i^{n - j} with x = (-1,-2,3,4).

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>

int
main (void)
{
  double data[] = { -1.0, 1.0, -1.0, 1.0,
                    -8.0, 4.0, -2.0, 1.0,
                    27.0, 9.0, 3.0, 1.0,
                    64.0, 16.0, 4.0, 1.0 };

  gsl_matrix_view m 
    = gsl_matrix_view_array (data, 4, 4);

  gsl_vector_complex *eval = gsl_vector_complex_alloc (4);
  gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4);

  gsl_eigen_nonsymmv_workspace * w = 
    gsl_eigen_nonsymmv_alloc (4);
  
  gsl_eigen_nonsymmv (&m.matrix, eval, evec, w);

  gsl_eigen_nonsymmv_free (w);

  gsl_eigen_nonsymmv_sort (eval, evec, 
                           GSL_EIGEN_SORT_ABS_DESC);
  
  {
    int i, j;

    for (i = 0; i < 4; i++)
      {
        gsl_complex eval_i 
           = gsl_vector_complex_get (eval, i);
        gsl_vector_complex_view evec_i 
           = gsl_matrix_complex_column (evec, i);

        printf ("eigenvalue = %g + %gi\n",
                GSL_REAL(eval_i), GSL_IMAG(eval_i));
        printf ("eigenvector = \n");
        for (j = 0; j < 4; ++j)
          {
            gsl_complex z = 
              gsl_vector_complex_get(&evec_i.vector, j);
            printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));
          }
      }
  }

  gsl_vector_complex_free(eval);
  gsl_matrix_complex_free(evec);

  return 0;
}

Here is the beginning of the output from the program,

$ ./a.out 
eigenvalue = -6.41391 + 0i
eigenvector = 
-0.0998822 + 0i
-0.111251 + 0i
0.292501 + 0i
0.944505 + 0i
eigenvalue = 5.54555 + 3.08545i
eigenvector = 
-0.043487 + -0.0076308i
0.0642377 + -0.142127i
-0.515253 + 0.0405118i
-0.840592 + -0.00148565i
...

This can be compared with the corresponding output from GNU OCTAVE,

octave> [v,d] = eig(vander([-1 -2 3 4]));
octave> diag(d)
ans =

  -6.4139 + 0.0000i
   5.5456 + 3.0854i
   5.5456 - 3.0854i
   2.3228 + 0.0000i

octave> v
v =

 Columns 1 through 3:

  -0.09988 + 0.00000i  -0.04350 - 0.00755i  -0.04350 + 0.00755i
  -0.11125 + 0.00000i   0.06399 - 0.14224i   0.06399 + 0.14224i
   0.29250 + 0.00000i  -0.51518 + 0.04142i  -0.51518 - 0.04142i
   0.94451 + 0.00000i  -0.84059 + 0.00000i  -0.84059 - 0.00000i

 Column 4:

  -0.14493 + 0.00000i
   0.35660 + 0.00000i
   0.91937 + 0.00000i
   0.08118 + 0.00000i

Note that the eigenvectors corresponding to the eigenvalue 5.54555 + 3.08545i differ by the multiplicative constant 0.9999984 + 0.0017674i which is an arbitrary phase factor of magnitude 1.


Previous: Eigenvalue and Eigenvector Examples, Up: Eigensystems   [Index]

15.9 References and Further Reading

Further information on the algorithms described in this section can be found in the following book,

Further information on the generalized eigensystems QZ algorithm can be found in this paper,

Eigensystem routines for very large matrices can be found in the Fortran library LAPACK. 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.


Next: , Previous: Eigensystems, Up: Top   [Index]

16 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 of 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 FFT References and Further Reading)


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16.1 Mathematical Definitions

Fast Fourier Transforms are efficient algorithms for calculating the discrete Fourier transform (DFT),

x_j = \sum_{k=0}^{n-1} z_k \exp(-2\pi i j k / n) 

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_m 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,

x_j = \sum_{k=0}^{n-1} z_k \exp(-2\pi i j k / n) 

and the definition of the inverse Fourier transform, x = IFFT(z), is,

z_j = {1 \over n} \sum_{k=0}^{n-1} x_k \exp(2\pi i j k / n).

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,

z^{backwards}_j = \sum_{k=0}^{n-1} x_k \exp(2\pi i j k / n).

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.


Next: , Previous: Mathematical Definitions, Up: Fast Fourier Transforms   [Index]

16.2 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,

double x[3*2];
gsl_complex_packed_array data = x;

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])

The array indices for the data have the same ordering as those in the definition of the DFT—i.e. there are no index transformations or permutations of the data.

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.

To perform an FFT on a vector argument, such as gsl_vector_complex * v, use the following definitions (or their equivalents) when calling the functions described in this chapter:

gsl_complex_packed_array data = v->data;
size_t stride = v->stride;
size_t n = v->size;

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.


Next: , Previous: Overview of complex data FFTs, Up: Fast Fourier Transforms   [Index]

16.3 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 the functions described in this section 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, gsl_fft_direction sign)
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. For the transform version of the function the sign argument can be either forward (-1) or backward (+1).

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, gsl_fft_direction sign)
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 (-1010), 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 sine function.


Next: , Previous: Radix-2 FFT routines for complex data, Up: Fast Fourier Transforms   [Index]

16.4 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 Paul Swarztrauber’s Fortran FFTPACK library. 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 the functions described in this section 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 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, gsl_fft_direction sign)
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. For the transform version of the function the sign argument can be either forward (-1) or backward (+1).

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] = 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.


Next: , Previous: Mixed-radix FFT routines for complex data, Up: Fast Fourier Transforms   [Index]

16.5 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:

z_k = z_{n-k}^*

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,

c_k = \sum_{j=0}^{n-1} x_j \exp(-2 \pi i j k /n)

Functions in gsl_fft_halfcomplex compute inverse or backwards transforms. They reconstruct real sequences by Fourier synthesis from their half-complex frequency coefficients, c,

x_j = {1 \over n} \sum_{k=0}^{n-1} c_k \exp(2 \pi i j k /n)

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 constrains 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 (which is desirable for better locality of memory accesses).


Next: , Previous: Overview of real data FFTs, Up: Fast Fourier Transforms   [Index]

16.6 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 (assuming stride=1),

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].imag  =    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]

Note that the output data can be converted into the full complex sequence using the function gsl_fft_halfcomplex_radix2_unpack described below.

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.

Function: int gsl_fft_halfcomplex_radix2_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_radix2_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[i*stride];
    double hc_imag 
      = halfcomplex_coefficient[(n-i)*stride];
    complex_coefficient[i*stride].real = hc_real;
    complex_coefficient[i*stride].imag = hc_imag;
    complex_coefficient[(n - i)*stride].real = hc_real;
    complex_coefficient[(n - i)*stride].imag = -hc_imag;
  }

if (i == n - i)
  {
    complex_coefficient[i*stride].real 
      = halfcomplex_coefficient[(n - 1)*stride];
    complex_coefficient[i*stride].imag 
      = 0.0;
  }

Next: , Previous: Radix-2 FFT routines for real data, Up: Fast Fourier Transforms   [Index]

16.7 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=6. In the even case 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 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 can be 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*stride].real 
      = real_coefficient[i*stride];
    complex_coefficient[i*stride].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)*stride];
    double hc_imag 
      = halfcomplex_coefficient[(2 * i)*stride];
    complex_coefficient[i*stride].real = hc_real;
    complex_coefficient[i*stride].imag = hc_imag;
    complex_coefficient[(n - i)*stride].real = hc_real;
    complex_coefficient[(n - i)*stride].imag = -hc_imag;
  }

if (i == n - i)
  {
    complex_coefficient[i*stride].real 
      = halfcomplex_coefficient[(n - 1)*stride];
    complex_coefficient[i*stride].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;
}

Previous: Mixed-radix FFT routines for real data, Up: Fast Fourier Transforms   [Index]

16.8 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 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 large-scale 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.

The source code for FFTPACK is available from Netlib,


Next: , Previous: Fast Fourier Transforms, Up: Top   [Index]

17 Numerical Integration

This chapter describes routines for performing numerical integration (quadrature) of a function in one dimension. There are routines for adaptive and non-adaptive integration of general functions, with specialised routines for specific cases. These include integration over infinite and semi-infinite ranges, singular integrals, including logarithmic singularities, computation of Cauchy principal values and oscillatory integrals. The library reimplements the algorithms used in QUADPACK, a numerical integration package written by Piessens, de Doncker-Kapenga, Ueberhuber and Kahaner. Fortran code for QUADPACK is available on Netlib. Also included are non-adaptive, fixed-order Gauss-Legendre integration routines with high precision coefficients by Pavel Holoborodko.

The functions described in this chapter are declared in the header file gsl_integration.h.


Next: , Up: Numerical Integration   [Index]

17.1 Introduction

Each algorithm computes an approximation to a definite integral of the form,

I = \int_a^b f(x) w(x) dx

where w(x) is a weight function (for general integrands w(x)=1). The user provides absolute and relative error bounds (epsabs, epsrel) which specify the following accuracy requirement,

|RESULT - I|  <= max(epsabs, epsrel |I|)

where RESULT is the numerical approximation obtained by the algorithm. The algorithms attempt to estimate the absolute error ABSERR = |RESULT - I| in such a way that the following inequality holds,

|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)

In short, the routines return the first approximation which has an absolute error smaller than epsabs or a relative error smaller than epsrel.

Note that this is an either-or constraint, not simultaneous. To compute to a specified absolute error, set epsrel to zero. To compute to a specified relative error, set epsabs to zero. The routines will fail to converge if the error bounds are too stringent, but always return the best approximation obtained up to that stage.

The algorithms in QUADPACK use a naming convention based on the following letters,

Q - quadrature routine

N - non-adaptive integrator
A - adaptive integrator

G - general integrand (user-defined)
W - weight function with integrand

S - singularities can be more readily integrated
P - points of special difficulty can be supplied
I - infinite range of integration
O - oscillatory weight function, cos or sin
F - Fourier integral
C - Cauchy principal value

The algorithms are built on pairs of quadrature rules, a higher order rule and a lower order rule. The higher order rule is used to compute the best approximation to an integral over a small range. The difference between the results of the higher order rule and the lower order rule gives an estimate of the error in the approximation.


Next: , Up: Numerical Integration Introduction   [Index]

17.1.1 Integrands without weight functions

The algorithms for general functions (without a weight function) are based on Gauss-Kronrod rules.

A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of order m. This is extended with additional points between each of the abscissae to give a higher order Kronrod rule of order 2m+1. The Kronrod rule is efficient because it reuses existing function evaluations from the Gaussian rule.

The higher order Kronrod rule is used as the best approximation to the integral, and the difference between the two rules is used as an estimate of the error in the approximation.


Next: , Previous: Integrands without weight functions, Up: Numerical Integration Introduction   [Index]

17.1.2 Integrands with weight functions

For integrands with weight functions the algorithms use Clenshaw-Curtis quadrature rules.

A Clenshaw-Curtis rule begins with an n-th order Chebyshev polynomial approximation to the integrand. This polynomial can be integrated exactly to give an approximation to the integral of the original function. The Chebyshev expansion can be extended to higher orders to improve the approximation and provide an estimate of the error.


Previous: Integrands with weight functions, Up: Numerical Integration Introduction   [Index]

17.1.3 Integrands with singular weight functions

The presence of singularities (or other behavior) in the integrand can cause slow convergence in the Chebyshev approximation. The modified Clenshaw-Curtis rules used in QUADPACK separate out several common weight functions which cause slow convergence.

These weight functions are integrated analytically against the Chebyshev polynomials to precompute modified Chebyshev moments. Combining the moments with the Chebyshev approximation to the function gives the desired integral. The use of analytic integration for the singular part of the function allows exact cancellations and substantially improves the overall convergence behavior of the integration.


Next: , Previous: Numerical Integration Introduction, Up: Numerical Integration   [Index]

17.2 QNG non-adaptive Gauss-Kronrod integration

The QNG algorithm is a non-adaptive procedure which uses fixed Gauss-Kronrod-Patterson abscissae to sample the integrand at a maximum of 87 points. It is provided for fast integration of smooth functions.

Function: int gsl_integration_qng (const gsl_function * f, double a, double b, double epsabs, double epsrel, double * result, double * abserr, size_t * neval)

This function applies the Gauss-Kronrod 10-point, 21-point, 43-point and 87-point integration rules in succession until an estimate of the integral of f over (a,b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The function returns the final approximation, result, an estimate of the absolute error, abserr and the number of function evaluations used, neval. The Gauss-Kronrod rules are designed in such a way that each rule uses all the results of its predecessors, in order to minimize the total number of function evaluations.


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17.3 QAG adaptive integration

The QAG algorithm is a simple adaptive integration procedure. The integration region is divided into subintervals, and on each iteration the subinterval with the largest estimated error is bisected. This reduces the overall error rapidly, as the subintervals become concentrated around local difficulties in the integrand. These subintervals are managed by a gsl_integration_workspace struct, which handles the memory for the subinterval ranges, results and error estimates.

Function: gsl_integration_workspace * gsl_integration_workspace_alloc (size_t n)

This function allocates a workspace sufficient to hold n double precision intervals, their integration results and error estimates. One workspace may be used multiple times as all necessary reinitialization is performed automatically by the integration routines.

Function: void gsl_integration_workspace_free (gsl_integration_workspace * w)

This function frees the memory associated with the workspace w.

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)

This function applies an integration rule adaptively until an estimate of the integral of f over (a,b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The function returns the final approximation, result, and an estimate of the absolute error, abserr. The integration rule is determined by the value of key, which should be chosen from the following symbolic names,

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)

corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod rules. The higher-order rules give better accuracy for smooth functions, while lower-order rules save time when the function contains local difficulties, such as discontinuities.

On each iteration the adaptive integration strategy bisects the interval with the largest error estimate. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace.


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17.4 QAGS adaptive integration with singularities

The presence of an integrable singularity in the integration region causes an adaptive routine to concentrate new subintervals around the singularity. As the subintervals decrease in size the successive approximations to the integral converge in a limiting fashion. This approach to the limit can be accelerated using an extrapolation procedure. The QAGS algorithm combines adaptive bisection with the Wynn epsilon-algorithm to speed up the integration of many types of integrable singularities.

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)

This function applies the Gauss-Kronrod 21-point integration rule adaptively until an estimate of the integral of f over (a,b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The results are extrapolated using the epsilon-algorithm, which accelerates the convergence of the integral in the presence of discontinuities and integrable singularities. The function returns the final approximation from the extrapolation, result, and an estimate of the absolute error, abserr. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace.


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17.5 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)

This function applies the adaptive integration algorithm QAGS taking account of the user-supplied locations of singular points. The array pts of length npts should contain the endpoints of the integration ranges defined by the integration region and locations of the singularities. For example, to integrate over the region (a,b) with break-points at x_1, x_2, x_3 (where a < x_1 < x_2 < x_3 < b) the following pts array should be used

pts[0] = a
pts[1] = x_1
pts[2] = x_2
pts[3] = x_3
pts[4] = b

with npts = 5.

If you know the locations of the singular points in the integration region then this routine will be faster than QAGS.


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17.6 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)

This function computes the integral of the function f over the infinite interval (-\infty,+\infty). The integral is mapped onto the semi-open interval (0,1] using the transformation x = (1-t)/t,

\int_{-\infty}^{+\infty} dx f(x) = 
     \int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2.

It is then integrated using the QAGS algorithm. The normal 21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point rule, because the transformation can generate an integrable singularity at the origin. In this case a lower-order rule is more efficient.

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)

This function computes the integral of the function f over the semi-infinite interval (a,+\infty). The integral is mapped onto the semi-open interval (0,1] using the transformation x = a + (1-t)/t,

\int_{a}^{+\infty} dx f(x) = 
     \int_0^1 dt f(a + (1-t)/t)/t^2

and then integrated using the QAGS algorithm.

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)

This function computes the integral of the function f over the semi-infinite interval (-\infty,b). The integral is mapped onto the semi-open interval (0,1] using the transformation x = b - (1-t)/t,

\int_{-\infty}^{b} dx f(x) = 
     \int_0^1 dt f(b - (1-t)/t)/t^2

and then integrated using the QAGS algorithm.


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17.7 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)

This function computes the Cauchy principal value of the integral of f over (a,b), with a singularity at c,

I = \int_a^b dx f(x) / (x - c)

The adaptive bisection algorithm of QAG is used, with modifications to ensure that subdivisions do not occur at the singular point x = c. When a subinterval contains the point x = c or is close to it then a special 25-point modified Clenshaw-Curtis rule is used to control the singularity. Further away from the singularity the algorithm uses an ordinary 15-point Gauss-Kronrod integration rule.


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17.8 QAWS adaptive integration for singular functions

The QAWS algorithm is designed for integrands with algebraic-logarithmic singularities at the end-points of an integration region. In order to work efficiently the algorithm requires a precomputed table of Chebyshev moments.

Function: gsl_integration_qaws_table * gsl_integration_qaws_table_alloc (double alpha, double beta, int mu, int nu)

This function allocates space for a gsl_integration_qaws_table struct describing a singular weight function W(x) with the parameters (\alpha, \beta, \mu, \nu),

W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)

where \alpha > -1, \beta > -1, and \mu = 0, 1, \nu = 0, 1. The weight function can take four different forms depending on the values of \mu and \nu,

W(x) = (x-a)^alpha (b-x)^beta                   (mu = 0, nu = 0)
W(x) = (x-a)^alpha (b-x)^beta log(x-a)          (mu = 1, nu = 0)
W(x) = (x-a)^alpha (b-x)^beta log(b-x)          (mu = 0, nu = 1)
W(x) = (x-a)^alpha (b-x)^beta log(x-a) log(b-x) (mu = 1, nu = 1)

The singular points (a,b) do not have to be specified until the integral is computed, where they are the endpoints of the integration range.

The function returns a pointer to the newly allocated table gsl_integration_qaws_table if no errors were detected, and 0 in the case of error.

Function: int gsl_integration_qaws_table_set (gsl_integration_qaws_table * t, double alpha, double beta, int mu, int nu)

This function modifies the parameters (\alpha, \beta, \mu, \nu) of an existing gsl_integration_qaws_table struct t.

Function: void gsl_integration_qaws_table_free (gsl_integration_qaws_table * t)

This function frees all the memory associated with the gsl_integration_qaws_table struct t.

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)

This function computes the integral of the function f(x) over the interval (a,b) with the singular weight function (x-a)^\alpha (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x). The parameters of the weight function (\alpha, \beta, \mu, \nu) are taken from the table t. The integral is,

I = \int_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x).

The adaptive bisection algorithm of QAG is used. When a subinterval contains one of the endpoints then a special 25-point modified Clenshaw-Curtis rule is used to control the singularities. For subintervals which do not include the endpoints an ordinary 15-point Gauss-Kronrod integration rule is used.


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17.9 QAWO adaptive integration for oscillatory functions

The QAWO algorithm is designed for integrands with an oscillatory factor, \sin(\omega x) or \cos(\omega x). In order to work efficiently the algorithm requires a table of Chebyshev moments which must be pre-computed with calls to the functions below.

Function: gsl_integration_qawo_table * gsl_integration_qawo_table_alloc (double omega, double L, enum gsl_integration_qawo_enum sine, size_t n)

This function allocates space for a gsl_integration_qawo_table struct and its associated workspace describing a sine or cosine weight function W(x) with the parameters (\omega, L),

W(x) = sin(omega x)
W(x) = cos(omega x)

The parameter L must be the length of the interval over which the function will be integrated L = b - a. The choice of sine or cosine is made with the parameter sine which should be chosen from one of the two following symbolic values:

GSL_INTEG_COSINE
GSL_INTEG_SINE

The gsl_integration_qawo_table is a table of the trigonometric coefficients required in the integration process. The parameter n determines the number of levels of coefficients that are computed. Each level corresponds to one bisection of the interval L, so that n levels are sufficient for subintervals down to the length L/2^n. The integration routine gsl_integration_qawo returns the error GSL_ETABLE if the number of levels is insufficient for the requested accuracy.

Function: int gsl_integration_qawo_table_set (gsl_integration_qawo_table * t, double omega, double L, enum gsl_integration_qawo_enum sine)

This function changes the parameters omega, L and sine of the existing workspace t.

Function: int gsl_integration_qawo_table_set_length (gsl_integration_qawo_table * t, double L)

This function allows the length parameter L of the workspace t to be changed.

Function: void gsl_integration_qawo_table_free (gsl_integration_qawo_table * t)

This function frees all the memory associated with the workspace t.

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)

This function uses an adaptive algorithm to compute the integral of f over (a,b) with the weight function \sin(\omega x) or \cos(\omega x) defined by the table wf,

I = \int_a^b dx f(x) sin(omega x)
I = \int_a^b dx f(x) cos(omega x)

The results are extrapolated using the epsilon-algorithm to accelerate the convergence of the integral. The function returns the final approximation from the extrapolation, result, and an estimate of the absolute error, abserr. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace.

Those subintervals with “large” widths d where d\omega > 4 are computed using a 25-point Clenshaw-Curtis integration rule, which handles the oscillatory behavior. Subintervals with a “small” widths where d\omega < 4 are computed using a 15-point Gauss-Kronrod integration.


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17.10 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)

This function attempts to compute a Fourier integral of the function f over the semi-infinite interval [a,+\infty).

I = \int_a^{+\infty} dx f(x) sin(omega x)
I = \int_a^{+\infty} dx f(x) cos(omega x)

The parameter \omega and choice of \sin or \cos is taken from the table wf (the length L can take any value, since it is overridden by this function to a value appropriate for the Fourier integration). The integral is computed using the QAWO algorithm over each of the subintervals,

C_1 = [a, a + c]
C_2 = [a + c, a + 2 c]
... = ...
C_k = [a + (k-1) c, a + k c]

where c = (2 floor(|\omega|) + 1) \pi/|\omega|. The width c is chosen to cover an odd number of periods so that the contributions from the intervals alternate in sign and are monotonically decreasing when f is positive and monotonically decreasing. The sum of this sequence of contributions is accelerated using the epsilon-algorithm.

This function works to an overall absolute tolerance of abserr. The following strategy is used: on each interval C_k the algorithm tries to achieve the tolerance

TOL_k = u_k abserr

where u_k = (1 - p)p^{k-1} and p = 9/10. The sum of the geometric series of contributions from each interval gives an overall tolerance of abserr.

If the integration of a subinterval leads to difficulties then the accuracy requirement for subsequent intervals is relaxed,

TOL_k = u_k max(abserr, max_{i<k}{E_i})

where E_k is the estimated error on the interval C_k.

The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace. The integration over each subinterval uses the memory provided by cycle_workspace as workspace for the QAWO algorithm.


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17.11 CQUAD doubly-adaptive integration

CQUAD is a new doubly-adaptive general-purpose quadrature routine which can handle most types of singularities, non-numerical function values such as Inf or NaN, as well as some divergent integrals. It generally requires more function evaluations than the integration routines in QUADPACK, yet fails less often for difficult integrands.

The underlying algorithm uses a doubly-adaptive scheme in which Clenshaw-Curtis quadrature rules of increasing degree are used to compute the integral in each interval. The L_2-norm of the difference between the underlying interpolatory polynomials of two successive rules is used as an error estimate. The interval is subdivided if the difference between two successive rules is too large or a rule of maximum degree has been reached.

Function: gsl_integration_cquad_workspace * gsl_integration_cquad_workspace_alloc (size_t n)

This function allocates a workspace sufficient to hold the data for n intervals. The number n is not the maximum number of intervals that will be evaluated. If the workspace is full, intervals with smaller error estimates will be discarded. A minimum of 3 intervals is required and for most functions, a workspace of size 100 is sufficient.

Function: void gsl_integration_cquad_workspace_free (gsl_integration_cquad_workspace * w)

This function frees the memory associated with the workspace w.

Function: int gsl_integration_cquad (const gsl_function * f, double a, double b, double epsabs, double epsrel, gsl_integration_cquad_workspace * workspace, double * result, double * abserr, size_t * nevals)

This function computes the integral of f over (a,b) within the desired absolute and relative error limits, epsabs and epsrel using the CQUAD algorithm. The function returns the final approximation, result, an estimate of the absolute error, abserr, and the number of function evaluations required, nevals.

The CQUAD algorithm divides the integration region into subintervals, and in each iteration, the subinterval with the largest estimated error is processed. The algorithm uses Clenshaw-Curits quadrature rules of degree 4, 8, 16 and 32 over 5, 9, 17 and 33 nodes respectively. Each interval is initialized with the lowest-degree rule. When an interval is processed, the next-higher degree rule is evaluated and an error estimate is computed based on the L_2-norm of the difference between the underlying interpolating polynomials of both rules. If the highest-degree rule has already been used, or the interpolatory polynomials differ significantly, the interval is bisected.

The subintervals and their results are stored in the memory provided by workspace. If the error estimate or the number of function evaluations is not needed, the pointers abserr and nevals can be set to NULL.


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17.12 Gauss-Legendre integration

The fixed-order Gauss-Legendre integration routines are provided for fast integration of smooth functions with known polynomial order. The n-point Gauss-Legendre rule is exact for polynomials of order 2*n-1 or less. For example, these rules are useful when integrating basis functions to form mass matrices for the Galerkin method. Unlike other numerical integration routines within the library, these routines do not accept absolute or relative error bounds.

Function: gsl_integration_glfixed_table * gsl_integration_glfixed_table_alloc (size_t n)

This function determines the Gauss-Legendre abscissae and weights necessary for an n-point fixed order integration scheme. If possible, high precision precomputed coefficients are used. If precomputed weights are not available, lower precision coefficients are computed on the fly.

Function: double gsl_integration_glfixed (const gsl_function * f, double a, double b, const gsl_integration_glfixed_table * t)

This function applies the Gauss-Legendre integration rule contained in table t and returns the result.

Function: int gsl_integration_glfixed_point (double a, double b, size_t i, double * xi, double * wi, const gsl_integration_glfixed_table * t)

For i in [0, …, t->n - 1], this function obtains the i-th Gauss-Legendre point xi and weight wi on the interval [a,b]. The points and weights are ordered by increasing point value. A function f may be integrated on [a,b] by summing wi * f(xi) over i.

Function: void gsl_integration_glfixed_table_free (gsl_integration_glfixed_table * t)

This function frees the memory associated with the table t.


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17.13 Error codes

In addition to the standard error codes for invalid arguments the functions can return the following values,

GSL_EMAXITER

the maximum number of subdivisions was exceeded.

GSL_EROUND

cannot reach tolerance because of roundoff error, or roundoff error was detected in the extrapolation table.

GSL_ESING

a non-integrable singularity or other bad integrand behavior was found in the integration interval.

GSL_EDIVERGE

the integral is divergent, or too slowly convergent to be integrated numerically.


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17.14 Examples

The integrator QAGS will handle a large class of definite integrals. For example, consider the following integral, which has an algebraic-logarithmic singularity at the origin,

\int_0^1 x^{-1/2} log(x) dx = -4

The program below computes this integral to a relative accuracy bound of 1e-7.

#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);

  gsl_integration_workspace_free (w);

  return 0;
}

The results below show that the desired accuracy is achieved after 8 subdivisions.

$ ./a.out 
result          = -3.999999999999973799
exact result    = -4.000000000000000000
estimated error =  0.000000000000246025
actual error    =  0.000000000000026201
intervals =  8

In fact, the extrapolation procedure used by QAGS produces an accuracy of almost twice as many digits. The error estimate returned by the extrapolation procedure is larger than the actual error, giving a margin of safety of one order of magnitude.


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17.15 References and Further Reading

The following book is the definitive reference for QUADPACK, and was written by the original authors. It provides descriptions of the algorithms, program listings, test programs and examples. It also includes useful advice on numerical integration and many references to the numerical integration literature used in developing QUADPACK.

The CQUAD integration algorithm is described in the following paper:


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18 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.


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18.1 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 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”.


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18.2 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.


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18.3 Random number generator initialization

Function: 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 Random number environment variables).

The details of the available generator types are described later in this chapter.

Function: 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 >= 1 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.

When using multiple seeds with the same generator, choose seed values greater than zero to avoid collisions with the default setting.

Note that the most generators only accept 32-bit seeds, with higher values being reduced modulo 2^32. For generators with smaller ranges the maximum seed value will typically be lower.

Function: void gsl_rng_free (gsl_rng * r)

This function frees all the memory associated with the generator r.


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18.4 Sampling from a random number generator

The following functions return uniformly distributed random numbers, either as integers or double precision floating point numbers. Inline versions of these functions are used when HAVE_INLINE is defined. To obtain non-uniform distributions see Random Number Distributions.

Function: 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 be determined using the auxiliary functions gsl_rng_max (r) and gsl_rng_min (r).

Function: 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).

Function: 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.

Function: 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 by scaling down and/or discarding samples from the generator r. All integers in the range [0,n-1] are produced with equal probability. For generators with a non-zero minimum value an offset is applied so that zero is returned with the correct probability.

Note that this function is designed for sampling from ranges smaller than the range of the underlying generator. The parameter n must be less than or equal to the range of the generator r. 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.

In particular, this function is not intended for generating the full range of unsigned integer values [0,2^32-1]. Instead choose a generator with the maximal integer range and zero minimum value, such as gsl_rng_ranlxd1, gsl_rng_mt19937 or gsl_rng_taus, and sample it directly using gsl_rng_get. The range of each generator can be found using the auxiliary functions described in the next section.


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18.5 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.

Function: 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.

Function: unsigned long int gsl_rng_max (const gsl_rng * r)

gsl_rng_max returns the largest value that gsl_rng_get can return.

Function: 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.

Function: void * gsl_rng_state (const gsl_rng * r)
Function: size_t gsl_rng_size (const gsl_rng * r)

These functions 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);
Function: 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);
  }

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18.6 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 = %lu\n", gsl_rng_default_seed);
  printf ("first value = %lu\n", gsl_rng_get (r));

  gsl_rng_free (r);
  return 0;
}

Running the program without any environment variables uses the initial defaults, an mt19937 generator with a seed of 0,

$ ./a.out 
generator type: mt19937
seed = 0
first value = 4293858116

By setting the two variables on the command line we can change the default generator and the seed,

$ 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

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18.7 Copying random number generator state

The above methods do not expose 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:

Function: 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.

Function: 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.


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18.8 Reading and writing random number generator state

The library provides functions for reading and writing the random number state to a file as binary data.

Function: int gsl_rng_fwrite (FILE * stream, const gsl_rng * r)

This function writes the random number state of the random number generator r 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_rng_fread (FILE * stream, gsl_rng * r)

This function reads the random number state into the random number generator r from the open stream stream in binary format. The random number generator r must be preinitialized with the correct random number generator type since type information is not saved. 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.


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18.9 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. For the most reliable source of uncorrelated numbers, the second-generation RANLUX generators have the strongest proof of randomness.

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. Later versions switched to 5489 as the default seed, you can choose this explicitly via gsl_rng_set instead if you require it.

For more information see,

The generator gsl_rng_mt19937 uses the second revision of the seeding procedure published by the two authors above in 2002. The original seeding procedures could cause spurious artifacts for some seed values. They are still available through the alternative generators gsl_rng_mt19937_1999 and 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 Lüscher, which produces “luxury random numbers”. This generator provides single precision output (24 bits) at three luxury levels ranlxs0, ranlxs1 and ranlxs2, in increasing order of strength. 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 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 increased decorrelation between samples as an additional safety margin.

Note that the range of allowed seeds for this generator is [0,2^31-1]. Higher seed values are wrapped modulo 2^31.

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, in increasing order of strength.

Generator: gsl_rng_ranlux
Generator: gsl_rng_ranlux389

The ranlux generator is an implementation of the original algorithm developed by Lüscher. 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 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 Lüscher 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,

z_n = (x_n - y_n) mod m_1

where the two underlying generators x_n and y_n are,

x_n = (a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3}) mod m_1
y_n = (b_1 y_{n-1} + b_2 y_{n-2} + b_3 y_{n-3}) mod m_2

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 lcm(m_1^3-1, m_2^3-1), which is approximately 2^185 (about 10^56). 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,

x_n = (a_1 x_{n-1} + a_5 x_{n-5}) mod m

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
Generator: gsl_rng_taus2

This is a maximally equidistributed combined Tausworthe generator by L’Ecuyer. The sequence is,

x_n = (s1_n ^^ s2_n ^^ s3_n) 

where,

s1_{n+1} = (((s1_n&4294967294)<<12)^^(((s1_n<<13)^^s1_n)>>19))
s2_{n+1} = (((s2_n&4294967288)<< 4)^^(((s2_n<< 2)^^s2_n)>>25))
s3_{n+1} = (((s3_n&4294967280)<<17)^^(((s3_n<< 3)^^s3_n)>>11))

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 2^88 (about 10^26). It uses 3 words of state per generator. For more information see,

The generator gsl_rng_taus2 uses the same algorithm as gsl_rng_taus but with an improved seeding procedure described in the paper,

The generator gsl_rng_taus2 should now be used in preference to gsl_rng_taus.

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.

r_n = r_{n-A} ^^ r_{n-B} ^^ r_{n-C} ^^ r_{n-D}

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 example on p392 of Ziff’s article: A=471, B=1586, C=6988, D=9689.

If the offsets are appropriately chosen (such as the one ones in this implementation), then the sequence is said to be maximal; that means that the period is 2^D - 1, where D is the longest lag. (It is one less than 2^D because it is not permitted to have all zeros in the ra[] array.) For this implementation with D=9689 that works out to about 10^2917.

Note that the implementation of this generator using a 32-bit integer amounts to 32 parallel implementations of one-bit generators. One consequence of this is that the period of this 32-bit generator is the same as for the one-bit generator. Moreover, this independence means that all 32-bit patterns are equally likely, and in particular that 0 is an allowed random value. (We are grateful to Heiko Bauke for clarifying for us these properties of GFSR random number generators.)

For more information see,


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18.10 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

x_{n+1} = (a x_n + c) mod m

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

x_{n+1} = (a x_n + c) mod m

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).


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18.11 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

x_{n+1} = (a x_n) mod m

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 compatible 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 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 based on the recurrence

x_n = x_{n-103} ^^ x_{n-250}

where ^^ denotes “exclusive-or”, defined on 32-bit words. The period of this generator is about 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,

x_{n+1} = (a x_n + c) mod m

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,

x_{n+1} = (a x_n) mod m

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

x_{n+1} = (a x_n) mod m

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,

x_{n+1} = (a x_n) mod m

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 was used in the IMSL Library (subroutine RNUN) and in MATLAB (the RAND function) in the past. 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,

t = u_{n-273} + u_{n-607}
u_n  = t - floor(t)

The original source code is available from NETLIB. For more information see,

Generator: gsl_rng_knuthran2

This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,

x_n = (a_1 x_{n-1} + a_2 x_{n-2}) mod m

with a_1 = 271828183, a_2 = 314159269, and m = 2^31 - 1.

Generator: gsl_rng_knuthran2002
Generator: gsl_rng_knuthran

This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., Section 3.6. Knuth provides its C code. The updated routine gsl_rng_knuthran2002 is from the revised 9th printing and corrects some weaknesses in the earlier version, which is implemented as gsl_rng_knuthran.

Generator: gsl_rng_borosh13
Generator: gsl_rng_fishman18
Generator: gsl_rng_fishman20
Generator: gsl_rng_lecuyer21
Generator: gsl_rng_waterman14

These multiplicative generators are taken from Knuth’s Seminumerical Algorithms, 3rd Ed., pages 106–108. Their sequence is,

x_{n+1} = (a x_n) mod m

where the seed specifies the initial value, x_1. The parameters a and m are as follows, Borosh-Niederreiter: a = 1812433253, m = 2^32, Fishman18: a = 62089911, m = 2^31 - 1, Fishman20: a = 48271, m = 2^31 - 1, L’Ecuyer: a = 40692, m = 2^31 - 249, Waterman: a = 1566083941, m = 2^32.

Generator: gsl_rng_fishman2x

This is the L’Ecuyer–Fishman random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,

z_{n+1} = (x_n - y_n) mod m

with m = 2^31 - 1. x_n and y_n are given by the fishman20 and lecuyer21 algorithms. The seed specifies the initial value, x_1.

Generator: gsl_rng_coveyou

This is the Coveyou random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., Section 3.2.2. Its sequence is,

x_{n+1} = (x_n (x_n + 1)) mod m

with m = 2^32. The seed specifies the initial value, x_1.


Next: , Previous: Other random number generators, Up: Random Number Generation   [Index]

18.12 Performance

The following table shows the relative performance of a selection the available random number generators. The fastest simulation quality generators are taus, gfsr4 and mt19937. The generators which offer the best mathematically-proven quality are those based on the RANLUX algorithm.

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

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18.13 Examples

The following program demonstrates the use of a random number generator to produce uniform random numbers in the range [0.0, 1.0),

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

int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;

  int i, n = 10;

  gsl_rng_env_setup();

  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

  for (i = 0; i < n; i++) 
    {
      double u = gsl_rng_uniform (r);
      printf ("%.5f\n", u);
    }

  gsl_rng_free (r);

  return 0;
}

Here is the output of the program,

$ ./a.out 
0.99974
0.16291
0.28262
0.94720
0.23166
0.48497
0.95748
0.74431
0.54004
0.73995

The numbers depend on the seed used by the generator. The default seed can be changed with the GSL_RNG_SEED environment variable to produce a different stream of numbers. The generator itself can be changed using the environment variable GSL_RNG_TYPE. Here is the output of the program using a seed value of 123 and the multiple-recursive generator mrg,

$ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out 
GSL_RNG_TYPE=mrg
GSL_RNG_SEED=123
0.33050
0.86631
0.32982
0.67620
0.53391
0.06457
0.16847
0.70229
0.04371
0.86374

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18.14 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,

The source code for the DIEHARD random number generator tests is also available online,

A comprehensive set of random number generator tests is available from NIST,


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18.15 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 Lüscher for providing notes and source code for the RANLXS and RANLXD generators.


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19 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, except that seeding is not required—each generator produces a single sequence.

The functions described in this section are declared in the header file gsl_qrng.h.


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19.1 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 (gsl_qrng * q)

This function reinitializes the generator q to its starting point. Note that quasi-random sequences do not use a seed and always produce the same set of values.


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19.2 Sampling from a quasi-random number generator

Function: int gsl_qrng_get (const gsl_qrng * q, double x[])

This function stores the next point from the sequence generator q in the array x. 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. An inline version of this function is used when HAVE_INLINE is defined.


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19.3 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 functions 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);

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19.4 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 q.


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19.5 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.

Generator: gsl_qrng_halton
Generator: gsl_qrng_reversehalton

These generators use the Halton and reverse Halton sequences described in J.H. Halton, Numerische Mathematik 2, 84-90 (1960) and B. Vandewoestyne and R. Cools Computational and Applied Mathematics 189, 1&2, 341-361 (2006). They are valid up to 1229 dimensions.


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19.6 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.


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19.7 References

The implementations of the quasi-random sequence routines are based on the algorithms described in the following paper,


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20 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 library also provides cumulative distribution functions and inverse cumulative distribution functions, sometimes referred to as quantile functions. The cumulative distribution functions and their inverses are computed separately for the upper and lower tails of the distribution, allowing full accuracy to be retained for small results.

The functions for random variates and probability density functions described in this section are declared in gsl_randist.h. The corresponding cumulative distribution functions are declared in gsl_cdf.h.

Note that the discrete random variate functions always return a value of type unsigned int, and on most platforms this has a maximum value of 2^32-1 ~=~ 4.29e9. They should only be called with a safe range of parameters (where there is a negligible probability of a variate exceeding this limit) to prevent incorrect results due to overflow.


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20.1 Introduction

Continuous random number distributions are defined by a probability density function, p(x), such that the probability of x occurring in the infinitesimal range x to x+dx is p dx.

The cumulative distribution function for the lower tail P(x) is defined by the integral,

P(x) = \int_{-\infty}^{x} dx' p(x')

and gives the probability of a variate taking a value less than x.

The cumulative distribution function for the upper tail Q(x) is defined by the integral,

Q(x) = \int_{x}^{+\infty} dx' p(x')

and gives the probability of a variate taking a value greater than x.

The upper and lower cumulative distribution functions are related by P(x) + Q(x) = 1 and satisfy 0 <= P(x) <= 1, 0 <= Q(x) <= 1.

The inverse cumulative distributions, x=P^{-1}(P) and x=Q^{-1}(Q) give the values of x which correspond to a specific value of P or Q. They can be used to find confidence limits from probability values.

For discrete distributions the probability of sampling the integer value k is given by p(k), where \sum_k p(k) = 1. The cumulative distribution for the lower tail P(k) of a discrete distribution is defined as,

P(k) = \sum_{i <= k} p(i)

where the sum is over the allowed range of the distribution less than or equal to k.

The cumulative distribution for the upper tail of a discrete distribution Q(k) is defined as

Q(k) = \sum_{i > k} p(i)

giving the sum of probabilities for all values greater than k. These two definitions satisfy the identity P(k)+Q(k)=1.

If the range of the distribution is 1 to n inclusive then P(n)=1, Q(n)=0 while P(1) = p(1), Q(1)=1-p(1).


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20.2 The Gaussian Distribution

Function: 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,

p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx

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-Muller algorithm which requires two calls to 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_ziggurat (const gsl_rng * r, double sigma)
Function: double gsl_ran_gaussian_ratio_method (const gsl_rng * r, double sigma)

This function computes a Gaussian random variate using the alternative Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva ratio methods. The Ziggurat algorithm is the fastest available algorithm in most cases.

Function: double gsl_ran_ugaussian (const gsl_rng * r)
Function: double gsl_ran_ugaussian_pdf (double x)
Function: double gsl_ran_ugaussian_ratio_method (const gsl_rng * r)

These functions compute results for the unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one, sigma = 1.

Function: double gsl_cdf_gaussian_P (double x, double sigma)
Function: double gsl_cdf_gaussian_Q (double x, double sigma)
Function: double gsl_cdf_gaussian_Pinv (double P, double sigma)
Function: double gsl_cdf_gaussian_Qinv (double Q, double sigma)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Gaussian distribution with standard deviation sigma.

Function: double gsl_cdf_ugaussian_P (double x)
Function: double gsl_cdf_ugaussian_Q (double x)
Function: double gsl_cdf_ugaussian_Pinv (double P)
Function: double gsl_cdf_ugaussian_Qinv (double Q)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the unit Gaussian distribution.


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20.3 The Gaussian Tail Distribution

Function: 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,

p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2/(2 \sigma^2)) dx

for x > a where N(a;\sigma) is the normalization constant,

N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).
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.


Function: double gsl_ran_ugaussian_tail (const gsl_rng * r, double a)
Function: 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.


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20.4 The Bivariate Gaussian Distribution

Function: 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,

p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy

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.



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20.5 The Exponential Distribution

Function: 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,

p(x) dx = {1 \over \mu} \exp(-x/\mu) dx

for 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.


Function: double gsl_cdf_exponential_P (double x, double mu)
Function: double gsl_cdf_exponential_Q (double x, double mu)
Function: double gsl_cdf_exponential_Pinv (double P, double mu)
Function: double gsl_cdf_exponential_Qinv (double Q, double mu)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the exponential distribution with mean mu.


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20.6 The Laplace Distribution

Function: double gsl_ran_laplace (const gsl_rng * r, double a)

This function returns a random variate from the Laplace distribution with width a. The distribution is,

p(x) dx = {1 \over 2 a}  \exp(-|x/a|) dx

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 width a, using the formula given above.


Function: double gsl_cdf_laplace_P (double x, double a)
Function: double gsl_cdf_laplace_Q (double x, double a)
Function: double gsl_cdf_laplace_Pinv (double P, double a)
Function: double gsl_cdf_laplace_Qinv (double Q, double a)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Laplace distribution with width a.


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20.7 The Exponential Power Distribution

Function: 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,

p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx

for 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 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.


Function: double gsl_cdf_exppow_P (double x, double a, double b)
Function: double gsl_cdf_exppow_Q (double x, double a, double b)

These functions compute the cumulative distribution functions P(x), Q(x) for the exponential power distribution with parameters a and b.


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20.8 The Cauchy Distribution

Function: 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,

p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx

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.


Function: double gsl_cdf_cauchy_P (double x, double a)
Function: double gsl_cdf_cauchy_Q (double x, double a)
Function: double gsl_cdf_cauchy_Pinv (double P, double a)
Function: double gsl_cdf_cauchy_Qinv (double Q, double a)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Cauchy distribution with scale parameter a.


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20.9 The Rayleigh Distribution

Function: 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,

p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx

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.


Function: double gsl_cdf_rayleigh_P (double x, double sigma)
Function: double gsl_cdf_rayleigh_Q (double x, double sigma)
Function: double gsl_cdf_rayleigh_Pinv (double P, double sigma)
Function: double gsl_cdf_rayleigh_Qinv (double Q, double sigma)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Rayleigh distribution with scale parameter sigma.


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20.10 The Rayleigh Tail Distribution

Function: 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,

p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx

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.



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20.11 The Landau Distribution

Function: 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,

p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s) 

For numerical purposes it is more convenient to use the following equivalent form of the integral,

p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
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.



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20.12 The Levy alpha-Stable Distributions

Function: 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,

p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)

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 \sigma = \sqrt{2} c. For \alpha < 1 the tails of the distribution become extremely wide.

The algorithm only works for 0 < alpha <= 2.



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20.13 The Levy skew alpha-Stable Distribution

Function: 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,

p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))

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 \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 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).



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20.14 The Gamma Distribution

Function: 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,

p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx

for x > 0.

The gamma distribution with an integer parameter a is known as the Erlang distribution.

The variates are computed using the Marsaglia-Tsang fast gamma method. This function for this method was previously called gsl_ran_gamma_mt and can still be accessed using this name.

Function: double gsl_ran_gamma_knuth (const gsl_rng * r, double a, double b)

This function returns a gamma variate using the algorithms from Knuth (vol 2).

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.


Function: double gsl_cdf_gamma_P (double x, double a, double b)
Function: double gsl_cdf_gamma_Q (double x, double a, double b)
Function: double gsl_cdf_gamma_Pinv (double P, double a, double b)
Function: double gsl_cdf_gamma_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the gamma distribution with parameters a and b.


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20.15 The Flat (Uniform) Distribution

Function: 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,

p(x) dx = {1 \over (b-a)} dx

if 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.


Function: double gsl_cdf_flat_P (double x, double a, double b)
Function: double gsl_cdf_flat_Q (double x, double a, double b)
Function: double gsl_cdf_flat_Pinv (double P, double a, double b)
Function: double gsl_cdf_flat_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for a uniform distribution from a to b.


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20.16 The Lognormal Distribution

Function: 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,

p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2} } \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx

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.


Function: double gsl_cdf_lognormal_P (double x, double zeta, double sigma)
Function: double gsl_cdf_lognormal_Q (double x, double zeta, double sigma)
Function: double gsl_cdf_lognormal_Pinv (double P, double zeta, double sigma)
Function: double gsl_cdf_lognormal_Qinv (double Q, double zeta, double sigma)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the lognormal distribution with parameters zeta and sigma.


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20.17 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,

X_i = \sum_i Y_i^2

has a chi-squared distribution with n degrees of freedom.

Function: 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,

p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx

for 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.


Function: double gsl_cdf_chisq_P (double x, double nu)
Function: double gsl_cdf_chisq_Q (double x, double nu)
Function: double gsl_cdf_chisq_Pinv (double P, double nu)
Function: double gsl_cdf_chisq_Qinv (double Q, double nu)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the chi-squared distribution with nu degrees of freedom.


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20.18 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,

X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }

has an F-distribution F(x;\nu_1,\nu_2).

Function: 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,

p(x) dx = 
   { \Gamma((\nu_1 + \nu_2)/2)
        \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) } 
   \nu_1^{\nu_1/2} \nu_2^{\nu_2/2} 
   x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}

for 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.


Function: double gsl_cdf_fdist_P (double x, double nu1, double nu2)
Function: double gsl_cdf_fdist_Q (double x, double nu1, double nu2)
Function: double gsl_cdf_fdist_Pinv (double P, double nu1, double nu2)
Function: double gsl_cdf_fdist_Qinv (double Q, double nu1, double nu2)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the F-distribution with nu1 and nu2 degrees of freedom.


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20.19 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,

X = { Y_1 \over \sqrt{Y_2 / \nu} }

has a t-distribution t(x;\nu) with \nu degrees of freedom.

Function: double gsl_ran_tdist (const gsl_rng * r, double nu)

This function returns a random variate from the t-distribution. The distribution function is,

p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
   (1 + x^2/\nu)^{-(\nu + 1)/2} dx

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.


Function: double gsl_cdf_tdist_P (double x, double nu)
Function: double gsl_cdf_tdist_Q (double x, double nu)
Function: double gsl_cdf_tdist_Pinv (double P, double nu)
Function: double gsl_cdf_tdist_Qinv (double Q, double nu)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the t-distribution with nu degrees of freedom.


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20.20 The Beta Distribution

Function: 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,

p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx

for 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.


Function: double gsl_cdf_beta_P (double x, double a, double b)
Function: double gsl_cdf_beta_Q (double x, double a, double b)
Function: double gsl_cdf_beta_Pinv (double P, double a, double b)
Function: double gsl_cdf_beta_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the beta distribution with parameters a and b.


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20.21 The Logistic Distribution

Function: double gsl_ran_logistic (const gsl_rng * r, double a)

This function returns a random variate from the logistic distribution. The distribution function is,

p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx

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.


Function: double gsl_cdf_logistic_P (double x, double a)
Function: double gsl_cdf_logistic_Q (double x, double a)
Function: double gsl_cdf_logistic_Pinv (double P, double a)
Function: double gsl_cdf_logistic_Qinv (double Q, double a)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the logistic distribution with scale parameter a.


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20.22 The Pareto Distribution

Function: 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,

p(x) dx = (a/b) / (x/b)^{a+1} dx

for 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.


Function: double gsl_cdf_pareto_P (double x, double a, double b)
Function: double gsl_cdf_pareto_Q (double x, double a, double b)
Function: double gsl_cdf_pareto_Pinv (double P, double a, double b)
Function: double gsl_cdf_pareto_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Pareto distribution with exponent a and scale b.


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20.23 Spherical Vector Distributions

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.

Function: void gsl_ran_dir_2d (const gsl_rng * r, double * x, double * y)
Function: 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 the Pentium (but not the case for the Sun Sparcstation). One 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 \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=2uv/(u^2+v^2).

Function: 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 3 dimensions).

Function: void gsl_ran_dir_nd (const gsl_rng * r, size_t 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).


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20.24 The Weibull Distribution

Function: 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,

p(x) dx = {b \over a^b} x^{b-1}  \exp(-(x/a)^b) dx

for 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.


Function: double gsl_cdf_weibull_P (double x, double a, double b)
Function: double gsl_cdf_weibull_Q (double x, double a, double b)
Function: double gsl_cdf_weibull_Pinv (double P, double a, double b)
Function: double gsl_cdf_weibull_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Weibull distribution with scale a and exponent b.


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20.25 The Type-1 Gumbel Distribution

Function: 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,

p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx

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.


Function: double gsl_cdf_gumbel1_P (double x, double a, double b)
Function: double gsl_cdf_gumbel1_Q (double x, double a, double b)
Function: double gsl_cdf_gumbel1_Pinv (double P, double a, double b)
Function: double gsl_cdf_gumbel1_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Type-1 Gumbel distribution with parameters a and b.


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20.26 The Type-2 Gumbel Distribution

Function: 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,

p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx

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.


Function: double gsl_cdf_gumbel2_P (double x, double a, double b)
Function: double gsl_cdf_gumbel2_Q (double x, double a, double b)
Function: double gsl_cdf_gumbel2_Pinv (double P, double a, double b)
Function: double gsl_cdf_gumbel2_Qinv (double Q, double a, double b)

These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Type-2 Gumbel distribution with parameters a and b.


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20.27 The Dirichlet Distribution

Function: void gsl_ran_dirichlet (const gsl_rng * r, size_t K, const double alpha[], double theta[])

This function returns an array of K random variates from a Dirichlet distribution of order K-1. The distribution function is

p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K = 
  (1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K

for theta_i >= 0 and alpha_i > 0. The delta function ensures that \sum \theta_i = 1. The normalization factor Z is

Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}

The random variates are generated by sampling K values from gamma distributions with parameters a=alpha_i, b=1, and renormalizing. See A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).

Function: double gsl_ran_dirichlet_pdf (size_t K, const double alpha[], const double theta[])

This function computes the probability density p(\theta_1, ... , \theta_K) at theta[K] for a Dirichlet distribution with parameters alpha[K], using the formula given above.

Function: double gsl_ran_dirichlet_lnpdf (size_t K, const double alpha[], const double theta[])

This function computes the logarithm of the probability density p(\theta_1, ... , \theta_K) for a Dirichlet distribution with parameters alpha[K].


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20.28 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:

  C[0] = 0 
C[k+1] = C[k]+P[k].

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[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 variables 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 it for something like the Poisson Distribution, a modification would have to be made, since it only takes a finite set of K outcomes.

Function: 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.

Function: 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.

Function: 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.

Function: void gsl_ran_discrete_free (gsl_ran_discrete_t * g)

De-allocates the lookup table pointed to by g.


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20.29 The Poisson Distribution

Function: 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,

p(k) = {\mu^k \over k!} \exp(-\mu)

for 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.


Function: double gsl_cdf_poisson_P (unsigned int k, double mu)
Function: double gsl_cdf_poisson_Q (unsigned int k, double mu)

These functions compute the cumulative distribution functions P(k), Q(k) for the Poisson distribution with parameter mu.


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20.30 The Bernoulli Distribution

Function: 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,

p(0) = 1 - p
p(1) = p
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.



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20.31 The Binomial Distribution

Function: 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,

p(k) = {n! \over k! (n-k)! } p^k (1-p)^{n-k}

for 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.


Function: double gsl_cdf_binomial_P (unsigned int k, double p, unsigned int n)
Function: double gsl_cdf_binomial_Q (unsigned int k, double p, unsigned int n)

These functions compute the cumulative distribution functions P(k), Q(k) for the binomial distribution with parameters p and n.


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20.32 The Multinomial Distribution

Function: void gsl_ran_multinomial (const gsl_rng * r, size_t K, unsigned int N, const double p[], unsigned int n[])

This function computes a random sample n[] from the multinomial distribution formed by N trials from an underlying distribution p[K]. The distribution function for n[] is,

P(n_1, n_2, ..., n_K) = 
  (N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K

where (n_1, n_2, ..., n_K) are nonnegative integers with sum_{k=1}^K n_k = N, and (p_1, p_2, ..., p_K) is a probability distribution with \sum p_i = 1. If the array p[K] is not normalized then its entries will be treated as weights and normalized appropriately. The arrays n[] and p[] must both be of length K.

Random variates are generated using the conditional binomial method (see C.S. Davis, The computer generation of multinomial random variates, Comp. Stat. Data Anal. 16 (1993) 205–217 for details).

Function: double gsl_ran_multinomial_pdf (size_t K, const double p[], const unsigned int n[])

This function computes the probability P(n_1, n_2, ..., n_K) of sampling n[K] from a multinomial distribution with parameters p[K], using the formula given above.

Function: double gsl_ran_multinomial_lnpdf (size_t K, const double p[], const unsigned int n[])

This function returns the logarithm of the probability for the multinomial distribution P(n_1, n_2, ..., n_K) with parameters p[K].


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