This is ./gsl-ref.info, produced by makeinfo version 4.8 from
./gsl-ref.texi.

INFO-DIR-SECTION Scientific software
START-INFO-DIR-ENTRY
* gsl-ref: (gsl-ref).                   GNU Scientific Library - Reference
END-INFO-DIR-ENTRY

   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004,
2005, 2006 The GSL Team.

   Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2 or
any later version published by the Free Software Foundation; with the
Invariant Sections being "GNU General Public License" and "Free Software
Needs Free Documentation", the Front-Cover text being "A GNU Manual",
and with the Back-Cover Text being (a) (see below).  A copy of the
license is included in the section entitled "GNU Free Documentation
License".

   (a) The Back-Cover Text is: "You have freedom to copy and modify this
GNU Manual, like GNU software."


File: gsl-ref.info,  Node: Top,  Next: Introduction,  Prev: (dir),  Up: (dir)

GSL
***

This file documents the GNU Scientific Library (GSL), a collection of
numerical routines for scientific computing.  It corresponds to release
1.8 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.

   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004,
2005, 2006 The GSL Team.

   Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2 or
any later version published by the Free Software Foundation; with the
Invariant Sections being "GNU General Public License" and "Free Software
Needs Free Documentation", the Front-Cover text being "A GNU Manual",
and with the Back-Cover Text being (a) (see below).  A copy of the
license is included in the section entitled "GNU Free Documentation
License".

   (a) The Back-Cover Text is: "You have freedom to copy and modify this
GNU Manual, like GNU software."


* Menu:

* Introduction::
* Using the library::
* Error Handling::
* Mathematical Functions::
* Complex Numbers::
* Polynomials::
* Special Functions::
* Vectors and Matrices::
* Permutations::
* Combinations::
* Sorting::
* BLAS Support::
* Linear Algebra::
* Eigensystems::
* Fast Fourier Transforms::
* Numerical Integration::
* Random Number Generation::
* Quasi-Random Sequences::
* Random Number Distributions::
* Statistics::
* Histograms::
* N-tuples::
* Monte Carlo Integration::
* Simulated Annealing::
* Ordinary Differential Equations::
* Interpolation::
* Numerical Differentiation::
* Chebyshev Approximations::
* Series Acceleration::
* Wavelet Transforms::
* Discrete Hankel Transforms::
* One dimensional Root-Finding::
* One dimensional Minimization::
* Multidimensional Root-Finding::
* Multidimensional Minimization::
* Least-Squares Fitting::
* Nonlinear Least-Squares Fitting::
* Physical Constants::
* IEEE floating-point arithmetic::
* Debugging Numerical Programs::
* Contributors to GSL::
* Autoconf Macros::
* GSL CBLAS Library::
* Free Software Needs Free Documentation::
* GNU General Public License::
* GNU Free Documentation License::
* Function Index::
* Variable Index::
* Type Index::
* Concept Index::


File: gsl-ref.info,  Node: Introduction,  Next: Using the library,  Prev: Top,  Up: Top

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.

* Menu:

* Routines available in GSL::
* GSL is Free Software::
* Obtaining GSL::
* No Warranty::
* Reporting Bugs::
* Further Information::
* Conventions used in this manual::


File: gsl-ref.info,  Node: Routines available in GSL,  Next: GSL is Free Software,  Up: Introduction

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 Numbers                  Roots of Polynomials             
     Special Functions                Vectors and Matrices             
     Permutations                     Combinations                     
     Sorting                          BLAS Support                     
     Linear Algebra                   CBLAS Library                    
     Fast Fourier Transforms          Eigensystems                     
     Random Numbers                   Quadrature                       
     Random Distributions             Quasi-Random Sequences           
     Histograms                       Statistics                       
     Monte Carlo Integration          N-Tuples                         
     Differential Equations           Simulated Annealing              
     Numerical Differentiation        Interpolation                    
     Series Acceleration              Chebyshev Approximations         
     Root-Finding                     Discrete Hankel Transforms       
     Least-Squares Fitting            Minimization                     
     IEEE Floating-Point              Physical Constants               
     Wavelets                                                          

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.


File: gsl-ref.info,  Node: GSL is Free Software,  Next: Obtaining GSL,  Prev: Routines available in GSL,  Up: Introduction

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
(*note GNU General Public License::).  Further information about this
license is available from the GNU Project webpage `Frequently Asked
Questions about the GNU GPL',

     `http://www.gnu.org/copyleft/gpl-faq.html'

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


File: gsl-ref.info,  Node: Obtaining GSL,  Next: No Warranty,  Prev: GSL is Free Software,  Up: Introduction

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,

     `http://www.gnu.org/software/gsl/'

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.  Precompiled versions of the library
can be purchased from commercial redistributors listed on the website
above.

   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.


File: gsl-ref.info,  Node: No Warranty,  Next: Reporting Bugs,  Prev: Obtaining GSL,  Up: Introduction

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 (*note GNU General
Public License::).


File: gsl-ref.info,  Node: Reporting Bugs,  Next: Further Information,  Prev: No Warranty,  Up: Introduction

1.5 Reporting Bugs
==================

A list of known bugs can be found in the `BUGS' file included in the
GSL distribution.  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:

   * The version number of GSL

   * The hardware and operating system

   * The compiler used, including version number and compilation options

   * A description of the bug behavior

   * A short program which exercises the bug

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.


File: gsl-ref.info,  Node: Further Information,  Next: Conventions used in this manual,  Prev: Reporting Bugs,  Up: Introduction

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 (2nd Ed.), ISBN
0954161734'.

   If you want to give a url, use "`http://www.gnu.org/software/gsl/'".


File: gsl-ref.info,  Node: Conventions used in this manual,  Prev: Further Information,  Up: Introduction

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').


File: gsl-ref.info,  Node: Using the library,  Next: Error Handling,  Prev: Introduction,  Up: Top

2 Using the library
*******************

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

* Menu:

* An Example Program::
* Compiling and Linking::
* Shared Libraries::
* ANSI C Compliance::
* Inline functions::
* Long double::
* Portability functions::
* Alternative optimized functions::
* Support for different numeric types::
* Compatibility with C++::
* Aliasing of arrays::
* Thread-safety::
* Deprecated Functions::
* Code Reuse::


File: gsl-ref.info,  Node: An Example Program,  Next: Compiling and Linking,  Up: Using the library

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,

     J0(5) = -1.775967713143382920e-01

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


File: gsl-ref.info,  Node: Compiling and Linking,  Next: Shared Libraries,  Prev: An Example Program,  Up: Using the library

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.

* Menu:

* Linking programs with the library::
* Linking with an alternative BLAS library::


File: gsl-ref.info,  Node: Linking programs with the library,  Next: Linking with an alternative BLAS library,  Up: Compiling and Linking

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.


File: gsl-ref.info,  Node: Linking with an alternative BLAS library,  Prev: Linking programs with the library,  Up: Compiling and Linking

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 called `libcblas',

     $ 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

For more information see *Note BLAS Support::.


File: gsl-ref.info,  Node: Shared Libraries,  Next: ANSI C Compliance,  Prev: Compiling and Linking,  Up: Using the library

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, 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:$LD_LIBRARY_PATH
     $ 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:$LD_LIBRARY_PATH

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 should 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


File: gsl-ref.info,  Node: ANSI C Compliance,  Next: Inline functions,  Prev: Shared Libraries,  Up: Using the library

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_'.


File: gsl-ref.info,  Node: Inline functions,  Next: Long double,  Prev: ANSI C Compliance,  Up: Using the library

2.5 Inline functions
====================

The `inline' keyword is not part of the original ANSI C standard (C89)
and the library does not export any inline function definitions by
default. However, the library provides optional inline versions of
performance-critical functions by conditional compilation.  The inline
versions of these functions can be included by defining the macro
`HAVE_INLINE' when compiling an application,

     $ gcc -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.

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


File: gsl-ref.info,  Node: Long double,  Next: Portability functions,  Prev: Inline functions,  Up: Using the library

2.6 Long double
===============

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

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

     checking whether printf works with long double... no

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

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


File: gsl-ref.info,  Node: Portability functions,  Next: Alternative optimized functions,  Prev: Long double,  Up: Using the library

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 *Note 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.


File: gsl-ref.info,  Node: Alternative optimized functions,  Next: Support for different numeric types,  Prev: Portability functions,  Up: Using the library

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.


File: gsl-ref.info,  Node: Support for different numeric types,  Next: Compatibility with C++,  Prev: Alternative optimized functions,  Up: Using the library

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


File: gsl-ref.info,  Node: Compatibility with C++,  Next: Aliasing of arrays,  Prev: Support for different numeric types,  Up: Using the library

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'.


File: gsl-ref.info,  Node: Aliasing of arrays,  Next: Thread-safety,  Prev: Compatibility with C++,  Up: Using the library

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.


File: gsl-ref.info,  Node: Thread-safety,  Next: Deprecated Functions,  Prev: Aliasing of arrays,  Up: Using the library

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.


File: gsl-ref.info,  Node: Deprecated Functions,  Next: Code Reuse,  Prev: Thread-safety,  Up: Using the library

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.


File: gsl-ref.info,  Node: Code Reuse,  Prev: Deprecated Functions,  Up: Using the library

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.


File: gsl-ref.info,  Node: Error Handling,  Next: Mathematical Functions,  Prev: Using the library,  Up: Top

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'.

* Menu:

* Error Reporting::
* Error Codes::
* Error Handlers::
* Using GSL error reporting in your own functions::
* Error Reporting Examples::


File: gsl-ref.info,  Node: Error Reporting,  Next: Error Codes,  Up: Error Handling

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.


File: gsl-ref.info,  Node: Error Codes,  Next: Error Handlers,  Prev: Error Reporting,  Up: Error Handling

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.  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'.


File: gsl-ref.info,  Node: Error Handlers,  Next: Using GSL error reporting in your own functions,  Prev: Error Codes,  Up: Error Handling

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'.


File: gsl-ref.info,  Node: Using GSL error reporting in your own functions,  Next: Error Reporting Examples,  Prev: Error Handlers,  Up: Error Handling

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);
       }


File: gsl-ref.info,  Node: Error Reporting Examples,  Prev: Using GSL error reporting in your own functions,  Up: Error Handling

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, 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.


File: gsl-ref.info,  Node: Mathematical Functions,  Next: Complex Numbers,  Prev: Error Handling,  Up: Top

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'.

* Menu:

* Mathematical Constants::
* Infinities and Not-a-number::
* Elementary Functions::
* Small integer powers::
* Testing the Sign of Numbers::
* Testing for Odd and Even Numbers::
* Maximum and Minimum functions::
* Approximate Comparison of Floating Point Numbers::


File: gsl-ref.info,  Node: Mathematical Constants,  Next: Infinities and Not-a-number,  Up: Mathematical Functions

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



File: gsl-ref.info,  Node: Infinities and Not-a-number,  Next: Elementary Functions,  Prev: Mathematical Constants,  Up: Mathematical Functions

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.

 -- 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.


File: gsl-ref.info,  Node: Elementary Functions,  Next: Small integer powers,  Prev: Infinities and Not-a-number,  Up: Mathematical Functions

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
(*note Portability functions::).

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

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

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

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

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

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

 -- 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)'.


File: gsl-ref.info,  Node: Small integer powers,  Next: Testing the Sign of Numbers,  Prev: Elementary Functions,  Up: Mathematical Functions

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 a simple
functions to fill this gap.  For reasons of efficiency, these functions
do not check for overflow or underflow conditions.

 -- Function: double gsl_pow_int (double X, int N)
     This routine computes the power x^n for integer N.  The power is
     computed 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 possible
     so that use of these functions should be as efficient as
     explicitly writing the corresponding product expression.

     #include <gsl/gsl_math.h>
     double y = gsl_pow_4 (3.141)  /* compute 3.141**4 */


File: gsl-ref.info,  Node: Testing the Sign of Numbers,  Next: Testing for Odd and Even Numbers,  Prev: Small integer powers,  Up: Mathematical Functions

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).


File: gsl-ref.info,  Node: Testing for Odd and Even Numbers,  Next: Maximum and Minimum functions,  Prev: Testing the Sign of Numbers,  Up: Mathematical Functions

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.


File: gsl-ref.info,  Node: Maximum and Minimum functions,  Next: Approximate Comparison of Floating Point Numbers,  Prev: Testing for Odd and Even Numbers,  Up: Mathematical Functions

4.7 Maximum and Minimum functions
=================================

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

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

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

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

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

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


File: gsl-ref.info,  Node: Approximate Comparison of Floating Point Numbers,  Prev: Maximum and Minimum functions,  Up: Mathematical Functions

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.

     The implementation is based on the package `fcmp' by T.C. Belding.


File: gsl-ref.info,  Node: Complex Numbers,  Next: Polynomials,  Prev: Mathematical Functions,  Up: Top

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)'(1) 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'.

* Menu:

* Complex numbers::
* Properties of complex numbers::
* Complex arithmetic operators::
* Elementary Complex Functions::
* Complex Trigonometric Functions::
* Inverse Complex Trigonometric Functions::
* Complex Hyperbolic Functions::
* Inverse Complex Hyperbolic Functions::
* Complex Number References and Further Reading::

   ---------- Footnotes ----------

   (1) Note that the first edition uses different definitions.


File: gsl-ref.info,  Node: Complex numbers,  Next: Properties of complex numbers,  Up: Complex Numbers

5.1 Complex numbers
===================

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

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

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

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

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

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

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

 -- Macro: GSL_SET_COMPLEX (ZP, X, Y)
     This macro uses the cartesian components (X,Y) to set the real and
     imaginary parts of the complex number pointed to by ZP.  For
     example,

          GSL_SET_COMPLEX(&z, 3, 4)

     sets Z to be 3 + 4i.

 -- Macro: GSL_SET_REAL (ZP,X)
 -- Macro: GSL_SET_IMAG (ZP,Y)
     These macros allow the real and imaginary parts of the complex
     number pointed to by ZP to be set independently.


File: gsl-ref.info,  Node: Properties of complex numbers,  Next: Complex arithmetic operators,  Prev: Complex numbers,  Up: Complex Numbers

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.


File: gsl-ref.info,  Node: Complex arithmetic operators,  Next: Elementary Complex Functions,  Prev: Properties of complex numbers,  Up: Complex Numbers

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).


File: gsl-ref.info,  Node: Elementary Complex Functions,  Next: Complex Trigonometric Functions,  Prev: Complex arithmetic operators,  Up: Complex Numbers

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).


File: gsl-ref.info,  Node: Complex Trigonometric Functions,  Next: Inverse Complex Trigonometric Functions,  Prev: Elementary Complex Functions,  Up: Complex Numbers

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).


File: gsl-ref.info,  Node: Inverse Complex Trigonometric Functions,  Next: Complex Hyperbolic Functions,  Prev: Complex Trigonometric Functions,  Up: Complex Numbers

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).


File: gsl-ref.info,  Node: Complex Hyperbolic Functions,  Next: Inverse Complex Hyperbolic Functions,  Prev: Inverse Complex Trigonometric Functions,  Up: Complex Numbers

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).


File: gsl-ref.info,  Node: Inverse Complex Hyperbolic Functions,  Next: Complex Number References and Further Reading,  Prev: Complex Hyperbolic Functions,  Up: Complex Numbers

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.

 -- 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).


File: gsl-ref.info,  Node: Complex Number References and Further Reading,  Prev: Inverse Complex Hyperbolic Functions,  Up: Complex Numbers

5.9 References and Further Reading
==================================

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

     T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
     "Implementing Complex Elementary Functions Using Exception
     Handling", `ACM Transactions on Mathematical Software', Volume 20
     (1994), pp 215-244, Corrigenda, p553

     T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
     "Implementing the complex arcsin and arccosine functions using
     exception handling", `ACM Transactions on Mathematical Software',
     Volume 23 (1997) pp 299-335

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

     Abramowitz and Stegun, `Handbook of Mathematical Functions',
     "Circular Functions in Terms of Real and Imaginary Parts", Formulas
     4.3.55-58, "Inverse Circular Functions in Terms of Real and
     Imaginary Parts", Formulas 4.4.37-39, "Hyperbolic Functions in
     Terms of Real and Imaginary Parts", Formulas 4.5.49-52, "Inverse
     Hyperbolic Functions--relation to Inverse Circular Functions",
     Formulas 4.6.14-19.

     Dave Gillespie, `Calc Manual', Free Software Foundation, ISBN
     1-882114-18-3


File: gsl-ref.info,  Node: Polynomials,  Next: Special Functions,  Prev: Complex Numbers,  Up: Top

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'.

* Menu:

* Polynomial Evaluation::
* Divided Difference Representation of Polynomials::
* Quadratic Equations::
* Cubic Equations::
* General Polynomial Equations::
* Roots of Polynomials Examples::
* Roots of Polynomials References and Further Reading::


File: gsl-ref.info,  Node: Polynomial Evaluation,  Next: Divided Difference Representation of Polynomials,  Up: Polynomials

6.1 Polynomial Evaluation
=========================

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


File: gsl-ref.info,  Node: Divided Difference Representation of Polynomials,  Next: Quadratic Equations,  Prev: Polynomial Evaluation,  Up: Polynomials

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.

 -- 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 (XA, YA) 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.

 -- 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.

 -- 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.


File: gsl-ref.info,  Node: Quadratic Equations,  Next: Cubic Equations,  Prev: Divided Difference Representation of Polynomials,  Up: Polynomials

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.



File: gsl-ref.info,  Node: Cubic Equations,  Next: General Polynomial Equations,  Prev: Quadratic Equations,  Up: Polynomials

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.


 -- 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.



File: gsl-ref.info,  Node: General Polynomial Equations,  Next: Roots of Polynomials Examples,  Prev: Cubic Equations,  Up: Polynomials

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 and
     `GSL_EFAILED' if the QR reduction does not converge.  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).


File: gsl-ref.info,  Node: Roots of Polynomials Examples,  Next: Roots of Polynomials References and Further Reading,  Prev: General Polynomial Equations,  Up: Polynomials

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).


File: gsl-ref.info,  Node: Roots of Polynomials References and Further Reading,  Prev: Roots of Polynomials Examples,  Up: Polynomials

6.7 References and Further Reading
==================================

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

     R.S. Martin, G. Peters and J.H. Wilkinson, "The QR Algorithm for
     Real Hessenberg Matrices", `Numerische Mathematik', 14 (1970),
     219-231.

     B.N. Parlett and C. Reinsch, "Balancing a Matrix for Calculation of
     Eigenvalues and Eigenvectors", `Numerische Mathematik', 13 (1969),
     293-304.

     A. Edelman and H. Murakami, "Polynomial roots from companion matrix
     eigenvalues", `Mathematics of Computation', Vol. 64, No. 210
     (1995), 763-776.

The formulas for divided differences are given in Abramowitz and Stegun,

     Abramowitz and Stegun, `Handbook of Mathematical Functions',
     Sections 25.1.4 and 25.2.26.


File: gsl-ref.info,  Node: Special Functions,  Next: Vectors and Matrices,  Prev: Polynomials,  Up: Top

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'.

* Menu:

* Special Function Usage::
* The gsl_sf_result struct::
* Special Function Modes::
* Airy Functions and Derivatives::
* Bessel Functions::
* Clausen Functions::
* Coulomb Functions::
* Coupling Coefficients::
* Dawson Function::
* Debye Functions::
* Dilogarithm::
* Elementary Operations::
* Elliptic Integrals::
* Elliptic Functions (Jacobi)::
* Error Functions::
* Exponential Functions::
* Exponential Integrals::
* Fermi-Dirac Function::
* Gamma and Beta Functions::
* Gegenbauer Functions::
* Hypergeometric Functions::
* Laguerre Functions::
* Lambert W Functions::
* Legendre Functions and Spherical Harmonics::
* Logarithm and Related Functions::
* Power Function::
* Psi (Digamma) Function::
* Synchrotron Functions::
* Transport Functions::
* Trigonometric Functions::
* Zeta Functions::
* Special Functions Examples::
* Special Functions References and Further Reading::


File: gsl-ref.info,  Node: Special Function Usage,  Next: The gsl_sf_result struct,  Up: Special Functions

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'.


File: gsl-ref.info,  Node: The gsl_sf_result struct,  Next: Special Function Modes,  Prev: Special Function Usage,  Up: Special Functions

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;


File: gsl-ref.info,  Node: Special Function Modes,  Next: Airy Functions and Derivatives,  Prev: The gsl_sf_result struct,  Up: Special Functions

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.


File: gsl-ref.info,  Node: Airy Functions and Derivatives,  Next: Bessel Functions,  Prev: Special Function Modes,  Up: Special Functions

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) + \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'.

* Menu:

* Airy Functions::
* Derivatives of Airy Functions::
* Zeros of Airy Functions::
* Zeros of Derivatives of Airy Functions::


File: gsl-ref.info,  Node: Airy Functions,  Next: Derivatives of Airy Functions,  Up: Airy Functions and Derivatives

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.


File: gsl-ref.info,  Node: Derivatives of Airy Functions,  Next: Zeros of Airy Functions,  Prev: Airy Functions,  Up: Airy Functions and Derivatives

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.


File: gsl-ref.info,  Node: Zeros of Airy Functions,  Next: Zeros of Derivatives of Airy Functions,  Prev: Derivatives of Airy Functions,  Up: Airy Functions and Derivatives

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).


File: gsl-ref.info,  Node: Zeros of Derivatives of Airy Functions,  Prev: Zeros of Airy Functions,  Up: Airy Functions and Derivatives

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).


File: gsl-ref.info,  Node: Bessel Functions,  Next: Clausen Functions,  Prev: Airy Functions and Derivatives,  Up: Special Functions

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'.

* Menu:

* Regular Cylindrical Bessel Functions::
* Irregular Cylindrical Bessel Functions::
* Regular Modified Cylindrical Bessel Functions::
* Irregular Modified Cylindrical Bessel Functions::
* Regular Spherical Bessel Functions::
* Irregular Spherical Bessel Functions::
* Regular Modified Spherical Bessel Functions::
* Irregular Modified Spherical Bessel Functions::
* Regular Bessel Function - Fractional Order::
* Irregular Bessel Functions - Fractional Order::
* Regular Modified Bessel Functions - Fractional Order::
* Irregular Modified Bessel Functions - Fractional Order::
* Zeros of Regular Bessel Functions::


File: gsl-ref.info,  Node: Regular Cylindrical Bessel Functions,  Next: Irregular Cylindrical Bessel Functions,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Irregular Cylindrical Bessel Functions,  Next: Regular Modified Cylindrical Bessel Functions,  Prev: Regular Cylindrical Bessel Functions,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Regular Modified Cylindrical Bessel Functions,  Next: Irregular Modified Cylindrical Bessel Functions,  Prev: Irregular Cylindrical Bessel Functions,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Irregular Modified Cylindrical Bessel Functions,  Next: Regular Spherical Bessel Functions,  Prev: Regular Modified Cylindrical Bessel Functions,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Regular Spherical Bessel Functions,  Next: Irregular Spherical Bessel Functions,  Prev: Irregular Modified Cylindrical Bessel Functions,  Up: Bessel Functions

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 * JL_X_ARRAY)
     This routine uses Steed's method to compute the values of the
     regular spherical Bessel functions j_l(x) for l from 0 to LMAX
     inclusive for 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.


File: gsl-ref.info,  Node: Irregular Spherical Bessel Functions,  Next: Regular Modified Spherical Bessel Functions,  Prev: Regular Spherical Bessel Functions,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Regular Modified Spherical Bessel Functions,  Next: Irregular Modified Spherical Bessel Functions,  Prev: Irregular Spherical Bessel Functions,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Irregular Modified Spherical Bessel Functions,  Next: Regular Bessel Function - Fractional Order,  Prev: Regular Modified Spherical Bessel Functions,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Regular Bessel Function - Fractional Order,  Next: Irregular Bessel Functions - Fractional Order,  Prev: Irregular Modified Spherical Bessel Functions,  Up: Bessel Functions

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).


File: gsl-ref.info,  Node: Irregular Bessel Functions - Fractional Order,  Next: Regular Modified Bessel Functions - Fractional Order,  Prev: Regular Bessel Function - Fractional Order,  Up: Bessel Functions

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).


File: gsl-ref.info,  Node: Regular Modified Bessel Functions - Fractional Order,  Next: Irregular Modified Bessel Functions - Fractional Order,  Prev: Irregular Bessel Functions - Fractional Order,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Irregular Modified Bessel Functions - Fractional Order,  Next: Zeros of Regular Bessel Functions,  Prev: Regular Modified Bessel Functions - Fractional Order,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Zeros of Regular Bessel Functions,  Prev: Irregular Modified Bessel Functions - Fractional Order,  Up: Bessel Functions

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.


File: gsl-ref.info,  Node: Clausen Functions,  Next: Coulomb Functions,  Prev: Bessel Functions,  Up: Special Functions

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).


File: gsl-ref.info,  Node: Coulomb Functions,  Next: Coupling Coefficients,  Prev: Clausen Functions,  Up: Special Functions

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.

* Menu:

* Normalized Hydrogenic Bound States::
* Coulomb Wave Functions::
* Coulomb Wave Function Normalization Constant::


File: gsl-ref.info,  Node: Normalized Hydrogenic Bound States,  Next: Coulomb Wave Functions,  Up: Coulomb Functions

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 (*note
     Laguerre Functions::).  The normalization is chosen such that the
     wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.


File: gsl-ref.info,  Node: Coulomb Wave Functions,  Next: Coulomb Wave Function Normalization Constant,  Prev: Normalized Hydrogenic Bound States,  Up: Coulomb Functions

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.


File: gsl-ref.info,  Node: Coulomb Wave Function Normalization Constant,  Prev: Coulomb Wave Functions,  Up: Coulomb Functions

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.


File: gsl-ref.info,  Node: Coupling Coefficients,  Next: Dawson Function,  Prev: Coulomb Functions,  Up: Special Functions

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'.

* Menu:

* 3-j Symbols::
* 6-j Symbols::
* 9-j Symbols::


File: gsl-ref.info,  Node: 3-j Symbols,  Next: 6-j Symbols,  Up: Coupling Coefficients

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.


File: gsl-ref.info,  Node: 6-j Symbols,  Next: 9-j Symbols,  Prev: 3-j Symbols,  Up: Coupling Coefficients

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.


File: gsl-ref.info,  Node: 9-j Symbols,  Prev: 6-j Symbols,  Up: Coupling Coefficients

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.


File: gsl-ref.info,  Node: Dawson Function,  Next: Debye Functions,  Prev: Coupling Coefficients,  Up: Special Functions

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.


File: gsl-ref.info,  Node: Debye Functions,  Next: Dilogarithm,  Prev: Dawson Function,  Up: Special Functions

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 fourth-order Debye function D_6(x) =
     (6/x^6) \int_0^x dt (t^6/(e^t - 1)).


File: gsl-ref.info,  Node: Dilogarithm,  Next: Elementary Operations,  Prev: Debye Functions,  Up: Special Functions

7.11 Dilogarithm
================

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

* Menu:

* Real Argument::
* Complex Argument::


File: gsl-ref.info,  Node: Real Argument,  Next: Complex Argument,  Up: Dilogarithm

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.



File: gsl-ref.info,  Node: Complex Argument,  Prev: Real Argument,  Up: Dilogarithm

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.


File: gsl-ref.info,  Node: Elementary Operations,  Next: Elliptic Integrals,  Prev: Dilogarithm,  Up: Special Functions

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.


File: gsl-ref.info,  Node: Elliptic Integrals,  Next: Elliptic Functions (Jacobi),  Prev: Elementary Operations,  Up: Special Functions

7.13 Elliptic Integrals
=======================

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

* Menu:

* Definition of Legendre Forms::
* Definition of Carlson Forms::
* Legendre Form of Complete Elliptic Integrals::
* Legendre Form of Incomplete Elliptic Integrals::
* Carlson Forms::


File: gsl-ref.info,  Node: Definition of Legendre Forms,  Next: Definition of Carlson Forms,  Up: Elliptic Integrals

7.13.1 Definition of Legendre Forms
-----------------------------------

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

     P(\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).  Further information on the Legendre forms of elliptic
integrals can be found in Abramowitz & Stegun, Chapter 17.  The
notation used here is based on Carlson, `Numerische Mathematik' 33
(1979) 1 and differs slightly from that used by Abramowitz & Stegun.


File: gsl-ref.info,  Node: Definition of Carlson Forms,  Next: Legendre Form of Complete Elliptic Integrals,  Prev: Definition of Legendre Forms,  Up: Elliptic Integrals

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)


File: gsl-ref.info,  Node: Legendre Form of Complete Elliptic Integrals,  Next: Legendre Form of Incomplete Elliptic Integrals,  Prev: Definition of Carlson Forms,  Up: Elliptic Integrals

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.

 -- 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.


File: gsl-ref.info,  Node: Legendre Form of Incomplete Elliptic Integrals,  Next: Carlson Forms,  Prev: Legendre Form of Complete Elliptic Integrals,  Up: Elliptic Integrals

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.

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

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

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

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



File: gsl-ref.info,  Node: Carlson Forms,  Prev: Legendre Form of Incomplete Elliptic Integrals,  Up: Elliptic Integrals

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.


File: gsl-ref.info,  Node: Elliptic Functions (Jacobi),  Next: Error Functions,  Prev: Elliptic Integrals,  Up: Special Functions

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.


File: gsl-ref.info,  Node: Error Functions,  Next: Exponential Functions,  Prev: Elliptic Functions (Jacobi),  Up: Special Functions

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'.

* Menu:

* Error Function::
* Complementary Error Function::
* Log Complementary Error Function::
* Probability functions::


File: gsl-ref.info,  Node: Error Function,  Next: Complementary Error Function,  Up: Error Functions

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).


File: gsl-ref.info,  Node: Complementary Error Function,  Next: Log Complementary Error Function,  Prev: Error Function,  Up: Error Functions

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).


File: gsl-ref.info,  Node: Log Complementary Error Function,  Next: Probability functions,  Prev: Complementary Error Function,  Up: Error Functions

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)).


File: gsl-ref.info,  Node: Probability functions,  Prev: Log Complementary Error Function,  Up: Error Functions

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 Mill's 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.


File: gsl-ref.info,  Node: Exponential Functions,  Next: Exponential Integrals,  Prev: Error Functions,  Up: Special Functions

7.16 Exponential Functions
==========================

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

* Menu:

* Exponential Function::
* Relative Exponential Functions::
* Exponentiation With Error Estimate::


File: gsl-ref.info,  Node: Exponential Function,  Next: Relative Exponential Functions,  Up: Exponential Functions

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.


File: gsl-ref.info,  Node: Relative Exponential Functions,  Next: Exponentiation With Error Estimate,  Prev: Exponential Function,  Up: Exponential Functions

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_exprel2'.  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)


File: gsl-ref.info,  Node: Exponentiation With Error Estimate,  Prev: Relative Exponential Functions,  Up: Exponential Functions

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.


File: gsl-ref.info,  Node: Exponential Integrals,  Next: Fermi-Dirac Function,  Prev: Exponential Functions,  Up: Special Functions

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'.

* Menu:

* Exponential Integral::
* Ei(x)::
* Hyperbolic Integrals::
* Ei_3(x)::
* Trigonometric Integrals::
* Arctangent Integral::


File: gsl-ref.info,  Node: Exponential Integral,  Next: Ei(x),  Up: Exponential Integrals

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.




File: gsl-ref.info,  Node: Ei(x),  Next: Hyperbolic Integrals,  Prev: Exponential Integral,  Up: Exponential Integrals

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.


File: gsl-ref.info,  Node: Hyperbolic Integrals,  Next: Ei_3(x),  Prev: Ei(x),  Up: Exponential Integrals

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').


File: gsl-ref.info,  Node: Ei_3(x),  Next: Trigonometric Integrals,  Prev: Hyperbolic Integrals,  Up: Exponential Integrals

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.


File: gsl-ref.info,  Node: Trigonometric Integrals,  Next: Arctangent Integral,  Prev: Ei_3(x),  Up: Exponential Integrals

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.


File: gsl-ref.info,  Node: Arctangent Integral,  Prev: Trigonometric Integrals,  Up: Exponential Integrals

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.


File: gsl-ref.info,  Node: Fermi-Dirac Function,  Next: Gamma and Beta Functions,  Prev: Exponential Integrals,  Up: Special Functions

7.18 Fermi-Dirac Function
=========================

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

* Menu:

* Complete Fermi-Dirac Integrals::
* Incomplete Fermi-Dirac Integrals::


File: gsl-ref.info,  Node: Complete Fermi-Dirac Integrals,  Next: Incomplete Fermi-Dirac Integrals,  Up: Fermi-Dirac Function

7.18.1 Complete Fermi-Dirac Integrals
-------------------------------------

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

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

 -- 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).


File: gsl-ref.info,  Node: Incomplete Fermi-Dirac Integrals,  Prev: Complete Fermi-Dirac Integrals,  Up: Fermi-Dirac Function

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).


File: gsl-ref.info,  Node: Gamma and Beta Functions,  Next: Gegenbauer Functions,  Prev: Fermi-Dirac Function,  Up: Special Functions

7.19 Gamma and Beta Functions
=============================

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

* Menu:

* Gamma Functions::
* Factorials::
* Pochhammer Symbol::
* Incomplete Gamma Functions::
* Beta Functions::
* Incomplete Beta Function::


File: gsl-ref.info,  Node: Gamma Functions,  Next: Factorials,  Up: Gamma and Beta Functions

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

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

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

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

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

          \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, 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.


File: gsl-ref.info,  Node: Factorials,  Next: Pochhammer Symbol,  Prev: Gamma Functions,  Up: Gamma and Beta Functions

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.


File: gsl-ref.info,  Node: Pochhammer Symbol,  Next: Incomplete Gamma Functions,  Prev: Factorials,  Up: Gamma and Beta Functions

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), subject to a and a+x not being negative integers.
     The Pochhammer symbol is also known as the Apell symbol and
     sometimes written as (a,x).

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

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

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


File: gsl-ref.info,  Node: Incomplete Gamma Functions,  Next: Beta Functions,  Prev: Pochhammer Symbol,  Up: Gamma and Beta Functions

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).


File: gsl-ref.info,  Node: Beta Functions,  Next: Incomplete Beta Function,  Prev: Incomplete Gamma Functions,  Up: Gamma and Beta Functions

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) for a > 0, b > 0.

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


File: gsl-ref.info,  Node: Incomplete Beta Function,  Prev: Beta Functions,  Up: Gamma and Beta Functions

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
     B_x(a,b)/B(a,b) where B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt
     for a > 0, b > 0, and 0 <= x <= 1.


File: gsl-ref.info,  Node: Gegenbauer Functions,  Next: Hypergeometric Functions,  Prev: Gamma and Beta Functions,  Up: Special Functions

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.


File: gsl-ref.info,  Node: Hypergeometric Functions,  Next: Laguerre Functions,  Prev: Gegenbauer Functions,  Up: Special Functions

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)
     These routines compute the confluent hypergeometric function
     U(a,b,x).

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

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

     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.
     exceptions:

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

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

 -- Function: double gsl_sf_hyperg_2F0 (double A, double B, double X)
 -- Function: int gsl_sf_hyperg_2F0_e (double A, double B, double X,
          gsl_sf_result * RESULT)
     These routines compute the hypergeometric function 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)


File: gsl-ref.info,  Node: Laguerre Functions,  Next: Lambert W Functions,  Prev: Hypergeometric Functions,  Up: Special Functions

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.



File: gsl-ref.info,  Node: Lambert W Functions,  Next: Legendre Functions and Spherical Harmonics,  Prev: Laguerre Functions,  Up: Special Functions

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).


File: gsl-ref.info,  Node: Legendre Functions and Spherical Harmonics,  Next: Logarithm and Related Functions,  Prev: Lambert W Functions,  Up: Special Functions

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'.

* Menu:

* Legendre Polynomials::
* Associated Legendre Polynomials and Spherical Harmonics::
* Conical Functions::
* Radial Functions for Hyperbolic Space::


File: gsl-ref.info,  Node: Legendre Polynomials,  Next: Associated Legendre Polynomials and Spherical Harmonics,  Up: Legendre Functions and Spherical Harmonics

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 an array of Legendre polynomials P_l(x),
     and optionally their 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.


File: gsl-ref.info,  Node: Associated Legendre Polynomials and Spherical Harmonics,  Next: Conical Functions,  Prev: Legendre Polynomials,  Up: Legendre Functions and Spherical Harmonics

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 an array of Legendre polynomials P_l^m(x),
     and optionally their 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 an array of normalized associated Legendre
     functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$,
     and optionally their 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.


File: gsl-ref.info,  Node: Conical Functions,  Next: Radial Functions for Hyperbolic Space,  Prev: Associated Legendre Polynomials and Spherical Harmonics,  Up: Legendre Functions and Spherical Harmonics

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.


File: gsl-ref.info,  Node: Radial Functions for Hyperbolic Space,  Prev: Conical Functions,  Up: Legendre Functions and Spherical Harmonics

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.


File: gsl-ref.info,  Node: Logarithm and Related Functions,  Next: Power Function,  Prev: Legendre Functions and Spherical Harmonics,  Up: Special Functions

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.


File: gsl-ref.info,  Node: Power Function,  Next: Psi (Digamma) Function,  Prev: Logarithm and Related Functions,  Up: Special Functions

7.26 Power Function
===================

The following functions are equivalent to the function `gsl_pow_int'
(*note 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);


File: gsl-ref.info,  Node: Psi (Digamma) Function,  Next: Synchrotron Functions,  Prev: Power Function,  Up: Special Functions

7.27 Psi (Digamma) Function
===========================

The polygamma functions of order m are defined by

     \psi^{(m)}(x) = (d/dx)^m \psi(x) = (d/dx)^{m+1} \log(\Gamma(x))

where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function.
These functions are declared in the header file `gsl_sf_psi.h'.

* Menu:

* Digamma Function::
* Trigamma Function::
* Polygamma Function::


File: gsl-ref.info,  Node: Digamma Function,  Next: Trigamma Function,  Up: Psi (Digamma) Function

7.27.1 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)].


File: gsl-ref.info,  Node: Trigamma Function,  Next: Polygamma Function,  Prev: Digamma Function,  Up: Psi (Digamma) Function

7.27.2 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.


File: gsl-ref.info,  Node: Polygamma Function,  Prev: Trigamma Function,  Up: Psi (Digamma) Function

7.27.3 Polygamma Function
-------------------------

 -- Function: double gsl_sf_psi_n (int M, double X)
 -- Function: int gsl_sf_psi_n_e (int M, double X, gsl_sf_result *
          RESULT)
     These routines compute the polygamma function \psi^{(m)}(x) for m
     >= 0, x > 0.


File: gsl-ref.info,  Node: Synchrotron Functions,  Next: Transport Functions,  Prev: Psi (Digamma) Function,  Up: Special Functions

7.28 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.


File: gsl-ref.info,  Node: Transport Functions,  Next: Trigonometric Functions,  Prev: Synchrotron Functions,  Up: Special Functions

7.29 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).


File: gsl-ref.info,  Node: Trigonometric Functions,  Next: Zeta Functions,  Prev: Transport Functions,  Up: Special Functions

7.30 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'.

* Menu:

* Circular Trigonometric Functions::
* Trigonometric Functions for Complex Arguments::
* Hyperbolic Trigonometric Functions::
* Conversion Functions::
* Restriction Functions::
* Trigonometric Functions With Error Estimates::


File: gsl-ref.info,  Node: Circular Trigonometric Functions,  Next: Trigonometric Functions for Complex Arguments,  Up: Trigonometric Functions

7.30.1 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.


File: gsl-ref.info,  Node: Trigonometric Functions for Complex Arguments,  Next: Hyperbolic Trigonometric Functions,  Prev: Circular Trigonometric Functions,  Up: Trigonometric Functions

7.30.2 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 SZR, SZI.

 -- Function: int gsl_sf_complex_logsin_e (double ZR, double ZI,
          gsl_sf_result * LSZR, gsl_sf_result * LSZI)
     This function computes the logarithm of the complex sine,
     \log(\sin(z_r + i z_i)) storing the real and imaginary parts in
     SZR, SZI.


File: gsl-ref.info,  Node: Hyperbolic Trigonometric Functions,  Next: Conversion Functions,  Prev: Trigonometric Functions for Complex Arguments,  Up: Trigonometric Functions

7.30.3 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.


File: gsl-ref.info,  Node: Conversion Functions,  Next: Restriction Functions,  Prev: Hyperbolic Trigonometric Functions,  Up: Trigonometric Functions

7.30.4 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].


File: gsl-ref.info,  Node: Restriction Functions,  Next: Trigonometric Functions With Error Estimates,  Prev: Conversion Functions,  Up: Trigonometric Functions

7.30.5 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].

 -- 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).


File: gsl-ref.info,  Node: Trigonometric Functions With Error Estimates,  Prev: Restriction Functions,  Up: Trigonometric Functions

7.30.6 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.


File: gsl-ref.info,  Node: Zeta Functions,  Next: Special Functions Examples,  Prev: Trigonometric Functions,  Up: Special Functions

7.31 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'.

* Menu:

* Riemann Zeta Function::
* Riemann Zeta Function Minus One::
* Hurwitz Zeta Function::
* Eta Function::


File: gsl-ref.info,  Node: Riemann Zeta Function,  Next: Riemann Zeta Function Minus One,  Up: Zeta Functions

7.31.1 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.


File: gsl-ref.info,  Node: Riemann Zeta Function Minus One,  Next: Hurwitz Zeta Function,  Prev: Riemann Zeta Function,  Up: Zeta Functions

7.31.2 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.


File: gsl-ref.info,  Node: Hurwitz Zeta Function,  Next: Eta Function,  Prev: Riemann Zeta Function Minus One,  Up: Zeta Functions

7.31.3 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.


File: gsl-ref.info,  Node: Eta Function,  Prev: Hurwitz Zeta Function,  Up: Zeta Functions

7.31.4 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.


File: gsl-ref.info,  Node: Special Functions Examples,  Next: Special Functions References and Further Reading,  Prev: Zeta Functions,  Up: Special Functions

7.32 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


File: gsl-ref.info,  Node: Special Functions References and Further Reading,  Prev: Special Functions Examples,  Up: Special Functions

7.33 References and Further Reading
===================================

The library follows the conventions of `Abramowitz & Stegun' where
possible,
     Abramowitz & Stegun (eds.), `Handbook of Mathematical Functions'

The following papers contain information on the algorithms used to
compute the special functions, 
     MISCFUN: A software package to compute uncommon special functions.
     `ACM Trans. Math. Soft.', vol. 22, 1996, 288-301

     G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd
     Edition (Cambridge University Press, 1944).

     G. Nemeth, Mathematical Approximations of Special Functions, Nova
     Science Publishers, ISBN 1-56072-052-2

     B.C. Carlson, Special Functions of Applied Mathematics (1977)

     W.J. Thompson, Atlas for Computing Mathematical Functions, John
     Wiley & Sons, New York (1997).

     Y.Y. Luke, Algorithms for the Computation of Mathematical
     Functions, Academic Press, New York (1977).



File: gsl-ref.info,  Node: Vectors and Matrices,  Next: Permutations,  Prev: Special Functions,  Up: Top

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.

* Menu:

* Data types::
* Blocks::
* Vectors::
* Matrices::
* Vector and Matrix References and Further Reading::


File: gsl-ref.info,  Node: Data types,  Next: Blocks,  Up: Vectors and Matrices

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.


File: gsl-ref.info,  Node: Blocks,  Next: Vectors,  Prev: Data types,  Up: Vectors and Matrices

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'

* Menu:

* Block allocation::
* Reading and writing blocks::
* Example programs for blocks::


File: gsl-ref.info,  Node: Block allocation,  Next: Reading and writing blocks,  Up: Blocks

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'.


File: gsl-ref.info,  Node: Reading and writing blocks,  Next: Example programs for blocks,  Prev: Block allocation,  Up: Blocks

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.


File: gsl-ref.info,  Node: Example programs for blocks,  Prev: Reading and writing blocks,  Up: Blocks

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


File: gsl-ref.info,  Node: Vectors,  Next: Matrices,  Prev: Blocks,  Up: Vectors and Matrices

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'

* Menu:

* Vector allocation::
* Accessing vector elements::
* Initializing vector elements::
* Reading and writing vectors::
* Vector views::
* Copying vectors::
* Exchanging elements::
* Vector operations::
* Finding maximum and minimum elements of vectors::
* Vector properties::
* Example programs for vectors::


File: gsl-ref.info,  Node: Vector allocation,  Next: Accessing vector elements,  Up: Vectors

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.


File: gsl-ref.info,  Node: Accessing vector elements,  Next: Initializing vector elements,  Prev: Vector allocation,  Up: Vectors

8.3.2 Accessing vector elements
-------------------------------

Unlike FORTRAN compilers, C compilers do not usually provide support
for range checking of vectors and matrices.  Range checking is
available in the GNU C Compiler bounds-checking extension, but it is not
part of the default installation of GCC.  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 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.

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

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

 -- Function: double * gsl_vector_ptr (gsl_vector * V, size_t I)
 -- Function: const double * gsl_vector_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.


File: gsl-ref.info,  Node: Initializing vector elements,  Next: Reading and writing vectors,  Prev: Accessing vector elements,  Up: Vectors

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.


File: gsl-ref.info,  Node: Reading and writing vectors,  Next: Vector views,  Prev: Initializing vector elements,  Up: Vectors

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.


File: gsl-ref.info,  Node: Vector views,  Next: Copying vectors,  Prev: Reading and writing vectors,  Up: Vectors

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'.


File: gsl-ref.info,  Node: Copying vectors,  Next: Exchanging elements,  Prev: Vector views,  Up: Vectors

8.3.6 Copying vectors
---------------------

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


File: gsl-ref.info,  Node: Exchanging elements,  Next: Vector operations,  Prev: Copying vectors,  Up: Vectors

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.


File: gsl-ref.info,  Node: Vector operations,  Next: Finding maximum and minimum elements of vectors,  Prev: Exchanging elements,  Up: Vectors

8.3.8 Vector operations
-----------------------

The following operations are only defined for real vectors.

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

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

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

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

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

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


File: gsl-ref.info,  Node: Finding maximum and minimum elements of vectors,  Next: Vector properties,  Prev: Vector operations,  Up: Vectors

8.3.9 Finding maximum and minimum elements of vectors
-----------------------------------------------------

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

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

 -- Function: void gsl_vector_minmax (const gsl_vector * V, double *
          MIN_OUT, double * MAX_OUT)
     This function returns the minimum and maximum values in the vector
     V, storing them in MIN_OUT and MAX_OUT.

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

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

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


File: gsl-ref.info,  Node: Vector properties,  Next: Example programs for vectors,  Prev: Finding maximum and minimum elements of vectors,  Up: Vectors

8.3.10 Vector properties
------------------------

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


File: gsl-ref.info,  Node: Example programs for vectors,  Prev: Vector properties,  Up: Vectors

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++)
         {
           printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
         }

       return 0;
     }

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

     $ ./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);
       }
       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));
         }

       return 0;
     }


File: gsl-ref.info,  Node: Matrices,  Next: Vector and Matrix References and Further Reading,  Prev: Vectors,  Up: Vectors and Matrices

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'

* Menu:

* Matrix allocation::
* Accessing matrix elements::
* Initializing matrix elements::
* Reading and writing matrices::
* Matrix views::
* Creating row and column views::
* Copying matrices::
* Copying rows and columns::
* Exchanging rows and columns::
* Matrix operations::
* Finding maximum and minimum elements of matrices::
* Matrix properties::
* Example programs for matrices::


File: gsl-ref.info,  Node: Matrix allocation,  Next: Accessing matrix elements,  Up: Matrices

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 vector then the
functions call the GSL error handler (with an error number of
`GSL_ENOMEM') in addition to returning a null pointer.  Thus if you use
the library error handler to abort your program then it isn't necessary
to check every `alloc'.

 -- Function: gsl_matrix * gsl_matrix_alloc (size_t N1, size_t N2)
     This function creates a matrix of size N1 rows by N2 columns,
     returning a pointer to a newly initialized matrix struct. A new
     block is allocated for the elements of the matrix, and stored in
     the BLOCK component of the matrix struct.  The block is "owned" by
     the matrix, and will be deallocated when the matrix is deallocated.

 -- Function: gsl_matrix * gsl_matrix_calloc (size_t N1, size_t N2)
     This function allocates memory for a matrix of size N1 rows by N2
     columns and initializes all the elements of the matrix to zero.

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


File: gsl-ref.info,  Node: Accessing matrix elements,  Next: Initializing matrix elements,  Prev: Matrix allocation,  Up: Matrices

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.

 -- Function: void gsl_matrix_set (gsl_matrix * M, size_t I, size_t J,
          double X)
     This function sets the value of the (i,j)-th element of a matrix M
     to X.  If I or J lies outside the allowed range of 0 to N1-1 and 0
     to N2-1 then the error handler is invoked.

 -- Function: double * gsl_matrix_ptr (gsl_matrix * M, size_t I, size_t
          J)
 -- Function: const double * gsl_matrix_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.


File: gsl-ref.info,  Node: Initializing matrix elements,  Next: Reading and writing matrices,  Prev: Accessing matrix elements,  Up: Matrices

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.


File: gsl-ref.info,  Node: Reading and writing matrices,  Next: Matrix views,  Prev: Initializing matrix elements,  Up: Matrices

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.


File: gsl-ref.info,  Node: Matrix views,  Next: Creating row and column views,  Prev: Reading and writing matrices,  Up: Matrices

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'.


File: gsl-ref.info,  Node: Creating row and column views,  Next: Copying matrices,  Prev: Matrix views,  Up: Matrices

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

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

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

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

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

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


File: gsl-ref.info,  Node: Copying matrices,  Next: Copying rows and columns,  Prev: Creating row and column views,  Up: Matrices

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.


File: gsl-ref.info,  Node: Copying rows and columns,  Next: Exchanging rows and columns,  Prev: Copying matrices,  Up: Matrices

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.


File: gsl-ref.info,  Node: Exchanging rows and columns,  Next: Matrix operations,  Prev: Copying rows and columns,  Up: Matrices

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.


File: gsl-ref.info,  Node: Matrix operations,  Next: Finding maximum and minimum elements of matrices,  Prev: Exchanging rows and columns,  Up: Matrices

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, a'(i,j) = a(i,j) + b(i,j). The two matrices must have the
     same dimensions.

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

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

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

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

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


File: gsl-ref.info,  Node: Finding maximum and minimum elements of matrices,  Next: Matrix properties,  Prev: Matrix operations,  Up: Matrices

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.


File: gsl-ref.info,  Node: Matrix properties,  Next: Example programs for matrices,  Prev: Finding maximum and minimum elements of matrices,  Up: Matrices

8.4.12 Matrix properties
------------------------

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


File: gsl-ref.info,  Node: Example programs for matrices,  Prev: Matrix properties,  Up: Matrices

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++)
         for (j = 0; j < 3; j++)
           printf ("m(%d,%d) = %g\n", i, j,
                   gsl_matrix_get (m, i, j));

       return 0;
     }

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

     $ ./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++;
           }

       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


File: gsl-ref.info,  Node: Vector and Matrix References and Further Reading,  Prev: Matrices,  Up: Vectors and Matrices

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,

     B. Stroustrup, `The C++ Programming Language' (3rd Ed), Section
     22.4 Vector Arithmetic.  Addison-Wesley 1997, ISBN 0-201-88954-4.


File: gsl-ref.info,  Node: Permutations,  Next: Combinations,  Prev: Vectors and Matrices,  Up: Top

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'.

* Menu:

* The Permutation struct::
* Permutation allocation::
* Accessing permutation elements::
* Permutation properties::
* Permutation functions::
* Applying Permutations::
* Reading and writing permutations::
* Permutations in cyclic form::
* Permutation Examples::
* Permutation References and Further Reading::


File: gsl-ref.info,  Node: The Permutation struct,  Next: Permutation allocation,  Up: Permutations

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;



File: gsl-ref.info,  Node: Permutation allocation,  Next: Accessing permutation elements,  Prev: The Permutation struct,  Up: Permutations

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.


File: gsl-ref.info,  Node: Accessing permutation elements,  Next: Permutation properties,  Prev: Permutation allocation,  Up: Permutations

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.

 -- 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.


File: gsl-ref.info,  Node: Permutation properties,  Next: Permutation functions,  Prev: Accessing permutation elements,  Up: Permutations

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 (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.


File: gsl-ref.info,  Node: Permutation functions,  Next: Applying Permutations,  Prev: Permutation properties,  Up: Permutations

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.


File: gsl-ref.info,  Node: Applying Permutations,  Next: Reading and writing permutations,  Prev: Permutation functions,  Up: Permutations

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.


File: gsl-ref.info,  Node: Reading and writing permutations,  Next: Permutations in cyclic form,  Prev: Applying Permutations,  Up: Permutations

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.  On a GNU
     system the type modifier `Z' represents `size_t', so `"%Zu\n"' is
     a suitable format.  The function returns `GSL_EFAILED' if there
     was a problem writing to the file.

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


File: gsl-ref.info,  Node: Permutations in cyclic form,  Next: Permutation Examples,  Prev: Reading and writing permutations,  Up: Permutations

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.


File: gsl-ref.info,  Node: Permutation Examples,  Next: Permutation References and Further Reading,  Prev: Permutations in cyclic form,  Up: Permutations

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");

       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);

       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'.


File: gsl-ref.info,  Node: Permutation References and Further Reading,  Prev: Permutation Examples,  Up: Permutations

9.10 References and Further Reading
===================================

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

     Donald E. Knuth, `The Art of Computer Programming: Sorting and
     Searching' (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.

For the definition of the "canonical form" see,

     Donald E. Knuth, `The Art of Computer Programming: Fundamental
     Algorithms' (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
     Section 1.3.3, `An Unusual Correspondence', p.178-179.


File: gsl-ref.info,  Node: Combinations,  Next: Sorting,  Prev: Permutations,  Up: Top

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'.

* Menu:

* The Combination struct::
* Combination allocation::
* Accessing combination elements::
* Combination properties::
* Combination functions::
* Reading and writing combinations::
* Combination Examples::
* Combination References and Further Reading::


File: gsl-ref.info,  Node: The Combination struct,  Next: Combination allocation,  Up: Combinations

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;



File: gsl-ref.info,  Node: Combination allocation,  Next: Accessing combination elements,  Prev: The Combination struct,  Up: Combinations

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.


File: gsl-ref.info,  Node: Accessing combination elements,  Next: Combination properties,  Prev: Combination allocation,  Up: Combinations

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.


File: gsl-ref.info,  Node: Combination properties,  Next: Combination functions,  Prev: Accessing combination elements,  Up: Combinations

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.


File: gsl-ref.info,  Node: Combination functions,  Next: Reading and writing combinations,  Prev: Combination properties,  Up: Combinations

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.


File: gsl-ref.info,  Node: Reading and writing combinations,  Next: Combination Examples,  Prev: Combination functions,  Up: Combinations

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.  On a GNU
     system the type modifier `Z' represents `size_t', so `"%Zu\n"' is
     a suitable format.  The function returns `GSL_EFAILED' if there
     was a problem writing to the file.

 -- Function: int gsl_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.


File: gsl-ref.info,  Node: Combination Examples,  Next: Combination References and Further Reading,  Prev: Reading and writing combinations,  Up: Combinations

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.


File: gsl-ref.info,  Node: Combination References and Further Reading,  Prev: Combination Examples,  Up: Combinations

10.8 References and Further Reading
===================================

Further information on combinations can be found in,

     Donald L. Kreher, Douglas R. Stinson, `Combinatorial Algorithms:
     Generation, Enumeration and Search', 1998, CRC Press LLC, ISBN
     084933988X



File: gsl-ref.info,  Node: Sorting,  Next: BLAS Support,  Prev: Combinations,  Up: Top

11 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.

* Menu:

* Sorting objects::
* Sorting vectors::
* Selecting the k smallest or largest elements::
* Computing the rank::
* Sorting Examples::
* Sorting References and Further Reading::


File: gsl-ref.info,  Node: Sorting objects,  Next: Sorting vectors,  Up: Sorting

11.1 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.


File: gsl-ref.info,  Node: Sorting vectors,  Next: Selecting the k smallest or largest elements,  Prev: Sorting objects,  Up: Sorting

11.2 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, size_t STRIDE, size_t N)
     This function sorts the N elements of the array DATA with stride
     STRIDE into ascending numerical order.

 -- Function: void gsl_sort_vector (gsl_vector * V)
     This function sorts the elements of the vector V into ascending
     numerical order.

 -- Function: int gsl_sort_index (size_t * P, const double * DATA,
          size_t STRIDE, size_t N)
     This function indirectly sorts the N elements of the array DATA
     with stride STRIDE into ascending order, storing the resulting
     permutation in P.  The array P must be allocated 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.


File: gsl-ref.info,  Node: Selecting the k smallest or largest elements,  Next: Computing the rank,  Prev: Sorting vectors,  Up: Sorting

11.3 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: void 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: void 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: void gsl_sort_vector_smallest (double * DEST, size_t K,
          const gsl_vector * V)
 -- Function: void gsl_sort_vector_largest (double * DEST, size_t K,
          const gsl_vector * V)
     These functions copy the K 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: void gsl_sort_smallest_index (size_t * P, size_t K, const
          double * SRC, size_t STRIDE, size_t N)
     This function stores the indices of the K 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: void gsl_sort_largest_index (size_t * P, size_t K, const
          double * SRC, size_t STRIDE, size_t N)
     This function stores the indices of the K 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: void gsl_sort_vector_smallest_index (size_t * P, size_t
          K, const gsl_vector * V)
 -- Function: void gsl_sort_vector_largest_index (size_t * P, size_t K,
          const gsl_vector * V)
     These functions store the indices of 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.


File: gsl-ref.info,  Node: Computing the rank,  Next: Sorting Examples,  Prev: Selecting the k smallest or largest elements,  Up: Sorting

11.4 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);
     }


File: gsl-ref.info,  Node: Sorting Examples,  Next: Sorting References and Further Reading,  Prev: Computing the rank,  Up: Sorting

11.5 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]);
         }
       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


File: gsl-ref.info,  Node: Sorting References and Further Reading,  Prev: Sorting Examples,  Up: Sorting

11.6 References and Further Reading
===================================

The subject of sorting is covered extensively in Knuth's `Sorting and
Searching',

     Donald E. Knuth, `The Art of Computer Programming: Sorting and
     Searching' (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.

The Heapsort algorithm is described in the following book,

     Robert Sedgewick, `Algorithms in C', Addison-Wesley, ISBN
     0201514257.


File: gsl-ref.info,  Node: BLAS Support,  Next: Linear Algebra,  Prev: Sorting,  Up: Top

12 BLAS Support
***************

The Basic Linear Algebra Subprograms (BLAS) define a set of fundamental
operations on vectors and matrices which can be used to create optimized
higher-level linear algebra functionality.

   The library provides a low-level layer which corresponds directly to
the C-language BLAS standard, referred to here as "CBLAS", and a
higher-level interface for operations on GSL vectors and matrices.
Users who are interested in simple operations on GSL vector and matrix
objects should use the high-level layer, which is declared in the file
`gsl_blas.h'.  This should satisfy the needs of most users.  Note that
GSL matrices are implemented using dense-storage so the interface only
includes the corresponding dense-storage BLAS functions.  The full BLAS
functionality for band-format and packed-format matrices is available
through the low-level CBLAS interface.

   The interface for the `gsl_cblas' layer is specified in the file
`gsl_cblas.h'.  This interface corresponds to the BLAS Technical
Forum's draft standard for the C interface to legacy BLAS
implementations. Users who have access to other conforming CBLAS
implementations can use these in place of the version provided by the
library.  Note that users who have only a Fortran BLAS library can use
a CBLAS conformant wrapper to convert it into a CBLAS library.  A
reference CBLAS wrapper for legacy Fortran implementations exists as
part of the draft CBLAS standard and can be obtained from Netlib.  The
complete set of CBLAS functions is listed in an appendix (*note 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".

* Menu:

* GSL BLAS Interface::
* BLAS Examples::
* BLAS References and Further Reading::


File: gsl-ref.info,  Node: GSL BLAS Interface,  Next: BLAS Examples,  Up: BLAS Support

12.1 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'.

* Menu:

* Level 1 GSL BLAS Interface::
* Level 2 GSL BLAS Interface::
* Level 3 GSL BLAS Interface::


File: gsl-ref.info,  Node: Level 1 GSL BLAS Interface,  Next: Level 2 GSL BLAS Interface,  Up: GSL BLAS Interface

12.1.1 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 RESULT

 -- Function: int gsl_blas_cdotc (const gsl_vector_complex_float * X,
          const gsl_vector_complex_float * Y, gsl_complex_float * DOTC)
 -- Function: int gsl_blas_zdotc (const gsl_vector_complex * X, const
          gsl_vector_complex * Y, gsl_complex * DOTC)
     These functions compute the complex conjugate scalar product x^H y
     for the vectors X and Y, returning the result in RESULT

 -- Function: float gsl_blas_snrm2 (const gsl_vector_float * X)
 -- Function: double gsl_blas_dnrm2 (const gsl_vector * X)
     These functions compute the Euclidean norm ||x||_2 = \sqrt {\sum
     x_i^2} of the vector X.

 -- Function: float gsl_blas_scnrm2 (const gsl_vector_complex_float * X)
 -- Function: double gsl_blas_dznrm2 (const gsl_vector_complex * X)
     These functions compute the Euclidean norm of the complex vector X,

          ||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.


File: gsl-ref.info,  Node: Level 2 GSL BLAS Interface,  Next: Level 3 GSL BLAS Interface,  Prev: Level 1 GSL BLAS Interface,  Up: GSL BLAS Interface

12.1.2 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 =\alpha 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.


File: gsl-ref.info,  Node: Level 3 GSL BLAS Interface,  Prev: Level 2 GSL BLAS Interface,  Up: GSL BLAS Interface

12.1.3 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 `CblasTrans'.  Since the
     matrix C is hermitian only its upper half or lower half need to be
     stored.  When UPLO is `CblasUpper' then the upper triangle and
     diagonal of C are used, and when UPLO is `CblasLower' then the
     lower triangle and diagonal of C are used.  The imaginary elements
     of the diagonal are automatically set to zero.

 -- Function: int gsl_blas_ssyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, float ALPHA, const gsl_matrix_float * A, const
          gsl_matrix_float * B, float BETA, gsl_matrix_float * C)
 -- Function: int gsl_blas_dsyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, double ALPHA, const gsl_matrix * A, const gsl_matrix *
          B, double BETA, gsl_matrix * C)
 -- Function: int gsl_blas_csyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, const gsl_complex_float ALPHA, const
          gsl_matrix_complex_float * A, const gsl_matrix_complex_float
          * B, const gsl_complex_float BETA, gsl_matrix_complex_float *
          C)
 -- Function: int gsl_blas_zsyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, const gsl_complex ALPHA, const gsl_matrix_complex * A,
          const gsl_matrix_complex * B, const gsl_complex BETA,
          gsl_matrix_complex * C)
     These functions compute a rank-2k update of the symmetric matrix C,
     C = \alpha A B^T + \alpha B A^T + \beta C when TRANS is
     `CblasNoTrans' and C = \alpha A^T B + \alpha B^T A + \beta C when
     TRANS is `CblasTrans'.  Since the matrix C is symmetric only its
     upper half or lower half need to be stored.  When UPLO is
     `CblasUpper' then the upper triangle and diagonal of C are used,
     and when UPLO is `CblasLower' then the lower triangle and diagonal
     of C are used.

 -- Function: int gsl_blas_cher2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, const gsl_complex_float ALPHA, const
          gsl_matrix_complex_float * A, const gsl_matrix_complex_float
          * B, float BETA, gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zher2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, const gsl_complex ALPHA, const gsl_matrix_complex * A,
          const gsl_matrix_complex * B, double BETA, gsl_matrix_complex
          * C)
     These functions compute a rank-2k update of the hermitian matrix C,
     C = \alpha A B^H + \alpha^* B A^H + \beta C when TRANS is
     `CblasNoTrans' and C = \alpha A^H B + \alpha^* B^H A + \beta C when
     TRANS is `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.


File: gsl-ref.info,  Node: BLAS Examples,  Next: BLAS References and Further Reading,  Prev: GSL BLAS Interface,  Up: BLAS Support

12.2 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 ]


File: gsl-ref.info,  Node: BLAS References and Further Reading,  Prev: BLAS Examples,  Up: BLAS Support

12.3 References and Further Reading
===================================

Information on the BLAS standards, including both the legacy and draft
interface standards, is available online from the BLAS Homepage and
BLAS Technical Forum web-site.

     `BLAS Homepage'
     `http://www.netlib.org/blas/'

     `BLAS Technical Forum'
     `http://www.netlib.org/cgi-bin/checkout/blast/blast.pl'

The following papers contain the specifications for Level 1, Level 2 and
Level 3 BLAS.

     C. Lawson, R. Hanson, D. Kincaid, F. Krogh, "Basic Linear Algebra
     Subprograms for Fortran Usage", `ACM Transactions on Mathematical
     Software', Vol. 5 (1979), Pages 308-325.

     J.J. Dongarra, J. DuCroz, S. Hammarling, R. Hanson, "An Extended
     Set of Fortran Basic Linear Algebra Subprograms", `ACM
     Transactions on Mathematical Software', Vol. 14, No. 1 (1988),
     Pages 1-32.

     J.J. Dongarra, I. Duff, J. DuCroz, S. Hammarling, "A Set of Level
     3 Basic Linear Algebra Subprograms", `ACM Transactions on
     Mathematical Software', Vol. 16 (1990), Pages 1-28.

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.


File: gsl-ref.info,  Node: Linear Algebra,  Next: Eigensystems,  Prev: BLAS Support,  Up: Top

13 Linear Algebra
*****************

This chapter describes functions for solving linear systems.  The
library provides simple linear algebra operations which operate directly
on the `gsl_vector' and `gsl_matrix' objects.  These are intended for
use with "small" systems where simple algorithms are acceptable.

   Anyone interested in large systems will want to use the sophisticated
routines found in LAPACK. The Fortran version of LAPACK is recommended
as the standard package for large-scale linear algebra.  It supports
blocked algorithms, specialized data representations and other
optimizations.

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

* Menu:

* LU Decomposition::
* QR Decomposition::
* QR Decomposition with Column Pivoting::
* Singular Value Decomposition::
* Cholesky Decomposition::
* Tridiagonal Decomposition of Real Symmetric Matrices::
* Tridiagonal Decomposition of Hermitian Matrices::
* Bidiagonalization::
* Householder Transformations::
* Householder solver for linear systems::
* Tridiagonal Systems::
* Linear Algebra Examples::
* Linear Algebra References and Further Reading::


File: gsl-ref.info,  Node: LU Decomposition,  Next: QR Decomposition,  Up: Linear Algebra

13.1 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'.

 -- 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
     LU decomposition of A into (LU,P). On input X should contain the
     right-hand side b, which is replaced by the solution on output.

 -- Function: int gsl_linalg_LU_refine (const gsl_matrix * A, const
          gsl_matrix * LU, const gsl_permutation * P, const gsl_vector
          * B, gsl_vector * X, gsl_vector * RESIDUAL)
 -- Function: int gsl_linalg_complex_LU_refine (const
          gsl_matrix_complex * A, const gsl_matrix_complex * LU, const
          gsl_permutation * P, const gsl_vector_complex * B,
          gsl_vector_complex * X, gsl_vector_complex * RESIDUAL)
     These functions apply an iterative improvement to X, the solution
     of A x = b, using the LU decomposition of A into (LU,P). The
     initial residual r = A x - b is also computed and stored in
     RESIDUAL.

 -- Function: int gsl_linalg_LU_invert (const gsl_matrix * LU, const
          gsl_permutation * P, gsl_matrix * INVERSE)
 -- Function: int gsl_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.


File: gsl-ref.info,  Node: QR Decomposition,  Next: QR Decomposition with Column Pivoting,  Prev: LU Decomposition,  Up: Linear Algebra

13.2 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 into (QR, TAU) given by `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 into (QR,TAU) given by
     `gsl_linalg_QR_decomp'. On input X should contain the right-hand
     side b, which is replaced by the solution on output.

 -- Function: int gsl_linalg_QR_lssolve (const gsl_matrix * QR, const
          gsl_vector * TAU, const gsl_vector * B, gsl_vector * X,
          gsl_vector * RESIDUAL)
     This function finds the least squares solution to the
     overdetermined system A x = b where the matrix A has more rows than
     columns.  The least squares solution minimizes the Euclidean norm
     of the residual, ||Ax - b||.The routine uses the QR decomposition
     of A into (QR, TAU) given by `gsl_linalg_QR_decomp'.  The solution
     is returned in X.  The residual is computed as a by-product and
     stored in RESIDUAL.

 -- Function: int gsl_linalg_QR_QTvec (const gsl_matrix * QR, const
          gsl_vector * TAU, gsl_vector * V)
     This function applies the matrix Q^T encoded in the decomposition
     (QR,TAU) to the vector V, storing the result Q^T v in V.  The
     matrix multiplication is carried out directly using the encoding
     of the Householder vectors without needing to form the full matrix
     Q^T.

 -- Function: int gsl_linalg_QR_Qvec (const gsl_matrix * QR, const
          gsl_vector * TAU, gsl_vector * V)
     This function applies the matrix Q encoded in the decomposition
     (QR,TAU) to the vector V, storing the result Q v in V.  The matrix
     multiplication is carried out directly using the encoding of the
     Householder vectors without needing to form the full matrix Q.

 -- Function: int gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const
          gsl_vector * B, gsl_vector * X)
     This function solves the triangular system R x = b for X. It may
     be useful if the product b' = Q^T b has already been computed
     using `gsl_linalg_QR_QTvec'.

 -- Function: int gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector
          * X)
     This function solves the triangular system R x = b for X in-place.
     On input X should contain the right-hand side b and is replaced by
     the solution on output. This function may be useful if the product
     b' = Q^T b has already been computed using `gsl_linalg_QR_QTvec'.

 -- Function: int gsl_linalg_QR_unpack (const gsl_matrix * QR, const
          gsl_vector * TAU, gsl_matrix * Q, gsl_matrix * R)
     This function unpacks the encoded QR decomposition (QR,TAU) into
     the matrices Q and R, where Q is M-by-M and R is M-by-N.

 -- Function: int gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix *
          R, const gsl_vector * B, gsl_vector * X)
     This function solves the system R x = Q^T b for X. It can be used
     when the QR decomposition of a matrix is available in unpacked
     form as (Q, R).

 -- Function: int gsl_linalg_QR_update (gsl_matrix * Q, gsl_matrix * R,
          gsl_vector * W, const gsl_vector * V)
     This function performs a rank-1 update w v^T of the QR
     decomposition (Q, R). The update is given by Q'R' = Q R + w v^T
     where the output matrices Q' and R' are also orthogonal and right
     triangular. Note that W is destroyed by the update.

 -- Function: int gsl_linalg_R_solve (const gsl_matrix * R, const
          gsl_vector * B, gsl_vector * X)
     This function solves the triangular system R x = b for the N-by-N
     matrix R.

 -- Function: int gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector *
          X)
     This function solves the triangular system R x = b in-place. On
     input X should contain the right-hand side b, which is replaced by
     the solution on output.


File: gsl-ref.info,  Node: QR Decomposition with Column Pivoting,  Next: Singular Value Decomposition,  Prev: QR Decomposition,  Up: Linear Algebra

13.3 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 into (QR, TAU, P) given by
     `gsl_linalg_QRPT_decomp'.

 -- Function: int gsl_linalg_QRPT_svx (const gsl_matrix * QR, const
          gsl_vector * TAU, const gsl_permutation * P, gsl_vector * X)
     This function solves the square system A x = b in-place using the
     QRP^T decomposition of A into (QR,TAU,P). On input X should
     contain the right-hand side b, which is replaced by the solution
     on output.

 -- Function: int gsl_linalg_QRPT_QRsolve (const gsl_matrix * Q, const
          gsl_matrix * R, const gsl_permutation * P, const gsl_vector *
          B, gsl_vector * X)
     This function solves the 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 * U, const
          gsl_vector * V)
     This function performs a rank-1 update w v^T of the QRP^T
     decomposition (Q, R, P). The update is given by Q'R' = Q R + w v^T
     where the output matrices Q' and R' are also orthogonal and right
     triangular. Note that W is destroyed by the update. The
     permutation P is not changed.

 -- Function: int gsl_linalg_QRPT_Rsolve (const gsl_matrix * QR, const
          gsl_permutation * P, const gsl_vector * B, gsl_vector * X)
     This function solves the triangular system R P^T x = b for the
     N-by-N matrix R contained in QR.

 -- Function: int gsl_linalg_QRPT_Rsvx (const gsl_matrix * QR, const
          gsl_permutation * P, gsl_vector * X)
     This function solves the triangular system R P^T x = b in-place
     for the N-by-N matrix R contained in QR. On input X should contain
     the right-hand side b, which is replaced by the solution on output.


File: gsl-ref.info,  Node: Singular Value Decomposition,  Next: Cholesky Decomposition,  Prev: QR Decomposition with Column Pivoting,  Up: Linear Algebra

13.4 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.

 -- 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 (gsl_matrix * U, gsl_matrix * V,
          gsl_vector * S, const gsl_vector * B, gsl_vector * X)
     This function solves the system A x = b using the singular value
     decomposition (U, S, V) of A given by `gsl_linalg_SV_decomp'.

     Only non-zero singular values are used in computing the solution.
     The parts of the solution corresponding to singular values of zero
     are ignored.  Other singular values can be edited out by setting
     them to zero before calling this function.

     In the over-determined case where A has more rows than columns the
     system is solved in the least squares sense, returning the solution
     X which minimizes ||A x - b||_2.


File: gsl-ref.info,  Node: Cholesky Decomposition,  Next: Tridiagonal Decomposition of Real Symmetric Matrices,  Prev: Singular Value Decomposition,  Up: Linear Algebra

13.5 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)
     This function factorizes the positive-definite symmetric square
     matrix A into the Cholesky decomposition A = L L^T. On output the
     diagonal and lower triangular part of the input matrix A contain
     the matrix L. The upper triangular part of the input matrix
     contains L^T, the diagonal terms being identical for both L and
     L^T.  If the matrix is not positive-definite then the
     decomposition will fail, returning the error code `GSL_EDOM'.

 -- Function: int gsl_linalg_cholesky_solve (const gsl_matrix *
          CHOLESKY, const gsl_vector * B, gsl_vector * X)
     This function solves the system A x = b using the Cholesky
     decomposition of A into the matrix CHOLESKY given by
     `gsl_linalg_cholesky_decomp'.

 -- Function: int gsl_linalg_cholesky_svx (const gsl_matrix * CHOLESKY,
          gsl_vector * X)
     This function solves the system A x = b in-place using the
     Cholesky decomposition of A into the matrix CHOLESKY given by
     `gsl_linalg_cholesky_decomp'. On input X should contain the
     right-hand side b, which is replaced by the solution on output.


File: gsl-ref.info,  Node: Tridiagonal Decomposition of Real Symmetric Matrices,  Next: Tridiagonal Decomposition of Hermitian Matrices,  Prev: Cholesky Decomposition,  Up: Linear Algebra

13.6 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.


File: gsl-ref.info,  Node: Tridiagonal Decomposition of Hermitian Matrices,  Next: Bidiagonalization,  Prev: Tridiagonal Decomposition of Real Symmetric Matrices,  Up: Linear Algebra

13.7 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 orthogonal matrix
     Q. This storage scheme is the same as used by LAPACK.  The upper
     triangular part of A and imaginary parts of the diagonal are not
     referenced.

 -- Function: int gsl_linalg_hermtd_unpack (const gsl_matrix_complex *
          A, const gsl_vector_complex * TAU, gsl_matrix_complex * Q,
          gsl_vector * 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.


File: gsl-ref.info,  Node: Bidiagonalization,  Next: Householder Transformations,  Prev: Tridiagonal Decomposition of Hermitian Matrices,  Up: Linear Algebra

13.8 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 given by
     `gsl_linalg_bidiag_decomp', (A, TAU_U, TAU_V) into the separate
     orthogonal matrices U, V and the diagonal vector DIAG and
     superdiagonal SUPERDIAG.  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 given by
     `gsl_linalg_bidiag_decomp', (A, TAU_U, TAU_V) into the separate
     orthogonal matrices U, V and the diagonal vector DIAG and
     superdiagonal SUPERDIAG.  The matrix U is stored in-place in A.

 -- Function: int gsl_linalg_bidiag_unpack_B (const gsl_matrix * A,
          gsl_vector * DIAG, gsl_vector * SUPERDIAG)
     This function unpacks the diagonal and superdiagonal of the
     bidiagonal decomposition of A given by `gsl_linalg_bidiag_decomp',
     into the diagonal vector DIAG and superdiagonal vector SUPERDIAG.


File: gsl-ref.info,  Node: Householder Transformations,  Next: Householder solver for linear systems,  Prev: Bidiagonalization,  Up: Linear Algebra

13.9 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)
     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)
     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)
     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)
     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.


File: gsl-ref.info,  Node: Householder solver for linear systems,  Next: Tridiagonal Systems,  Prev: Householder Transformations,  Up: Linear Algebra

13.10 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.


File: gsl-ref.info,  Node: Tridiagonal Systems,  Next: Linear Algebra Examples,  Prev: Householder solver for linear systems,  Up: Linear Algebra

13.11 Tridiagonal Systems
=========================

 -- 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 )


File: gsl-ref.info,  Node: Linear Algebra Examples,  Next: Linear Algebra References and Further Reading,  Prev: Tridiagonal Systems,  Up: Linear Algebra

13.12 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);
       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.


File: gsl-ref.info,  Node: Linear Algebra References and Further Reading,  Prev: Linear Algebra Examples,  Up: Linear Algebra

13.13 References and Further Reading
====================================

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

     G. H. Golub, C. F. Van Loan, `Matrix Computations' (3rd Ed, 1996),
     Johns Hopkins University Press, ISBN 0-8018-5414-8.

The LAPACK library is described in the following manual,

     `LAPACK Users' Guide' (Third Edition, 1999), Published by SIAM,
     ISBN 0-89871-447-8.

     `http://www.netlib.org/lapack'

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,

     T.F. Chan, "An Improved Algorithm for Computing the Singular Value
     Decomposition", `ACM Transactions on Mathematical Software', 8
     (1982), pp 72-83.

The Jacobi algorithm for singular value decomposition is described in
the following papers,

     J.C. Nash, "A one-sided transformation method for the singular
     value decomposition and algebraic eigenproblem", `Computer
     Journal', Volume 18, Number 1 (1973), p 74-76

     James Demmel, Kresimir Veselic, "Jacobi's Method is more accurate
     than QR", `Lapack Working Note 15' (LAWN-15), October 1989.
     Available from netlib, `http://www.netlib.org/lapack/' in the
     `lawns' or `lawnspdf' directories.


File: gsl-ref.info,  Node: Eigensystems,  Next: Fast Fourier Transforms,  Prev: Linear Algebra,  Up: Top

14 Eigensystems
***************

This chapter describes functions for computing eigenvalues and
eigenvectors of matrices.  There are routines for real symmetric and
complex hermitian matrices, and eigenvalues can be computed with or
without eigenvectors.  The algorithms used are symmetric
bidiagonalization followed by QR reduction.

   These routines are intended for "small" systems where simple
algorithms are acceptable.  Anyone interested in finding eigenvalues
and eigenvectors of large matrices will want to use the sophisticated
routines found in LAPACK. The Fortran version of LAPACK is recommended
as the standard package for large-scale linear algebra.

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

* Menu:

* Real Symmetric Matrices::
* Complex Hermitian Matrices::
* Sorting Eigenvalues and Eigenvectors::
* Eigenvalue and Eigenvector Examples::
* Eigenvalue and Eigenvector References::


File: gsl-ref.info,  Node: Real Symmetric Matrices,  Next: Complex Hermitian Matrices,  Up: Eigensystems

14.1 Real Symmetric Matrices
============================

 -- Function: gsl_eigen_symm_workspace * gsl_eigen_symm_alloc (const
          size_t N)
     This function allocates a workspace for computing eigenvalues of
     N-by-N real symmetric matrices.  The size of the workspace is
     O(2n).

 -- Function: void gsl_eigen_symm_free (gsl_eigen_symm_workspace * W)
     This function frees the memory associated with the workspace W.

 -- Function: int gsl_eigen_symm (gsl_matrix * A, gsl_vector * EVAL,
          gsl_eigen_symm_workspace * W)
     This function computes the eigenvalues of the real symmetric matrix
     A.  Additional workspace of the appropriate size must be provided
     in W.  The diagonal and lower triangular part of A are destroyed
     during the computation, but the strict upper triangular part is
     not referenced.  The eigenvalues are stored in the vector EVAL and
     are unordered.

 -- Function: gsl_eigen_symmv_workspace * gsl_eigen_symmv_alloc (const
          size_t N)
     This function allocates a workspace for computing eigenvalues and
     eigenvectors of N-by-N real symmetric matrices.  The size of the
     workspace is O(4n).

 -- Function: void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * W)
     This function frees the memory associated with the workspace W.

 -- Function: int gsl_eigen_symmv (gsl_matrix * A, gsl_vector * EVAL,
          gsl_matrix * EVEC, gsl_eigen_symmv_workspace * W)
     This function computes the eigenvalues and eigenvectors of the real
     symmetric matrix A.  Additional workspace of the appropriate size
     must be provided in W.  The diagonal and lower triangular part of
     A are destroyed during the computation, but the strict upper
     triangular part is not referenced.  The eigenvalues are stored in
     the vector EVAL and are unordered.  The corresponding eigenvectors
     are stored in the columns of the matrix EVEC.  For example, the
     eigenvector in the first column corresponds to the first
     eigenvalue.  The eigenvectors are guaranteed to be mutually
     orthogonal and normalised to unit magnitude.


File: gsl-ref.info,  Node: Complex Hermitian Matrices,  Next: Sorting Eigenvalues and Eigenvectors,  Prev: Real Symmetric Matrices,  Up: Eigensystems

14.2 Complex Hermitian Matrices
===============================

 -- Function: gsl_eigen_herm_workspace * gsl_eigen_herm_alloc (const
          size_t N)
     This function allocates a workspace for computing eigenvalues of
     N-by-N complex hermitian matrices.  The size of the workspace is
     O(3n).

 -- Function: void gsl_eigen_herm_free (gsl_eigen_herm_workspace * W)
     This function frees the memory associated with the workspace W.

 -- Function: int gsl_eigen_herm (gsl_matrix_complex * A, gsl_vector *
          EVAL, gsl_eigen_herm_workspace * W)
     This function computes the eigenvalues of the complex hermitian
     matrix A.  Additional workspace of the appropriate size must be
     provided in W.  The diagonal and lower triangular part of A are
     destroyed during the computation, but the strict upper triangular
     part is not referenced.  The imaginary parts of the diagonal are
     assumed to be zero and are not referenced. The eigenvalues are
     stored in the vector EVAL and are unordered.

 -- Function: gsl_eigen_hermv_workspace * gsl_eigen_hermv_alloc (const
          size_t N)
     This function allocates a workspace for computing eigenvalues and
     eigenvectors of N-by-N complex hermitian matrices.  The size of
     the workspace is O(5n).

 -- Function: void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * W)
     This function frees the memory associated with the workspace W.

 -- Function: int gsl_eigen_hermv (gsl_matrix_complex * A, gsl_vector *
          EVAL, gsl_matrix_complex * EVEC, gsl_eigen_hermv_workspace *
          W)
     This function computes the eigenvalues and eigenvectors of the
     complex hermitian matrix A.  Additional workspace of the
     appropriate size must be provided in W.  The diagonal and lower
     triangular part of A are destroyed during the computation, but the
     strict upper triangular part is not referenced. The imaginary
     parts of the diagonal are assumed to be zero and are not
     referenced.  The eigenvalues are stored in the vector EVAL and are
     unordered.  The corresponding complex eigenvectors are stored in
     the columns of the matrix EVEC.  For example, the eigenvector in
     the first column corresponds to the first eigenvalue.  The
     eigenvectors are guaranteed to be mutually orthogonal and
     normalised to unit magnitude.


File: gsl-ref.info,  Node: Sorting Eigenvalues and Eigenvectors,  Next: Eigenvalue and Eigenvector Examples,  Prev: Complex Hermitian Matrices,  Up: Eigensystems

14.3 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.


File: gsl-ref.info,  Node: Eigenvalue and Eigenvector Examples,  Next: Eigenvalue and Eigenvector References,  Prev: Sorting Eigenvalues and Eigenvectors,  Up: Eigensystems

14.4 Examples
=============

The following program computes the eigenvalues and eigenvectors of the
4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).

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

     int
     main (void)
     {
       double data[] = { 1.0  , 1/2.0, 1/3.0, 1/4.0,
                         1/2.0, 1/3.0, 1/4.0, 1/5.0,
                         1/3.0, 1/4.0, 1/5.0, 1/6.0,
                         1/4.0, 1/5.0, 1/6.0, 1/7.0 };

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

       gsl_vector *eval = gsl_vector_alloc (4);
       gsl_matrix *evec = gsl_matrix_alloc (4, 4);

       gsl_eigen_symmv_workspace * w =
         gsl_eigen_symmv_alloc (4);

       gsl_eigen_symmv (&m.matrix, eval, evec, w);

       gsl_eigen_symmv_free (w);

       gsl_eigen_symmv_sort (eval, evec,
                             GSL_EIGEN_SORT_ABS_ASC);

       {
         int i;

         for (i = 0; i < 4; i++)
           {
             double eval_i
                = gsl_vector_get (eval, i);
             gsl_vector_view evec_i
                = gsl_matrix_column (evec, i);

             printf ("eigenvalue = %g\n", eval_i);
             printf ("eigenvector = \n");
             gsl_vector_fprintf (stdout,
                                 &evec_i.vector, "%g");
           }
       }

       return 0;
     }

Here is the beginning of the output from the program,

     $ ./a.out
     eigenvalue = 9.67023e-05
     eigenvector =
     -0.0291933
     0.328712
     -0.791411
     0.514553
     ...

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

     octave> [v,d] = eig(hilb(4));
     octave> diag(d)
     ans =

        9.6702e-05
        6.7383e-03
        1.6914e-01
        1.5002e+00

     octave> v
     v =

        0.029193   0.179186  -0.582076   0.792608
       -0.328712  -0.741918   0.370502   0.451923
        0.791411   0.100228   0.509579   0.322416
       -0.514553   0.638283   0.514048   0.252161

Note that the eigenvectors can differ by a change of sign, since the
sign of an eigenvector is arbitrary.


File: gsl-ref.info,  Node: Eigenvalue and Eigenvector References,  Prev: Eigenvalue and Eigenvector Examples,  Up: Eigensystems

14.5 References and Further Reading
===================================

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

     G. H. Golub, C. F. Van Loan, `Matrix Computations' (3rd Ed, 1996),
     Johns Hopkins University Press, ISBN 0-8018-5414-8.

The LAPACK library is described in,

     `LAPACK Users' Guide' (Third Edition, 1999), Published by SIAM,
     ISBN 0-89871-447-8.

     `http://www.netlib.org/lapack'

The LAPACK source code can be found at the website above along with an
online copy of the users guide.


File: gsl-ref.info,  Node: Fast Fourier Transforms,  Next: Numerical Integration,  Prev: Eigensystems,  Up: Top

15 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' (*note FFT References and Further
Reading::)

* Menu:

* Mathematical Definitions::
* Overview of complex data FFTs::
* Radix-2 FFT routines for complex data::
* Mixed-radix FFT routines for complex data::
* Overview of real data FFTs::
* Radix-2 FFT routines for real data::
* Mixed-radix FFT routines for real data::
* FFT References and Further Reading::


File: gsl-ref.info,  Node: Mathematical Definitions,  Next: Overview of complex data FFTs,  Up: Fast Fourier Transforms

15.1 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_n then the DFT can be computed in O(N \sum f_i)
operations.  For a radix-2 FFT this gives an operation count of O(N
\log_2 N).

   All the FFT functions offer three types of transform: forwards,
inverse and backwards, based on the same mathematical definitions.  The
definition of the "forward fourier transform", x = FFT(z), is,

     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.


File: gsl-ref.info,  Node: Overview of complex data FFTs,  Next: Radix-2 FFT routines for complex data,  Prev: Mathematical Definitions,  Up: Fast Fourier Transforms

15.2 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_complex_vector
* 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.


File: gsl-ref.info,  Node: Radix-2 FFT routines for complex data,  Next: Mixed-radix FFT routines for complex data,  Prev: Overview of complex data FFTs,  Up: Fast Fourier Transforms

15.3 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 (-10
... 10), where the negative times wrap around the end of the array.

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

     #define REAL(z,i) ((z)[2*(i)])
     #define IMAG(z,i) ((z)[2*(i)+1])

     int
     main (void)
     {
       int i; double data[2*128];

       for (i = 0; i < 128; i++)
         {
            REAL(data,i) = 0.0; IMAG(data,i) = 0.0;
         }

       REAL(data,0) = 1.0;

       for (i = 1; i <= 10; i++)
         {
            REAL(data,i) = REAL(data,128-i) = 1.0;
         }

       for (i = 0; i < 128; i++)
         {
           printf ("%d %e %e\n", i,
                   REAL(data,i), IMAG(data,i));
         }
       printf ("\n");

       gsl_fft_complex_radix2_forward (data, 1, 128);

       for (i = 0; i < 128; i++)
         {
           printf ("%d %e %e\n", i,
                   REAL(data,i)/sqrt(128),
                   IMAG(data,i)/sqrt(128));
         }

       return 0;
     }

Note that we have assumed that the program is using the default error
handler (which calls `abort' for any errors).  If you are not using a
safe error handler you would need to check the return status of
`gsl_fft_complex_radix2_forward'.

   The transformed data is rescaled by 1/\sqrt N so that it fits on the
same plot as the input.  Only the real part is shown, by the choice of
the input data the imaginary part is zero.  Allowing for the
wrap-around of negative times at t=128, and working in units of k/N,
the DFT approximates the continuum fourier transform, giving a
modulated sine function.


File: gsl-ref.info,  Node: Mixed-radix FFT routines for complex data,  Next: Overview of real data FFTs,  Prev: Radix-2 FFT routines for complex data,  Up: Fast Fourier Transforms

15.4 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.


File: gsl-ref.info,  Node: Overview of real data FFTs,  Next: Radix-2 FFT routines for real data,  Prev: Mixed-radix FFT routines for complex data,  Up: Fast Fourier Transforms

15.5 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).


File: gsl-ref.info,  Node: Radix-2 FFT routines for real data,  Next: Mixed-radix FFT routines for real data,  Prev: Overview of real data FFTs,  Up: Fast Fourier Transforms

15.6 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,

          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]

   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.



File: gsl-ref.info,  Node: Mixed-radix FFT routines for real data,  Next: FFT References and Further Reading,  Prev: Radix-2 FFT routines for real data,  Up: Fast Fourier Transforms

15.7 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].real
                = real_coefficient[i];
              complex_coefficient[i].imag
                = 0.0;
            }

 -- Function: int gsl_fft_halfcomplex_unpack (const double
          HALFCOMPLEX_COEFFICIENT[], gsl_complex_packed_array
          COMPLEX_COEFFICIENT, size_t STRIDE, size_t N)
     This function converts HALFCOMPLEX_COEFFICIENT, an array of
     half-complex coefficients as returned by `gsl_fft_real_transform',
     into an ordinary complex array, COMPLEX_COEFFICIENT.  It fills in
     the complex array using the symmetry z_k = z_{N-k}^* to
     reconstruct the redundant elements.  The algorithm for the
     conversion is,

          complex_coefficient[0].real
            = halfcomplex_coefficient[0];
          complex_coefficient[0].imag
            = 0.0;

          for (i = 1; i < n - i; i++)
            {
              double hc_real
                = halfcomplex_coefficient[2 * i - 1];
              double hc_imag
                = halfcomplex_coefficient[2 * i];
              complex_coefficient[i].real = hc_real;
              complex_coefficient[i].imag = hc_imag;
              complex_coefficient[n - i].real = hc_real;
              complex_coefficient[n - i].imag = -hc_imag;
            }

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

   Here is an example program using `gsl_fft_real_transform' and
`gsl_fft_halfcomplex_inverse'.  It generates a real signal in the shape
of a square pulse.  The pulse is fourier transformed to frequency
space, and all but the lowest ten frequency components are removed from
the array of fourier coefficients returned by `gsl_fft_real_transform'.

   The remaining fourier coefficients are transformed back to the
time-domain, to give a filtered version of the square pulse.  Since
fourier coefficients are stored using the half-complex symmetry both
positive and negative frequencies are removed and the final filtered
signal is also real.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_real.h>
     #include <gsl/gsl_fft_halfcomplex.h>

     int
     main (void)
     {
       int i, n = 100;
       double data[n];

       gsl_fft_real_wavetable * real;
       gsl_fft_halfcomplex_wavetable * hc;
       gsl_fft_real_workspace * work;

       for (i = 0; i < n; i++)
         {
           data[i] = 0.0;
         }

       for (i = n / 3; i < 2 * n / 3; i++)
         {
           data[i] = 1.0;
         }

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e\n", i, data[i]);
         }
       printf ("\n");

       work = gsl_fft_real_workspace_alloc (n);
       real = gsl_fft_real_wavetable_alloc (n);

       gsl_fft_real_transform (data, 1, n,
                               real, work);

       gsl_fft_real_wavetable_free (real);

       for (i = 11; i < n; i++)
         {
           data[i] = 0;
         }

       hc = gsl_fft_halfcomplex_wavetable_alloc (n);

       gsl_fft_halfcomplex_inverse (data, 1, n,
                                    hc, work);
       gsl_fft_halfcomplex_wavetable_free (hc);

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e\n", i, data[i]);
         }

       gsl_fft_real_workspace_free (work);
       return 0;
     }


File: gsl-ref.info,  Node: FFT References and Further Reading,  Prev: Mixed-radix FFT routines for real data,  Up: Fast Fourier Transforms

15.8 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,

     P. Duhamel and M. Vetterli.  Fast fourier transforms: A tutorial
     review and a state of the art.  `Signal Processing', 19:259-299,
     1990.

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,

     E. Oran Brigham.  `The Fast Fourier Transform'.  Prentice Hall,
     1974.

     C. S. Burrus and T. W. Parks.  `DFT/FFT and Convolution
     Algorithms'.  Wiley, 1984.

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,

     Clive Temperton.  Self-sorting mixed-radix fast fourier transforms.
     `Journal of Computational Physics', 52(1):1-23, 1983.

The derivation of FFTs for real-valued data is explained in the
following two articles,

     Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C.
     Sidney Burrus.  Real-valued fast fourier transform algorithms.
     `IEEE Transactions on Acoustics, Speech, and Signal Processing',
     ASSP-35(6):849-863, 1987.

     Clive Temperton.  Fast mixed-radix real fourier transforms.
     `Journal of Computational Physics', 52:340-350, 1983.

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,

     Digital Signal Processing Committee and IEEE Acoustics, Speech,
     and Signal Processing Committee, editors.  `Programs for Digital
     Signal Processing'.  IEEE Press, 1979.

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.

     FFTW Website, `http://www.fftw.org/'

The source code for FFTPACK is available from Netlib,

     FFTPACK, `http://www.netlib.org/fftpack/'


File: gsl-ref.info,  Node: Numerical Integration,  Next: Random Number Generation,  Prev: Fast Fourier Transforms,  Up: Top

16 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,
Doncker-Kapenga, Uberhuber and Kahaner.  Fortran code for QUADPACK is
available on Netlib.

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

* Menu:

* Numerical Integration Introduction::
* QNG non-adaptive Gauss-Kronrod integration::
* QAG adaptive integration::
* QAGS adaptive integration with singularities::
* QAGP adaptive integration with known singular points::
* QAGI adaptive integration on infinite intervals::
* QAWC adaptive integration for Cauchy principal values::
* QAWS adaptive integration for singular functions::
* QAWO adaptive integration for oscillatory functions::
* QAWF adaptive integration for Fourier integrals::
* Numerical integration error codes::
* Numerical integration examples::
* Numerical integration References and Further Reading::


File: gsl-ref.info,  Node: Numerical Integration Introduction,  Next: QNG non-adaptive Gauss-Kronrod integration,  Up: Numerical Integration

16.1 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|)

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.

* Menu:

* Integrands without weight functions::
* Integrands with weight functions::
* Integrands with singular weight functions::


File: gsl-ref.info,  Node: Integrands without weight functions,  Next: Integrands with weight functions,  Up: Numerical Integration Introduction

16.1.1 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.


File: gsl-ref.info,  Node: Integrands with weight functions,  Next: Integrands with singular weight functions,  Prev: Integrands without weight functions,  Up: Numerical Integration Introduction

16.1.2 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.


File: gsl-ref.info,  Node: Integrands with singular weight functions,  Prev: Integrands with weight functions,  Up: Numerical Integration Introduction

16.1.3 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.


File: gsl-ref.info,  Node: QNG non-adaptive Gauss-Kronrod integration,  Next: QAG adaptive integration,  Prev: Numerical Integration Introduction,  Up: Numerical Integration

16.2 QNG non-adaptive Gauss-Kronrod integration
===============================================

The QNG algorithm is a non-adaptive procedure which uses fixed
Gauss-Kronrod 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.


File: gsl-ref.info,  Node: QAG adaptive integration,  Next: QAGS adaptive integration with singularities,  Prev: QNG non-adaptive Gauss-Kronrod integration,  Up: Numerical Integration

16.3 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.

 -- 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.


File: gsl-ref.info,  Node: QAGS adaptive integration with singularities,  Next: QAGP adaptive integration with known singular points,  Prev: QAG adaptive integration,  Up: Numerical Integration

16.4 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.



File: gsl-ref.info,  Node: QAGP adaptive integration with known singular points,  Next: QAGI adaptive integration on infinite intervals,  Prev: QAGS adaptive integration with singularities,  Up: Numerical Integration

16.5 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'.



File: gsl-ref.info,  Node: QAGI adaptive integration on infinite intervals,  Next: QAWC adaptive integration for Cauchy principal values,  Prev: QAGP adaptive integration with known singular points,  Up: Numerical Integration

16.6 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.


File: gsl-ref.info,  Node: QAWC adaptive integration for Cauchy principal values,  Next: QAWS adaptive integration for singular functions,  Prev: QAGI adaptive integration on infinite intervals,  Up: Numerical Integration

16.7 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.



File: gsl-ref.info,  Node: QAWS adaptive integration for singular functions,  Next: QAWO adaptive integration for oscillatory functions,  Prev: QAWC adaptive integration for Cauchy principal values,  Up: Numerical Integration

16.8 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 and associated workspace 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
     `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.



File: gsl-ref.info,  Node: QAWO adaptive integration for oscillatory functions,  Next: QAWF adaptive integration for Fourier integrals,  Prev: QAWS adaptive integration for singular functions,  Up: Numerical Integration

16.9 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.



File: gsl-ref.info,  Node: QAWF adaptive integration for Fourier integrals,  Next: Numerical integration error codes,  Prev: QAWO adaptive integration for oscillatory functions,  Up: Numerical Integration

16.10 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.



File: gsl-ref.info,  Node: Numerical integration error codes,  Next: Numerical integration examples,  Prev: QAWF adaptive integration for Fourier integrals,  Up: Numerical Integration

16.11 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.


File: gsl-ref.info,  Node: Numerical integration examples,  Next: Numerical integration References and Further Reading,  Prev: Numerical integration error codes,  Up: Numerical Integration

16.12 Examples
==============

The integrator `QAGS' will handle a large class of definite integrals.
For example, consider the following integral, which has a
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);

       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.


File: gsl-ref.info,  Node: Numerical integration References and Further Reading,  Prev: Numerical integration examples,  Up: Numerical Integration

16.13 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.

     R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, D.K. Kahaner.
     `QUADPACK A subroutine package for automatic integration' Springer
     Verlag, 1983.



File: gsl-ref.info,  Node: Random Number Generation,  Next: Quasi-Random Sequences,  Prev: Numerical Integration,  Up: Top

17 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'.

* Menu:

* General comments on random numbers::
* The Random Number Generator Interface::
* Random number generator initialization::
* Sampling from a random number generator::
* Auxiliary random number generator functions::
* Random number environment variables::
* Copying random number generator state::
* Reading and writing random number generator state::
* Random number generator algorithms::
* Unix random number generators::
* Other random number generators::
* Random Number Generator Performance::
* Random Number Generator Examples::
* Random Number References and Further Reading::
* Random Number Acknowledgements::


File: gsl-ref.info,  Node: General comments on random numbers,  Next: The Random Number Generator Interface,  Up: Random Number Generation

17.1 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".


File: gsl-ref.info,  Node: The Random Number Generator Interface,  Next: Random number generator initialization,  Prev: General comments on random numbers,  Up: Random Number Generation

17.2 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'.


File: gsl-ref.info,  Node: Random number generator initialization,  Next: Sampling from a random number generator,  Prev: The Random Number Generator Interface,  Up: Random Number Generation

17.3 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'
     (*note 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 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'.

 -- Function: void gsl_rng_free (gsl_rng * R)
     This function frees all the memory associated with the generator R.


File: gsl-ref.info,  Node: Sampling from a random number generator,  Next: Auxiliary random number generator functions,  Prev: Random number generator initialization,  Up: Random Number Generation

17.4 Sampling from a random number generator
============================================

The following functions return uniformly distributed random numbers,
either as integers or double precision floating point numbers.  To
obtain non-uniform distributions *note 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 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 mimimum 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.


File: gsl-ref.info,  Node: Auxiliary random number generator functions,  Next: Random number environment variables,  Prev: Sampling from a random number generator,  Up: Random Number Generation

17.5 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);
            }


File: gsl-ref.info,  Node: Random number environment variables,  Next: Copying random number generator state,  Prev: Auxiliary random number generator functions,  Up: Random Number Generation

17.6 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));
       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


File: gsl-ref.info,  Node: Copying random number generator state,  Next: Reading and writing random number generator state,  Prev: Random number environment variables,  Up: Random Number Generation

17.7 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.


File: gsl-ref.info,  Node: Reading and writing random number generator state,  Next: Random number generator algorithms,  Prev: Copying random number generator state,  Up: Random Number Generation

17.8 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 or formatted text.

 -- 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.


File: gsl-ref.info,  Node: Random number generator algorithms,  Next: Unix random number generators,  Prev: Reading and writing random number generator state,  Up: Random Number Generation

17.9 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.

 -- Generator: gsl_rng_mt19937
     The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
     variant of the twisted generalized feedback shift-register
     algorithm, and is known as the "Mersenne Twister" generator.  It
     has a Mersenne prime period of 2^19937 - 1 (about 10^6000) and is
     equi-distributed in 623 dimensions.  It has passed the DIEHARD
     statistical tests.  It uses 624 words of state per generator and is
     comparable in speed to the other generators.  The original
     generator used a default seed of 4357 and choosing S equal to zero
     in `gsl_rng_set' reproduces this.

     For more information see,
          Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
          623-dimensionally equidistributed uniform pseudorandom number
          generator". `ACM Transactions on Modeling and Computer
          Simulation', Vol. 8, No. 1 (Jan. 1998), Pages 3-30

     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 Lu"scher, which produces "luxury random
     numbers".  This generator provides single precision output (24
     bits) at three luxury levels `ranlxs0', `ranlxs1' and `ranlxs2'.
     It uses double-precision floating point arithmetic internally and
     can be significantly faster than the integer version of `ranlux',
     particularly on 64-bit architectures.  The period of the generator
     is about 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.

 -- Generator: gsl_rng_ranlxd1
 -- Generator: gsl_rng_ranlxd2
     These generators produce double precision output (48 bits) from the
     RANLXS generator.  The library provides two luxury levels
     `ranlxd1' and `ranlxd2'.

 -- Generator: gsl_rng_ranlux
 -- Generator: gsl_rng_ranlux389
     The `ranlux' generator is an implementation of the original
     algorithm developed by Lu"scher.  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 Lu"scher 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,
          M. Lu"scher, "A portable high-quality random number generator
          for lattice field theory calculations", `Computer Physics
          Communications', 79 (1994) 100-110.

          F. James, "RANLUX: A Fortran implementation of the
          high-quality pseudo-random number generator of Lu"scher",
          `Computer Physics Communications', 79 (1994) 111-114

 -- 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,

          P. L'Ecuyer, "Combined Multiple Recursive Random Number
          Generators", `Operations Research', 44, 5 (1996), 816-822.

 -- 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,
          P. L'Ecuyer, F. Blouin, and R. Coutre, "A search for good
          multiple recursive random number generators", `ACM
          Transactions on Modeling and Computer Simulation' 3, 87-98
          (1993).

 -- 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,

          P. L'Ecuyer, "Maximally Equidistributed Combined Tausworthe
          Generators", `Mathematics of Computation', 65, 213 (1996),
          203-213.

     The generator `gsl_rng_taus2' uses the same algorithm as
     `gsl_rng_taus' but with an improved seeding procedure described in
     the paper,

          P. L'Ecuyer, "Tables of Maximally Equidistributed Combined
          LFSR Generators", `Mathematics of Computation', 68, 225
          (1999), 261-269

     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,
          Robert M. Ziff, "Four-tap shift-register-sequence
          random-number generators", `Computers in Physics', 12(4),
          Jul/Aug 1998, pp 385-392.


File: gsl-ref.info,  Node: Unix random number generators,  Next: Other random number generators,  Prev: Random number generator algorithms,  Up: Random Number Generation

17.10 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).


File: gsl-ref.info,  Node: Other random number generators,  Next: Random Number Generator Performance,  Prev: Unix random number generators,  Up: Random Number Generation

17.11 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,
          S. Kirkpatrick and E. Stoll, "A very fast shift-register
          sequence random number generator", `Journal of Computational
          Physics', 40, 517-526 (1981)

 -- 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,
          Makoto Matsumoto and Yoshiharu Kurita, "Twisted GFSR
          Generators II", `ACM Transactions on Modelling and Computer
          Simulation', Vol. 4, No. 3, 1994, pages 254-266.

 -- 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 is used in the IMSL Library (subroutine RNUN) and in
     MATLAB (the RAND function).  It is also sometimes known by the
     acronym "GGL" (I'm not sure what that stands for).

     For more information see,
          Park and Miller, "Random Number Generators: Good ones are
          hard to find", `Communications of the ACM', October 1988,
          Volume 31, No 10, pages 1192-1201.

 -- 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,
          W. Petersen, "Lagged Fibonacci Random Number Generators for
          the NEC SX-3", `International Journal of High Speed
          Computing' (1994).

 -- Generator: gsl_rng_borosh13
     This is the Borosh-Niederreiter random number generator. It is
     taken from Knuth's `Seminumerical Algorithms', 3rd Ed., pages
     106-108. Its sequence is,

          x_{n+1} = (a x_n) mod m

     with a = 1812433253 and m = 2^32.  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.

 -- Generator: gsl_rng_fishman18
     This is the Fishman, Moore III random number generator. It is
     taken from Knuth's `Seminumerical Algorithms', 3rd Ed., pages
     106-108. Its sequence is,

          x_{n+1} = (a x_n) mod m

     with a = 62089911 and m = 2^31 - 1.  The seed specifies the
     initial value, x_1.

 -- Generator: gsl_rng_fishman20
     This is the Fishman random number generator. It is taken from
     Knuth's `Seminumerical Algorithms', 3rd Ed., page 108. Its
     sequence is,

          x_{n+1} = (a x_n) mod m

     with a = 48271 and m = 2^31 - 1.  The seed specifies the initial
     value, x_1.

 -- 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_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_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.

 -- Generator: gsl_rng_lecuyer21
     This is the L'Ecuyer random number generator. It is taken from
     Knuth's `Seminumerical Algorithms', 3rd Ed., page 106-108. Its
     sequence is,

          x_{n+1} = (a x_n) mod m

     with a = 40692 and m = 2^31 - 249.  The seed specifies the initial
     value, x_1.

 -- Generator: gsl_rng_waterman14
     This is the Waterman random number generator. It is taken from
     Knuth's `Seminumerical Algorithms', 3rd Ed., page 106-108. Its
     sequence is,

          x_{n+1} = (a x_n) mod m

     with a = 1566083941 and m = 2^32.  The seed specifies the initial
     value, x_1.


File: gsl-ref.info,  Node: Random Number Generator Performance,  Next: Random Number Generator Examples,  Prev: Other random number generators,  Up: Random Number Generation

17.12 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

     1852 k ints/sec,    935 k doubles/sec, ran3
      813 k ints/sec,    575 k doubles/sec, ran0
      787 k ints/sec,    476 k doubles/sec, ran1
      379 k ints/sec,    292 k doubles/sec, ran2


File: gsl-ref.info,  Node: Random Number Generator Examples,  Next: Random Number References and Further Reading,  Prev: Random Number Generator Performance,  Up: Random Number Generation

17.13 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


File: gsl-ref.info,  Node: Random Number References and Further Reading,  Next: Random Number Acknowledgements,  Prev: Random Number Generator Examples,  Up: Random Number Generation

17.14 References and Further Reading
====================================

The subject of random number generation and testing is reviewed
extensively in Knuth's `Seminumerical Algorithms'.

     Donald E. Knuth, `The Art of Computer Programming: Seminumerical
     Algorithms' (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.

Further information is available in the review paper written by Pierre
L'Ecuyer,

     P. L'Ecuyer, "Random Number Generation", Chapter 4 of the Handbook
     on Simulation, Jerry Banks Ed., Wiley, 1998, 93-137.

     `http://www.iro.umontreal.ca/~lecuyer/papers.html' in the file
     `handsim.ps'.

The source code for the DIEHARD random number generator tests is also
available online,

     `DIEHARD source code' G. Marsaglia,

     `http://stat.fsu.edu/pub/diehard/'

A comprehensive set of random number generator tests is available from
NIST,

     NIST Special Publication 800-22, "A Statistical Test Suite for the
     Validation of Random Number Generators and Pseudo Random Number
     Generators for Cryptographic Applications".

     `http://csrc.nist.gov/rng/'


File: gsl-ref.info,  Node: Random Number Acknowledgements,  Prev: Random Number References and Further Reading,  Up: Random Number Generation

17.15 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
Lu"scher for providing notes and source code for the RANLXS and RANLXD
generators.


File: gsl-ref.info,  Node: Quasi-Random Sequences,  Next: Random Number Distributions,  Prev: Random Number Generation,  Up: Top

18 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'.

* Menu:

* Quasi-random number generator initialization::
* Sampling from a quasi-random number generator::
* Auxiliary quasi-random number generator functions::
* Saving and resorting quasi-random number generator state::
* Quasi-random number generator algorithms::
* Quasi-random number generator examples::
* Quasi-random number references::


File: gsl-ref.info,  Node: Quasi-random number generator initialization,  Next: Sampling from a quasi-random number generator,  Up: Quasi-Random Sequences

18.1 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.


File: gsl-ref.info,  Node: Sampling from a quasi-random number generator,  Next: Auxiliary quasi-random number generator functions,  Prev: Quasi-random number generator initialization,  Up: Quasi-Random Sequences

18.2 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.


File: gsl-ref.info,  Node: Auxiliary quasi-random number generator functions,  Next: Saving and resorting quasi-random number generator state,  Prev: Sampling from a quasi-random number generator,  Up: Quasi-Random Sequences

18.3 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);


File: gsl-ref.info,  Node: Saving and resorting quasi-random number generator state,  Next: Quasi-random number generator algorithms,  Prev: Auxiliary quasi-random number generator functions,  Up: Quasi-Random Sequences

18.4 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.


File: gsl-ref.info,  Node: Quasi-random number generator algorithms,  Next: Quasi-random number generator examples,  Prev: Saving and resorting quasi-random number generator state,  Up: Quasi-Random Sequences

18.5 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.


File: gsl-ref.info,  Node: Quasi-random number generator examples,  Next: Quasi-random number references,  Prev: Quasi-random number generator algorithms,  Up: Quasi-Random Sequences

18.6 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.


File: gsl-ref.info,  Node: Quasi-random number references,  Prev: Quasi-random number generator examples,  Up: Quasi-Random Sequences

18.7 References
===============

The implementations of the quasi-random sequence routines are based on
the algorithms described in the following paper,

     P. Bratley and B.L. Fox and H. Niederreiter, "Algorithm 738:
     Programs to Generate Niederreiter's Low-discrepancy Sequences",
     `ACM Transactions on Mathematical Software', Vol. 20, No. 4,
     December, 1994, p. 494-495.


File: gsl-ref.info,  Node: Random Number Distributions,  Next: Statistics,  Prev: Quasi-Random Sequences,  Up: Top

19 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.

* Menu:

* Random Number Distribution Introduction::
* The Gaussian Distribution::
* The Gaussian Tail Distribution::
* The Bivariate Gaussian Distribution::
* The Exponential Distribution::
* The Laplace Distribution::
* The Exponential Power Distribution::
* The Cauchy Distribution::
* The Rayleigh Distribution::
* The Rayleigh Tail Distribution::
* The Landau Distribution::
* The Levy alpha-Stable Distributions::
* The Levy skew alpha-Stable Distribution::
* The Gamma Distribution::
* The Flat (Uniform) Distribution::
* The Lognormal Distribution::
* The Chi-squared Distribution::
* The F-distribution::
* The t-distribution::
* The Beta Distribution::
* The Logistic Distribution::
* The Pareto Distribution::
* Spherical Vector Distributions::
* The Weibull Distribution::
* The Type-1 Gumbel Distribution::
* The Type-2 Gumbel Distribution::
* The Dirichlet Distribution::
* General Discrete Distributions::
* The Poisson Distribution::
* The Bernoulli Distribution::
* The Binomial Distribution::
* The Multinomial Distribution::
* The Negative Binomial Distribution::
* The Pascal Distribution::
* The Geometric Distribution::
* The Hypergeometric Distribution::
* The Logarithmic Distribution::
* Shuffling and Sampling::
* Random Number Distribution Examples::
* Random Number Distribution References and Further Reading::


File: gsl-ref.info,  Node: Random Number Distribution Introduction,  Next: The Gaussian Distribution,  Up: Random Number Distributions

19.1 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).


File: gsl-ref.info,  Node: The Gaussian Distribution,  Next: The Gaussian Tail Distribution,  Prev: Random Number Distribution Introduction,  Up: Random Number Distributions

19.2 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-Mueller 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.


File: gsl-ref.info,  Node: The Gaussian Tail Distribution,  Next: The Bivariate Gaussian Distribution,  Prev: The Gaussian Distribution,  Up: Random Number Distributions

19.3 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.


File: gsl-ref.info,  Node: The Bivariate Gaussian Distribution,  Next: The Exponential Distribution,  Prev: The Gaussian Tail Distribution,  Up: Random Number Distributions

19.4 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.



File: gsl-ref.info,  Node: The Exponential Distribution,  Next: The Laplace Distribution,  Prev: The Bivariate Gaussian Distribution,  Up: Random Number Distributions

19.5 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.


File: gsl-ref.info,  Node: The Laplace Distribution,  Next: The Exponential Power Distribution,  Prev: The Exponential Distribution,  Up: Random Number Distributions

19.6 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.


File: gsl-ref.info,  Node: The Exponential Power Distribution,  Next: The Cauchy Distribution,  Prev: The Laplace Distribution,  Up: Random Number Distributions

19.7 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.


File: gsl-ref.info,  Node: The Cauchy Distribution,  Next: The Rayleigh Distribution,  Prev: The Exponential Power Distribution,  Up: Random Number Distributions

19.8 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.


File: gsl-ref.info,  Node: The Rayleigh Distribution,  Next: The Rayleigh Tail Distribution,  Prev: The Cauchy Distribution,  Up: Random Number Distributions

19.9 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.


File: gsl-ref.info,  Node: The Rayleigh Tail Distribution,  Next: The Landau Distribution,  Prev: The Rayleigh Distribution,  Up: Random Number Distributions

19.10 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.



File: gsl-ref.info,  Node: The Landau Distribution,  Next: The Levy alpha-Stable Distributions,  Prev: The Rayleigh Tail Distribution,  Up: Random Number Distributions

19.11 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.



File: gsl-ref.info,  Node: The Levy alpha-Stable Distributions,  Next: The Levy skew alpha-Stable Distribution,  Prev: The Landau Distribution,  Up: Random Number Distributions

19.12 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.



File: gsl-ref.info,  Node: The Levy skew alpha-Stable Distribution,  Next: The Gamma Distribution,  Prev: The Levy alpha-Stable Distributions,  Up: Random Number Distributions

19.13 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).



File: gsl-ref.info,  Node: The Gamma Distribution,  Next: The Flat (Uniform) Distribution,  Prev: The Levy skew alpha-Stable Distribution,  Up: Random Number Distributions

19.14 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
     algorithms from Knuth (vol 2).

 -- Function: double gsl_ran_gamma_mt (const gsl_rng * R, double A,
          double B)
     This function returns a gamma variate using the Marsaglia-Tsang
     fast gamma method.

 -- 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.


File: gsl-ref.info,  Node: The Flat (Uniform) Distribution,  Next: The Lognormal Distribution,  Prev: The Gamma Distribution,  Up: Random Number Distributions

19.15 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.


File: gsl-ref.info,  Node: The Lognormal Distribution,  Next: The Chi-squared Distribution,  Prev: The Flat (Uniform) Distribution,  Up: Random Number Distributions

19.16 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.


File: gsl-ref.info,  Node: The Chi-squared Distribution,  Next: The F-distribution,  Prev: The Lognormal Distribution,  Up: Random Number Distributions

19.17 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.


File: gsl-ref.info,  Node: The F-distribution,  Next: The t-distribution,  Prev: The Chi-squared Distribution,  Up: Random Number Distributions

19.18 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.


File: gsl-ref.info,  Node: The t-distribution,  Next: The Beta Distribution,  Prev: The F-distribution,  Up: Random Number Distributions

19.19 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.


File: gsl-ref.info,  Node: The Beta Distribution,  Next: The Logistic Distribution,  Prev: The t-distribution,  Up: Random Number Distributions

19.20 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.


File: gsl-ref.info,  Node: The Logistic Distribution,  Next: The Pareto Distribution,  Prev: The Beta Distribution,  Up: Random Number Distributions

19.21 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.


File: gsl-ref.info,  Node: The Pareto Distribution,  Next: Spherical Vector Distributions,  Prev: The Logistic Distribution,  Up: Random Number Distributions

19.22 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.


File: gsl-ref.info,  Node: Spherical Vector Distributions,  Next: The Weibull Distribution,  Prev: The Pareto Distribution,  Up: Random Number Distributions

19.23 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).


File: gsl-ref.info,  Node: The Weibull Distribution,  Next: The Type-1 Gumbel Distribution,  Prev: Spherical Vector Distributions,  Up: Random Number Distributions

19.24 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.


File: gsl-ref.info,  Node: The Type-1 Gumbel Distribution,  Next: The Type-2 Gumbel Distribution,  Prev: The Weibull Distribution,  Up: Random Number Distributions

19.25 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.


File: gsl-ref.info,  Node: The Type-2 Gumbel Distribution,  Next: The Dirichlet Distribution,  Prev: The Type-1 Gumbel Distribution,  Up: Random Number Distributions

19.26 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.


File: gsl-ref.info,  Node: The Dirichlet Distribution,  Next: General Discrete Distributions,  Prev: The Type-2 Gumbel Distribution,  Up: Random Number Distributions

19.27 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].


File: gsl-ref.info,  Node: General Discrete Distributions,  Next: The Poisson Distribution,  Prev: The Dirichlet Distribution,  Up: Random Number Distributions

19.28 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 numbers in a computer, Comm ACM 6, 37-38 (1963)) is very clever,
and readers interested in examples of good algorithm design are
directed to this short and well-written paper.  Unfortunately, for
large K, Marsaglia's lookup table can be quite large.

   A much better approach is due to Alastair J. Walker (An efficient
method for generating discrete random variables with general
distributions, ACM Trans on Mathematical Software 3, 253-256 (1977);
see also Knuth, v2, 3rd ed, p120-121,139).  This requires two lookup
tables, one floating point and one integer, but both only of size K.
After preprocessing, the random numbers are generated in O(1) time,
even for large K.  The preprocessing suggested by Walker requires
O(K^2) effort, but that is not actually necessary, and the
implementation provided here only takes O(K) effort.  In general, more
preprocessing leads to faster generation of the individual random
numbers, but a diminishing return is reached pretty early.  Knuth points
out that the optimal preprocessing is combinatorially difficult for
large K.

   This method can be used to speed up some of the discrete random
number generators below, such as the binomial distribution.  To use 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.


File: gsl-ref.info,  Node: The Poisson Distribution,  Next: The Bernoulli Distribution,  Prev: General Discrete Distributions,  Up: Random Number Distributions

19.29 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.


File: gsl-ref.info,  Node: The Bernoulli Distribution,  Next: The Binomial Distribution,  Prev: The Poisson Distribution,  Up: Random Number Distributions

19.30 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.



File: gsl-ref.info,  Node: The Binomial Distribution,  Next: The Multinomial Distribution,  Prev: The Bernoulli Distribution,  Up: Random Number Distributions

19.31 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.


File: gsl-ref.info,  Node: The Multinomial Distribution,  Next: The Negative Binomial Distribution,  Prev: The Binomial Distribution,  Up: Random Number Distributions

19.32 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 returns an array of K random variates from a
     multinomial distribution. The distribution function 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.

     Random variates are generated using the conditional binomial
     method (see C.S. David, `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].


File: gsl-ref.info,  Node: The Negative Binomial Distribution,  Next: The Pascal Distribution,  Prev: The Multinomial Distribution,  Up: Random Number Distributions

19.33 The Negative Binomial Distribution
========================================

 -- Function: unsigned int gsl_ran_negative_binomial (const gsl_rng *
          R, double P, double N)
     This function returns a random integer from the negative binomial
     distribution, the number of failures occurring before N successes
     in independent trials with probability P of success.  The
     probability distribution for negative binomial variates is,

          p(k) = {\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) } p^n (1-p)^k

     Note that n is not required to be an integer.

 -- Function: double gsl_ran_negative_binomial_pdf (unsigned int K,
          double P, double N)
     This function computes the probability p(k) of obtaining K from a
     negative binomial distribution with parameters P and N, using the
     formula given above.


 -- Function: double gsl_cdf_negative_binomial_P (unsigned int K,
          double P, double N)
 -- Function: double gsl_cdf_negative_binomial_Q (unsigned int K,
          double P, double N)
     These functions compute the cumulative distribution functions
     P(k), Q(k) for the negative binomial distribution with parameters
     P and N.


File: gsl-ref.info,  Node: The Pascal Distribution,  Next: The Geometric Distribution,  Prev: The Negative Binomial Distribution,  Up: Random Number Distributions

19.34 The Pascal Distribution
=============================

 -- Function: unsigned int gsl_ran_pascal (const gsl_rng * R, double P,
          unsigned int N)
     This function returns a random integer from the Pascal
     distribution.  The Pascal distribution is simply a negative
     binomial distribution with an integer value of n.

          p(k) = {(n + k - 1)! \over k! (n - 1)! } p^n (1-p)^k

     for k >= 0

 -- Function: double gsl_ran_pascal_pdf (unsigned int K, double P,
          unsigned int N)
     This function computes the probability p(k) of obtaining K from a
     Pascal distribution with parameters P and N, using the formula
     given above.


 -- Function: double gsl_cdf_pascal_P (unsigned int K, double P,
          unsigned int N)
 -- Function: double gsl_cdf_pascal_Q (unsigned int K, double P,
          unsigned int N)
     These functions compute the cumulative distribution functions
     P(k), Q(k) for the Pascal distribution with parameters P and N.


File: gsl-ref.info,  Node: The Geometric Distribution,  Next: The Hypergeometric Distribution,  Prev: The Pascal Distribution,  Up: Random Number Distributions

19.35 The Geometric Distribution
================================

 -- Function: unsigned int gsl_ran_geometric (const gsl_rng * R, double
          P)
     This function returns a random integer from the geometric
     distribution, the number of independent trials with probability P
     until the first success.  The probability distribution for
     geometric variates is,

          p(k) =  p (1-p)^(k-1)

     for k >= 1.  Note that the distribution begins with k=1 with this
     definition.  There is another convention in which the exponent k-1
     is replaced by k.

 -- Function: double gsl_ran_geometric_pdf (unsigned int K, double P)
     This function computes the probability p(k) of obtaining K from a
     geometric distribution with probability parameter P, using the
     formula given above.


 -- Function: double gsl_cdf_geometric_P (unsigned int K, double P)
 -- Function: double gsl_cdf_geometric_Q (unsigned int K, double P)
     These functions compute the cumulative distribution functions
     P(k), Q(k) for the geometric distribution with parameter P.


File: gsl-ref.info,  Node: The Hypergeometric Distribution,  Next: The Logarithmic Distribution,  Prev: The Geometric Distribution,  Up: Random Number Distributions

19.36 The Hypergeometric Distribution
=====================================

 -- Function: unsigned int gsl_ran_hypergeometric (const gsl_rng * R,
          unsigned int N1, unsigned int N2, unsigned int T)
     This function returns a random integer from the hypergeometric
     distribution.  The probability distribution for hypergeometric
     random variates is,

          p(k) =  C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)

     where C(a,b) = a!/(b!(a-b)!) and t <= n_1 + n_2.  The domain of k
     is max(0,t-n_2), ..., min(t,n_1).

     If a population contains n_1 elements of "type 1" and n_2 elements
     of "type 2" then the hypergeometric distribution gives the
     probability of obtaining k elements of "type 1" in t samples from
     the population without replacement.

 -- Function: double gsl_ran_hypergeometric_pdf (unsigned int K,
          unsigned int N1, unsigned int N2, unsigned int T)
     This function computes the probability p(k) of obtaining K from a
     hypergeometric distribution with parameters N1, N2, T, using the
     formula given above.


 -- Function: double gsl_cdf_hypergeometric_P (unsigned int K, unsigned
          int N1, unsigned int N2, unsigned int T)
 -- Function: double gsl_cdf_hypergeometric_Q (unsigned int K, unsigned
          int N1, unsigned int N2, unsigned int T)
     These functions compute the cumulative distribution functions
     P(k), Q(k) for the hypergeometric distribution with parameters N1,
     N2 and T.


File: gsl-ref.info,  Node: The Logarithmic Distribution,  Next: Shuffling and Sampling,  Prev: The Hypergeometric Distribution,  Up: Random Number Distributions

19.37 The Logarithmic Distribution
==================================

 -- Function: unsigned int gsl_ran_logarithmic (const gsl_rng * R,
          double P)
     This function returns a random integer from the logarithmic
     distribution.  The probability distribution for logarithmic random
     variates is,

          p(k) = {-1 \over \log(1-p)} {(p^k \over k)}

     for k >= 1.

 -- Function: double gsl_ran_logarithmic_pdf (unsigned int K, double P)
     This function computes the probability p(k) of obtaining K from a
     logarithmic distribution with probability parameter P, using the
     formula given above.



File: gsl-ref.info,  Node: Shuffling and Sampling,  Next: Random Number Distribution Examples,  Prev: The Logarithmic Distribution,  Up: Random Number Distributions

19.38 Shuffling and Sampling
============================

The following functions allow the shuffling and sampling of a set of
objects.  The algorithms rely on a random number generator as a source
of randomness and a poor quality generator can lead to correlations in
the output.  In particular it is important to avoid generators with a
short period.  For more information see Knuth, v2, 3rd ed, Section
3.4.2, "Random Sampling and Shuffling".

 -- Function: void gsl_ran_shuffle (const gsl_rng * R, void * BASE,
          size_t N, size_t SIZE)
     This function randomly shuffles the order of N objects, each of
     size SIZE, stored in the array BASE[0..N-1].  The output of the
     random number generator R is used to produce the permutation.  The
     algorithm generates all possible n!  permutations with equal
     probability, assuming a perfect source of random numbers.

     The following code shows how to shuffle the numbers from 0 to 51,

          int a[52];

          for (i = 0; i < 52; i++)
            {
              a[i] = i;
            }

          gsl_ran_shuffle (r, a, 52, sizeof (int));


 -- Function: int gsl_ran_choose (const gsl_rng * R, void * DEST,
          size_t K, void * SRC, size_t N, size_t SIZE)
     This function fills the array DEST[k] with K objects taken
     randomly from the N elements of the array SRC[0..N-1].  The
     objects are each of size SIZE.  The output of the random number
     generator R is used to make the selection.  The algorithm ensures
     all possible samples are equally likely, assuming a perfect source
     of randomness.

     The objects are sampled _without_ replacement, thus each object can
     only appear once in DEST[k].  It is required that K be less than
     or equal to `n'.  The objects in DEST will be in the same relative
     order as those in SRC.  You will need to call `gsl_ran_shuffle(r,
     dest, n, size)' if you want to randomize the order.

     The following code shows how to select a random sample of three
     unique numbers from the set 0 to 99,

          double a[3], b[100];

          for (i = 0; i < 100; i++)
            {
              b[i] = (double) i;
            }

          gsl_ran_choose (r, a, 3, b, 100, sizeof (double));


 -- Function: void gsl_ran_sample (const gsl_rng * R, void * DEST,
          size_t K, void * SRC, size_t N, size_t SIZE)
     This function is like `gsl_ran_choose' but samples K items from
     the original array of N items SRC with replacement, so the same
     object can appear more than once in the output sequence DEST.
     There is no requirement that K be less than N in this case.


File: gsl-ref.info,  Node: Random Number Distribution Examples,  Next: Random Number Distribution References and Further Reading,  Prev: Shuffling and Sampling,  Up: Random Number Distributions

19.39 Examples
==============

The following program demonstrates the use of a random number generator
to produce variates from a distribution.  It prints 10 samples from the
Poisson distribution with a mean of 3.

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

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       int i, n = 10;
       double mu = 3.0;

       /* create a generator chosen by the
          environment variable GSL_RNG_TYPE */

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       /* print n random variates chosen from
          the poisson distribution with mean
          parameter mu */

       for (i = 0; i < n; i++)
         {
           unsigned int k = gsl_ran_poisson (r, mu);
           printf (" %u", k);
         }

       printf ("\n");
       return 0;
     }

If the library and header files are installed under `/usr/local' (the
default location) then the program can be compiled with these options,

     $ gcc -Wall demo.c -lgsl -lgslcblas -lm

Here is the output of the program,

     $ ./a.out
      2 5 5 2 1 0 3 4 1 1

The variates depend on the seed used by the generator.  The seed for the
default generator type `gsl_rng_default' can be changed with the
`GSL_RNG_SEED' environment variable to produce a different stream of
variates,

     $ GSL_RNG_SEED=123 ./a.out
     GSL_RNG_SEED=123
      4 5 6 3 3 1 4 2 5 5

The following program generates a random walk in two dimensions.

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

     int
     main (void)
     {
       int i;
       double x = 0, y = 0, dx, dy;

       const gsl_rng_type * T;
       gsl_rng * r;

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

       printf ("%g %g\n", x, y);

       for (i = 0; i < 10; i++)
         {
           gsl_ran_dir_2d (r, &dx, &dy);
           x += dx; y += dy;
           printf ("%g %g\n", x, y);
         }
       return 0;
     }

Here is the output from the program, three 10-step random walks from
the origin,

   The following program computes the upper and lower cumulative
distribution functions for the standard normal distribution at x=2.

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

     int
     main (void)
     {
       double P, Q;
       double x = 2.0;

       P = gsl_cdf_ugaussian_P (x);
       printf ("prob(x < %f) = %f\n", x, P);

       Q = gsl_cdf_ugaussian_Q (x);
       printf ("prob(x > %f) = %f\n", x, Q);

       x = gsl_cdf_ugaussian_Pinv (P);
       printf ("Pinv(%f) = %f\n", P, x);

       x = gsl_cdf_ugaussian_Qinv (Q);
       printf ("Qinv(%f) = %f\n", Q, x);

       return 0;
     }

Here is the output of the program,

     prob(x < 2.000000) = 0.977250
     prob(x > 2.000000) = 0.022750
     Pinv(0.977250) = 2.000000
     Qinv(0.022750) = 2.000000


File: gsl-ref.info,  Node: Random Number Distribution References and Further Reading,  Prev: Random Number Distribution Examples,  Up: Random Number Distributions

19.40 References and Further Reading
====================================

For an encyclopaedic coverage of the subject readers are advised to
consult the book `Non-Uniform Random Variate Generation' by Luc
Devroye.  It covers every imaginable distribution and provides hundreds
of algorithms.

     Luc Devroye, `Non-Uniform Random Variate Generation',
     Springer-Verlag, ISBN 0-387-96305-7.

The subject of random variate generation is also reviewed by Knuth, who
describes algorithms for all the major distributions.

     Donald E. Knuth, `The Art of Computer Programming: Seminumerical
     Algorithms' (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.

The Particle Data Group provides a short review of techniques for
generating distributions of random numbers in the "Monte Carlo" section
of its Annual Review of Particle Physics.

     `Review of Particle Properties' R.M. Barnett et al., Physical
     Review D54, 1 (1996) `http://pdg.lbl.gov/'.

The Review of Particle Physics is available online in postscript and pdf
format.

An overview of methods used to compute cumulative distribution functions
can be found in `Statistical Computing' by W.J. Kennedy and J.E.
Gentle. Another general reference is `Elements of Statistical
Computing' by R.A. Thisted.

     William E. Kennedy and James E. Gentle, `Statistical Computing'
     (1980), Marcel Dekker, ISBN 0-8247-6898-1.

     Ronald A. Thisted, `Elements of Statistical Computing' (1988),
     Chapman & Hall, ISBN 0-412-01371-1.

The cumulative distribution functions for the Gaussian distribution are
based on the following papers,

     `Rational Chebyshev Approximations Using Linear Equations', W.J.
     Cody, W. Fraser, J.F. Hart. Numerische Mathematik 12, 242-251
     (1968).

     `Rational Chebyshev Approximations for the Error Function', W.J.
     Cody. Mathematics of Computation 23, n107, 631-637 (July 1969).


File: gsl-ref.info,  Node: Statistics,  Next: Histograms,  Prev: Random Number Distributions,  Up: Top

20 Statistics
*************

This chapter describes the statistical functions in the library.  The
basic statistical functions include routines to compute the mean,
variance and standard deviation.  More advanced functions allow you to
calculate absolute deviations, skewness, and kurtosis as well as the
median and arbitrary percentiles.  The algorithms use recurrence
relations to compute average quantities in a stable way, without large
intermediate values that might overflow.

   The functions are available in versions for datasets in the standard
floating-point and integer types.  The versions for double precision
floating-point data have the prefix `gsl_stats' and are declared in the
header file `gsl_statistics_double.h'.  The versions for integer data
have the prefix `gsl_stats_int' and are declared in the header file
`gsl_statistics_int.h'.

* Menu:

* Mean and standard deviation and variance::
* Absolute deviation::
* Higher moments (skewness and kurtosis)::
* Autocorrelation::
* Covariance::
* Weighted Samples::
* Maximum and Minimum values::
* Median and Percentiles::
* Example statistical programs::
* Statistics References and Further Reading::


File: gsl-ref.info,  Node: Mean and standard deviation and variance,  Next: Absolute deviation,  Up: Statistics

20.1 Mean, Standard Deviation and Variance
==========================================

 -- Function: double gsl_stats_mean (const double DATA[], size_t
          STRIDE, size_t N)
     This function returns the arithmetic mean of DATA, a dataset of
     length N with stride STRIDE.  The arithmetic mean, or "sample
     mean", is denoted by \Hat\mu and defined as,

          \Hat\mu = (1/N) \sum x_i

     where x_i are the elements of the dataset DATA.  For samples drawn
     from a gaussian distribution the variance of \Hat\mu is \sigma^2 /
     N.

 -- Function: double gsl_stats_variance (const double DATA[], size_t
          STRIDE, size_t N)
     This function returns the estimated, or "sample", variance of
     DATA, a dataset of length N with stride STRIDE.  The estimated
     variance is denoted by \Hat\sigma^2 and is defined by,

          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2

     where x_i are the elements of the dataset DATA.  Note that the
     normalization factor of 1/(N-1) results from the derivation of
     \Hat\sigma^2 as an unbiased estimator of the population variance
     \sigma^2.  For samples drawn from a gaussian distribution the
     variance of \Hat\sigma^2 itself is 2 \sigma^4 / N.

     This function computes the mean via a call to `gsl_stats_mean'.  If
     you have already computed the mean then you can pass it directly to
     `gsl_stats_variance_m'.

 -- Function: double gsl_stats_variance_m (const double DATA[], size_t
          STRIDE, size_t N, double MEAN)
     This function returns the sample variance of DATA relative to the
     given value of MEAN.  The function is computed with \Hat\mu
     replaced by the value of MEAN that you supply,

          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2

 -- Function: double gsl_stats_sd (const double DATA[], size_t STRIDE,
          size_t N)
 -- Function: double gsl_stats_sd_m (const double DATA[], size_t
          STRIDE, size_t N, double MEAN)
     The standard deviation is defined as the square root of the
     variance.  These functions return the square root of the
     corresponding variance functions above.

 -- Function: double gsl_stats_variance_with_fixed_mean (const double
          DATA[], size_t STRIDE, size_t N, double MEAN)
     This function computes an unbiased estimate of the variance of
     DATA when the population mean MEAN of the underlying distribution
     is known _a priori_.  In this case the estimator for the variance
     uses the factor 1/N and the sample mean \Hat\mu is replaced by the
     known population mean \mu,

          \Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2

 -- Function: double gsl_stats_sd_with_fixed_mean (const double DATA[],
          size_t STRIDE, size_t N, double MEAN)
     This function calculates the standard deviation of DATA for a
     fixed population mean MEAN.  The result is the square root of the
     corresponding variance function.


File: gsl-ref.info,  Node: Absolute deviation,  Next: Higher moments (skewness and kurtosis),  Prev: Mean and standard deviation and variance,  Up: Statistics

20.2 Absolute deviation
=======================

 -- Function: double gsl_stats_absdev (const double DATA[], size_t
          STRIDE, size_t N)
     This function computes the absolute deviation from the mean of
     DATA, a dataset of length N with stride STRIDE.  The absolute
     deviation from the mean is defined as,

          absdev  = (1/N) \sum |x_i - \Hat\mu|

     where x_i are the elements of the dataset DATA.  The absolute
     deviation from the mean provides a more robust measure of the
     width of a distribution than the variance.  This function computes
     the mean of DATA via a call to `gsl_stats_mean'.

 -- Function: double gsl_stats_absdev_m (const double DATA[], size_t
          STRIDE, size_t N, double MEAN)
     This function computes the absolute deviation of the dataset DATA
     relative to the given value of MEAN,

          absdev  = (1/N) \sum |x_i - mean|

     This function is useful if you have already computed the mean of
     DATA (and want to avoid recomputing it), or wish to calculate the
     absolute deviation relative to another value (such as zero, or the
     median).


File: gsl-ref.info,  Node: Higher moments (skewness and kurtosis),  Next: Autocorrelation,  Prev: Absolute deviation,  Up: Statistics

20.3 Higher moments (skewness and kurtosis)
===========================================

 -- Function: double gsl_stats_skew (const double DATA[], size_t
          STRIDE, size_t N)
     This function computes the skewness of DATA, a dataset of length N
     with stride STRIDE.  The skewness is defined as,

          skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3

     where x_i are the elements of the dataset DATA.  The skewness
     measures the asymmetry of the tails of a distribution.

     The function computes the mean and estimated standard deviation of
     DATA via calls to `gsl_stats_mean' and `gsl_stats_sd'.

 -- Function: double gsl_stats_skew_m_sd (const double DATA[], size_t
          STRIDE, size_t N, double MEAN, double SD)
     This function computes the skewness of the dataset DATA using the
     given values of the mean MEAN and standard deviation SD,

          skew = (1/N) \sum ((x_i - mean)/sd)^3

     These functions are useful if you have already computed the mean
     and standard deviation of DATA and want to avoid recomputing them.

 -- Function: double gsl_stats_kurtosis (const double DATA[], size_t
          STRIDE, size_t N)
     This function computes the kurtosis of DATA, a dataset of length N
     with stride STRIDE.  The kurtosis is defined as,

          kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4)  - 3

     The kurtosis measures how sharply peaked a distribution is,
     relative to its width.  The kurtosis is normalized to zero for a
     gaussian distribution.

 -- Function: double gsl_stats_kurtosis_m_sd (const double DATA[],
          size_t STRIDE, size_t N, double MEAN, double SD)
     This function computes the kurtosis of the dataset DATA using the
     given values of the mean MEAN and standard deviation SD,

          kurtosis = ((1/N) \sum ((x_i - mean)/sd)^4) - 3

     This function is useful if you have already computed the mean and
     standard deviation of DATA and want to avoid recomputing them.


File: gsl-ref.info,  Node: Autocorrelation,  Next: Covariance,  Prev: Higher moments (skewness and kurtosis),  Up: Statistics

20.4 Autocorrelation
====================

 -- Function: double gsl_stats_lag1_autocorrelation (const double
          DATA[], const size_t STRIDE, const size_t N)
     This function computes the lag-1 autocorrelation of the dataset
     DATA.

          a_1 = {\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu)
                 \over
                 \sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}

 -- Function: double gsl_stats_lag1_autocorrelation_m (const double
          DATA[], const size_t STRIDE, const size_t N, const double
          MEAN)
     This function computes the lag-1 autocorrelation of the dataset
     DATA using the given value of the mean MEAN.



File: gsl-ref.info,  Node: Covariance,  Next: Weighted Samples,  Prev: Autocorrelation,  Up: Statistics

20.5 Covariance
===============

 -- Function: double gsl_stats_covariance (const double DATA1[], const
          size_t STRIDE1, const double DATA2[], const size_t STRIDE2,
          const size_t N)
     This function computes the covariance of the datasets DATA1 and
     DATA2 which must both be of the same length N.

          covar = (1/(n - 1)) \sum_{i = 1}^{n} (x_i - \Hat x) (y_i - \Hat y)

 -- Function: double gsl_stats_covariance_m (const double DATA1[],
          const size_t STRIDE1, const double DATA2[], const size_t N,
          const double MEAN1, const double MEAN2)
     This function computes the covariance of the datasets DATA1 and
     DATA2 using the given values of the means, MEAN1 and MEAN2.  This
     is useful if you have already computed the means of DATA1 and
     DATA2 and want to avoid recomputing them.


File: gsl-ref.info,  Node: Weighted Samples,  Next: Maximum and Minimum values,  Prev: Covariance,  Up: Statistics

20.6 Weighted Samples
=====================

The functions described in this section allow the computation of
statistics for weighted samples.  The functions accept an array of
samples, x_i, with associated weights, w_i.  Each sample x_i is
considered as having been drawn from a Gaussian distribution with
variance \sigma_i^2.  The sample weight w_i is defined as the
reciprocal of this variance, w_i = 1/\sigma_i^2.  Setting a weight to
zero corresponds to removing a sample from a dataset.

 -- Function: double gsl_stats_wmean (const double W[], size_t WSTRIDE,
          const double DATA[], size_t STRIDE, size_t N)
     This function returns the weighted mean of the dataset DATA with
     stride STRIDE and length N, using the set of weights W with stride
     WSTRIDE and length N.  The weighted mean is defined as,

          \Hat\mu = (\sum w_i x_i) / (\sum w_i)

 -- Function: double gsl_stats_wvariance (const double W[], size_t
          WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
     This function returns the estimated variance of the dataset DATA
     with stride STRIDE and length N, using the set of weights W with
     stride WSTRIDE and length N.  The estimated variance of a weighted
     dataset is defined as,

          \Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2)))
                          \sum w_i (x_i - \Hat\mu)^2

     Note that this expression reduces to an unweighted variance with
     the familiar 1/(N-1) factor when there are N equal non-zero
     weights.

 -- Function: double gsl_stats_wvariance_m (const double W[], size_t
          WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
          WMEAN)
     This function returns the estimated variance of the weighted
     dataset DATA using the given weighted mean WMEAN.

 -- Function: double gsl_stats_wsd (const double W[], size_t WSTRIDE,
          const double DATA[], size_t STRIDE, size_t N)
     The standard deviation is defined as the square root of the
     variance.  This function returns the square root of the
     corresponding variance function `gsl_stats_wvariance' above.

 -- Function: double gsl_stats_wsd_m (const double W[], size_t WSTRIDE,
          const double DATA[], size_t STRIDE, size_t N, double WMEAN)
     This function returns the square root of the corresponding variance
     function `gsl_stats_wvariance_m' above.

 -- Function: double gsl_stats_wvariance_with_fixed_mean (const double
          W[], size_t WSTRIDE, const double DATA[], size_t STRIDE,
          size_t N, const double MEAN)
     This function computes an unbiased estimate of the variance of
     weighted dataset DATA when the population mean MEAN of the
     underlying distribution is known _a priori_.  In this case the
     estimator for the variance replaces the sample mean \Hat\mu by the
     known population mean \mu,

          \Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) / (\sum w_i)

 -- Function: double gsl_stats_wsd_with_fixed_mean (const double W[],
          size_t WSTRIDE, const double DATA[], size_t STRIDE, size_t N,
          const double MEAN)
     The standard deviation is defined as the square root of the
     variance.  This function returns the square root of the
     corresponding variance function above.

 -- Function: double gsl_stats_wabsdev (const double W[], size_t
          WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
     This function computes the weighted absolute deviation from the
     weighted mean of DATA.  The absolute deviation from the mean is
     defined as,

          absdev = (\sum w_i |x_i - \Hat\mu|) / (\sum w_i)

 -- Function: double gsl_stats_wabsdev_m (const double W[], size_t
          WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
          WMEAN)
     This function computes the absolute deviation of the weighted
     dataset DATA about the given weighted mean WMEAN.

 -- Function: double gsl_stats_wskew (const double W[], size_t WSTRIDE,
          const double DATA[], size_t STRIDE, size_t N)
     This function computes the weighted skewness of the dataset DATA.

          skew = (\sum w_i ((x_i - xbar)/\sigma)^3) / (\sum w_i)

 -- Function: double gsl_stats_wskew_m_sd (const double W[], size_t
          WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
          WMEAN, double WSD)
     This function computes the weighted skewness of the dataset DATA
     using the given values of the weighted mean and weighted standard
     deviation, WMEAN and WSD.

 -- Function: double gsl_stats_wkurtosis (const double W[], size_t
          WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
     This function computes the weighted kurtosis of the dataset DATA.

          kurtosis = ((\sum w_i ((x_i - xbar)/sigma)^4) / (\sum w_i)) - 3

 -- Function: double gsl_stats_wkurtosis_m_sd (const double W[], size_t
          WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
          WMEAN, double WSD)
     This function computes the weighted kurtosis of the dataset DATA
     using the given values of the weighted mean and weighted standard
     deviation, WMEAN and WSD.


File: gsl-ref.info,  Node: Maximum and Minimum values,  Next: Median and Percentiles,  Prev: Weighted Samples,  Up: Statistics

20.7 Maximum and Minimum values
===============================

The following functions find the maximum and minimum values of a
dataset (or their indices).  If the data contains `NaN's then a `NaN'
will be returned, since the maximum or minimum value is undefined.  For
functions which return an index, the location of the first `NaN' in the
array is returned.

 -- Function: double gsl_stats_max (const double DATA[], size_t STRIDE,
          size_t N)
     This function returns the maximum value in DATA, a dataset of
     length N with stride STRIDE.  The maximum value is defined as the
     value of the element x_i which satisfies x_i >= x_j for all j.

     If you want instead to find the element with the largest absolute
     magnitude you will need to apply `fabs' or `abs' to your data
     before calling this function.

 -- Function: double gsl_stats_min (const double DATA[], size_t STRIDE,
          size_t N)
     This function returns the minimum value in DATA, a dataset of
     length N with stride STRIDE.  The minimum value is defined as the
     value of the element x_i which satisfies x_i <= x_j for all j.

     If you want instead to find the element with the smallest absolute
     magnitude you will need to apply `fabs' or `abs' to your data
     before calling this function.

 -- Function: void gsl_stats_minmax (double * MIN, double * MAX, const
          double DATA[], size_t STRIDE, size_t N)
     This function finds both the minimum and maximum values MIN, MAX
     in DATA in a single pass.

 -- Function: size_t gsl_stats_max_index (const double DATA[], size_t
          STRIDE, size_t N)
     This function returns the index of the maximum value in DATA, a
     dataset of length N with stride STRIDE.  The maximum value is
     defined as the value of the element x_i which satisfies x_i >= x_j
     for all j.  When there are several equal maximum elements then the
     first one is chosen.

 -- Function: size_t gsl_stats_min_index (const double DATA[], size_t
          STRIDE, size_t N)
     This function returns the index of the minimum value in DATA, a
     dataset of length N with stride STRIDE.  The minimum value is
     defined as the value of the element x_i which satisfies x_i >= x_j
     for all j.  When there are several equal minimum elements then the
     first one is chosen.

 -- Function: void gsl_stats_minmax_index (size_t * MIN_INDEX, size_t *
          MAX_INDEX, const double DATA[], size_t STRIDE, size_t N)
     This function returns the indexes MIN_INDEX, MAX_INDEX of the
     minimum and maximum values in DATA in a single pass.


File: gsl-ref.info,  Node: Median and Percentiles,  Next: Example statistical programs,  Prev: Maximum and Minimum values,  Up: Statistics

20.8 Median and Percentiles
===========================

The median and percentile functions described in this section operate on
sorted data.  For convenience we use "quantiles", measured on a scale
of 0 to 1, instead of percentiles (which use a scale of 0 to 100).

 -- Function: double gsl_stats_median_from_sorted_data (const double
          SORTED_DATA[], size_t STRIDE, size_t N)
     This function returns the median value of SORTED_DATA, a dataset
     of length N with stride STRIDE.  The elements of the array must be
     in ascending numerical order.  There are no checks to see whether
     the data are sorted, so the function `gsl_sort' should always be
     used first.

     When the dataset has an odd number of elements the median is the
     value of element (n-1)/2.  When the dataset has an even number of
     elements the median is the mean of the two nearest middle values,
     elements (n-1)/2 and n/2.  Since the algorithm for computing the
     median involves interpolation this function always returns a
     floating-point number, even for integer data types.

 -- Function: double gsl_stats_quantile_from_sorted_data (const double
          SORTED_DATA[], size_t STRIDE, size_t N, double F)
     This function returns a quantile value of SORTED_DATA, a
     double-precision array of length N with stride STRIDE.  The
     elements of the array must be in ascending numerical order.  The
     quantile is determined by the F, a fraction between 0 and 1.  For
     example, to compute the value of the 75th percentile F should have
     the value 0.75.

     There are no checks to see whether the data are sorted, so the
     function `gsl_sort' should always be used first.

     The quantile is found by interpolation, using the formula

          quantile = (1 - \delta) x_i + \delta x_{i+1}

     where i is `floor'((n - 1)f) and \delta is (n-1)f - i.

     Thus the minimum value of the array (`data[0*stride]') is given by
     F equal to zero, the maximum value (`data[(n-1)*stride]') is given
     by F equal to one and the median value is given by F equal to 0.5.
     Since the algorithm for computing quantiles involves
     interpolation this function always returns a floating-point
     number, even for integer data types.


File: gsl-ref.info,  Node: Example statistical programs,  Next: Statistics References and Further Reading,  Prev: Median and Percentiles,  Up: Statistics

20.9 Examples
=============

Here is a basic example of how to use the statistical functions:

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

     int
     main(void)
     {
       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
       double mean, variance, largest, smallest;

       mean     = gsl_stats_mean(data, 1, 5);
       variance = gsl_stats_variance(data, 1, 5);
       largest  = gsl_stats_max(data, 1, 5);
       smallest = gsl_stats_min(data, 1, 5);

       printf ("The dataset is %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       printf ("The sample mean is %g\n", mean);
       printf ("The estimated variance is %g\n", variance);
       printf ("The largest value is %g\n", largest);
       printf ("The smallest value is %g\n", smallest);
       return 0;
     }

   The program should produce the following output,

     The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
     The sample mean is 16.54
     The estimated variance is 4.2984
     The largest value is 18.3
     The smallest value is 12.6

   Here is an example using sorted data,

     #include <stdio.h>
     #include <gsl/gsl_sort.h>
     #include <gsl/gsl_statistics.h>

     int
     main(void)
     {
       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
       double median, upperq, lowerq;

       printf ("Original dataset:  %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       gsl_sort (data, 1, 5);

       printf ("Sorted dataset: %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       median
         = gsl_stats_median_from_sorted_data (data,
                                              1, 5);

       upperq
         = gsl_stats_quantile_from_sorted_data (data,
                                                1, 5,
                                                0.75);
       lowerq
         = gsl_stats_quantile_from_sorted_data (data,
                                                1, 5,
                                                0.25);

       printf ("The median is %g\n", median);
       printf ("The upper quartile is %g\n", upperq);
       printf ("The lower quartile is %g\n", lowerq);
       return 0;
     }

   This program should produce the following output,

     Original dataset: 17.2, 18.1, 16.5, 18.3, 12.6
     Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3
     The median is 17.2
     The upper quartile is 18.1
     The lower quartile is 16.5


File: gsl-ref.info,  Node: Statistics References and Further Reading,  Prev: Example statistical programs,  Up: Statistics

20.10 References and Further Reading
====================================

The standard reference for almost any topic in statistics is the
multi-volume `Advanced Theory of Statistics' by Kendall and Stuart.

     Maurice Kendall, Alan Stuart, and J. Keith Ord.  `The Advanced
     Theory of Statistics' (multiple volumes) reprinted as `Kendall's
     Advanced Theory of Statistics'.  Wiley, ISBN 047023380X.

Many statistical concepts can be more easily understood by a Bayesian
approach.  The following book by Gelman, Carlin, Stern and Rubin gives a
comprehensive coverage of the subject.

     Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin.
     `Bayesian Data Analysis'.  Chapman & Hall, ISBN 0412039915.

For physicists the Particle Data Group provides useful reviews of
Probability and Statistics in the "Mathematical Tools" section of its
Annual Review of Particle Physics.

     `Review of Particle Properties' R.M. Barnett et al., Physical
     Review D54, 1 (1996)

The Review of Particle Physics is available online at the website
`http://pdg.lbl.gov/'.


File: gsl-ref.info,  Node: Histograms,  Next: N-tuples,  Prev: Statistics,  Up: Top

21 Histograms
*************

This chapter describes functions for creating histograms.  Histograms
provide a convenient way of summarizing the distribution of a set of
data. A histogram consists of a set of "bins" which count the number of
events falling into a given range of a continuous variable x.  In GSL
the bins of a histogram contain floating-point numbers, so they can be
used to record both integer and non-integer distributions.  The bins
can use arbitrary sets of ranges (uniformly spaced bins are the
default).  Both one and two-dimensional histograms are supported.

   Once a histogram has been created it can also be converted into a
probability distribution function.  The library provides efficient
routines for selecting random samples from probability distributions.
This can be useful for generating simulations based on real data.

   The functions are declared in the header files `gsl_histogram.h' and
`gsl_histogram2d.h'.

* Menu:

* The histogram struct::
* Histogram allocation::
* Copying Histograms::
* Updating and accessing histogram elements::
* Searching histogram ranges::
* Histogram Statistics::
* Histogram Operations::
* Reading and writing histograms::
* Resampling from histograms::
* The histogram probability distribution struct::
* Example programs for histograms::
* Two dimensional histograms::
* The 2D histogram struct::
* 2D Histogram allocation::
* Copying 2D Histograms::
* Updating and accessing 2D histogram elements::
* Searching 2D histogram ranges::
* 2D Histogram Statistics::
* 2D Histogram Operations::
* Reading and writing 2D histograms::
* Resampling from 2D histograms::
* Example programs for 2D histograms::


File: gsl-ref.info,  Node: The histogram struct,  Next: Histogram allocation,  Up: Histograms

21.1 The histogram struct
=========================

A histogram is defined by the following struct,

 -- Data Type: gsl_histogram
    `size_t n'
          This is the number of histogram bins

    `double * range'
          The ranges of the bins are stored in an array of N+1 elements
          pointed to by RANGE.

    `double * bin'
          The counts for each bin are stored in an array of N elements
          pointed to by BIN.  The bins are floating-point numbers, so
          you can increment them by non-integer values if necessary.

The range for BIN[i] is given by RANGE[i] to RANGE[i+1].  For n bins
there are n+1 entries in the array RANGE.  Each bin is inclusive at the
lower end and exclusive at the upper end.  Mathematically this means
that the bins are defined by the following inequality,
     bin[i] corresponds to range[i] <= x < range[i+1]

Here is a diagram of the correspondence between ranges and bins on the
number-line for x,


          [ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[4] )
       ---|---------|---------|---------|---------|---------|---  x
        r[0]      r[1]      r[2]      r[3]      r[4]      r[5]

In this picture the values of the RANGE array are denoted by r.  On the
left-hand side of each bin the square bracket `[' denotes an inclusive
lower bound (r <= x), and the round parentheses `)' on the right-hand
side denote an exclusive upper bound (x < r).  Thus any samples which
fall on the upper end of the histogram are excluded.  If you want to
include this value for the last bin you will need to add an extra bin
to your histogram.

   The `gsl_histogram' struct and its associated functions are defined
in the header file `gsl_histogram.h'.


File: gsl-ref.info,  Node: Histogram allocation,  Next: Copying Histograms,  Prev: The histogram struct,  Up: Histograms

21.2 Histogram allocation
=========================

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

 -- Function: gsl_histogram * gsl_histogram_alloc (size_t N)
     This function allocates memory for a histogram with N bins, and
     returns a pointer to a newly created `gsl_histogram' struct.  If
     insufficient memory is available a null pointer is returned and the
     error handler is invoked with an error code of `GSL_ENOMEM'. The
     bins and ranges are not initialized, and should be prepared using
     one of the range-setting functions below in order to make the
     histogram ready for use.

 -- Function: int gsl_histogram_set_ranges (gsl_histogram * H, const
          double RANGE[], size_t SIZE)
     This function sets the ranges of the existing histogram H using
     the array RANGE of size SIZE.  The values of the histogram bins
     are reset to zero.  The `range' array should contain the desired
     bin limits.  The ranges can be arbitrary, subject to the
     restriction that they are monotonically increasing.

     The following example shows how to create a histogram with
     logarithmic bins with ranges [1,10), [10,100) and [100,1000).

          gsl_histogram * h = gsl_histogram_alloc (3);

          /* bin[0] covers the range 1 <= x < 10 */
          /* bin[1] covers the range 10 <= x < 100 */
          /* bin[2] covers the range 100 <= x < 1000 */

          double range[4] = { 1.0, 10.0, 100.0, 1000.0 };

          gsl_histogram_set_ranges (h, range, 4);

     Note that the size of the RANGE array should be defined to be one
     element bigger than the number of bins.  The additional element is
     required for the upper value of the final bin.

 -- Function: int gsl_histogram_set_ranges_uniform (gsl_histogram * H,
          double XMIN, double XMAX)
     This function sets the ranges of the existing histogram H to cover
     the range XMIN to XMAX uniformly.  The values of the histogram
     bins are reset to zero.  The bin ranges are shown in the table
     below,
          bin[0] corresponds to xmin <= x < xmin + d
          bin[1] corresponds to xmin + d <= x < xmin + 2 d
          ......
          bin[n-1] corresponds to xmin + (n-1)d <= x < xmax

     where d is the bin spacing, d = (xmax-xmin)/n.

 -- Function: void gsl_histogram_free (gsl_histogram * H)
     This function frees the histogram H and all of the memory
     associated with it.


File: gsl-ref.info,  Node: Copying Histograms,  Next: Updating and accessing histogram elements,  Prev: Histogram allocation,  Up: Histograms

21.3 Copying Histograms
=======================

 -- Function: int gsl_histogram_memcpy (gsl_histogram * DEST, const
          gsl_histogram * SRC)
     This function copies the histogram SRC into the pre-existing
     histogram DEST, making DEST into an exact copy of SRC.  The two
     histograms must be of the same size.

 -- Function: gsl_histogram * gsl_histogram_clone (const gsl_histogram
          * SRC)
     This function returns a pointer to a newly created histogram which
     is an exact copy of the histogram SRC.


File: gsl-ref.info,  Node: Updating and accessing histogram elements,  Next: Searching histogram ranges,  Prev: Copying Histograms,  Up: Histograms

21.4 Updating and accessing histogram elements
==============================================

There are two ways to access histogram bins, either by specifying an x
coordinate or by using the bin-index directly.  The functions for
accessing the histogram through x coordinates use a binary search to
identify the bin which covers the appropriate range.

 -- Function: int gsl_histogram_increment (gsl_histogram * H, double X)
     This function updates the histogram H by adding one (1.0) to the
     bin whose range contains the coordinate X.

     If X lies in the valid range of the histogram then the function
     returns zero to indicate success.  If X is less than the lower
     limit of the histogram then the function returns `GSL_EDOM', and
     none of bins are modified.  Similarly, if the value of X is greater
     than or equal to the upper limit of the histogram then the function
     returns `GSL_EDOM', and none of the bins are modified.  The error
     handler is not called, however, since it is often necessary to
     compute histograms for a small range of a larger dataset, ignoring
     the values outside the range of interest.

 -- Function: int gsl_histogram_accumulate (gsl_histogram * H, double
          X, double WEIGHT)
     This function is similar to `gsl_histogram_increment' but increases
     the value of the appropriate bin in the histogram H by the
     floating-point number WEIGHT.

 -- Function: double gsl_histogram_get (const gsl_histogram * H, size_t
          I)
     This function returns the contents of the I-th bin of the histogram
     H.  If I lies outside the valid range of indices for the histogram
     then the error handler is called with an error code of `GSL_EDOM'
     and the function returns 0.

 -- Function: int gsl_histogram_get_range (const gsl_histogram * H,
          size_t I, double * LOWER, double * UPPER)
     This function finds the upper and lower range limits of the I-th
     bin of the histogram H.  If the index I is valid then the
     corresponding range limits are stored in LOWER and UPPER.  The
     lower limit is inclusive (i.e. events with this coordinate are
     included in the bin) and the upper limit is exclusive (i.e. events
     with the coordinate of the upper limit are excluded and fall in the
     neighboring higher bin, if it exists).  The function returns 0 to
     indicate success.  If I lies outside the valid range of indices for
     the histogram then the error handler is called and the function
     returns an error code of `GSL_EDOM'.

 -- Function: double gsl_histogram_max (const gsl_histogram * H)
 -- Function: double gsl_histogram_min (const gsl_histogram * H)
 -- Function: size_t gsl_histogram_bins (const gsl_histogram * H)
     These functions return the maximum upper and minimum lower range
     limits and the number of bins of the histogram H.  They provide a
     way of determining these values without accessing the
     `gsl_histogram' struct directly.

 -- Function: void gsl_histogram_reset (gsl_histogram * H)
     This function resets all the bins in the histogram H to zero.


File: gsl-ref.info,  Node: Searching histogram ranges,  Next: Histogram Statistics,  Prev: Updating and accessing histogram elements,  Up: Histograms

21.5 Searching histogram ranges
===============================

The following functions are used by the access and update routines to
locate the bin which corresponds to a given x coordinate.

 -- Function: int gsl_histogram_find (const gsl_histogram * H, double
          X, size_t * I)
     This function finds and sets the index I to the bin number which
     covers the coordinate X in the histogram H.  The bin is located
     using a binary search. The search includes an optimization for
     histograms with uniform range, and will return the correct bin
     immediately in this case.  If X is found in the range of the
     histogram then the function sets the index I and returns
     `GSL_SUCCESS'.  If X lies outside the valid range of the histogram
     then the function returns `GSL_EDOM' and the error handler is
     invoked.


File: gsl-ref.info,  Node: Histogram Statistics,  Next: Histogram Operations,  Prev: Searching histogram ranges,  Up: Histograms

21.6 Histogram Statistics
=========================

 -- Function: double gsl_histogram_max_val (const gsl_histogram * H)
     This function returns the maximum value contained in the histogram
     bins.

 -- Function: size_t gsl_histogram_max_bin (const gsl_histogram * H)
     This function returns the index of the bin containing the maximum
     value. In the case where several bins contain the same maximum
     value the smallest index is returned.

 -- Function: double gsl_histogram_min_val (const gsl_histogram * H)
     This function returns the minimum value contained in the histogram
     bins.

 -- Function: size_t gsl_histogram_min_bin (const gsl_histogram * H)
     This function returns the index of the bin containing the minimum
     value. In the case where several bins contain the same maximum
     value the smallest index is returned.

 -- Function: double gsl_histogram_mean (const gsl_histogram * H)
     This function returns the mean of the histogrammed variable, where
     the histogram is regarded as a probability distribution. Negative
     bin values are ignored for the purposes of this calculation.  The
     accuracy of the result is limited by the bin width.

 -- Function: double gsl_histogram_sigma (const gsl_histogram * H)
     This function returns the standard deviation of the histogrammed
     variable, where the histogram is regarded as a probability
     distribution. Negative bin values are ignored for the purposes of
     this calculation. The accuracy of the result is limited by the bin
     width.

 -- Function: double gsl_histogram_sum (const gsl_histogram * H)
     This function returns the sum of all bin values. Negative bin
     values are included in the sum.


File: gsl-ref.info,  Node: Histogram Operations,  Next: Reading and writing histograms,  Prev: Histogram Statistics,  Up: Histograms

21.7 Histogram Operations
=========================

 -- Function: int gsl_histogram_equal_bins_p (const gsl_histogram * H1,
          const gsl_histogram * H2)
     This function returns 1 if the all of the individual bin ranges of
     the two histograms are identical, and 0 otherwise.

 -- Function: int gsl_histogram_add (gsl_histogram * H1, const
          gsl_histogram * H2)
     This function adds the contents of the bins in histogram H2 to the
     corresponding bins of histogram H1,  i.e. h'_1(i) = h_1(i) +
     h_2(i).  The two histograms must have identical bin ranges.

 -- Function: int gsl_histogram_sub (gsl_histogram * H1, const
          gsl_histogram * H2)
     This function subtracts the contents of the bins in histogram H2
     from the corresponding bins of histogram H1, i.e. h'_1(i) = h_1(i)
     - h_2(i).  The two histograms must have identical bin ranges.

 -- Function: int gsl_histogram_mul (gsl_histogram * H1, const
          gsl_histogram * H2)
     This function multiplies the contents of the bins of histogram H1
     by the contents of the corresponding bins in histogram H2, i.e.
     h'_1(i) = h_1(i) * h_2(i).  The two histograms must have identical
     bin ranges.

 -- Function: int gsl_histogram_div (gsl_histogram * H1, const
          gsl_histogram * H2)
     This function divides the contents of the bins of histogram H1 by
     the contents of the corresponding bins in histogram H2, i.e.
     h'_1(i) = h_1(i) / h_2(i).  The two histograms must have identical
     bin ranges.

 -- Function: int gsl_histogram_scale (gsl_histogram * H, double SCALE)
     This function multiplies the contents of the bins of histogram H
     by the constant SCALE, i.e. h'_1(i) = h_1(i) * scale.

 -- Function: int gsl_histogram_shift (gsl_histogram * H, double OFFSET)
     This function shifts the contents of the bins of histogram H by
     the constant OFFSET, i.e. h'_1(i) = h_1(i) + offset.


File: gsl-ref.info,  Node: Reading and writing histograms,  Next: Resampling from histograms,  Prev: Histogram Operations,  Up: Histograms

21.8 Reading and writing histograms
===================================

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

 -- Function: int gsl_histogram_fwrite (FILE * STREAM, const
          gsl_histogram * H)
     This function writes the ranges and bins of the histogram H to the
     stream STREAM in binary format.  The return value is 0 for success
     and `GSL_EFAILED' if there was a problem writing to the file.
     Since the data is written in the native binary format it may not
     be portable between different architectures.

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

 -- Function: int gsl_histogram_fprintf (FILE * STREAM, const
          gsl_histogram * H, const char * RANGE_FORMAT, const char *
          BIN_FORMAT)
     This function writes the ranges and bins of the histogram H
     line-by-line to the stream STREAM using the format specifiers
     RANGE_FORMAT and BIN_FORMAT.  These should be one of the `%g',
     `%e' or `%f' formats for floating point numbers.  The function
     returns 0 for success and `GSL_EFAILED' if there was a problem
     writing to the file.  The histogram output is formatted in three
     columns, and the columns are separated by spaces, like this,

          range[0] range[1] bin[0]
          range[1] range[2] bin[1]
          range[2] range[3] bin[2]
          ....
          range[n-1] range[n] bin[n-1]

     The values of the ranges are formatted using RANGE_FORMAT and the
     value of the bins are formatted using BIN_FORMAT.  Each line
     contains the lower and upper limit of the range of the bins and the
     value of the bin itself.  Since the upper limit of one bin is the
     lower limit of the next there is duplication of these values
     between lines but this allows the histogram to be manipulated with
     line-oriented tools.

 -- Function: int gsl_histogram_fscanf (FILE * STREAM, gsl_histogram *
          H)
     This function reads formatted data from the stream STREAM into the
     histogram H.  The data is assumed to be in the three-column format
     used by `gsl_histogram_fprintf'.  The histogram H must be
     preallocated with the correct length since the function uses the
     size of H to determine how many numbers to read.  The function
     returns 0 for success and `GSL_EFAILED' if there was a problem
     reading from the file.


File: gsl-ref.info,  Node: Resampling from histograms,  Next: The histogram probability distribution struct,  Prev: Reading and writing histograms,  Up: Histograms

21.9 Resampling from histograms
===============================

A histogram made by counting events can be regarded as a measurement of
a probability distribution.  Allowing for statistical error, the height
of each bin represents the probability of an event where the value of x
falls in the range of that bin.  The probability distribution function
has the one-dimensional form p(x)dx where,

     p(x) = n_i/ (N w_i)

In this equation n_i is the number of events in the bin which contains
x, w_i is the width of the bin and N is the total number of events.
The distribution of events within each bin is assumed to be uniform.


File: gsl-ref.info,  Node: The histogram probability distribution struct,  Next: Example programs for histograms,  Prev: Resampling from histograms,  Up: Histograms

21.10 The histogram probability distribution struct
===================================================

The probability distribution function for a histogram consists of a set
of "bins" which measure the probability of an event falling into a
given range of a continuous variable x. A probability distribution
function is defined by the following struct, which actually stores the
cumulative probability distribution function.  This is the natural
quantity for generating samples via the inverse transform method,
because there is a one-to-one mapping between the cumulative
probability distribution and the range [0,1].  It can be shown that by
taking a uniform random number in this range and finding its
corresponding coordinate in the cumulative probability distribution we
obtain samples with the desired probability distribution.

 -- Data Type: gsl_histogram_pdf
    `size_t n'
          This is the number of bins used to approximate the probability
          distribution function.

    `double * range'
          The ranges of the bins are stored in an array of N+1 elements
          pointed to by RANGE.

    `double * sum'
          The cumulative probability for the bins is stored in an array
          of N elements pointed to by SUM.

The following functions allow you to create a `gsl_histogram_pdf'
struct which represents this probability distribution and generate
random samples from it.

 -- Function: gsl_histogram_pdf * gsl_histogram_pdf_alloc (size_t N)
     This function allocates memory for a probability distribution with
     N bins and returns a pointer to a newly initialized
     `gsl_histogram_pdf' struct. If insufficient memory is available a
     null pointer is returned and the error handler is invoked with an
     error code of `GSL_ENOMEM'.

 -- Function: int gsl_histogram_pdf_init (gsl_histogram_pdf * P, const
          gsl_histogram * H)
     This function initializes the probability distribution P with the
     contents of the histogram H. If any of the bins of H are negative
     then the error handler is invoked with an error code of `GSL_EDOM'
     because a probability distribution cannot contain negative values.

 -- Function: void gsl_histogram_pdf_free (gsl_histogram_pdf * P)
     This function frees the probability distribution function P and
     all of the memory associated with it.

 -- Function: double gsl_histogram_pdf_sample (const gsl_histogram_pdf
          * P, double R)
     This function uses R, a uniform random number between zero and
     one, to compute a single random sample from the probability
     distribution P.  The algorithm used to compute the sample s is
     given by the following formula,

          s = range[i] + delta * (range[i+1] - range[i])

     where i is the index which satisfies sum[i] <=  r < sum[i+1] and
     delta is (r - sum[i])/(sum[i+1] - sum[i]).


File: gsl-ref.info,  Node: Example programs for histograms,  Next: Two dimensional histograms,  Prev: The histogram probability distribution struct,  Up: Histograms

21.11 Example programs for histograms
=====================================

The following program shows how to make a simple histogram of a column
of numerical data supplied on `stdin'.  The program takes three
arguments, specifying the upper and lower bounds of the histogram and
the number of bins.  It then reads numbers from `stdin', one line at a
time, and adds them to the histogram.  When there is no more data to
read it prints out the accumulated histogram using
`gsl_histogram_fprintf'.

     #include <stdio.h>
     #include <stdlib.h>
     #include <gsl/gsl_histogram.h>

     int
     main (int argc, char **argv)
     {
       double a, b;
       size_t n;

       if (argc != 4)
         {
           printf ("Usage: gsl-histogram xmin xmax n\n"
                   "Computes a histogram of the data "
                   "on stdin using n bins from xmin "
                   "to xmax\n");
           exit (0);
         }

       a = atof (argv[1]);
       b = atof (argv[2]);
       n = atoi (argv[3]);

       {
         double x;
         gsl_histogram * h = gsl_histogram_alloc (n);
         gsl_histogram_set_ranges_uniform (h, a, b);

         while (fscanf (stdin, "%lg", &x) == 1)
           {
             gsl_histogram_increment (h, x);
           }
         gsl_histogram_fprintf (stdout, h, "%g", "%g");
         gsl_histogram_free (h);
       }
       exit (0);
     }

Here is an example of the program in use.  We generate 10000 random
samples from a Cauchy distribution with a width of 30 and histogram
them over the range -100 to 100, using 200 bins.

     $ gsl-randist 0 10000 cauchy 30
        | gsl-histogram -100 100 200 > histogram.dat

A plot of the resulting histogram shows the familiar shape of the
Cauchy distribution and the fluctuations caused by the finite sample
size.

     $ awk '{print $1, $3 ; print $2, $3}' histogram.dat
        | graph -T X


File: gsl-ref.info,  Node: Two dimensional histograms,  Next: The 2D histogram struct,  Prev: Example programs for histograms,  Up: Histograms

21.12 Two dimensional histograms
================================

A two dimensional histogram consists of a set of "bins" which count the
number of events falling in a given area of the (x,y) plane.  The
simplest way to use a two dimensional histogram is to record
two-dimensional position information, n(x,y).  Another possibility is
to form a "joint distribution" by recording related variables.  For
example a detector might record both the position of an event (x) and
the amount of energy it deposited E.  These could be histogrammed as
the joint distribution n(x,E).


File: gsl-ref.info,  Node: The 2D histogram struct,  Next: 2D Histogram allocation,  Prev: Two dimensional histograms,  Up: Histograms

21.13 The 2D histogram struct
=============================

Two dimensional histograms are defined by the following struct,

 -- Data Type: gsl_histogram2d
    `size_t nx, ny'
          This is the number of histogram bins in the x and y
          directions.

    `double * xrange'
          The ranges of the bins in the x-direction are stored in an
          array of NX + 1 elements pointed to by XRANGE.

    `double * yrange'
          The ranges of the bins in the y-direction are stored in an
          array of NY + 1 elements pointed to by YRANGE.

    `double * bin'
          The counts for each bin are stored in an array pointed to by
          BIN.  The bins are floating-point numbers, so you can
          increment them by non-integer values if necessary.  The array
          BIN stores the two dimensional array of bins in a single
          block of memory according to the mapping `bin(i,j)' = `bin[i
          * ny + j]'.

The range for `bin(i,j)' is given by `xrange[i]' to `xrange[i+1]' in
the x-direction and `yrange[j]' to `yrange[j+1]' in the y-direction.
Each bin is inclusive at the lower end and exclusive at the upper end.
Mathematically this means that the bins are defined by the following
inequality,
     bin(i,j) corresponds to xrange[i] <= x < xrange[i+1]
                         and yrange[j] <= y < yrange[j+1]

Note that any samples which fall on the upper sides of the histogram are
excluded.  If you want to include these values for the side bins you
will need to add an extra row or column to your histogram.

   The `gsl_histogram2d' struct and its associated functions are
defined in the header file `gsl_histogram2d.h'.


File: gsl-ref.info,  Node: 2D Histogram allocation,  Next: Copying 2D Histograms,  Prev: The 2D histogram struct,  Up: Histograms

21.14 2D Histogram allocation
=============================

The functions for allocating memory to a 2D histogram follow the style
of `malloc' and `free'.  In addition they also perform their own error
checking.  If there is insufficient memory available to allocate a
histogram then the functions call the error handler (with an error
number of `GSL_ENOMEM') in addition to returning a null pointer.  Thus
if you use the library error handler to abort your program then it
isn't necessary to check every 2D histogram `alloc'.

 -- Function: gsl_histogram2d * gsl_histogram2d_alloc (size_t NX,
          size_t NY)
     This function allocates memory for a two-dimensional histogram with
     NX bins in the x direction and NY bins in the y direction.  The
     function returns a pointer to a newly created `gsl_histogram2d'
     struct. If insufficient memory is available a null pointer is
     returned and the error handler is invoked with an error code of
     `GSL_ENOMEM'. The bins and ranges must be initialized with one of
     the functions below before the histogram is ready for use.

 -- Function: int gsl_histogram2d_set_ranges (gsl_histogram2d * H,
          const double XRANGE[], size_t XSIZE, const double YRANGE[],
          size_t YSIZE)
     This function sets the ranges of the existing histogram H using
     the arrays XRANGE and YRANGE of size XSIZE and YSIZE respectively.
     The values of the histogram bins are reset to zero.

 -- Function: int gsl_histogram2d_set_ranges_uniform (gsl_histogram2d *
          H, double XMIN, double XMAX, double YMIN, double YMAX)
     This function sets the ranges of the existing histogram H to cover
     the ranges XMIN to XMAX and YMIN to YMAX uniformly.  The values of
     the histogram bins are reset to zero.

 -- Function: void gsl_histogram2d_free (gsl_histogram2d * H)
     This function frees the 2D histogram H and all of the memory
     associated with it.


File: gsl-ref.info,  Node: Copying 2D Histograms,  Next: Updating and accessing 2D histogram elements,  Prev: 2D Histogram allocation,  Up: Histograms

21.15 Copying 2D Histograms
===========================

 -- Function: int gsl_histogram2d_memcpy (gsl_histogram2d * DEST, const
          gsl_histogram2d * SRC)
     This function copies the histogram SRC into the pre-existing
     histogram DEST, making DEST into an exact copy of SRC.  The two
     histograms must be of the same size.

 -- Function: gsl_histogram2d * gsl_histogram2d_clone (const
          gsl_histogram2d * SRC)
     This function returns a pointer to a newly created histogram which
     is an exact copy of the histogram SRC.


File: gsl-ref.info,  Node: Updating and accessing 2D histogram elements,  Next: Searching 2D histogram ranges,  Prev: Copying 2D Histograms,  Up: Histograms

21.16 Updating and accessing 2D histogram elements
==================================================

You can access the bins of a two-dimensional histogram either by
specifying a pair of (x,y) coordinates or by using the bin indices
(i,j) directly.  The functions for accessing the histogram through
(x,y) coordinates use binary searches in the x and y directions to
identify the bin which covers the appropriate range.

 -- Function: int gsl_histogram2d_increment (gsl_histogram2d * H,
          double X, double Y)
     This function updates the histogram H by adding one (1.0) to the
     bin whose x and y ranges contain the coordinates (X,Y).

     If the point (x,y) lies inside the valid ranges of the histogram
     then the function returns zero to indicate success.  If (x,y) lies
     outside the limits of the histogram then the function returns
     `GSL_EDOM', and none of the bins are modified.  The error handler
     is not called, since it is often necessary to compute histograms
     for a small range of a larger dataset, ignoring any coordinates
     outside the range of interest.

 -- Function: int gsl_histogram2d_accumulate (gsl_histogram2d * H,
          double X, double Y, double WEIGHT)
     This function is similar to `gsl_histogram2d_increment' but
     increases the value of the appropriate bin in the histogram H by
     the floating-point number WEIGHT.

 -- Function: double gsl_histogram2d_get (const gsl_histogram2d * H,
          size_t I, size_t J)
     This function returns the contents of the (I,J)-th bin of the
     histogram H.  If (I,J) lies outside the valid range of indices for
     the histogram then the error handler is called with an error code
     of `GSL_EDOM' and the function returns 0.

 -- Function: int gsl_histogram2d_get_xrange (const gsl_histogram2d *
          H, size_t I, double * XLOWER, double * XUPPER)
 -- Function: int gsl_histogram2d_get_yrange (const gsl_histogram2d *
          H, size_t J, double * YLOWER, double * YUPPER)
     These functions find the upper and lower range limits of the I-th
     and J-th bins in the x and y directions of the histogram H.  The
     range limits are stored in XLOWER and XUPPER or YLOWER and YUPPER.
     The lower limits are inclusive (i.e. events with these
     coordinates are included in the bin) and the upper limits are
     exclusive (i.e. events with the value of the upper limit are not
     included and fall in the neighboring higher bin, if it exists).
     The functions return 0 to indicate success.  If I or J lies
     outside the valid range of indices for the histogram then the
     error handler is called with an error code of `GSL_EDOM'.

 -- Function: double gsl_histogram2d_xmax (const gsl_histogram2d * H)
 -- Function: double gsl_histogram2d_xmin (const gsl_histogram2d * H)
 -- Function: size_t gsl_histogram2d_nx (const gsl_histogram2d * H)
 -- Function: double gsl_histogram2d_ymax (const gsl_histogram2d * H)
 -- Function: double gsl_histogram2d_ymin (const gsl_histogram2d * H)
 -- Function: size_t gsl_histogram2d_ny (const gsl_histogram2d * H)
     These functions return the maximum upper and minimum lower range
     limits and the number of bins for the x and y directions of the
     histogram H.  They provide a way of determining these values
     without accessing the `gsl_histogram2d' struct directly.

 -- Function: void gsl_histogram2d_reset (gsl_histogram2d * H)
     This function resets all the bins of the histogram H to zero.


File: gsl-ref.info,  Node: Searching 2D histogram ranges,  Next: 2D Histogram Statistics,  Prev: Updating and accessing 2D histogram elements,  Up: Histograms

21.17 Searching 2D histogram ranges
===================================

The following functions are used by the access and update routines to
locate the bin which corresponds to a given (x,y) coordinate.

 -- Function: int gsl_histogram2d_find (const gsl_histogram2d * H,
          double X, double Y, size_t * I, size_t * J)
     This function finds and sets the indices I and J to the to the bin
     which covers the coordinates (X,Y). The bin is located using a
     binary search.  The search includes an optimization for histograms
     with uniform ranges, and will return the correct bin immediately
     in this case. If (x,y) is found then the function sets the indices
     (I,J) and returns `GSL_SUCCESS'.  If (x,y) lies outside the valid
     range of the histogram then the function returns `GSL_EDOM' and
     the error handler is invoked.


File: gsl-ref.info,  Node: 2D Histogram Statistics,  Next: 2D Histogram Operations,  Prev: Searching 2D histogram ranges,  Up: Histograms

21.18 2D Histogram Statistics
=============================

 -- Function: double gsl_histogram2d_max_val (const gsl_histogram2d * H)
     This function returns the maximum value contained in the histogram
     bins.

 -- Function: void gsl_histogram2d_max_bin (const gsl_histogram2d * H,
          size_t * I, size_t * J)
     This function finds the indices of the bin containing the maximum
     value in the histogram H and stores the result in (I,J). In the
     case where several bins contain the same maximum value the first
     bin found is returned.

 -- Function: double gsl_histogram2d_min_val (const gsl_histogram2d * H)
     This function returns the minimum value contained in the histogram
     bins.

 -- Function: void gsl_histogram2d_min_bin (const gsl_histogram2d * H,
          size_t * I, size_t * J)
     This function finds the indices of the bin containing the minimum
     value in the histogram H and stores the result in (I,J). In the
     case where several bins contain the same maximum value the first
     bin found is returned.

 -- Function: double gsl_histogram2d_xmean (const gsl_histogram2d * H)
     This function returns the mean of the histogrammed x variable,
     where the histogram is regarded as a probability distribution.
     Negative bin values are ignored for the purposes of this
     calculation.

 -- Function: double gsl_histogram2d_ymean (const gsl_histogram2d * H)
     This function returns the mean of the histogrammed y variable,
     where the histogram is regarded as a probability distribution.
     Negative bin values are ignored for the purposes of this
     calculation.

 -- Function: double gsl_histogram2d_xsigma (const gsl_histogram2d * H)
     This function returns the standard deviation of the histogrammed x
     variable, where the histogram is regarded as a probability
     distribution. Negative bin values are ignored for the purposes of
     this calculation.

 -- Function: double gsl_histogram2d_ysigma (const gsl_histogram2d * H)
     This function returns the standard deviation of the histogrammed y
     variable, where the histogram is regarded as a probability
     distribution. Negative bin values are ignored for the purposes of
     this calculation.

 -- Function: double gsl_histogram2d_cov (const gsl_histogram2d * H)
     This function returns the covariance of the histogrammed x and y
     variables, where the histogram is regarded as a probability
     distribution. Negative bin values are ignored for the purposes of
     this calculation.

 -- Function: double gsl_histogram2d_sum (const gsl_histogram2d * H)
     This function returns the sum of all bin values. Negative bin
     values are included in the sum.


File: gsl-ref.info,  Node: 2D Histogram Operations,  Next: Reading and writing 2D histograms,  Prev: 2D Histogram Statistics,  Up: Histograms

21.19 2D Histogram Operations
=============================

 -- Function: int gsl_histogram2d_equal_bins_p (const gsl_histogram2d *
          H1, const gsl_histogram2d * H2)
     This function returns 1 if all the individual bin ranges of the two
     histograms are identical, and 0 otherwise.

 -- Function: int gsl_histogram2d_add (gsl_histogram2d * H1, const
          gsl_histogram2d * H2)
     This function adds the contents of the bins in histogram H2 to the
     corresponding bins of histogram H1, i.e. h'_1(i,j) = h_1(i,j) +
     h_2(i,j).  The two histograms must have identical bin ranges.

 -- Function: int gsl_histogram2d_sub (gsl_histogram2d * H1, const
          gsl_histogram2d * H2)
     This function subtracts the contents of the bins in histogram H2
     from the corresponding bins of histogram H1, i.e. h'_1(i,j) =
     h_1(i,j) - h_2(i,j).  The two histograms must have identical bin
     ranges.

 -- Function: int gsl_histogram2d_mul (gsl_histogram2d * H1, const
          gsl_histogram2d * H2)
     This function multiplies the contents of the bins of histogram H1
     by the contents of the corresponding bins in histogram H2, i.e.
     h'_1(i,j) = h_1(i,j) * h_2(i,j).  The two histograms must have
     identical bin ranges.

 -- Function: int gsl_histogram2d_div (gsl_histogram2d * H1, const
          gsl_histogram2d * H2)
     This function divides the contents of the bins of histogram H1 by
     the contents of the corresponding bins in histogram H2, i.e.
     h'_1(i,j) = h_1(i,j) / h_2(i,j).  The two histograms must have
     identical bin ranges.

 -- Function: int gsl_histogram2d_scale (gsl_histogram2d * H, double
          SCALE)
     This function multiplies the contents of the bins of histogram H
     by the constant SCALE, i.e. h'_1(i,j) = h_1(i,j) scale.

 -- Function: int gsl_histogram2d_shift (gsl_histogram2d * H, double
          OFFSET)
     This function shifts the contents of the bins of histogram H by
     the constant OFFSET, i.e. h'_1(i,j) = h_1(i,j) + offset.


File: gsl-ref.info,  Node: Reading and writing 2D histograms,  Next: Resampling from 2D histograms,  Prev: 2D Histogram Operations,  Up: Histograms

21.20 Reading and writing 2D histograms
=======================================

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

 -- Function: int gsl_histogram2d_fwrite (FILE * STREAM, const
          gsl_histogram2d * H)
     This function writes the ranges and bins of the histogram H to the
     stream STREAM in binary format.  The return value is 0 for success
     and `GSL_EFAILED' if there was a problem writing to the file.
     Since the data is written in the native binary format it may not
     be portable between different architectures.

 -- Function: int gsl_histogram2d_fread (FILE * STREAM, gsl_histogram2d
          * H)
     This function reads into the histogram H from the stream STREAM in
     binary format.  The histogram H must be preallocated with the
     correct size since the function uses the number of x and y bins in
     H to determine how many bytes to read.  The return value is 0 for
     success and `GSL_EFAILED' if there was a problem reading from the
     file.  The data is assumed to have been written in the native
     binary format on the same architecture.

 -- Function: int gsl_histogram2d_fprintf (FILE * STREAM, const
          gsl_histogram2d * H, const char * RANGE_FORMAT, const char *
          BIN_FORMAT)
     This function writes the ranges and bins of the histogram H
     line-by-line to the stream STREAM using the format specifiers
     RANGE_FORMAT and BIN_FORMAT.  These should be one of the `%g',
     `%e' or `%f' formats for floating point numbers.  The function
     returns 0 for success and `GSL_EFAILED' if there was a problem
     writing to the file.  The histogram output is formatted in five
     columns, and the columns are separated by spaces, like this,

          xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0)
          xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1)
          xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2)
          ....
          xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1)

          xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0)
          xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1)
          xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2)
          ....
          xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1)

          ....

          xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0)
          xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1)
          xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2)
          ....
          xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)

     Each line contains the lower and upper limits of the bin and the
     contents of the bin.  Since the upper limits of the each bin are
     the lower limits of the neighboring bins there is duplication of
     these values but this allows the histogram to be manipulated with
     line-oriented tools.

 -- Function: int gsl_histogram2d_fscanf (FILE * STREAM,
          gsl_histogram2d * H)
     This function reads formatted data from the stream STREAM into the
     histogram H.  The data is assumed to be in the five-column format
     used by `gsl_histogram_fprintf'.  The histogram H must be
     preallocated with the correct lengths since the function uses the
     sizes of H to determine how many numbers to read.  The function
     returns 0 for success and `GSL_EFAILED' if there was a problem
     reading from the file.


File: gsl-ref.info,  Node: Resampling from 2D histograms,  Next: Example programs for 2D histograms,  Prev: Reading and writing 2D histograms,  Up: Histograms

21.21 Resampling from 2D histograms
===================================

As in the one-dimensional case, a two-dimensional histogram made by
counting events can be regarded as a measurement of a probability
distribution.  Allowing for statistical error, the height of each bin
represents the probability of an event where (x,y) falls in the range
of that bin.  For a two-dimensional histogram the probability
distribution takes the form p(x,y) dx dy where,

     p(x,y) = n_{ij}/ (N A_{ij})

In this equation n_{ij} is the number of events in the bin which
contains (x,y), A_{ij} is the area of the bin and N is the total number
of events.  The distribution of events within each bin is assumed to be
uniform.

 -- Data Type: gsl_histogram2d_pdf
    `size_t nx, ny'
          This is the number of histogram bins used to approximate the
          probability distribution function in the x and y directions.

    `double * xrange'
          The ranges of the bins in the x-direction are stored in an
          array of NX + 1 elements pointed to by XRANGE.

    `double * yrange'
          The ranges of the bins in the y-direction are stored in an
          array of NY + 1 pointed to by YRANGE.

    `double * sum'
          The cumulative probability for the bins is stored in an array
          of NX*NY elements pointed to by SUM.

The following functions allow you to create a `gsl_histogram2d_pdf'
struct which represents a two dimensional probability distribution and
generate random samples from it.

 -- Function: gsl_histogram2d_pdf * gsl_histogram2d_pdf_alloc (size_t
          NX, size_t NY)
     This function allocates memory for a two-dimensional probability
     distribution of size NX-by-NY and returns a pointer to a newly
     initialized `gsl_histogram2d_pdf' struct. If insufficient memory
     is available a null pointer is returned and the error handler is
     invoked with an error code of `GSL_ENOMEM'.

 -- Function: int gsl_histogram2d_pdf_init (gsl_histogram2d_pdf * P,
          const gsl_histogram2d * H)
     This function initializes the two-dimensional probability
     distribution calculated P from the histogram H.  If any of the
     bins of H are negative then the error handler is invoked with an
     error code of `GSL_EDOM' because a probability distribution cannot
     contain negative values.

 -- Function: void gsl_histogram2d_pdf_free (gsl_histogram2d_pdf * P)
     This function frees the two-dimensional probability distribution
     function P and all of the memory associated with it.

 -- Function: int gsl_histogram2d_pdf_sample (const gsl_histogram2d_pdf
          * P, double R1, double R2, double * X, double * Y)
     This function uses two uniform random numbers between zero and one,
     R1 and R2, to compute a single random sample from the
     two-dimensional probability distribution P.


File: gsl-ref.info,  Node: Example programs for 2D histograms,  Prev: Resampling from 2D histograms,  Up: Histograms

21.22 Example programs for 2D histograms
========================================

This program demonstrates two features of two-dimensional histograms.
First a 10-by-10 two-dimensional histogram is created with x and y
running from 0 to 1.  Then a few sample points are added to the
histogram, at (0.3,0.3) with a height of 1, at (0.8,0.1) with a height
of 5 and at (0.7,0.9) with a height of 0.5.  This histogram with three
events is used to generate a random sample of 1000 simulated events,
which are printed out.

     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_histogram2d.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10);

       gsl_histogram2d_set_ranges_uniform (h,
                                           0.0, 1.0,
                                           0.0, 1.0);

       gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
       gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
       gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       {
         int i;
         gsl_histogram2d_pdf * p
           = gsl_histogram2d_pdf_alloc (h->nx, h->ny);

         gsl_histogram2d_pdf_init (p, h);

         for (i = 0; i < 1000; i++) {
           double x, y;
           double u = gsl_rng_uniform (r);
           double v = gsl_rng_uniform (r);

           gsl_histogram2d_pdf_sample (p, u, v, &x, &y);

           printf ("%g %g\n", x, y);
         }
       }

      return 0;
     }



File: gsl-ref.info,  Node: N-tuples,  Next: Monte Carlo Integration,  Prev: Histograms,  Up: Top

22 N-tuples
***********

This chapter describes functions for creating and manipulating
"ntuples", sets of values associated with events.  The ntuples are
stored in files. Their values can be extracted in any combination and
"booked" in a histogram using a selection function.

   The values to be stored are held in a user-defined data structure,
and an ntuple is created associating this data structure with a file.
The values are then written to the file (normally inside a loop) using
the ntuple functions described below.

   A histogram can be created from ntuple data by providing a selection
function and a value function.  The selection function specifies whether
an event should be included in the subset to be analyzed or not. The
value function computes the entry to be added to the histogram for each
event.

   All the ntuple functions are defined in the header file
`gsl_ntuple.h'

* Menu:

* The ntuple struct::
* Creating ntuples::
* Opening an existing ntuple file::
* Writing ntuples::
* Reading ntuples ::
* Closing an ntuple file::
* Histogramming ntuple values::
* Example ntuple programs::
* Ntuple References and Further Reading::


File: gsl-ref.info,  Node: The ntuple struct,  Next: Creating ntuples,  Up: N-tuples

22.1 The ntuple struct
======================

Ntuples are manipulated using the `gsl_ntuple' struct. This struct
contains information on the file where the ntuple data is stored, a
pointer to the current ntuple data row and the size of the user-defined
ntuple data struct.

     typedef struct {
         FILE * file;
         void * ntuple_data;
         size_t size;
     } gsl_ntuple;


File: gsl-ref.info,  Node: Creating ntuples,  Next: Opening an existing ntuple file,  Prev: The ntuple struct,  Up: N-tuples

22.2 Creating ntuples
=====================

 -- Function: gsl_ntuple * gsl_ntuple_create (char * FILENAME, void *
          NTUPLE_DATA, size_t SIZE)
     This function creates a new write-only ntuple file FILENAME for
     ntuples of size SIZE and returns a pointer to the newly created
     ntuple struct.  Any existing file with the same name is truncated
     to zero length and overwritten.  A pointer to memory for the
     current ntuple row NTUPLE_DATA must be supplied--this is used to
     copy ntuples in and out of the file.


File: gsl-ref.info,  Node: Opening an existing ntuple file,  Next: Writing ntuples,  Prev: Creating ntuples,  Up: N-tuples

22.3 Opening an existing ntuple file
====================================

 -- Function: gsl_ntuple * gsl_ntuple_open (char * FILENAME, void *
          NTUPLE_DATA, size_t SIZE)
     This function opens an existing ntuple file FILENAME for reading
     and returns a pointer to a corresponding ntuple struct. The
     ntuples in the file must have size SIZE.  A pointer to memory for
     the current ntuple row NTUPLE_DATA must be supplied--this is used
     to copy ntuples in and out of the file.


File: gsl-ref.info,  Node: Writing ntuples,  Next: Reading ntuples,  Prev: Opening an existing ntuple file,  Up: N-tuples

22.4 Writing ntuples
====================

 -- Function: int gsl_ntuple_write (gsl_ntuple * NTUPLE)
     This function writes the current ntuple NTUPLE->NTUPLE_DATA of
     size NTUPLE->SIZE to the corresponding file.

 -- Function: int gsl_ntuple_bookdata (gsl_ntuple * NTUPLE)
     This function is a synonym for `gsl_ntuple_write'.


File: gsl-ref.info,  Node: Reading ntuples,  Next: Closing an ntuple file,  Prev: Writing ntuples,  Up: N-tuples

22.5 Reading ntuples
====================

 -- Function: int gsl_ntuple_read (gsl_ntuple * NTUPLE)
     This function reads the current row of the ntuple file for NTUPLE
     and stores the values in NTUPLE->DATA.


File: gsl-ref.info,  Node: Closing an ntuple file,  Next: Histogramming ntuple values,  Prev: Reading ntuples,  Up: N-tuples

22.6 Closing an ntuple file
===========================

 -- Function: int gsl_ntuple_close (gsl_ntuple * NTUPLE)
     This function closes the ntuple file NTUPLE and frees its
     associated allocated memory.


File: gsl-ref.info,  Node: Histogramming ntuple values,  Next: Example ntuple programs,  Prev: Closing an ntuple file,  Up: N-tuples

22.7 Histogramming ntuple values
================================

Once an ntuple has been created its contents can be histogrammed in
various ways using the function `gsl_ntuple_project'.  Two user-defined
functions must be provided, a function to select events and a function
to compute scalar values. The selection function and the value function
both accept the ntuple row as a first argument and other parameters as
a second argument.

   The "selection function" determines which ntuple rows are selected
for histogramming.  It is defined by the following struct,
     typedef struct {
       int (* function) (void * ntuple_data, void * params);
       void * params;
     } gsl_ntuple_select_fn;

The struct component FUNCTION should return a non-zero value for each
ntuple row that is to be included in the histogram.

   The "value function" computes scalar values for those ntuple rows
selected by the selection function,
     typedef struct {
       double (* function) (void * ntuple_data, void * params);
       void * params;
     } gsl_ntuple_value_fn;

In this case the struct component FUNCTION should return the value to
be added to the histogram for the ntuple row.

 -- Function: int gsl_ntuple_project (gsl_histogram * H, gsl_ntuple *
          NTUPLE, gsl_ntuple_value_fn * VALUE_FUNC,
          gsl_ntuple_select_fn * SELECT_FUNC)
     This function updates the histogram H from the ntuple NTUPLE using
     the functions VALUE_FUNC and SELECT_FUNC. For each ntuple row
     where the selection function SELECT_FUNC is non-zero the
     corresponding value of that row is computed using the function
     VALUE_FUNC and added to the histogram.  Those ntuple rows where
     SELECT_FUNC returns zero are ignored.  New entries are added to
     the histogram, so subsequent calls can be used to accumulate
     further data in the same histogram.


File: gsl-ref.info,  Node: Example ntuple programs,  Next: Ntuple References and Further Reading,  Prev: Histogramming ntuple values,  Up: N-tuples

22.8 Examples
=============

The following example programs demonstrate the use of ntuples in
managing a large dataset.  The first program creates a set of 10,000
simulated "events", each with 3 associated values (x,y,z).  These are
generated from a gaussian distribution with unit variance, for
demonstration purposes, and written to the ntuple file `test.dat'.

     #include <gsl/gsl_ntuple.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     struct data
     {
       double x;
       double y;
       double z;
     };

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       struct data ntuple_row;
       int i;

       gsl_ntuple *ntuple
         = gsl_ntuple_create ("test.dat", &ntuple_row,
                              sizeof (ntuple_row));

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (i = 0; i < 10000; i++)
         {
           ntuple_row.x = gsl_ran_ugaussian (r);
           ntuple_row.y = gsl_ran_ugaussian (r);
           ntuple_row.z = gsl_ran_ugaussian (r);

           gsl_ntuple_write (ntuple);
         }

       gsl_ntuple_close (ntuple);
       return 0;
     }

The next program analyses the ntuple data in the file `test.dat'.  The
analysis procedure is to compute the squared-magnitude of each event,
E^2=x^2+y^2+z^2, and select only those which exceed a lower limit of
1.5.  The selected events are then histogrammed using their E^2 values.

     #include <math.h>
     #include <gsl/gsl_ntuple.h>
     #include <gsl/gsl_histogram.h>

     struct data
     {
       double x;
       double y;
       double z;
     };

     int sel_func (void *ntuple_data, void *params);
     double val_func (void *ntuple_data, void *params);

     int
     main (void)
     {
       struct data ntuple_row;

       gsl_ntuple *ntuple
         = gsl_ntuple_open ("test.dat", &ntuple_row,
                            sizeof (ntuple_row));
       double lower = 1.5;

       gsl_ntuple_select_fn S;
       gsl_ntuple_value_fn V;

       gsl_histogram *h = gsl_histogram_alloc (100);
       gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);

       S.function = &sel_func;
       S.params = &lower;

       V.function = &val_func;
       V.params = 0;

       gsl_ntuple_project (h, ntuple, &V, &S);
       gsl_histogram_fprintf (stdout, h, "%f", "%f");
       gsl_histogram_free (h);
       gsl_ntuple_close (ntuple);

       return 0;
     }

     int
     sel_func (void *ntuple_data, void *params)
     {
       struct data * data = (struct data *) ntuple_data;
       double x, y, z, E2, scale;
       scale = *(double *) params;

       x = data->x;
       y = data->y;
       z = data->z;

       E2 = x * x + y * y + z * z;

       return E2 > scale;
     }

     double
     val_func (void *ntuple_data, void *params)
     {
       struct data * data = (struct data *) ntuple_data;
       double x, y, z;

       x = data->x;
       y = data->y;
       z = data->z;

       return x * x + y * y + z * z;
     }

   The following plot shows the distribution of the selected events.
Note the cut-off at the lower bound.


File: gsl-ref.info,  Node: Ntuple References and Further Reading,  Prev: Example ntuple programs,  Up: N-tuples

22.9 References and Further Reading
===================================

Further information on the use of ntuples can be found in the
documentation for the CERN packages PAW and HBOOK (available online).


File: gsl-ref.info,  Node: Monte Carlo Integration,  Next: Simulated Annealing,  Prev: N-tuples,  Up: Top

23 Monte Carlo Integration
**************************

This chapter describes routines for multidimensional Monte Carlo
integration.  These include the traditional Monte Carlo method and
adaptive algorithms such as VEGAS and MISER which use importance
sampling and stratified sampling techniques. Each algorithm computes an
estimate of a multidimensional definite integral of the form,

     I = \int_xl^xu dx \int_yl^yu  dy ...  f(x, y, ...)

over a hypercubic region ((x_l,x_u), (y_l,y_u), ...) using a fixed
number of function calls.  The routines also provide a statistical
estimate of the error on the result.  This error estimate should be
taken as a guide rather than as a strict error bound--random sampling
of the region may not uncover all the important features of the
function, resulting in an underestimate of the error.

   The functions are defined in separate header files for each routine,
`gsl_monte_plain.h', `gsl_monte_miser.h' and `gsl_monte_vegas.h'.

* Menu:

* Monte Carlo Interface::
* PLAIN Monte Carlo::
* MISER::
* VEGAS::
* Monte Carlo Examples::
* Monte Carlo Integration References and Further Reading::


File: gsl-ref.info,  Node: Monte Carlo Interface,  Next: PLAIN Monte Carlo,  Up: Monte Carlo Integration

23.1 Interface
==============

All of the Monte Carlo integration routines use the same general form of
interface.  There is an allocator to allocate memory for control
variables and workspace, a routine to initialize those control
variables, the integrator itself, and a function to free the space when
done.

   Each integration function requires a random number generator to be
supplied, and returns an estimate of the integral and its standard
deviation.  The accuracy of the result is determined by the number of
function calls specified by the user.  If a known level of accuracy is
required this can be achieved by calling the integrator several times
and averaging the individual results until the desired accuracy is
obtained.

   Random sample points used within the Monte Carlo routines are always
chosen strictly within the integration region, so that endpoint
singularities are automatically avoided.

   The function to be integrated has its own datatype, defined in the
header file `gsl_monte.h'.

 -- Data Type: gsl_monte_function
     This data type defines a general function with parameters for Monte
     Carlo integration.

    `double (* f) (double * X, size_t DIM, void * PARAMS)'
          this function should return the value f(x,params) for the
          argument X and parameters PARAMS, where X is an array of size
          DIM giving the coordinates of the point where the function is
          to be evaluated.

    `size_t dim'
          the number of dimensions for X.

    `void * params'
          a pointer to the parameters of the function.

Here is an example for a quadratic function in two dimensions,

     f(x,y) = a x^2 + b x y + c y^2

with a = 3, b = 2, c = 1.  The following code defines a
`gsl_monte_function' `F' which you could pass to an integrator:

     struct my_f_params { double a; double b; double c; };

     double
     my_f (double x[], size_t dim, void * p) {
        struct my_f_params * fp = (struct my_f_params *)p;

        if (dim != 2)
           {
             fprintf (stderr, "error: dim != 2");
             abort ();
           }

        return  fp->a * x[0] * x[0]
                  + fp->b * x[0] * x[1]
                    + fp->c * x[1] * x[1];
     }

     gsl_monte_function F;
     struct my_f_params params = { 3.0, 2.0, 1.0 };

     F.f = &my_f;
     F.dim = 2;
     F.params = &params;

The function f(x) can be evaluated using the following macro,

     #define GSL_MONTE_FN_EVAL(F,x)
         (*((F)->f))(x,(F)->dim,(F)->params)


File: gsl-ref.info,  Node: PLAIN Monte Carlo,  Next: MISER,  Prev: Monte Carlo Interface,  Up: Monte Carlo Integration

23.2 PLAIN Monte Carlo
======================

The plain Monte Carlo algorithm samples points randomly from the
integration region to estimate the integral and its error.  Using this
algorithm the estimate of the integral E(f; N) for N randomly
distributed points x_i is given by,

     E(f; N) = =  V <f> = (V / N) \sum_i^N f(x_i)

where V is the volume of the integration region.  The error on this
estimate \sigma(E;N) is calculated from the estimated variance of the
mean,

     \sigma^2 (E; N) = (V / N) \sum_i^N (f(x_i) -  <f>)^2.

For large N this variance decreases asymptotically as \Var(f)/N, where
\Var(f) is the true variance of the function over the integration
region.  The error estimate itself should decrease as
\sigma(f)/\sqrt{N}.  The familiar law of errors decreasing as
1/\sqrt{N} applies--to reduce the error by a factor of 10 requires a
100-fold increase in the number of sample points.

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

 -- Function: gsl_monte_plain_state * gsl_monte_plain_alloc (size_t DIM)
     This function allocates and initializes a workspace for Monte Carlo
     integration in DIM dimensions.

 -- Function: int gsl_monte_plain_init (gsl_monte_plain_state* S)
     This function initializes a previously allocated integration state.
     This allows an existing workspace to be reused for different
     integrations.

 -- Function: int gsl_monte_plain_integrate (gsl_monte_function * F,
          double * XL, double * XU, size_t DIM, size_t CALLS, gsl_rng *
          R, gsl_monte_plain_state * S, double * RESULT, double *
          ABSERR)
     This routines uses the plain Monte Carlo algorithm to integrate the
     function F over the DIM-dimensional hypercubic region defined by
     the lower and upper limits in the arrays XL and XU, each of size
     DIM.  The integration uses a fixed number of function calls CALLS,
     and obtains random sampling points using the random number
     generator R. A previously allocated workspace S must be supplied.
     The result of the integration is returned in RESULT, with an
     estimated absolute error ABSERR.

 -- Function: void gsl_monte_plain_free (gsl_monte_plain_state * S)
     This function frees the memory associated with the integrator state
     S.


File: gsl-ref.info,  Node: MISER,  Next: VEGAS,  Prev: PLAIN Monte Carlo,  Up: Monte Carlo Integration

23.3 MISER
==========

The MISER algorithm of Press and Farrar is based on recursive
stratified sampling.  This technique aims to reduce the overall
integration error by concentrating integration points in the regions of
highest variance.

   The idea of stratified sampling begins with the observation that for
two disjoint regions a and b with Monte Carlo estimates of the integral
E_a(f) and E_b(f) and variances \sigma_a^2(f) and \sigma_b^2(f), the
variance \Var(f) of the combined estimate E(f) = (1/2) (E_a(f) + E_b(f))
is given by,

     \Var(f) = (\sigma_a^2(f) / 4 N_a) + (\sigma_b^2(f) / 4 N_b).

It can be shown that this variance is minimized by distributing the
points such that,

     N_a / (N_a + N_b) = \sigma_a / (\sigma_a + \sigma_b).

Hence the smallest error estimate is obtained by allocating sample
points in proportion to the standard deviation of the function in each
sub-region.

   The MISER algorithm proceeds by bisecting the integration region
along one coordinate axis to give two sub-regions at each step.  The
direction is chosen by examining all d possible bisections and
selecting the one which will minimize the combined variance of the two
sub-regions.  The variance in the sub-regions is estimated by sampling
with a fraction of the total number of points available to the current
step.  The same procedure is then repeated recursively for each of the
two half-spaces from the best bisection. The remaining sample points are
allocated to the sub-regions using the formula for N_a and N_b.  This
recursive allocation of integration points continues down to a
user-specified depth where each sub-region is integrated using a plain
Monte Carlo estimate.  These individual values and their error
estimates are then combined upwards to give an overall result and an
estimate of its error.

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

 -- Function: gsl_monte_miser_state * gsl_monte_miser_alloc (size_t DIM)
     This function allocates and initializes a workspace for Monte Carlo
     integration in DIM dimensions.  The workspace is used to maintain
     the state of the integration.

 -- Function: int gsl_monte_miser_init (gsl_monte_miser_state* S)
     This function initializes a previously allocated integration state.
     This allows an existing workspace to be reused for different
     integrations.

 -- Function: int gsl_monte_miser_integrate (gsl_monte_function * F,
          double * XL, double * XU, size_t DIM, size_t CALLS, gsl_rng *
          R, gsl_monte_miser_state * S, double * RESULT, double *
          ABSERR)
     This routines uses the MISER Monte Carlo algorithm to integrate the
     function F over the DIM-dimensional hypercubic region defined by
     the lower and upper limits in the arrays XL and XU, each of size
     DIM.  The integration uses a fixed number of function calls CALLS,
     and obtains random sampling points using the random number
     generator R. A previously allocated workspace S must be supplied.
     The result of the integration is returned in RESULT, with an
     estimated absolute error ABSERR.

 -- Function: void gsl_monte_miser_free (gsl_monte_miser_state * S)
     This function frees the memory associated with the integrator state
     S.

   The MISER algorithm has several configurable parameters. The
following variables can be accessed through the `gsl_monte_miser_state'
struct,

 -- Variable: double estimate_frac
     This parameter specifies the fraction of the currently available
     number of function calls which are allocated to estimating the
     variance at each recursive step. The default value is 0.1.

 -- Variable: size_t min_calls
     This parameter specifies the minimum number of function calls
     required for each estimate of the variance. If the number of
     function calls allocated to the estimate using ESTIMATE_FRAC falls
     below MIN_CALLS then MIN_CALLS are used instead.  This ensures
     that each estimate maintains a reasonable level of accuracy.  The
     default value of MIN_CALLS is `16 * dim'.

 -- Variable: size_t min_calls_per_bisection
     This parameter specifies the minimum number of function calls
     required to proceed with a bisection step.  When a recursive step
     has fewer calls available than MIN_CALLS_PER_BISECTION it performs
     a plain Monte Carlo estimate of the current sub-region and
     terminates its branch of the recursion.  The default value of this
     parameter is `32 * min_calls'.

 -- Variable: double alpha
     This parameter controls how the estimated variances for the two
     sub-regions of a bisection are combined when allocating points.
     With recursive sampling the overall variance should scale better
     than 1/N, since the values from the sub-regions will be obtained
     using a procedure which explicitly minimizes their variance.  To
     accommodate this behavior the MISER algorithm allows the total
     variance to depend on a scaling parameter \alpha,

          \Var(f) = {\sigma_a \over N_a^\alpha} + {\sigma_b \over N_b^\alpha}.

     The authors of the original paper describing MISER recommend the
     value \alpha = 2 as a good choice, obtained from numerical
     experiments, and this is used as the default value in this
     implementation.

 -- Variable: double dither
     This parameter introduces a random fractional variation of size
     DITHER into each bisection, which can be used to break the
     symmetry of integrands which are concentrated near the exact
     center of the hypercubic integration region.  The default value of
     dither is zero, so no variation is introduced. If needed, a
     typical value of DITHER is 0.1.


File: gsl-ref.info,  Node: VEGAS,  Next: Monte Carlo Examples,  Prev: MISER,  Up: Monte Carlo Integration

23.4 VEGAS
==========

The VEGAS algorithm of Lepage is based on importance sampling.  It
samples points from the probability distribution described by the
function |f|, so that the points are concentrated in the regions that
make the largest contribution to the integral.

   In general, if the Monte Carlo integral of f is sampled with points
distributed according to a probability distribution described by the
function g, we obtain an estimate E_g(f; N),

     E_g(f; N) = E(f/g; N)

with a corresponding variance,

     \Var_g(f; N) = \Var(f/g; N).

If the probability distribution is chosen as g = |f|/I(|f|) then it can
be shown that the variance V_g(f; N) vanishes, and the error in the
estimate will be zero.  In practice it is not possible to sample from
the exact distribution g for an arbitrary function, so importance
sampling algorithms aim to produce efficient approximations to the
desired distribution.

   The VEGAS algorithm approximates the exact distribution by making a
number of passes over the integration region while histogramming the
function f. Each histogram is used to define a sampling distribution
for the next pass.  Asymptotically this procedure converges to the
desired distribution. In order to avoid the number of histogram bins
growing like K^d the probability distribution is approximated by a
separable function: g(x_1, x_2, ...) = g_1(x_1) g_2(x_2) ...  so that
the number of bins required is only Kd.  This is equivalent to locating
the peaks of the function from the projections of the integrand onto
the coordinate axes.  The efficiency of VEGAS depends on the validity
of this assumption.  It is most efficient when the peaks of the
integrand are well-localized.  If an integrand can be rewritten in a
form which is approximately separable this will increase the efficiency
of integration with VEGAS.

   VEGAS incorporates a number of additional features, and combines both
stratified sampling and importance sampling.  The integration region is
divided into a number of "boxes", with each box getting a fixed number
of points (the goal is 2).  Each box can then have a fractional number
of bins, but if the ratio of bins-per-box is less than two, Vegas
switches to a kind variance reduction (rather than importance sampling).

 -- Function: gsl_monte_vegas_state * gsl_monte_vegas_alloc (size_t DIM)
     This function allocates and initializes a workspace for Monte Carlo
     integration in DIM dimensions.  The workspace is used to maintain
     the state of the integration.

 -- Function: int gsl_monte_vegas_init (gsl_monte_vegas_state* S)
     This function initializes a previously allocated integration state.
     This allows an existing workspace to be reused for different
     integrations.

 -- Function: int gsl_monte_vegas_integrate (gsl_monte_function * F,
          double * XL, double * XU, size_t DIM, size_t CALLS, gsl_rng *
          R, gsl_monte_vegas_state * S, double * RESULT, double *
          ABSERR)
     This routines uses the VEGAS Monte Carlo algorithm to integrate the
     function F over the DIM-dimensional hypercubic region defined by
     the lower and upper limits in the arrays XL and XU, each of size
     DIM.  The integration uses a fixed number of function calls CALLS,
     and obtains random sampling points using the random number
     generator R. A previously allocated workspace S must be supplied.
     The result of the integration is returned in RESULT, with an
     estimated absolute error ABSERR.  The result and its error
     estimate are based on a weighted average of independent samples.
     The chi-squared per degree of freedom for the weighted average is
     returned via the state struct component, S->CHISQ, and must be
     consistent with 1 for the weighted average to be reliable.

 -- Function: void gsl_monte_vegas_free (gsl_monte_vegas_state * S)
     This function frees the memory associated with the integrator state
     S.

   The VEGAS algorithm computes a number of independent estimates of the
integral internally, according to the `iterations' parameter described
below, and returns their weighted average.  Random sampling of the
integrand can occasionally produce an estimate where the error is zero,
particularly if the function is constant in some regions. An estimate
with zero error causes the weighted average to break down and must be
handled separately. In the original Fortran implementations of VEGAS
the error estimate is made non-zero by substituting a small value
(typically `1e-30').  The implementation in GSL differs from this and
avoids the use of an arbitrary constant--it either assigns the value a
weight which is the average weight of the preceding estimates or
discards it according to the following procedure,

current estimate has zero error, weighted average has finite error
     The current estimate is assigned a weight which is the average
     weight of the preceding estimates.

current estimate has finite error, previous estimates had zero error
     The previous estimates are discarded and the weighted averaging
     procedure begins with the current estimate.

current estimate has zero error, previous estimates had zero error
     The estimates are averaged using the arithmetic mean, but no error
     is computed.

   The VEGAS algorithm is highly configurable. The following variables
can be accessed through the `gsl_monte_vegas_state' struct,

 -- Variable: double result
 -- Variable: double sigma
     These parameters contain the raw value of the integral RESULT and
     its error SIGMA from the last iteration of the algorithm.

 -- Variable: double chisq
     This parameter gives the chi-squared per degree of freedom for the
     weighted estimate of the integral.  The value of CHISQ should be
     close to 1.  A value of CHISQ which differs significantly from 1
     indicates that the values from different iterations are
     inconsistent.  In this case the weighted error will be
     under-estimated, and further iterations of the algorithm are
     needed to obtain reliable results.

 -- Variable: double alpha
     The parameter `alpha' controls the stiffness of the rebinning
     algorithm.  It is typically set between one and two. A value of
     zero prevents rebinning of the grid.  The default value is 1.5.

 -- Variable: size_t iterations
     The number of iterations to perform for each call to the routine.
     The default value is 5 iterations.

 -- Variable: int stage
     Setting this determines the "stage" of the calculation.  Normally,
     `stage = 0' which begins with a new uniform grid and empty weighted
     average.  Calling vegas with `stage = 1' retains the grid from the
     previous run but discards the weighted average, so that one can
     "tune" the grid using a relatively small number of points and then
     do a large run with `stage = 1' on the optimized grid.  Setting
     `stage = 2' keeps the grid and the weighted average from the
     previous run, but may increase (or decrease) the number of
     histogram bins in the grid depending on the number of calls
     available.  Choosing `stage = 3' enters at the main loop, so that
     nothing is changed, and is equivalent to performing additional
     iterations in a previous call.

 -- Variable: int mode
     The possible choices are `GSL_VEGAS_MODE_IMPORTANCE',
     `GSL_VEGAS_MODE_STRATIFIED', `GSL_VEGAS_MODE_IMPORTANCE_ONLY'.
     This determines whether VEGAS will use importance sampling or
     stratified sampling, or whether it can pick on its own.  In low
     dimensions VEGAS uses strict stratified sampling (more precisely,
     stratified sampling is chosen if there are fewer than 2 bins per
     box).

 -- Variable: int verbose
 -- Variable: FILE * ostream
     These parameters set the level of information printed by VEGAS. All
     information is written to the stream OSTREAM.  The default setting
     of VERBOSE is `-1', which turns off all output.  A VERBOSE value
     of `0' prints summary information about the weighted average and
     final result, while a value of `1' also displays the grid
     coordinates.  A value of `2' prints information from the rebinning
     procedure for each iteration.


File: gsl-ref.info,  Node: Monte Carlo Examples,  Next: Monte Carlo Integration References and Further Reading,  Prev: VEGAS,  Up: Monte Carlo Integration

23.5 Examples
=============

The example program below uses the Monte Carlo routines to estimate the
value of the following 3-dimensional integral from the theory of random
walks,

     I = \int_{-pi}^{+pi} {dk_x/(2 pi)}
         \int_{-pi}^{+pi} {dk_y/(2 pi)}
         \int_{-pi}^{+pi} {dk_z/(2 pi)}
          1 / (1 - cos(k_x)cos(k_y)cos(k_z)).

The analytic value of this integral can be shown to be I =
\Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859....  The integral gives
the mean time spent at the origin by a random walk on a body-centered
cubic lattice in three dimensions.

   For simplicity we will compute the integral over the region (0,0,0)
to (\pi,\pi,\pi) and multiply by 8 to obtain the full result.  The
integral is slowly varying in the middle of the region but has
integrable singularities at the corners (0,0,0), (0,\pi,\pi),
(\pi,0,\pi) and (\pi,\pi,0).  The Monte Carlo routines only select
points which are strictly within the integration region and so no
special measures are needed to avoid these singularities.

     #include <stdlib.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_monte.h>
     #include <gsl/gsl_monte_plain.h>
     #include <gsl/gsl_monte_miser.h>
     #include <gsl/gsl_monte_vegas.h>

     /* Computation of the integral,

           I = int (dx dy dz)/(2pi)^3  1/(1-cos(x)cos(y)cos(z))

        over (-pi,-pi,-pi) to (+pi, +pi, +pi).  The exact answer
        is Gamma(1/4)^4/(4 pi^3).  This example is taken from
        C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
        Volume 1", Section 1.1, p21, which cites the original
        paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
        1800 (1977) */

     /* For simplicity we compute the integral over the region
        (0,0,0) -> (pi,pi,pi) and multiply by 8 */

     double exact = 1.3932039296856768591842462603255;

     double
     g (double *k, size_t dim, void *params)
     {
       double A = 1.0 / (M_PI * M_PI * M_PI);
       return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
     }

     void
     display_results (char *title, double result, double error)
     {
       printf ("%s ==================\n", title);
       printf ("result = % .6f\n", result);
       printf ("sigma  = % .6f\n", error);
       printf ("exact  = % .6f\n", exact);
       printf ("error  = % .6f = %.1g sigma\n", result - exact,
               fabs (result - exact) / error);
     }

     int
     main (void)
     {
       double res, err;

       double xl[3] = { 0, 0, 0 };
       double xu[3] = { M_PI, M_PI, M_PI };

       const gsl_rng_type *T;
       gsl_rng *r;

       gsl_monte_function G = { &g, 3, 0 };

       size_t calls = 500000;

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       {
         gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
         gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s,
                                    &res, &err);
         gsl_monte_plain_free (s);

         display_results ("plain", res, err);
       }

       {
         gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
         gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
                                    &res, &err);
         gsl_monte_miser_free (s);

         display_results ("miser", res, err);
       }

       {
         gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);

         gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
                                    &res, &err);
         display_results ("vegas warm-up", res, err);

         printf ("converging...\n");

         do
           {
             gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
                                        &res, &err);
             printf ("result = % .6f sigma = % .6f "
                     "chisq/dof = %.1f\n", res, err, s->chisq);
           }
         while (fabs (s->chisq - 1.0) > 0.5);

         display_results ("vegas final", res, err);

         gsl_monte_vegas_free (s);
       }
       return 0;
     }

With 500,000 function calls the plain Monte Carlo algorithm achieves a
fractional error of 0.6%.  The estimated error `sigma' is consistent
with the actual error, and the computed result differs from the true
result by about one standard deviation,

     plain ==================
     result =  1.385867
     sigma  =  0.007938
     exact  =  1.393204
     error  = -0.007337 = 0.9 sigma

The MISER algorithm reduces the error by a factor of two, and also
correctly estimates the error,

     miser ==================
     result =  1.390656
     sigma  =  0.003743
     exact  =  1.393204
     error  = -0.002548 = 0.7 sigma

In the case of the VEGAS algorithm the program uses an initial warm-up
run of 10,000 function calls to prepare, or "warm up", the grid.  This
is followed by a main run with five iterations of 100,000 function
calls. The chi-squared per degree of freedom for the five iterations are
checked for consistency with 1, and the run is repeated if the results
have not converged. In this case the estimates are consistent on the
first pass.

     vegas warm-up ==================
     result =  1.386925
     sigma  =  0.002651
     exact  =  1.393204
     error  = -0.006278 = 2 sigma
     converging...
     result =  1.392957 sigma =  0.000452 chisq/dof = 1.1
     vegas final ==================
     result =  1.392957
     sigma  =  0.000452
     exact  =  1.393204
     error  = -0.000247 = 0.5 sigma

If the value of `chisq' had differed significantly from 1 it would
indicate inconsistent results, with a correspondingly underestimated
error.  The final estimate from VEGAS (using a similar number of
function calls) is significantly more accurate than the other two
algorithms.


File: gsl-ref.info,  Node: Monte Carlo Integration References and Further Reading,  Prev: Monte Carlo Examples,  Up: Monte Carlo Integration

23.6 References and Further Reading
===================================

The MISER algorithm is described in the following article by Press and
Farrar,

     W.H. Press, G.R. Farrar, `Recursive Stratified Sampling for
     Multidimensional Monte Carlo Integration', Computers in Physics,
     v4 (1990), pp190-195.

The VEGAS algorithm is described in the following papers,

     G.P. Lepage, `A New Algorithm for Adaptive Multidimensional
     Integration', Journal of Computational Physics 27, 192-203, (1978)

     G.P. Lepage, `VEGAS: An Adaptive Multi-dimensional Integration
     Program', Cornell preprint CLNS 80-447, March 1980


File: gsl-ref.info,  Node: Simulated Annealing,  Next: Ordinary Differential Equations,  Prev: Monte Carlo Integration,  Up: Top

24 Simulated Annealing
**********************

Stochastic search techniques are used when the structure of a space is
not well understood or is not smooth, so that techniques like Newton's
method (which requires calculating Jacobian derivative matrices) cannot
be used. In particular, these techniques are frequently used to solve
combinatorial optimization problems, such as the traveling salesman
problem.

   The goal is to find a point in the space at which a real valued
"energy function" (or "cost function") is minimized.  Simulated
annealing is a minimization technique which has given good results in
avoiding local minima; it is based on the idea of taking a random walk
through the space at successively lower temperatures, where the
probability of taking a step is given by a Boltzmann distribution.

   The functions described in this chapter are declared in the header
file `gsl_siman.h'.

* Menu:

* Simulated Annealing algorithm::
* Simulated Annealing functions::
* Examples with Simulated Annealing::
* Simulated Annealing References and Further Reading::


File: gsl-ref.info,  Node: Simulated Annealing algorithm,  Next: Simulated Annealing functions,  Up: Simulated Annealing

24.1 Simulated Annealing algorithm
==================================

The simulated annealing algorithm takes random walks through the problem
space, looking for points with low energies; in these random walks, the
probability of taking a step is determined by the Boltzmann
distribution,

     p = e^{-(E_{i+1} - E_i)/(kT)}

if E_{i+1} > E_i, and p = 1 when E_{i+1} <= E_i.

   In other words, a step will occur if the new energy is lower.  If
the new energy is higher, the transition can still occur, and its
likelihood is proportional to the temperature T and inversely
proportional to the energy difference E_{i+1} - E_i.

   The temperature T is initially set to a high value, and a random
walk is carried out at that temperature.  Then the temperature is
lowered very slightly according to a "cooling schedule", for example: T
-> T/mu_T where \mu_T is slightly greater than 1.  

   The slight probability of taking a step that gives higher energy is
what allows simulated annealing to frequently get out of local minima.


File: gsl-ref.info,  Node: Simulated Annealing functions,  Next: Examples with Simulated Annealing,  Prev: Simulated Annealing algorithm,  Up: Simulated Annealing

24.2 Simulated Annealing functions
==================================

 -- Function: void gsl_siman_solve (const gsl_rng * R, void * X0_P,
          gsl_siman_Efunc_t EF, gsl_siman_step_t TAKE_STEP,
          gsl_siman_metric_t DISTANCE, gsl_siman_print_t
          PRINT_POSITION, gsl_siman_copy_t COPYFUNC,
          gsl_siman_copy_construct_t COPY_CONSTRUCTOR,
          gsl_siman_destroy_t DESTRUCTOR, size_t ELEMENT_SIZE,
          gsl_siman_params_t PARAMS)
     This function performs a simulated annealing search through a given
     space.  The space is specified by providing the functions EF and
     DISTANCE.  The simulated annealing steps are generated using the
     random number generator R and the function TAKE_STEP.

     The starting configuration of the system should be given by X0_P.
     The routine offers two modes for updating configurations, a
     fixed-size mode and a variable-size mode.  In the fixed-size mode
     the configuration is stored as a single block of memory of size
     ELEMENT_SIZE.  Copies of this configuration are created, copied
     and destroyed internally using the standard library functions
     `malloc', `memcpy' and `free'.  The function pointers COPYFUNC,
     COPY_CONSTRUCTOR and DESTRUCTOR should be null pointers in
     fixed-size mode.  In the variable-size mode the functions
     COPYFUNC, COPY_CONSTRUCTOR and DESTRUCTOR are used to create, copy
     and destroy configurations internally.  The variable ELEMENT_SIZE
     should be zero in the variable-size mode.

     The PARAMS structure (described below) controls the run by
     providing the temperature schedule and other tunable parameters to
     the algorithm.

     On exit the best result achieved during the search is placed in
     `*X0_P'.  If the annealing process has been successful this should
     be a good approximation to the optimal point in the space.

     If the function pointer PRINT_POSITION is not null, a debugging
     log will be printed to `stdout' with the following columns:

          number_of_iterations temperature x x-(*x0_p) Ef(x)

     and the output of the function PRINT_POSITION itself.  If
     PRINT_POSITION is null then no information is printed.

The simulated annealing routines require several user-specified
functions to define the configuration space and energy function.  The
prototypes for these functions are given below.

 -- Data Type: gsl_siman_Efunc_t
     This function type should return the energy of a configuration XP.

          double (*gsl_siman_Efunc_t) (void *xp)

 -- Data Type: gsl_siman_step_t
     This function type should modify the configuration XP using a
     random step taken from the generator R, up to a maximum distance of
     STEP_SIZE.

          void (*gsl_siman_step_t) (const gsl_rng *r, void *xp,
                                    double step_size)

 -- Data Type: gsl_siman_metric_t
     This function type should return the distance between two
     configurations XP and YP.

          double (*gsl_siman_metric_t) (void *xp, void *yp)

 -- Data Type: gsl_siman_print_t
     This function type should print the contents of the configuration
     XP.

          void (*gsl_siman_print_t) (void *xp)

 -- Data Type: gsl_siman_copy_t
     This function type should copy the configuration SOURCE into DEST.

          void (*gsl_siman_copy_t) (void *source, void *dest)

 -- Data Type: gsl_siman_copy_construct_t
     This function type should create a new copy of the configuration
     XP.

          void * (*gsl_siman_copy_construct_t) (void *xp)

 -- Data Type: gsl_siman_destroy_t
     This function type should destroy the configuration XP, freeing its
     memory.

          void (*gsl_siman_destroy_t) (void *xp)

 -- Data Type: gsl_siman_params_t
     These are the parameters that control a run of `gsl_siman_solve'.
     This structure contains all the information needed to control the
     search, beyond the energy function, the step function and the
     initial guess.

    `int n_tries'
          The number of points to try for each step.

    `int iters_fixed_T'
          The number of iterations at each temperature.

    `double step_size'
          The maximum step size in the random walk.

    `double k, t_initial, mu_t, t_min'
          The parameters of the Boltzmann distribution and cooling
          schedule.


File: gsl-ref.info,  Node: Examples with Simulated Annealing,  Next: Simulated Annealing References and Further Reading,  Prev: Simulated Annealing functions,  Up: Simulated Annealing

24.3 Examples
=============

The simulated annealing package is clumsy, and it has to be because it
is written in C, for C callers, and tries to be polymorphic at the same
time.  But here we provide some examples which can be pasted into your
application with little change and should make things easier.

* Menu:

* Trivial example::
* Traveling Salesman Problem::


File: gsl-ref.info,  Node: Trivial example,  Next: Traveling Salesman Problem,  Up: Examples with Simulated Annealing

24.3.1 Trivial example
----------------------

The first example, in one dimensional cartesian space, sets up an energy
function which is a damped sine wave; this has many local minima, but
only one global minimum, somewhere between 1.0 and 1.5.  The initial
guess given is 15.5, which is several local minima away from the global
minimum.

     #include <math.h>
     #include <stdlib.h>
     #include <gsl/gsl_siman.h>

     /* set up parameters for this simulated annealing run */

     /* how many points do we try before stepping */
     #define N_TRIES 200

     /* how many iterations for each T? */
     #define ITERS_FIXED_T 10

     /* max step size in random walk */
     #define STEP_SIZE 10

     /* Boltzmann constant */
     #define K 1.0

     /* initial temperature */
     #define T_INITIAL 0.002

     /* damping factor for temperature */
     #define MU_T 1.005
     #define T_MIN 2.0e-6

     gsl_siman_params_t params
       = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
          K, T_INITIAL, MU_T, T_MIN};

     /* now some functions to test in one dimension */
     double E1(void *xp)
     {
       double x = * ((double *) xp);

       return exp(-pow((x-1.0),2.0))*sin(8*x);
     }

     double M1(void *xp, void *yp)
     {
       double x = *((double *) xp);
       double y = *((double *) yp);

       return fabs(x - y);
     }

     void S1(const gsl_rng * r, void *xp, double step_size)
     {
       double old_x = *((double *) xp);
       double new_x;

       double u = gsl_rng_uniform(r);
       new_x = u * 2 * step_size - step_size + old_x;

       memcpy(xp, &new_x, sizeof(new_x));
     }

     void P1(void *xp)
     {
       printf ("%12g", *((double *) xp));
     }

     int
     main(int argc, char *argv[])
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       double x_initial = 15.5;

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc(T);

       gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
                       NULL, NULL, NULL,
                       sizeof(double), params);
       return 0;
     }

   Here are a couple of plots that are generated by running
`siman_test' in the following way:

     $ ./siman_test | grep -v "^#"
      | xyplot -xyil -y -0.88 -0.83 -d "x...y"
      | xyps -d > siman-test.eps
     $ ./siman_test | grep -v "^#"
      | xyplot -xyil -xl "generation" -yl "energy" -d "x..y"
      | xyps -d > siman-energy.eps


File: gsl-ref.info,  Node: Traveling Salesman Problem,  Prev: Trivial example,  Up: Examples with Simulated Annealing

24.3.2 Traveling Salesman Problem
---------------------------------

The TSP ("Traveling Salesman Problem") is the classic combinatorial
optimization problem.  I have provided a very simple version of it,
based on the coordinates of twelve cities in the southwestern United
States.  This should maybe be called the "Flying Salesman Problem",
since I am using the great-circle distance between cities, rather than
the driving distance.  Also: I assume the earth is a sphere, so I don't
use geoid distances.

   The `gsl_siman_solve()' routine finds a route which is 3490.62
Kilometers long; this is confirmed by an exhaustive search of all
possible routes with the same initial city.

   The full code can be found in `siman/siman_tsp.c', but I include
here some plots generated in the following way:

     $ ./siman_tsp > tsp.output
     $ grep -v "^#" tsp.output
      | xyplot -xyil -d "x................y"
         -lx "generation" -ly "distance"
         -lt "TSP -- 12 southwest cities"
      | xyps -d > 12-cities.eps
     $ grep initial_city_coord tsp.output
      | awk '{print $2, $3, $4, $5}'
      | xyplot -xyil -lb0 -cs 0.8
         -lx "longitude (- means west)" -ly "latitude"
         -lt "TSP -- initial-order"
      | xyps -d > initial-route.eps
     $ grep final_city_coord tsp.output
      | awk '{print $2, $3, $4, $5}'
      | xyplot -xyil -lb0 -cs 0.8
         -lx "longitude (- means west)" -ly "latitude"
         -lt "TSP -- final-order"
      | xyps -d > final-route.eps

This is the output showing the initial order of the cities; longitude is
negative, since it is west and I want the plot to look like a map.

     # initial coordinates of cities (longitude and latitude)
     ###initial_city_coord: -105.95 35.68 Santa Fe
     ###initial_city_coord: -112.07 33.54 Phoenix
     ###initial_city_coord: -106.62 35.12 Albuquerque
     ###initial_city_coord: -103.2 34.41 Clovis
     ###initial_city_coord: -107.87 37.29 Durango
     ###initial_city_coord: -96.77 32.79 Dallas
     ###initial_city_coord: -105.92 35.77 Tesuque
     ###initial_city_coord: -107.84 35.15 Grants
     ###initial_city_coord: -106.28 35.89 Los Alamos
     ###initial_city_coord: -106.76 32.34 Las Cruces
     ###initial_city_coord: -108.58 37.35 Cortez
     ###initial_city_coord: -108.74 35.52 Gallup
     ###initial_city_coord: -105.95 35.68 Santa Fe

   The optimal route turns out to be:

     # final coordinates of cities (longitude and latitude)
     ###final_city_coord: -105.95 35.68 Santa Fe
     ###final_city_coord: -106.28 35.89 Los Alamos
     ###final_city_coord: -106.62 35.12 Albuquerque
     ###final_city_coord: -107.84 35.15 Grants
     ###final_city_coord: -107.87 37.29 Durango
     ###final_city_coord: -108.58 37.35 Cortez
     ###final_city_coord: -108.74 35.52 Gallup
     ###final_city_coord: -112.07 33.54 Phoenix
     ###final_city_coord: -106.76 32.34 Las Cruces
     ###final_city_coord: -96.77 32.79 Dallas
     ###final_city_coord: -103.2 34.41 Clovis
     ###final_city_coord: -105.92 35.77 Tesuque
     ###final_city_coord: -105.95 35.68 Santa Fe

Here's a plot of the cost function (energy) versus generation (point in
the calculation at which a new temperature is set) for this problem:


File: gsl-ref.info,  Node: Simulated Annealing References and Further Reading,  Prev: Examples with Simulated Annealing,  Up: Simulated Annealing

24.4 References and Further Reading
===================================

Further information is available in the following book,

     `Modern Heuristic Techniques for Combinatorial Problems', Colin R.
     Reeves (ed.), McGraw-Hill, 1995 (ISBN 0-07-709239-2).


File: gsl-ref.info,  Node: Ordinary Differential Equations,  Next: Interpolation,  Prev: Simulated Annealing,  Up: Top

25 Ordinary Differential Equations
**********************************

This chapter describes functions for solving ordinary differential
equation (ODE) initial value problems.  The library provides a variety
of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines,
and higher-level components for adaptive step-size control.  The
components can be combined by the user to achieve the desired solution,
with full access to any intermediate steps.

   These functions are declared in the header file `gsl_odeiv.h'.

* Menu:

* Defining the ODE System::
* Stepping Functions::
* Adaptive Step-size Control::
* Evolution::
* ODE Example programs::
* ODE References and Further Reading::


File: gsl-ref.info,  Node: Defining the ODE System,  Next: Stepping Functions,  Up: Ordinary Differential Equations

25.1 Defining the ODE System
============================

The routines solve the general n-dimensional first-order system,

     dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))

for i = 1, \dots, n.  The stepping functions rely on the vector of
derivatives f_i and the Jacobian matrix, J_{ij} = df_i(t,y(t)) / dy_j.
A system of equations is defined using the `gsl_odeiv_system' datatype.

 -- Data Type: gsl_odeiv_system
     This data type defines a general ODE system with arbitrary
     parameters.

    `int (* function) (double t, const double y[], double dydt[], void * params)'
          This function should store the vector elements
          f_i(t,y,params) in the array DYDT, for arguments (T,Y) and
          parameters PARAMS.  The function should return `GSL_SUCCESS'
          if the calculation was completed successfully. Any other
          return value indicates an error.

    `int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);'
          This function should store the vector of derivative elements
          df_i(t,y,params)/dt in the array DFDT and the Jacobian matrix
          J_{ij} in the array DFDY, regarded as a row-ordered matrix
          `J(i,j) = dfdy[i * dimension + j]' where `dimension' is the
          dimension of the system.  The function should return
          `GSL_SUCCESS' if the calculation was completed successfully.
          Any other return value indicates an error.

          Some of the simpler solver algorithms do not make use of the
          Jacobian matrix, so it is not always strictly necessary to
          provide it (the `jacobian' element of the struct can be
          replaced by a null pointer for those algorithms).  However,
          it is useful to provide the Jacobian to allow the solver
          algorithms to be interchanged--the best algorithms make use
          of the Jacobian.

    `size_t dimension;'
          This is the dimension of the system of equations.

    `void * params'
          This is a pointer to the arbitrary parameters of the system.


File: gsl-ref.info,  Node: Stepping Functions,  Next: Adaptive Step-size Control,  Prev: Defining the ODE System,  Up: Ordinary Differential Equations

25.2 Stepping Functions
=======================

The lowest level components are the "stepping functions" which advance
a solution from time t to t+h for a fixed step-size h and estimate the
resulting local error.

 -- Function: gsl_odeiv_step * gsl_odeiv_step_alloc (const
          gsl_odeiv_step_type * T, size_t DIM)
     This function returns a pointer to a newly allocated instance of a
     stepping function of type T for a system of DIM dimensions.

 -- Function: int gsl_odeiv_step_reset (gsl_odeiv_step * S)
     This function resets the stepping function S.  It should be used
     whenever the next use of S will not be a continuation of a
     previous step.

 -- Function: void gsl_odeiv_step_free (gsl_odeiv_step * S)
     This function frees all the memory associated with the stepping
     function S.

 -- Function: const char * gsl_odeiv_step_name (const gsl_odeiv_step *
          S)
     This function returns a pointer to the name of the stepping
     function.  For example,

          printf ("step method is '%s'\n",
                   gsl_odeiv_step_name (s));

     would print something like `step method is 'rk4''.

 -- Function: unsigned int gsl_odeiv_step_order (const gsl_odeiv_step *
          S)
     This function returns the order of the stepping function on the
     previous step.  This order can vary if the stepping function
     itself is adaptive.

 -- Function: int gsl_odeiv_step_apply (gsl_odeiv_step * S, double T,
          double H, double Y[], double YERR[], const double DYDT_IN[],
          double DYDT_OUT[], const gsl_odeiv_system * DYDT)
     This function applies the stepping function S to the system of
     equations defined by DYDT, using the step size H to advance the
     system from time T and state Y to time T+H.  The new state of the
     system is stored in Y on output, with an estimate of the absolute
     error in each component stored in YERR.  If the argument DYDT_IN
     is not null it should point an array containing the derivatives
     for the system at time T on input. This is optional as the
     derivatives will be computed internally if they are not provided,
     but allows the reuse of existing derivative information.  On
     output the new derivatives of the system at time T+H will be
     stored in DYDT_OUT if it is not null.

     If the user-supplied functions defined in the system DYDT return a
     status other than `GSL_SUCCESS' the step will be aborted.  In this
     case, the elements of Y will be restored to their pre-step values
     and the error code from the user-supplied function will be
     returned.  To distinguish between error codes from the
     user-supplied functions and those from `gsl_odeiv_step_apply'
     itself, any user-defined return values should be distinct from the
     standard GSL error codes.

   The following algorithms are available,

 -- Step Type: gsl_odeiv_step_rk2
     Embedded Runge-Kutta (2, 3) method.

 -- Step Type: gsl_odeiv_step_rk4
     4th order (classical) Runge-Kutta.

 -- Step Type: gsl_odeiv_step_rkf45
     Embedded Runge-Kutta-Fehlberg (4, 5) method.  This method is a good
     general-purpose integrator.

 -- Step Type: gsl_odeiv_step_rkck
     Embedded Runge-Kutta Cash-Karp (4, 5) method.

 -- Step Type: gsl_odeiv_step_rk8pd
     Embedded Runge-Kutta Prince-Dormand (8,9) method.

 -- Step Type: gsl_odeiv_step_rk2imp
     Implicit 2nd order Runge-Kutta at Gaussian points.

 -- Step Type: gsl_odeiv_step_rk4imp
     Implicit 4th order Runge-Kutta at Gaussian points.

 -- Step Type: gsl_odeiv_step_bsimp
     Implicit Bulirsch-Stoer method of Bader and Deuflhard.  This
     algorithm requires the Jacobian.

 -- Step Type: gsl_odeiv_step_gear1
     M=1 implicit Gear method.

 -- Step Type: gsl_odeiv_step_gear2
     M=2 implicit Gear method.


File: gsl-ref.info,  Node: Adaptive Step-size Control,  Next: Evolution,  Prev: Stepping Functions,  Up: Ordinary Differential Equations

25.3 Adaptive Step-size Control
===============================

The control function examines the proposed change to the solution
produced by a stepping function and attempts to determine the optimal
step-size for a user-specified level of error.

 -- Function: gsl_odeiv_control * gsl_odeiv_control_standard_new
          (double EPS_ABS, double EPS_REL, double A_Y, double A_DYDT)
     The standard control object is a four parameter heuristic based on
     absolute and relative errors EPS_ABS and EPS_REL, and scaling
     factors A_Y and A_DYDT for the system state y(t) and derivatives
     y'(t) respectively.

     The step-size adjustment procedure for this method begins by
     computing the desired error level D_i for each component,

          D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)

     and comparing it with the observed error E_i = |yerr_i|.  If the
     observed error E exceeds the desired error level D by more than
     10% for any component then the method reduces the step-size by an
     appropriate factor,

          h_new = h_old * S * (E/D)^(-1/q)

     where q is the consistency order of the method (e.g. q=4 for 4(5)
     embedded RK), and S is a safety factor of 0.9. The ratio E/D is
     taken to be the maximum of the ratios E_i/D_i.

     If the observed error E is less than 50% of the desired error
     level D for the maximum ratio E_i/D_i then the algorithm takes the
     opportunity to increase the step-size to bring the error in line
     with the desired level,

          h_new = h_old * S * (E/D)^(-1/(q+1))

     This encompasses all the standard error scaling methods. To avoid
     uncontrolled changes in the stepsize, the overall scaling factor is
     limited to the range 1/5 to 5.

 -- Function: gsl_odeiv_control * gsl_odeiv_control_y_new (double
          EPS_ABS, double EPS_REL)
     This function creates a new control object which will keep the
     local error on each step within an absolute error of EPS_ABS and
     relative error of EPS_REL with respect to the solution y_i(t).
     This is equivalent to the standard control object with A_Y=1 and
     A_DYDT=0.

 -- Function: gsl_odeiv_control * gsl_odeiv_control_yp_new (double
          EPS_ABS, double EPS_REL)
     This function creates a new control object which will keep the
     local error on each step within an absolute error of EPS_ABS and
     relative error of EPS_REL with respect to the derivatives of the
     solution y'_i(t).  This is equivalent to the standard control
     object with A_Y=0 and A_DYDT=1.

 -- Function: gsl_odeiv_control * gsl_odeiv_control_scaled_new (double
          EPS_ABS, double EPS_REL, double A_Y, double A_DYDT, const
          double SCALE_ABS[], size_t DIM)
     This function creates a new control object which uses the same
     algorithm as `gsl_odeiv_control_standard_new' but with an absolute
     error which is scaled for each component by the array SCALE_ABS.
     The formula for D_i for this control object is,

          D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)

     where s_i is the i-th component of the array SCALE_ABS.  The same
     error control heuristic is used by the Matlab ODE suite.

 -- Function: gsl_odeiv_control * gsl_odeiv_control_alloc (const
          gsl_odeiv_control_type * T)
     This function returns a pointer to a newly allocated instance of a
     control function of type T.  This function is only needed for
     defining new types of control functions.  For most purposes the
     standard control functions described above should be sufficient.

 -- Function: int gsl_odeiv_control_init (gsl_odeiv_control * C, double
          EPS_ABS, double EPS_REL, double A_Y, double A_DYDT)
     This function initializes the control function C with the
     parameters EPS_ABS (absolute error), EPS_REL (relative error), A_Y
     (scaling factor for y) and A_DYDT (scaling factor for derivatives).

 -- Function: void gsl_odeiv_control_free (gsl_odeiv_control * C)
     This function frees all the memory associated with the control
     function C.

 -- Function: int gsl_odeiv_control_hadjust (gsl_odeiv_control * C,
          gsl_odeiv_step * S, const double Y0[], const double YERR[],
          const double DYDT[], double * H)
     This function adjusts the step-size H using the control function
     C, and the current values of Y, YERR and DYDT.  The stepping
     function STEP is also needed to determine the order of the method.
     If the error in the y-values YERR is found to be too large then
     the step-size H is reduced and the function returns
     `GSL_ODEIV_HADJ_DEC'.  If the error is sufficiently small then H
     may be increased and `GSL_ODEIV_HADJ_INC' is returned.  The
     function returns `GSL_ODEIV_HADJ_NIL' if the step-size is
     unchanged.  The goal of the function is to estimate the largest
     step-size which satisfies the user-specified accuracy requirements
     for the current point.

 -- Function: const char * gsl_odeiv_control_name (const
          gsl_odeiv_control * C)
     This function returns a pointer to the name of the control
     function.  For example,

          printf ("control method is '%s'\n",
                  gsl_odeiv_control_name (c));

     would print something like `control method is 'standard''


File: gsl-ref.info,  Node: Evolution,  Next: ODE Example programs,  Prev: Adaptive Step-size Control,  Up: Ordinary Differential Equations

25.4 Evolution
==============

The highest level of the system is the evolution function which combines
the results of a stepping function and control function to reliably
advance the solution forward over an interval (t_0, t_1).  If the
control function signals that the step-size should be decreased the
evolution function backs out of the current step and tries the proposed
smaller step-size.  This process is continued until an acceptable
step-size is found.

 -- Function: gsl_odeiv_evolve * gsl_odeiv_evolve_alloc (size_t DIM)
     This function returns a pointer to a newly allocated instance of an
     evolution function for a system of DIM dimensions.

 -- Function: int gsl_odeiv_evolve_apply (gsl_odeiv_evolve * E,
          gsl_odeiv_control * CON, gsl_odeiv_step * STEP, const
          gsl_odeiv_system * DYDT, double * T, double T1, double * H,
          double Y[])
     This function advances the system (E, DYDT) from time T and
     position Y using the stepping function STEP.  The new time and
     position are stored in T and Y on output.  The initial step-size
     is taken as H, but this will be modified using the control
     function C to achieve the appropriate error bound if necessary.
     The routine may make several calls to STEP in order to determine
     the optimum step-size. If the step-size has been changed the value
     of H will be modified on output.  The maximum time T1 is
     guaranteed not to be exceeded by the time-step.  On the final
     time-step the value of T will be set to T1 exactly.

     If the user-supplied functions defined in the system DYDT return a
     status other than `GSL_SUCCESS' the step will be aborted.  In this
     case,  T and Y will be restored to their pre-step values and the
     error code from the user-supplied function will be returned.  To
     distinguish between error codes from the user-supplied functions
     and those from `gsl_odeiv_evolve_apply' itself, any user-defined
     return values should be distinct from the standard GSL error codes.

 -- Function: int gsl_odeiv_evolve_reset (gsl_odeiv_evolve * E)
     This function resets the evolution function E.  It should be used
     whenever the next use of E will not be a continuation of a
     previous step.

 -- Function: void gsl_odeiv_evolve_free (gsl_odeiv_evolve * E)
     This function frees all the memory associated with the evolution
     function E.


File: gsl-ref.info,  Node: ODE Example programs,  Next: ODE References and Further Reading,  Prev: Evolution,  Up: Ordinary Differential Equations

25.5 Examples
=============

The following program solves the second-order nonlinear Van der Pol
oscillator equation,

     x''(t) + \mu x'(t) (x(t)^2 - 1) + x(t) = 0

This can be converted into a first order system suitable for use with
the routines described in this chapter by introducing a separate
variable for the velocity, y = x'(t),

     x' = y
     y' = -x + \mu y (1-x^2)

The program begins by defining functions for these derivatives and
their Jacobian,

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

     int
     func (double t, const double y[], double f[],
           void *params)
     {
       double mu = *(double *)params;
       f[0] = y[1];
       f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
       return GSL_SUCCESS;
     }

     int
     jac (double t, const double y[], double *dfdy,
          double dfdt[], void *params)
     {
       double mu = *(double *)params;
       gsl_matrix_view dfdy_mat
         = gsl_matrix_view_array (dfdy, 2, 2);
       gsl_matrix * m = &dfdy_mat.matrix;
       gsl_matrix_set (m, 0, 0, 0.0);
       gsl_matrix_set (m, 0, 1, 1.0);
       gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
       gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
       dfdt[0] = 0.0;
       dfdt[1] = 0.0;
       return GSL_SUCCESS;
     }

     int
     main (void)
     {
       const gsl_odeiv_step_type * T
         = gsl_odeiv_step_rk8pd;

       gsl_odeiv_step * s
         = gsl_odeiv_step_alloc (T, 2);
       gsl_odeiv_control * c
         = gsl_odeiv_control_y_new (1e-6, 0.0);
       gsl_odeiv_evolve * e
         = gsl_odeiv_evolve_alloc (2);

       double mu = 10;
       gsl_odeiv_system sys = {func, jac, 2, &mu};

       double t = 0.0, t1 = 100.0;
       double h = 1e-6;
       double y[2] = { 1.0, 0.0 };

       while (t < t1)
         {
           int status = gsl_odeiv_evolve_apply (e, c, s,
                                                &sys,
                                                &t, t1,
                                                &h, y);

           if (status != GSL_SUCCESS)
               break;

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv_evolve_free (e);
       gsl_odeiv_control_free (c);
       gsl_odeiv_step_free (s);
       return 0;
     }

For functions with multiple parameters, the appropriate information can
be passed in through the PARAMS argument using a pointer to a struct.

   The main loop of the program evolves the solution from (y, y') = (1,
0) at t=0 to t=100.  The step-size h is automatically adjusted by the
controller to maintain an absolute accuracy of 10^{-6} in the function
values Y.

To obtain the values at regular intervals, rather than the variable
spacings chosen by the control function, the main loop can be modified
to advance the solution from one point to the next.  For example, the
following main loop prints the solution at the fixed points t = 0, 1,
2, \dots, 100,

       for (i = 1; i <= 100; i++)
         {
           double ti = i * t1 / 100.0;

           while (t < ti)
             {
               gsl_odeiv_evolve_apply (e, c, s,
                                       &sys,
                                       &t, ti, &h,
                                       y);
             }

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

It is also possible to work with a non-adaptive integrator, using only
the stepping function itself.  The following program uses the `rk4'
fourth-order Runge-Kutta stepping function with a fixed stepsize of
0.01,

     int
     main (void)
     {
       const gsl_odeiv_step_type * T
         = gsl_odeiv_step_rk4;

       gsl_odeiv_step * s
         = gsl_odeiv_step_alloc (T, 2);

       double mu = 10;
       gsl_odeiv_system sys = {func, jac, 2, &mu};

       double t = 0.0, t1 = 100.0;
       double h = 1e-2;
       double y[2] = { 1.0, 0.0 }, y_err[2];
       double dydt_in[2], dydt_out[2];

       /* initialise dydt_in from system parameters */
       GSL_ODEIV_FN_EVAL(&sys, t, y, dydt_in);

       while (t < t1)
         {
           int status = gsl_odeiv_step_apply (s, t, h,
                                              y, y_err,
                                              dydt_in,
                                              dydt_out,
                                              &sys);

           if (status != GSL_SUCCESS)
               break;

           dydt_in[0] = dydt_out[0];
           dydt_in[1] = dydt_out[1];

           t += h;

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv_step_free (s);
       return 0;
     }

The derivatives must be initialized for the starting point t=0 before
the first step is taken.  Subsequent steps use the output derivatives
DYDT_OUT as inputs to the next step by copying their values into
DYDT_IN.


File: gsl-ref.info,  Node: ODE References and Further Reading,  Prev: ODE Example programs,  Up: Ordinary Differential Equations

25.6 References and Further Reading
===================================

Many of the basic Runge-Kutta formulas can be found in the Handbook of
Mathematical Functions,

     Abramowitz & Stegun (eds.), `Handbook of Mathematical Functions',
     Section 25.5.

The implicit Bulirsch-Stoer algorithm `bsimp' is described in the
following paper,

     G. Bader and P. Deuflhard, "A Semi-Implicit Mid-Point Rule for
     Stiff Systems of Ordinary Differential Equations.", Numer. Math.
     41, 373-398, 1983.


File: gsl-ref.info,  Node: Interpolation,  Next: Numerical Differentiation,  Prev: Ordinary Differential Equations,  Up: Top

26 Interpolation
****************

This chapter describes functions for performing interpolation.  The
library provides a variety of interpolation methods, including Cubic
splines and Akima splines.  The interpolation types are interchangeable,
allowing different methods to be used without recompiling.
Interpolations can be defined for both normal and periodic boundary
conditions.  Additional functions are available for computing
derivatives and integrals of interpolating functions.

   The functions described in this section are declared in the header
files `gsl_interp.h' and `gsl_spline.h'.

* Menu:

* Introduction to Interpolation::
* Interpolation Functions::
* Interpolation Types::
* Index Look-up and Acceleration::
* Evaluation of Interpolating Functions::
* Higher-level Interface::
* Interpolation Example programs::
* Interpolation References and Further Reading::


File: gsl-ref.info,  Node: Introduction to Interpolation,  Next: Interpolation Functions,  Up: Interpolation

26.1 Introduction
=================

Given a set of data points (x_1, y_1) \dots (x_n, y_n) the routines
described in this section compute a continuous interpolating function
y(x) such that y(x_i) = y_i.  The interpolation is piecewise smooth,
and its behavior at the end-points is determined by the type of
interpolation used.


File: gsl-ref.info,  Node: Interpolation Functions,  Next: Interpolation Types,  Prev: Introduction to Interpolation,  Up: Interpolation

26.2 Interpolation Functions
============================

The interpolation function for a given dataset is stored in a
`gsl_interp' object.  These are created by the following functions.

 -- Function: gsl_interp * gsl_interp_alloc (const gsl_interp_type * T,
          size_t SIZE)
     This function returns a pointer to a newly allocated interpolation
     object of type T for SIZE data-points.

 -- Function: int gsl_interp_init (gsl_interp * INTERP, const double
          XA[], const double YA[], size_t SIZE)
     This function initializes the interpolation object INTERP for the
     data (XA,YA) where XA and YA are arrays of size SIZE.  The
     interpolation object (`gsl_interp') does not save the data arrays
     XA and YA and only stores the static state computed from the data.
     The XA data array is always assumed to be strictly ordered; the
     behavior for other arrangements is not defined.

 -- Function: void gsl_interp_free (gsl_interp * INTERP)
     This function frees the interpolation object INTERP.


File: gsl-ref.info,  Node: Interpolation Types,  Next: Index Look-up and Acceleration,  Prev: Interpolation Functions,  Up: Interpolation

26.3 Interpolation Types
========================

The interpolation library provides five interpolation types:

 -- Interpolation Type: gsl_interp_linear
     Linear interpolation.  This interpolation method does not require
     any additional memory.

 -- Interpolation Type: gsl_interp_polynomial
     Polynomial interpolation.  This method should only be used for
     interpolating small numbers of points because polynomial
     interpolation introduces large oscillations, even for well-behaved
     datasets.  The number of terms in the interpolating polynomial is
     equal to the number of points.

 -- Interpolation Type: gsl_interp_cspline
     Cubic spline with natural boundary conditions.  The resulting
     curve is piecewise cubic on each interval, with matching first and
     second derivatives at the supplied data-points.  The second
     derivative is chosen to be zero at the first point and last point.

 -- Interpolation Type: gsl_interp_cspline_periodic
     Cubic spline with periodic boundary conditions.  The resulting
     curve is piecewise cubic on each interval, with matching first and
     second derivatives at the supplied data-points.  The derivatives
     at the first and last points are also matched.  Note that the last
     point in the data must have the same y-value as the first point,
     otherwise the resulting periodic interpolation will have a
     discontinuity at the boundary.


 -- Interpolation Type: gsl_interp_akima
     Non-rounded Akima spline with natural boundary conditions.  This
     method uses the non-rounded corner algorithm of Wodicka.

 -- Interpolation Type: gsl_interp_akima_periodic
     Non-rounded Akima spline with periodic boundary conditions.  This
     method uses the non-rounded corner algorithm of Wodicka.

   The following related functions are available:

 -- Function: const char * gsl_interp_name (const gsl_interp * INTERP)
     This function returns the name of the interpolation type used by
     INTERP.  For example,

          printf ("interp uses '%s' interpolation.\n",
                  gsl_interp_name (interp));

     would print something like,

          interp uses 'cspline' interpolation.

 -- Function: unsigned int gsl_interp_min_size (const gsl_interp *
          INTERP)
     This function returns the minimum number of points required by the
     interpolation type of INTERP.  For example, Akima spline
     interpolation requires a minimum of 5 points.


File: gsl-ref.info,  Node: Index Look-up and Acceleration,  Next: Evaluation of Interpolating Functions,  Prev: Interpolation Types,  Up: Interpolation

26.4 Index Look-up and Acceleration
===================================

The state of searches can be stored in a `gsl_interp_accel' object,
which is a kind of iterator for interpolation lookups.  It caches the
previous value of an index lookup.  When the subsequent interpolation
point falls in the same interval its index value can be returned
immediately.

 -- Function: size_t gsl_interp_bsearch (const double X_ARRAY[], double
          X, size_t INDEX_LO, size_t INDEX_HI)
     This function returns the index i of the array X_ARRAY such that
     `x_array[i] <= x < x_array[i+1]'.  The index is searched for in
     the range [INDEX_LO,INDEX_HI].

 -- Function: gsl_interp_accel * gsl_interp_accel_alloc (void)
     This function returns a pointer to an accelerator object, which is
     a kind of iterator for interpolation lookups.  It tracks the state
     of lookups, thus allowing for application of various acceleration
     strategies.

 -- Function: size_t gsl_interp_accel_find (gsl_interp_accel * A, const
          double X_ARRAY[], size_t SIZE, double X)
     This function performs a lookup action on the data array X_ARRAY
     of size SIZE, using the given accelerator A.  This is how lookups
     are performed during evaluation of an interpolation.  The function
     returns an index i such that `x_array[i] <= x < x_array[i+1]'.

 -- Function: void gsl_interp_accel_free (gsl_interp_accel* ACC)
     This function frees the accelerator object ACC.


File: gsl-ref.info,  Node: Evaluation of Interpolating Functions,  Next: Higher-level Interface,  Prev: Index Look-up and Acceleration,  Up: Interpolation

26.5 Evaluation of Interpolating Functions
==========================================

 -- Function: double gsl_interp_eval (const gsl_interp * INTERP, const
          double XA[], const double YA[], double X, gsl_interp_accel *
          ACC)
 -- Function: int gsl_interp_eval_e (const gsl_interp * INTERP, const
          double XA[], const double YA[], double X, gsl_interp_accel *
          ACC, double * Y)
     These functions return the interpolated value of Y for a given
     point X, using the interpolation object INTERP, data arrays XA and
     YA and the accelerator ACC.

 -- Function: double gsl_interp_eval_deriv (const gsl_interp * INTERP,
          const double XA[], const double YA[], double X,
          gsl_interp_accel * ACC)
 -- Function: int gsl_interp_eval_deriv_e (const gsl_interp * INTERP,
          const double XA[], const double YA[], double X,
          gsl_interp_accel * ACC, double * D)
     These functions return the derivative D of an interpolated
     function for a given point X, using the interpolation object
     INTERP, data arrays XA and YA and the accelerator ACC.

 -- Function: double gsl_interp_eval_deriv2 (const gsl_interp * INTERP,
          const double XA[], const double YA[], double X,
          gsl_interp_accel * ACC)
 -- Function: int gsl_interp_eval_deriv2_e (const gsl_interp * INTERP,
          const double XA[], const double YA[], double X,
          gsl_interp_accel * ACC, double * D2)
     These functions return the second derivative D2 of an interpolated
     function for a given point X, using the interpolation object
     INTERP, data arrays XA and YA and the accelerator ACC.

 -- Function: double gsl_interp_eval_integ (const gsl_interp * INTERP,
          const double XA[], const double YA[], double A, double B,
          gsl_interp_accel * ACC)
 -- Function: int gsl_interp_eval_integ_e (const gsl_interp * INTERP,
          const double XA[], const double YA[], double A, double B,
          gsl_interp_accel * ACC, double * RESULT)
     These functions return the numerical integral RESULT of an
     interpolated function over the range [A, B], using the
     interpolation object INTERP, data arrays XA and YA and the
     accelerator ACC.


File: gsl-ref.info,  Node: Higher-level Interface,  Next: Interpolation Example programs,  Prev: Evaluation of Interpolating Functions,  Up: Interpolation

26.6 Higher-level Interface
===========================

The functions described in the previous sections required the user to
supply pointers to the x and y arrays on each call.  The following
functions are equivalent to the corresponding `gsl_interp' functions
but maintain a copy of this data in the `gsl_spline' object.  This
removes the need to pass both XA and YA as arguments on each
evaluation. These functions are defined in the header file
`gsl_spline.h'.

 -- Function: gsl_spline * gsl_spline_alloc (const gsl_interp_type * T,
          size_t SIZE)

 -- Function: int gsl_spline_init (gsl_spline * SPLINE, const double
          XA[], const double YA[], size_t SIZE)

 -- Function: void gsl_spline_free (gsl_spline * SPLINE)

 -- Function: const char * gsl_spline_name (const gsl_spline * SPLINE)

 -- Function: unsigned int gsl_spline_min_size (const gsl_spline *
          SPLINE)

 -- Function: double gsl_spline_eval (const gsl_spline * SPLINE, double
          X, gsl_interp_accel * ACC)
 -- Function: int gsl_spline_eval_e (const gsl_spline * SPLINE, double
          X, gsl_interp_accel * ACC, double * Y)

 -- Function: double gsl_spline_eval_deriv (const gsl_spline * SPLINE,
          double X, gsl_interp_accel * ACC)
 -- Function: int gsl_spline_eval_deriv_e (const gsl_spline * SPLINE,
          double X, gsl_interp_accel * ACC, double * D)

 -- Function: double gsl_spline_eval_deriv2 (const gsl_spline * SPLINE,
          double X, gsl_interp_accel * ACC)
 -- Function: int gsl_spline_eval_deriv2_e (const gsl_spline * SPLINE,
          double X, gsl_interp_accel * ACC, double * D2)

 -- Function: double gsl_spline_eval_integ (const gsl_spline * SPLINE,
          double A, double B, gsl_interp_accel * ACC)
 -- Function: int gsl_spline_eval_integ_e (const gsl_spline * SPLINE,
          double A, double B, gsl_interp_accel * ACC, double * RESULT)


File: gsl-ref.info,  Node: Interpolation Example programs,  Next: Interpolation References and Further Reading,  Prev: Higher-level Interface,  Up: Interpolation

26.7 Examples
=============

The following program demonstrates the use of the interpolation and
spline functions.  It computes a cubic spline interpolation of the
10-point dataset (x_i, y_i) where x_i = i + \sin(i)/2 and y_i = i +
\cos(i^2) for i = 0 \dots 9.

     #include <stdlib.h>
     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_spline.h>

     int
     main (void)
     {
       int i;
       double xi, yi, x[10], y[10];

       printf ("#m=0,S=2\n");

       for (i = 0; i < 10; i++)
         {
           x[i] = i + 0.5 * sin (i);
           y[i] = i + cos (i * i);
           printf ("%g %g\n", x[i], y[i]);
         }

       printf ("#m=1,S=0\n");

       {
         gsl_interp_accel *acc
           = gsl_interp_accel_alloc ();
         gsl_spline *spline
           = gsl_spline_alloc (gsl_interp_cspline, 10);

         gsl_spline_init (spline, x, y, 10);

         for (xi = x[0]; xi < x[9]; xi += 0.01)
           {
             yi = gsl_spline_eval (spline, xi, acc);
             printf ("%g %g\n", xi, yi);
           }
         gsl_spline_free (spline);
         gsl_interp_accel_free (acc);
       }
       return 0;
     }

The output is designed to be used with the GNU plotutils `graph'
program,

     $ ./a.out > interp.dat
     $ graph -T ps < interp.dat > interp.ps

The result shows a smooth interpolation of the original points.  The
interpolation method can changed simply by varying the first argument of
`gsl_spline_alloc'.

   The next program demonstrates a periodic cubic spline with 4 data
points.  Note that the first and last points must be supplied with the
same y-value for a periodic spline.

     #include <stdlib.h>
     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_spline.h>

     int
     main (void)
     {
       int N = 4;
       double x[4] = {0.00, 0.10,  0.27,  0.30};
       double y[4] = {0.15, 0.70, -0.10,  0.15}; /* Note: first = last
                                                    for periodic data */

       gsl_interp_accel *acc = gsl_interp_accel_alloc ();
       const gsl_interp_type *t = gsl_interp_cspline_periodic;
       gsl_spline *spline = gsl_spline_alloc (t, N);

       int i; double xi, yi;

       printf ("#m=0,S=5\n");
       for (i = 0; i < N; i++)
         {
           printf ("%g %g\n", x[i], y[i]);
         }

       printf ("#m=1,S=0\n");
       gsl_spline_init (spline, x, y, N);

       for (i = 0; i <= 100; i++)
         {
           xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
           yi = gsl_spline_eval (spline, xi, acc);
           printf ("%g %g\n", xi, yi);
         }

       gsl_spline_free (spline);
       gsl_interp_accel_free (acc);
       return 0;
     }

The output can be plotted with GNU `graph'.

     $ ./a.out > interp.dat
     $ graph -T ps < interp.dat > interp.ps

The result shows a periodic interpolation of the original points. The
slope of the fitted curve is the same at the beginning and end of the
data, and the second derivative is also.


File: gsl-ref.info,  Node: Interpolation References and Further Reading,  Prev: Interpolation Example programs,  Up: Interpolation

26.8 References and Further Reading
===================================

Descriptions of the interpolation algorithms and further references can
be found in the following books:

     C.W. Ueberhuber, `Numerical Computation (Volume 1), Chapter 9
     "Interpolation"', Springer (1997), ISBN 3-540-62058-3.

     D.M. Young, R.T. Gregory `A Survey of Numerical Mathematics
     (Volume 1), Chapter 6.8', Dover (1988), ISBN 0-486-65691-8.



File: gsl-ref.info,  Node: Numerical Differentiation,  Next: Chebyshev Approximations,  Prev: Interpolation,  Up: Top

27 Numerical Differentiation
****************************

The functions described in this chapter compute numerical derivatives by
finite differencing.  An adaptive algorithm is used to find the best
choice of finite difference and to estimate the error in the derivative.
These functions are declared in the header file `gsl_deriv.h'.

* Menu:

* Numerical Differentiation functions::
* Numerical Differentiation Examples::
* Numerical Differentiation References::


File: gsl-ref.info,  Node: Numerical Differentiation functions,  Next: Numerical Differentiation Examples,  Up: Numerical Differentiation

27.1 Functions
==============

 -- Function: int gsl_deriv_central (const gsl_function * F, double X,
          double H, double * RESULT, double * ABSERR)
     This function computes the numerical derivative of the function F
     at the point X using an adaptive central difference algorithm with
     a step-size of H.   The derivative is returned in RESULT and an
     estimate of its absolute error is returned in ABSERR.

     The initial value of H is used to estimate an optimal step-size,
     based on the scaling of the truncation error and round-off error
     in the derivative calculation.  The derivative is computed using a
     5-point rule for equally spaced abscissae at x-h, x-h/2, x, x+h/2,
     x, with an error estimate taken from the difference between the
     5-point rule and the corresponding 3-point rule x-h, x, x+h.  Note
     that the value of the function at x does not contribute to the
     derivative calculation, so only 4-points are actually used.

 -- Function: int gsl_deriv_forward (const gsl_function * F, double X,
          double H, double * RESULT, double * ABSERR)
     This function computes the numerical derivative of the function F
     at the point X using an adaptive forward difference algorithm with
     a step-size of H. The function is evaluated only at points greater
     than X, and never at X itself.  The derivative is returned in
     RESULT and an estimate of its absolute error is returned in
     ABSERR.  This function should be used if f(x) has a discontinuity
     at X, or is undefined for values less than X.

     The initial value of H is used to estimate an optimal step-size,
     based on the scaling of the truncation error and round-off error
     in the derivative calculation.  The derivative at x is computed
     using an "open" 4-point rule for equally spaced abscissae at x+h/4,
     x+h/2, x+3h/4, x+h, with an error estimate taken from the
     difference between the 4-point rule and the corresponding 2-point
     rule x+h/2, x+h.

 -- Function: int gsl_deriv_backward (const gsl_function * F, double X,
          double H, double * RESULT, double * ABSERR)
     This function computes the numerical derivative of the function F
     at the point X using an adaptive backward difference algorithm
     with a step-size of H. The function is evaluated only at points
     less than X, and never at X itself.  The derivative is returned in
     RESULT and an estimate of its absolute error is returned in
     ABSERR.  This function should be used if f(x) has a discontinuity
     at X, or is undefined for values greater than X.

     This function is equivalent to calling `gsl_deriv_forward' with a
     negative step-size.


File: gsl-ref.info,  Node: Numerical Differentiation Examples,  Next: Numerical Differentiation References,  Prev: Numerical Differentiation functions,  Up: Numerical Differentiation

27.2 Examples
=============

The following code estimates the derivative of the function f(x) =
x^{3/2} at x=2 and at x=0.  The function f(x) is undefined for x<0 so
the derivative at x=0 is computed using `gsl_deriv_forward'.

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

     double f (double x, void * params)
     {
       return pow (x, 1.5);
     }

     int
     main (void)
     {
       gsl_function F;
       double result, abserr;

       F.function = &f;
       F.params = 0;

       printf ("f(x) = x^(3/2)\n");

       gsl_deriv_central (&F, 2.0, 1e-8, &result, &abserr);
       printf ("x = 2.0\n");
       printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
       printf ("exact = %.10f\n\n", 1.5 * sqrt(2.0));

       gsl_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
       printf ("x = 0.0\n");
       printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
       printf ("exact = %.10f\n", 0.0);

       return 0;
     }

Here is the output of the program,

     $ ./a.out
     f(x) = x^(3/2)
     x = 2.0
     f'(x) = 2.1213203120 +/- 0.0000004064
     exact = 2.1213203436

     x = 0.0
     f'(x) = 0.0000000160 +/- 0.0000000339
     exact = 0.0000000000


File: gsl-ref.info,  Node: Numerical Differentiation References,  Prev: Numerical Differentiation Examples,  Up: Numerical Differentiation

27.3 References and Further Reading
===================================

The algorithms used by these functions are described in the following
sources:

     Abramowitz and Stegun, `Handbook of Mathematical Functions',
     Section 25.3.4, and Table 25.5 (Coefficients for Differentiation).

     S.D. Conte and Carl de Boor, `Elementary Numerical Analysis: An
     Algorithmic Approach', McGraw-Hill, 1972.


File: gsl-ref.info,  Node: Chebyshev Approximations,  Next: Series Acceleration,  Prev: Numerical Differentiation,  Up: Top

28 Chebyshev Approximations
***************************

This chapter describes routines for computing Chebyshev approximations
to univariate functions.  A Chebyshev approximation is a truncation of
the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials
T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials
on the interval [-1,1] with the weight function 1 / \sqrt{1-x^2}.  The
first few Chebyshev polynomials are, T_0(x) = 1, T_1(x) = x, T_2(x) = 2
x^2 - 1.  For further information see Abramowitz & Stegun, Chapter 22.

   The functions described in this chapter are declared in the header
file `gsl_chebyshev.h'.

* Menu:

* Chebyshev Definitions::
* Creation and Calculation of Chebyshev Series::
* Chebyshev Series Evaluation::
* Derivatives and Integrals::
* Chebyshev Approximation examples::
* Chebyshev Approximation References and Further Reading::


File: gsl-ref.info,  Node: Chebyshev Definitions,  Next: Creation and Calculation of Chebyshev Series,  Up: Chebyshev Approximations

28.1 Definitions
================

A Chebyshev series  is stored using the following structure,

     typedef struct
     {
       double * c;   /* coefficients  c[0] .. c[order] */
       int order;    /* order of expansion             */
       double a;     /* lower interval point           */
       double b;     /* upper interval point           */
       ...
     } gsl_cheb_series

The approximation is made over the range [a,b] using ORDER+1 terms,
including the coefficient c[0].  The series is computed using the
following convention,

     f(x) = (c_0 / 2) + \sum_{n=1} c_n T_n(x)

which is needed when accessing the coefficients directly.


File: gsl-ref.info,  Node: Creation and Calculation of Chebyshev Series,  Next: Chebyshev Series Evaluation,  Prev: Chebyshev Definitions,  Up: Chebyshev Approximations

28.2 Creation and Calculation of Chebyshev Series
=================================================

 -- Function: gsl_cheb_series * gsl_cheb_alloc (const size_t N)
     This function allocates space for a Chebyshev series of order N
     and returns a pointer to a new `gsl_cheb_series' struct.

 -- Function: void gsl_cheb_free (gsl_cheb_series * CS)
     This function frees a previously allocated Chebyshev series CS.

 -- Function: int gsl_cheb_init (gsl_cheb_series * CS, const
          gsl_function * F, const double A, const double B)
     This function computes the Chebyshev approximation CS for the
     function F over the range (a,b) to the previously specified order.
     The computation of the Chebyshev approximation is an O(n^2)
     process, and requires n function evaluations.


File: gsl-ref.info,  Node: Chebyshev Series Evaluation,  Next: Derivatives and Integrals,  Prev: Creation and Calculation of Chebyshev Series,  Up: Chebyshev Approximations

28.3 Chebyshev Series Evaluation
================================

 -- Function: double gsl_cheb_eval (const gsl_cheb_series * CS, double
          X)
     This function evaluates the Chebyshev series CS at a given point X.

 -- Function: int gsl_cheb_eval_err (const gsl_cheb_series * CS, const
          double X, double * RESULT, double * ABSERR)
     This function computes the Chebyshev series CS at a given point X,
     estimating both the series RESULT and its absolute error ABSERR.
     The error estimate is made from the first neglected term in the
     series.

 -- Function: double gsl_cheb_eval_n (const gsl_cheb_series * CS,
          size_t ORDER, double X)
     This function evaluates the Chebyshev series CS at a given point
     N, to (at most) the given order ORDER.

 -- Function: int gsl_cheb_eval_n_err (const gsl_cheb_series * CS,
          const size_t ORDER, const double X, double * RESULT, double *
          ABSERR)
     This function evaluates a Chebyshev series CS at a given point X,
     estimating both the series RESULT and its absolute error ABSERR,
     to (at most) the given order ORDER.  The error estimate is made
     from the first neglected term in the series.


File: gsl-ref.info,  Node: Derivatives and Integrals,  Next: Chebyshev Approximation examples,  Prev: Chebyshev Series Evaluation,  Up: Chebyshev Approximations

28.4 Derivatives and Integrals
==============================

The following functions allow a Chebyshev series to be differentiated or
integrated, producing a new Chebyshev series.  Note that the error
estimate produced by evaluating the derivative series will be
underestimated due to the contribution of higher order terms being
neglected.

 -- Function: int gsl_cheb_calc_deriv (gsl_cheb_series * DERIV, const
          gsl_cheb_series * CS)
     This function computes the derivative of the series CS, storing
     the derivative coefficients in the previously allocated DERIV.
     The two series CS and DERIV must have been allocated with the same
     order.

 -- Function: int gsl_cheb_calc_integ (gsl_cheb_series * INTEG, const
          gsl_cheb_series * CS)
     This function computes the integral of the series CS, storing the
     integral coefficients in the previously allocated INTEG.  The two
     series CS and INTEG must have been allocated with the same order.
     The lower limit of the integration is taken to be the left hand
     end of the range A.


File: gsl-ref.info,  Node: Chebyshev Approximation examples,  Next: Chebyshev Approximation References and Further Reading,  Prev: Derivatives and Integrals,  Up: Chebyshev Approximations

28.5 Examples
=============

The following example program computes Chebyshev approximations to a
step function.  This is an extremely difficult approximation to make,
due to the discontinuity, and was chosen as an example where
approximation error is visible.  For smooth functions the Chebyshev
approximation converges extremely rapidly and errors would not be
visible.

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

     double
     f (double x, void *p)
     {
       if (x < 0.5)
         return 0.25;
       else
         return 0.75;
     }

     int
     main (void)
     {
       int i, n = 10000;

       gsl_cheb_series *cs = gsl_cheb_alloc (40);

       gsl_function F;

       F.function = f;
       F.params = 0;

       gsl_cheb_init (cs, &F, 0.0, 1.0);

       for (i = 0; i < n; i++)
         {
           double x = i / (double)n;
           double r10 = gsl_cheb_eval_n (cs, 10, x);
           double r40 = gsl_cheb_eval (cs, x);
           printf ("%g %g %g %g\n",
                   x, GSL_FN_EVAL (&F, x), r10, r40);
         }

       gsl_cheb_free (cs);

       return 0;
     }

The output from the program gives the original function, 10-th order
approximation and 40-th order approximation, all sampled at intervals of
0.001 in x.


File: gsl-ref.info,  Node: Chebyshev Approximation References and Further Reading,  Prev: Chebyshev Approximation examples,  Up: Chebyshev Approximations

28.6 References and Further Reading
===================================

The following paper describes the use of Chebyshev series,

     R. Broucke, "Ten Subroutines for the Manipulation of Chebyshev
     Series [C1] (Algorithm 446)". `Communications of the ACM' 16(4),
     254-256 (1973)


File: gsl-ref.info,  Node: Series Acceleration,  Next: Wavelet Transforms,  Prev: Chebyshev Approximations,  Up: Top

29 Series Acceleration
**********************

The functions described in this chapter accelerate the convergence of a
series using the Levin u-transform.  This method takes a small number of
terms from the start of a series and uses a systematic approximation to
compute an extrapolated value and an estimate of its error.  The
u-transform works for both convergent and divergent series, including
asymptotic series.

   These functions are declared in the header file `gsl_sum.h'.

* Menu:

* Acceleration functions::
* Acceleration functions without error estimation::
* Example of accelerating a series::
* Series Acceleration References::


File: gsl-ref.info,  Node: Acceleration functions,  Next: Acceleration functions without error estimation,  Up: Series Acceleration

29.1 Acceleration functions
===========================

The following functions compute the full Levin u-transform of a series
with its error estimate.  The error estimate is computed by propagating
rounding errors from each term through to the final extrapolation.

   These functions are intended for summing analytic series where each
term is known to high accuracy, and the rounding errors are assumed to
originate from finite precision. They are taken to be relative errors of
order `GSL_DBL_EPSILON' for each term.

   The calculation of the error in the extrapolated value is an O(N^2)
process, which is expensive in time and memory.  A faster but less
reliable method which estimates the error from the convergence of the
extrapolated value is described in the next section.  For the method
described here a full table of intermediate values and derivatives
through to O(N) must be computed and stored, but this does give a
reliable error estimate.

 -- Function: gsl_sum_levin_u_workspace * gsl_sum_levin_u_alloc (size_t
          N)
     This function allocates a workspace for a Levin u-transform of N
     terms.  The size of the workspace is O(2n^2 + 3n).

 -- Function: int gsl_sum_levin_u_free (gsl_sum_levin_u_workspace * W)
     This function frees the memory associated with the workspace W.

 -- Function: int gsl_sum_levin_u_accel (const double * ARRAY, size_t
          ARRAY_SIZE, gsl_sum_levin_u_workspace * W, double *
          SUM_ACCEL, double * ABSERR)
     This function takes the terms of a series in ARRAY of size
     ARRAY_SIZE and computes the extrapolated limit of the series using
     a Levin u-transform.  Additional working space must be provided in
     W.  The extrapolated sum is stored in SUM_ACCEL, with an estimate
     of the absolute error stored in ABSERR.  The actual term-by-term
     sum is returned in `w->sum_plain'. The algorithm calculates the
     truncation error (the difference between two successive
     extrapolations) and round-off error (propagated from the individual
     terms) to choose an optimal number of terms for the extrapolation.


File: gsl-ref.info,  Node: Acceleration functions without error estimation,  Next: Example of accelerating a series,  Prev: Acceleration functions,  Up: Series Acceleration

29.2 Acceleration functions without error estimation
====================================================

The functions described in this section compute the Levin u-transform of
series and attempt to estimate the error from the "truncation error" in
the extrapolation, the difference between the final two approximations.
Using this method avoids the need to compute an intermediate table of
derivatives because the error is estimated from the behavior of the
extrapolated value itself. Consequently this algorithm is an O(N)
process and only requires O(N) terms of storage.  If the series
converges sufficiently fast then this procedure can be acceptable.  It
is appropriate to use this method when there is a need to compute many
extrapolations of series with similar convergence properties at
high-speed.  For example, when numerically integrating a function
defined by a parameterized series where the parameter varies only
slightly. A reliable error estimate should be computed first using the
full algorithm described above in order to verify the consistency of the
results.

 -- Function: gsl_sum_levin_utrunc_workspace *
gsl_sum_levin_utrunc_alloc (size_t N)
     This function allocates a workspace for a Levin u-transform of N
     terms, without error estimation.  The size of the workspace is
     O(3n).

 -- Function: int gsl_sum_levin_utrunc_free
          (gsl_sum_levin_utrunc_workspace * W)
     This function frees the memory associated with the workspace W.

 -- Function: int gsl_sum_levin_utrunc_accel (const double * ARRAY,
          size_t ARRAY_SIZE, gsl_sum_levin_utrunc_workspace * W, double
          * SUM_ACCEL, double * ABSERR_TRUNC)
     This function takes the terms of a series in ARRAY of size
     ARRAY_SIZE and computes the extrapolated limit of the series using
     a Levin u-transform.  Additional working space must be provided in
     W.  The extrapolated sum is stored in SUM_ACCEL.  The actual
     term-by-term sum is returned in `w->sum_plain'. The algorithm
     terminates when the difference between two successive
     extrapolations reaches a minimum or is sufficiently small. The
     difference between these two values is used as estimate of the
     error and is stored in ABSERR_TRUNC.  To improve the reliability
     of the algorithm the extrapolated values are replaced by moving
     averages when calculating the truncation error, smoothing out any
     fluctuations.


File: gsl-ref.info,  Node: Example of accelerating a series,  Next: Series Acceleration References,  Prev: Acceleration functions without error estimation,  Up: Series Acceleration

29.3 Examples
=============

The following code calculates an estimate of \zeta(2) = \pi^2 / 6 using
the series,

     \zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...

After N terms the error in the sum is O(1/N), making direct summation
of the series converge slowly.

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

     #define N 20

     int
     main (void)
     {
       double t[N];
       double sum_accel, err;
       double sum = 0;
       int n;

       gsl_sum_levin_u_workspace * w
         = gsl_sum_levin_u_alloc (N);

       const double zeta_2 = M_PI * M_PI / 6.0;

       /* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */

       for (n = 0; n < N; n++)
         {
           double np1 = n + 1.0;
           t[n] = 1.0 / (np1 * np1);
           sum += t[n];
         }

       gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);

       printf ("term-by-term sum = % .16f using %d terms\n",
               sum, N);

       printf ("term-by-term sum = % .16f using %d terms\n",
               w->sum_plain, w->terms_used);

       printf ("exact value      = % .16f\n", zeta_2);
       printf ("accelerated sum  = % .16f using %d terms\n",
               sum_accel, w->terms_used);

       printf ("estimated error  = % .16f\n", err);
       printf ("actual error     = % .16f\n",
               sum_accel - zeta_2);

       gsl_sum_levin_u_free (w);
       return 0;
     }

The output below shows that the Levin u-transform is able to obtain an
estimate of the sum to 1 part in 10^10 using the first eleven terms of
the series.  The error estimate returned by the function is also
accurate, giving the correct number of significant digits.

     $ ./a.out
     term-by-term sum =  1.5961632439130233 using 20 terms
     term-by-term sum =  1.5759958390005426 using 13 terms
     exact value      =  1.6449340668482264
     accelerated sum  =  1.6449340668166479 using 13 terms
     estimated error  =  0.0000000000508580
     actual error     = -0.0000000000315785

Note that a direct summation of this series would require 10^10 terms
to achieve the same precision as the accelerated sum does in 13 terms.


File: gsl-ref.info,  Node: Series Acceleration References,  Prev: Example of accelerating a series,  Up: Series Acceleration

29.4 References and Further Reading
===================================

The algorithms used by these functions are described in the following
papers,

     T. Fessler, W.F. Ford, D.A. Smith, HURRY: An acceleration
     algorithm for scalar sequences and series `ACM Transactions on
     Mathematical Software', 9(3):346-354, 1983.  and Algorithm 602
     9(3):355-357, 1983.

The theory of the u-transform was presented by Levin,

     D. Levin, Development of Non-Linear Transformations for Improving
     Convergence of Sequences, `Intern. J. Computer Math.' B3:371-388,
     1973.

A review paper on the Levin Transform is available online,
     Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations,
     `http://arxiv.org/abs/math/0005209'.


File: gsl-ref.info,  Node: Wavelet Transforms,  Next: Discrete Hankel Transforms,  Prev: Series Acceleration,  Up: Top

30 Wavelet Transforms
*********************

This chapter describes functions for performing Discrete Wavelet
Transforms (DWTs).  The library includes wavelets for real data in both
one and two dimensions.  The wavelet functions are declared in the
header files `gsl_wavelet.h' and `gsl_wavelet2d.h'.

* Menu:

* DWT Definitions::
* DWT Initialization::
* DWT Transform Functions::
* DWT Examples::
* DWT References::


File: gsl-ref.info,  Node: DWT Definitions,  Next: DWT Initialization,  Up: Wavelet Transforms

30.1 Definitions
================

The continuous wavelet transform and its inverse are defined by the
relations,

     w(s,\tau) = \int f(t) * \psi^*_{s,\tau}(t) dt

and,

     f(t) = \int \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau ds

where the basis functions \psi_{s,\tau} are obtained by scaling and
translation from a single function, referred to as the "mother wavelet".

   The discrete version of the wavelet transform acts on equally-spaced
samples, with fixed scaling and translation steps (s, \tau).  The
frequency and time axes are sampled "dyadically" on scales of 2^j
through a level parameter j.  The resulting family of functions
{\psi_{j,n}} constitutes an orthonormal basis for square-integrable
signals.

   The discrete wavelet transform is an O(N) algorithm, and is also
referred to as the "fast wavelet transform".


File: gsl-ref.info,  Node: DWT Initialization,  Next: DWT Transform Functions,  Prev: DWT Definitions,  Up: Wavelet Transforms

30.2 Initialization
===================

The `gsl_wavelet' structure contains the filter coefficients defining
the wavelet and any associated offset parameters.

 -- Function: gsl_wavelet * gsl_wavelet_alloc (const gsl_wavelet_type *
          T, size_t K)
     This function allocates and initializes a wavelet object of type
     T.  The parameter K selects the specific member of the wavelet
     family.  A null pointer is returned if insufficient memory is
     available or if a unsupported member is selected.

   The following wavelet types are implemented:

 -- Wavelet: gsl_wavelet_daubechies
 -- Wavelet: gsl_wavelet_daubechies_centered
     The is the Daubechies wavelet family of maximum phase with k/2
     vanishing moments.  The implemented wavelets are k=4, 6, ..., 20,
     with K even.

 -- Wavelet: gsl_wavelet_haar
 -- Wavelet: gsl_wavelet_haar_centered
     This is the Haar wavelet.  The only valid choice of k for the Haar
     wavelet is k=2.

 -- Wavelet: gsl_wavelet_bspline
 -- Wavelet: gsl_wavelet_bspline_centered
     This is the biorthogonal B-spline wavelet family of order (i,j).
     The implemented values of k = 100*i + j are 103, 105, 202, 204,
     206, 208, 301, 303, 305 307, 309.

The centered forms of the wavelets align the coefficients of the various
sub-bands on edges.  Thus the resulting visualization of the
coefficients of the wavelet transform in the phase plane is easier to
understand.

 -- Function: const char * gsl_wavelet_name (const gsl_wavelet * W)
     This function returns a pointer to the name of the wavelet family
     for W.

 -- Function: void gsl_wavelet_free (gsl_wavelet * W)
     This function frees the wavelet object W.

   The `gsl_wavelet_workspace' structure contains scratch space of the
same size as the input data and is used to hold intermediate results
during the transform.

 -- Function: gsl_wavelet_workspace * gsl_wavelet_workspace_alloc
          (size_t N)
     This function allocates a workspace for the discrete wavelet
     transform.  To perform a one-dimensional transform on N elements,
     a workspace of size N must be provided.  For two-dimensional
     transforms of N-by-N matrices it is sufficient to allocate a
     workspace of size N, since the transform operates on individual
     rows and columns.

 -- Function: void gsl_wavelet_workspace_free (gsl_wavelet_workspace *
          WORK)
     This function frees the allocated workspace WORK.


File: gsl-ref.info,  Node: DWT Transform Functions,  Next: DWT Examples,  Prev: DWT Initialization,  Up: Wavelet Transforms

30.3 Transform Functions
========================

This sections describes the actual functions performing the discrete
wavelet transform.  Note that the transforms use periodic boundary
conditions.  If the signal is not periodic in the sample length then
spurious coefficients will appear at the beginning and end of each level
of the transform.

* Menu:

* DWT in one dimension::
* DWT in two dimension::


File: gsl-ref.info,  Node: DWT in one dimension,  Next: DWT in two dimension,  Up: DWT Transform Functions

30.3.1 Wavelet transforms in one dimension
------------------------------------------

 -- Function: int gsl_wavelet_transform (const gsl_wavelet * W, double
          * DATA, size_t STRIDE, size_t N, gsl_wavelet_direction DIR,
          gsl_wavelet_workspace * WORK)
 -- Function: int gsl_wavelet_transform_forward (const gsl_wavelet * W,
          double * DATA, size_t STRIDE, size_t N, gsl_wavelet_workspace
          * WORK)
 -- Function: int gsl_wavelet_transform_inverse (const gsl_wavelet * W,
          double * DATA, size_t STRIDE, size_t N, gsl_wavelet_workspace
          * WORK)
     These functions compute in-place forward and inverse discrete
     wavelet transforms of length N with stride STRIDE on the array
     DATA. The length of the transform N is restricted to powers of
     two.  For the `transform' version of the function the argument DIR
     can be either `forward' (+1) or `backward' (-1).  A workspace WORK
     of length N must be provided.

     For the forward transform, the elements of the original array are
     replaced by the discrete wavelet transform f_i -> w_{j,k} in a
     packed triangular storage layout, where J is the index of the level
     j = 0 ... J-1 and K is the index of the coefficient within each
     level, k = 0 ... (2^j)-1.  The total number of levels is J =
     \log_2(n).  The output data has the following form,

          (s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0}, ...,
            d_{j,k}, ..., d_{J-1,2^{J-1}-1})

     where the first element is the smoothing coefficient s_{-1,0},
     followed by the detail coefficients d_{j,k} for each level j.  The
     backward transform inverts these coefficients to obtain the
     original data.

     These functions return a status of `GSL_SUCCESS' upon successful
     completion.  `GSL_EINVAL' is returned if N is not an integer power
     of 2 or if insufficient workspace is provided.


File: gsl-ref.info,  Node: DWT in two dimension,  Prev: DWT in one dimension,  Up: DWT Transform Functions

30.3.2 Wavelet transforms in two dimension
------------------------------------------

The library provides functions to perform two-dimensional discrete
wavelet transforms on square matrices.  The matrix dimensions must be an
integer power of two.  There are two possible orderings of the rows and
columns in the two-dimensional wavelet transform, referred to as the
"standard" and "non-standard" forms.

   The "standard" transform performs a complete discrete wavelet
transform on the rows of the matrix, followed by a separate complete
discrete wavelet transform on the columns of the resulting
row-transformed matrix.  This procedure uses the same ordering as a
two-dimensional fourier transform.

   The "non-standard" transform is performed in interleaved passes on
the rows and columns of the matrix for each level of the transform.  The
first level of the transform is applied to the matrix rows, and then to
the matrix columns.  This procedure is then repeated across the rows and
columns of the data for the subsequent levels of the transform, until
the full discrete wavelet transform is complete.  The non-standard form
of the discrete wavelet transform is typically used in image analysis.

   The functions described in this section are declared in the header
file `gsl_wavelet2d.h'.

 -- Function: int gsl_wavelet2d_transform (const gsl_wavelet * W,
          double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
          gsl_wavelet_direction DIR, gsl_wavelet_workspace * WORK)
 -- Function: int gsl_wavelet2d_transform_forward (const gsl_wavelet *
          W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
          gsl_wavelet_workspace * WORK)
 -- Function: int gsl_wavelet2d_transform_inverse (const gsl_wavelet *
          W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
          gsl_wavelet_workspace * WORK)
     These functions compute two-dimensional in-place forward and
     inverse discrete wavelet transforms in standard and non-standard
     forms on the array DATA stored in row-major form with dimensions
     SIZE1 and SIZE2 and physical row length TDA.  The dimensions must
     be equal (square matrix) and are restricted to powers of two.  For
     the `transform' version of the function the argument DIR can be
     either `forward' (+1) or `backward' (-1).  A workspace WORK of the
     appropriate size must be provided.  On exit, the appropriate
     elements of the array DATA are replaced by their two-dimensional
     wavelet transform.

     The functions return a status of `GSL_SUCCESS' upon successful
     completion.  `GSL_EINVAL' is returned if SIZE1 and SIZE2 are not
     equal and integer powers of 2, or if insufficient workspace is
     provided.

 -- Function: int gsl_wavelet2d_transform_matrix (const gsl_wavelet *
          W, gsl_matrix * M, gsl_wavelet_direction DIR,
          gsl_wavelet_workspace * WORK)
 -- Function: int gsl_wavelet2d_transform_matrix_forward (const
          gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
 -- Function: int gsl_wavelet2d_transform_matrix_inverse (const
          gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
     These functions compute the two-dimensional in-place wavelet
     transform on a matrix A.

 -- Function: int gsl_wavelet2d_nstransform (const gsl_wavelet * W,
          double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
          gsl_wavelet_direction DIR, gsl_wavelet_workspace * WORK)
 -- Function: int gsl_wavelet2d_nstransform_forward (const gsl_wavelet
          * W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
          gsl_wavelet_workspace * WORK)
 -- Function: int gsl_wavelet2d_nstransform_inverse (const gsl_wavelet
          * W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
          gsl_wavelet_workspace * WORK)
     These functions compute the two-dimensional wavelet transform in
     non-standard form.

 -- Function: int gsl_wavelet2d_nstransform_matrix (const gsl_wavelet *
          W, gsl_matrix * M, gsl_wavelet_direction DIR,
          gsl_wavelet_workspace * WORK)
 -- Function: int gsl_wavelet2d_nstransform_matrix_forward (const
          gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
 -- Function: int gsl_wavelet2d_nstransform_matrix_inverse (const
          gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
     These functions compute the non-standard form of the
     two-dimensional in-place wavelet transform on a matrix A.


File: gsl-ref.info,  Node: DWT Examples,  Next: DWT References,  Prev: DWT Transform Functions,  Up: Wavelet Transforms

30.4 Examples
=============

The following program demonstrates the use of the one-dimensional
wavelet transform functions.  It computes an approximation to an input
signal (of length 256) using the 20 largest components of the wavelet
transform, while setting the others to zero.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_sort.h>
     #include <gsl/gsl_wavelet.h>

     int
     main (int argc, char **argv)
     {
       int i, n = 256, nc = 20;
       double *data = malloc (n * sizeof (double));
       double *abscoeff = malloc (n * sizeof (double));
       size_t *p = malloc (n * sizeof (size_t));

       gsl_wavelet *w;
       gsl_wavelet_workspace *work;

       w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
       work = gsl_wavelet_workspace_alloc (n);

       FILE *f = fopen (argv[1], "r");
       for (i = 0; i < n; i++)
         {
           fscanf (f, "%lg", &data[i]);
         }
       fclose (f);

       gsl_wavelet_transform_forward (w, data, 1, n, work);

       for (i = 0; i < n; i++)
         {
           abscoeff[i] = fabs (data[i]);
         }

       gsl_sort_index (p, abscoeff, 1, n);

       for (i = 0; (i + nc) < n; i++)
         data[p[i]] = 0;

       gsl_wavelet_transform_inverse (w, data, 1, n, work);

       for (i = 0; i < n; i++)
         {
           printf ("%g\n", data[i]);
         }
     }

The output can be used with the GNU plotutils `graph' program,

     $ ./a.out ecg.dat > dwt.dat
     $ graph -T ps -x 0 256 32 -h 0.3 -a dwt.dat > dwt.ps


File: gsl-ref.info,  Node: DWT References,  Prev: DWT Examples,  Up: Wavelet Transforms

30.5 References and Further Reading
===================================

The mathematical background to wavelet transforms is covered in the
original lectures by Daubechies,

     Ingrid Daubechies.  Ten Lectures on Wavelets.  `CBMS-NSF Regional
     Conference Series in Applied Mathematics' (1992), SIAM, ISBN
     0898712742.

An easy to read introduction to the subject with an emphasis on the
application of the wavelet transform in various branches of science is,

     Paul S. Addison. `The Illustrated Wavelet Transform Handbook'.
     Institute of Physics Publishing (2002), ISBN 0750306920.

For extensive coverage of signal analysis by wavelets, wavelet packets
and local cosine bases see,

     S. G. Mallat.  `A wavelet tour of signal processing' (Second
     edition). Academic Press (1999), ISBN 012466606X.

The concept of multiresolution analysis underlying the wavelet transform
is described in,

     S. G. Mallat.  Multiresolution Approximations and Wavelet
     Orthonormal Bases of L^2(R).  `Transactions of the American
     Mathematical Society', 315(1), 1989, 69-87.

     S. G. Mallat.  A Theory for Multiresolution Signal
     Decomposition--The Wavelet Representation.  `IEEE Transactions on
     Pattern Analysis and Machine Intelligence', 11, 1989, 674-693.

The coefficients for the individual wavelet families implemented by the
library can be found in the following papers,

     I. Daubechies.  Orthonormal Bases of Compactly Supported Wavelets.
     `Communications on Pure and Applied Mathematics', 41 (1988)
     909-996.

     A. Cohen, I. Daubechies, and J.-C. Feauveau.  Biorthogonal Bases
     of Compactly Supported Wavelets.  `Communications on Pure and
     Applied Mathematics', 45 (1992) 485-560.

The PhysioNet archive of physiological datasets can be found online at
`http://www.physionet.org/' and is described in the following paper,

     Goldberger et al.  PhysioBank, PhysioToolkit, and PhysioNet:
     Components of a New Research Resource for Complex Physiologic
     Signals.  `Circulation' 101(23):e215-e220 2000.


File: gsl-ref.info,  Node: Discrete Hankel Transforms,  Next: One dimensional Root-Finding,  Prev: Wavelet Transforms,  Up: Top

31 Discrete Hankel Transforms
*****************************

This chapter describes functions for performing Discrete Hankel
Transforms (DHTs).  The functions are declared in the header file
`gsl_dht.h'.

* Menu:

* Discrete Hankel Transform Definition::
* Discrete Hankel Transform Functions::
* Discrete Hankel Transform References::


File: gsl-ref.info,  Node: Discrete Hankel Transform Definition,  Next: Discrete Hankel Transform Functions,  Up: Discrete Hankel Transforms

31.1 Definitions
================

The discrete Hankel transform acts on a vector of sampled data, where
the samples are assumed to have been taken at points related to the
zeroes of a Bessel function of fixed order; compare this to the case of
the discrete Fourier transform, where samples are taken at points
related to the zeroes of the sine or cosine function.

   Specifically, let f(t) be a function on the unit interval.  Then the
finite \nu-Hankel transform of f(t) is defined to be the set of numbers
g_m given by,
     g_m = \int_0^1 t dt J_\nu(j_(\nu,m)t) f(t),

so that,
     f(t) = \sum_{m=1}^\infty (2 J_\nu(j_(\nu,m)x) / J_(\nu+1)(j_(\nu,m))^2) g_m.

Suppose that f is band-limited in the sense that g_m=0 for m > M. Then
we have the following fundamental sampling theorem.
     g_m = (2 / j_(\nu,M)^2)
           \sum_{k=1}^{M-1} f(j_(\nu,k)/j_(\nu,M))
               (J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,k))^2).

It is this discrete expression which defines the discrete Hankel
transform. The kernel in the summation above defines the matrix of the
\nu-Hankel transform of size M-1.  The coefficients of this matrix,
being dependent on \nu and M, must be precomputed and stored; the
`gsl_dht' object encapsulates this data.  The allocation function
`gsl_dht_alloc' returns a `gsl_dht' object which must be properly
initialized with `gsl_dht_init' before it can be used to perform
transforms on data sample vectors, for fixed \nu and M, using the
`gsl_dht_apply' function. The implementation allows a scaling of the
fundamental interval, for convenience, so that one can assume the
function is defined on the interval [0,X], rather than the unit
interval.

   Notice that by assumption f(t) vanishes at the endpoints of the
interval, consistent with the inversion formula and the sampling
formula given above. Therefore, this transform corresponds to an
orthogonal expansion in eigenfunctions of the Dirichlet problem for the
Bessel differential equation.


File: gsl-ref.info,  Node: Discrete Hankel Transform Functions,  Next: Discrete Hankel Transform References,  Prev: Discrete Hankel Transform Definition,  Up: Discrete Hankel Transforms

31.2 Functions
==============

 -- Function: gsl_dht * gsl_dht_alloc (size_t SIZE)
     This function allocates a Discrete Hankel transform object of size
     SIZE.

 -- Function: int gsl_dht_init (gsl_dht * T, double NU, double XMAX)
     This function initializes the transform T for the given values of
     NU and X.

 -- Function: gsl_dht * gsl_dht_new (size_t SIZE, double NU, double
          XMAX)
     This function allocates a Discrete Hankel transform object of size
     SIZE and initializes it for the given values of NU and X.

 -- Function: void gsl_dht_free (gsl_dht * T)
     This function frees the transform T.

 -- Function: int gsl_dht_apply (const gsl_dht * T, double * F_IN,
          double * F_OUT)
     This function applies the transform T to the array F_IN whose size
     is equal to the size of the transform.  The result is stored in
     the array F_OUT which must be of the same length.

 -- Function: double gsl_dht_x_sample (const gsl_dht * T, int N)
     This function returns the value of the N-th sample point in the
     unit interval, (j_{\nu,n+1}/j_{\nu,M}) X. These are the points
     where the function f(t) is assumed to be sampled.

 -- Function: double gsl_dht_k_sample (const gsl_dht * T, int N)
     This function returns the value of the N-th sample point in
     "k-space", j_{\nu,n+1}/X.


File: gsl-ref.info,  Node: Discrete Hankel Transform References,  Prev: Discrete Hankel Transform Functions,  Up: Discrete Hankel Transforms

31.3 References and Further Reading
===================================

The algorithms used by these functions are described in the following
papers,

     H. Fisk Johnson, Comp. Phys. Comm. 43, 181 (1987).

     D. Lemoine, J. Chem. Phys. 101, 3936 (1994).


File: gsl-ref.info,  Node: One dimensional Root-Finding,  Next: One dimensional Minimization,  Prev: Discrete Hankel Transforms,  Up: Top

32 One dimensional Root-Finding
*******************************

This chapter describes routines for finding roots of arbitrary
one-dimensional functions.  The library provides low level components
for a variety of iterative solvers and convergence tests.  These can be
combined by the user to achieve the desired solution, with full access
to the intermediate steps of the iteration.  Each class of methods uses
the same framework, so that you can switch between solvers at runtime
without needing to recompile your program.  Each instance of a solver
keeps track of its own state, allowing the solvers to be used in
multi-threaded programs.

   The header file `gsl_roots.h' contains prototypes for the root
finding functions and related declarations.

* Menu:

* Root Finding Overview::
* Root Finding Caveats::
* Initializing the Solver::
* Providing the function to solve::
* Search Bounds and Guesses::
* Root Finding Iteration::
* Search Stopping Parameters::
* Root Bracketing Algorithms::
* Root Finding Algorithms using Derivatives::
* Root Finding Examples::
* Root Finding References and Further Reading::


File: gsl-ref.info,  Node: Root Finding Overview,  Next: Root Finding Caveats,  Up: One dimensional Root-Finding

32.1 Overview
=============

One-dimensional root finding algorithms can be divided into two classes,
"root bracketing" and "root polishing".  Algorithms which proceed by
bracketing a root are guaranteed to converge.  Bracketing algorithms
begin with a bounded region known to contain a root.  The size of this
bounded region is reduced, iteratively, until it encloses the root to a
desired tolerance.  This provides a rigorous error estimate for the
location of the root.

   The technique of "root polishing" attempts to improve an initial
guess to the root.  These algorithms converge only if started "close
enough" to a root, and sacrifice a rigorous error bound for speed.  By
approximating the behavior of a function in the vicinity of a root they
attempt to find a higher order improvement of an initial guess.  When
the behavior of the function is compatible with the algorithm and a good
initial guess is available a polishing algorithm can provide rapid
convergence.

   In GSL both types of algorithm are available in similar frameworks.
The user provides a high-level driver for the algorithms, and the
library provides the individual functions necessary for each of the
steps.  There are three main phases of the iteration.  The steps are,

   * initialize solver state, S, for algorithm T

   * update S using the iteration T

   * test S for convergence, and repeat iteration if necessary

The state for bracketing solvers is held in a `gsl_root_fsolver'
struct.  The updating procedure uses only function evaluations (not
derivatives).  The state for root polishing solvers is held in a
`gsl_root_fdfsolver' struct.  The updates require both the function and
its derivative (hence the name `fdf') to be supplied by the user.


File: gsl-ref.info,  Node: Root Finding Caveats,  Next: Initializing the Solver,  Prev: Root Finding Overview,  Up: One dimensional Root-Finding

32.2 Caveats
============

Note that root finding functions can only search for one root at a time.
When there are several roots in the search area, the first root to be
found will be returned; however it is difficult to predict which of the
roots this will be. _In most cases, no error will be reported if you
try to find a root in an area where there is more than one._

   Care must be taken when a function may have a multiple root (such as
f(x) = (x-x_0)^2 or f(x) = (x-x_0)^3).  It is not possible to use
root-bracketing algorithms on even-multiplicity roots.  For these
algorithms the initial interval must contain a zero-crossing, where the
function is negative at one end of the interval and positive at the
other end.  Roots with even-multiplicity do not cross zero, but only
touch it instantaneously.  Algorithms based on root bracketing will
still work for odd-multiplicity roots (e.g. cubic, quintic, ...).  Root
polishing algorithms generally work with higher multiplicity roots, but
at a reduced rate of convergence.  In these cases the "Steffenson
algorithm" can be used to accelerate the convergence of multiple roots.

   While it is not absolutely required that f have a root within the
search region, numerical root finding functions should not be used
haphazardly to check for the _existence_ of roots.  There are better
ways to do this.  Because it is easy to create situations where
numerical root finders can fail, it is a bad idea to throw a root
finder at a function you do not know much about.  In general it is best
to examine the function visually by plotting before searching for a
root.


File: gsl-ref.info,  Node: Initializing the Solver,  Next: Providing the function to solve,  Prev: Root Finding Caveats,  Up: One dimensional Root-Finding

32.3 Initializing the Solver
============================

 -- Function: gsl_root_fsolver * gsl_root_fsolver_alloc (const
          gsl_root_fsolver_type * T)
     This function returns a pointer to a newly allocated instance of a
     solver of type T.  For example, the following code creates an
     instance of a bisection solver,

          const gsl_root_fsolver_type * T
            = gsl_root_fsolver_bisection;
          gsl_root_fsolver * s
            = gsl_root_fsolver_alloc (T);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of `GSL_ENOMEM'.

 -- Function: gsl_root_fdfsolver * gsl_root_fdfsolver_alloc (const
          gsl_root_fdfsolver_type * T)
     This function returns a pointer to a newly allocated instance of a
     derivative-based solver of type T.  For example, the following
     code creates an instance of a Newton-Raphson solver,

          const gsl_root_fdfsolver_type * T
            = gsl_root_fdfsolver_newton;
          gsl_root_fdfsolver * s
            = gsl_root_fdfsolver_alloc (T);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of `GSL_ENOMEM'.

 -- Function: int gsl_root_fsolver_set (gsl_root_fsolver * S,
          gsl_function * F, double X_LOWER, double X_UPPER)
     This function initializes, or reinitializes, an existing solver S
     to use the function F and the initial search interval [X_LOWER,
     X_UPPER].

 -- Function: int gsl_root_fdfsolver_set (gsl_root_fdfsolver * S,
          gsl_function_fdf * FDF, double ROOT)
     This function initializes, or reinitializes, an existing solver S
     to use the function and derivative FDF and the initial guess ROOT.

 -- Function: void gsl_root_fsolver_free (gsl_root_fsolver * S)
 -- Function: void gsl_root_fdfsolver_free (gsl_root_fdfsolver * S)
     These functions free all the memory associated with the solver S.

 -- Function: const char * gsl_root_fsolver_name (const
          gsl_root_fsolver * S)
 -- Function: const char * gsl_root_fdfsolver_name (const
          gsl_root_fdfsolver * S)
     These functions return a pointer to the name of the solver.  For
     example,

          printf ("s is a '%s' solver\n",
                  gsl_root_fsolver_name (s));

     would print something like `s is a 'bisection' solver'.


File: gsl-ref.info,  Node: Providing the function to solve,  Next: Search Bounds and Guesses,  Prev: Initializing the Solver,  Up: One dimensional Root-Finding

32.4 Providing the function to solve
====================================

You must provide a continuous function of one variable for the root
finders to operate on, and, sometimes, its first derivative.  In order
to allow for general parameters the functions are defined by the
following data types:

 -- Data Type: gsl_function
     This data type defines a general function with parameters.

    `double (* function) (double X, void * PARAMS)'
          this function should return the value f(x,params) for
          argument X and parameters PARAMS

    `void * params'
          a pointer to the parameters of the function

   Here is an example for the general quadratic function,

     f(x) = a x^2 + b x + c

with a = 3, b = 2, c = 1.  The following code defines a `gsl_function'
`F' which you could pass to a root finder:

     struct my_f_params { double a; double b; double c; };

     double
     my_f (double x, void * p) {
        struct my_f_params * params
          = (struct my_f_params *)p;
        double a = (params->a);
        double b = (params->b);
        double c = (params->c);

        return  (a * x + b) * x + c;
     }

     gsl_function F;
     struct my_f_params params = { 3.0, 2.0, 1.0 };

     F.function = &my_f;
     F.params = &params;

The function f(x) can be evaluated using the following macro,

     #define GSL_FN_EVAL(F,x)
         (*((F)->function))(x,(F)->params)

 -- Data Type: gsl_function_fdf
     This data type defines a general function with parameters and its
     first derivative.

    `double (* f) (double X, void * PARAMS)'
          this function should return the value of f(x,params) for
          argument X and parameters PARAMS

    `double (* df) (double X, void * PARAMS)'
          this function should return the value of the derivative of F
          with respect to X, f'(x,params), for argument X and
          parameters PARAMS

    `void (* fdf) (double X, void * PARAMS, double * F, double * Df)'
          this function should set the values of the function F to
          f(x,params) and its derivative DF to f'(x,params) for
          argument X and parameters PARAMS.  This function provides an
          optimization of the separate functions for f(x) and f'(x)--it
          is always faster to compute the function and its derivative
          at the same time.

    `void * params'
          a pointer to the parameters of the function

   Here is an example where f(x) = 2\exp(2x):

     double
     my_f (double x, void * params)
     {
        return exp (2 * x);
     }

     double
     my_df (double x, void * params)
     {
        return 2 * exp (2 * x);
     }

     void
     my_fdf (double x, void * params,
             double * f, double * df)
     {
        double t = exp (2 * x);

        *f = t;
        *df = 2 * t;   /* uses existing value */
     }

     gsl_function_fdf FDF;

     FDF.f = &my_f;
     FDF.df = &my_df;
     FDF.fdf = &my_fdf;
     FDF.params = 0;

The function f(x) can be evaluated using the following macro,

     #define GSL_FN_FDF_EVAL_F(FDF,x)
          (*((FDF)->f))(x,(FDF)->params)

The derivative f'(x) can be evaluated using the following macro,

     #define GSL_FN_FDF_EVAL_DF(FDF,x)
          (*((FDF)->df))(x,(FDF)->params)

and both the function y = f(x) and its derivative dy = f'(x) can be
evaluated at the same time using the following macro,

     #define GSL_FN_FDF_EVAL_F_DF(FDF,x,y,dy)
          (*((FDF)->fdf))(x,(FDF)->params,(y),(dy))

The macro stores f(x) in its Y argument and f'(x) in its DY
argument--both of these should be pointers to `double'.


File: gsl-ref.info,  Node: Search Bounds and Guesses,  Next: Root Finding Iteration,  Prev: Providing the function to solve,  Up: One dimensional Root-Finding

32.5 Search Bounds and Guesses
==============================

You provide either search bounds or an initial guess; this section
explains how search bounds and guesses work and how function arguments
control them.

   A guess is simply an x value which is iterated until it is within
the desired precision of a root.  It takes the form of a `double'.

   Search bounds are the endpoints of a interval which is iterated until
the length of the interval is smaller than the requested precision.  The
interval is defined by two values, the lower limit and the upper limit.
Whether the endpoints are intended to be included in the interval or not
depends on the context in which the interval is used.


File: gsl-ref.info,  Node: Root Finding Iteration,  Next: Search Stopping Parameters,  Prev: Search Bounds and Guesses,  Up: One dimensional Root-Finding

32.6 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any solver of the
corresponding type.  The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.

 -- Function: int gsl_root_fsolver_iterate (gsl_root_fsolver * S)
 -- Function: int gsl_root_fdfsolver_iterate (gsl_root_fdfsolver * S)
     These functions perform a single iteration of the solver S.  If the
     iteration encounters an unexpected problem then an error code will
     be returned,

    `GSL_EBADFUNC'
          the iteration encountered a singular point where the function
          or its derivative evaluated to `Inf' or `NaN'.

    `GSL_EZERODIV'
          the derivative of the function vanished at the iteration
          point, preventing the algorithm from continuing without a
          division by zero.

   The solver maintains a current best estimate of the root at all
times.  The bracketing solvers also keep track of the current best
interval bounding the root.  This information can be accessed with the
following auxiliary functions,

 -- Function: double gsl_root_fsolver_root (const gsl_root_fsolver * S)
 -- Function: double gsl_root_fdfsolver_root (const gsl_root_fdfsolver
          * S)
     These functions return the current estimate of the root for the
     solver S.

 -- Function: double gsl_root_fsolver_x_lower (const gsl_root_fsolver *
          S)
 -- Function: double gsl_root_fsolver_x_upper (const gsl_root_fsolver *
          S)
     These functions return the current bracketing interval for the
     solver S.


File: gsl-ref.info,  Node: Search Stopping Parameters,  Next: Root Bracketing Algorithms,  Prev: Root Finding Iteration,  Up: One dimensional Root-Finding

32.7 Search Stopping Parameters
===============================

A root finding procedure should stop when one of the following
conditions is true:

   * A root has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result in
several standard ways.

 -- Function: int gsl_root_test_interval (double X_LOWER, double
          X_UPPER, double EPSABS, double EPSREL)
     This function tests for the convergence of the interval [X_LOWER,
     X_UPPER] with absolute error EPSABS and relative error EPSREL.
     The test returns `GSL_SUCCESS' if the following condition is
     achieved,

          |a - b| < epsabs + epsrel min(|a|,|b|)

     when the interval x = [a,b] does not include the origin.  If the
     interval includes the origin then \min(|a|,|b|) is replaced by
     zero (which is the minimum value of |x| over the interval).  This
     ensures that the relative error is accurately estimated for roots
     close to the origin.

     This condition on the interval also implies that any estimate of
     the root r in the interval satisfies the same condition with
     respect to the true root r^*,

          |r - r^*| < epsabs + epsrel r^*

     assuming that the true root r^* is contained within the interval.

 -- Function: int gsl_root_test_delta (double X1, double X0, double
          EPSABS, double EPSREL)
     This function tests for the convergence of the sequence ..., X0,
     X1 with absolute error EPSABS and relative error EPSREL.  The test
     returns `GSL_SUCCESS' if the following condition is achieved,

          |x_1 - x_0| < epsabs + epsrel |x_1|

     and returns `GSL_CONTINUE' otherwise.

 -- Function: int gsl_root_test_residual (double F, double EPSABS)
     This function tests the residual value F against the absolute
     error bound EPSABS.  The test returns `GSL_SUCCESS' if the
     following condition is achieved,

          |f| < epsabs

     and returns `GSL_CONTINUE' otherwise.  This criterion is suitable
     for situations where the precise location of the root, x, is
     unimportant provided a value can be found where the residual,
     |f(x)|, is small enough.


File: gsl-ref.info,  Node: Root Bracketing Algorithms,  Next: Root Finding Algorithms using Derivatives,  Prev: Search Stopping Parameters,  Up: One dimensional Root-Finding

32.8 Root Bracketing Algorithms
===============================

The root bracketing algorithms described in this section require an
initial interval which is guaranteed to contain a root--if a and b are
the endpoints of the interval then f(a) must differ in sign from f(b).
This ensures that the function crosses zero at least once in the
interval.  If a valid initial interval is used then these algorithm
cannot fail, provided the function is well-behaved.

   Note that a bracketing algorithm cannot find roots of even degree,
since these do not cross the x-axis.

 -- Solver: gsl_root_fsolver_bisection
     The "bisection algorithm" is the simplest method of bracketing the
     roots of a function.   It is the slowest algorithm provided by the
     library, with linear convergence.

     On each iteration, the interval is bisected and the value of the
     function at the midpoint is calculated.  The sign of this value is
     used to determine which half of the interval does not contain a
     root.  That half is discarded to give a new, smaller interval
     containing the root.  This procedure can be continued indefinitely
     until the interval is sufficiently small.

     At any time the current estimate of the root is taken as the
     midpoint of the interval.


 -- Solver: gsl_root_fsolver_falsepos
     The "false position algorithm" is a method of finding roots based
     on linear interpolation.  Its convergence is linear, but it is
     usually faster than bisection.

     On each iteration a line is drawn between the endpoints (a,f(a))
     and (b,f(b)) and the point where this line crosses the x-axis
     taken as a "midpoint".  The value of the function at this point is
     calculated and its sign is used to determine which side of the
     interval does not contain a root.  That side is discarded to give a
     new, smaller interval containing the root.  This procedure can be
     continued indefinitely until the interval is sufficiently small.

     The best estimate of the root is taken from the linear
     interpolation of the interval on the current iteration.


 -- Solver: gsl_root_fsolver_brent
     The "Brent-Dekker method" (referred to here as "Brent's method")
     combines an interpolation strategy with the bisection algorithm.
     This produces a fast algorithm which is still robust.

     On each iteration Brent's method approximates the function using an
     interpolating curve.  On the first iteration this is a linear
     interpolation of the two endpoints.  For subsequent iterations the
     algorithm uses an inverse quadratic fit to the last three points,
     for higher accuracy.  The intercept of the interpolating curve
     with the x-axis is taken as a guess for the root.  If it lies
     within the bounds of the current interval then the interpolating
     point is accepted, and used to generate a smaller interval.  If
     the interpolating point is not accepted then the algorithm falls
     back to an ordinary bisection step.

     The best estimate of the root is taken from the most recent
     interpolation or bisection.


File: gsl-ref.info,  Node: Root Finding Algorithms using Derivatives,  Next: Root Finding Examples,  Prev: Root Bracketing Algorithms,  Up: One dimensional Root-Finding

32.9 Root Finding Algorithms using Derivatives
==============================================

The root polishing algorithms described in this section require an
initial guess for the location of the root.  There is no absolute
guarantee of convergence--the function must be suitable for this
technique and the initial guess must be sufficiently close to the root
for it to work.  When these conditions are satisfied then convergence is
quadratic.

   These algorithms make use of both the function and its derivative.

 -- Derivative Solver: gsl_root_fdfsolver_newton
     Newton's Method is the standard root-polishing algorithm.  The
     algorithm begins with an initial guess for the location of the
     root.  On each iteration, a line tangent to the function f is
     drawn at that position.  The point where this line crosses the
     x-axis becomes the new guess.  The iteration is defined by the
     following sequence,

          x_{i+1} = x_i - f(x_i)/f'(x_i)

     Newton's method converges quadratically for single roots, and
     linearly for multiple roots.


 -- Derivative Solver: gsl_root_fdfsolver_secant
     The "secant method" is a simplified version of Newton's method
     which does not require the computation of the derivative on every
     step.

     On its first iteration the algorithm begins with Newton's method,
     using the derivative to compute a first step,

          x_1 = x_0 - f(x_0)/f'(x_0)

     Subsequent iterations avoid the evaluation of the derivative by
     replacing it with a numerical estimate, the slope of the line
     through the previous two points,

          x_{i+1} = x_i f(x_i) / f'_{est} where
           f'_{est} = (f(x_i) - f(x_{i-1})/(x_i - x_{i-1})

     When the derivative does not change significantly in the vicinity
     of the root the secant method gives a useful saving.
     Asymptotically the secant method is faster than Newton's method
     whenever the cost of evaluating the derivative is more than 0.44
     times the cost of evaluating the function itself.  As with all
     methods of computing a numerical derivative the estimate can
     suffer from cancellation errors if the separation of the points
     becomes too small.

     On single roots, the method has a convergence of order (1 + \sqrt
     5)/2 (approximately 1.62).  It converges linearly for multiple
     roots.


 -- Derivative Solver: gsl_root_fdfsolver_steffenson
     The "Steffenson Method" provides the fastest convergence of all the
     routines.  It combines the basic Newton algorithm with an Aitken
     "delta-squared" acceleration.  If the Newton iterates are x_i then
     the acceleration procedure generates a new sequence R_i,

          R_i = x_i - (x_{i+1} - x_i)^2 / (x_{i+2} - 2 x_{i+1} + x_{i})

     which converges faster than the original sequence under reasonable
     conditions.  The new sequence requires three terms before it can
     produce its first value so the method returns accelerated values
     on the second and subsequent iterations.  On the first iteration
     it returns the ordinary Newton estimate.  The Newton iterate is
     also returned if the denominator of the acceleration term ever
     becomes zero.

     As with all acceleration procedures this method can become
     unstable if the function is not well-behaved.


File: gsl-ref.info,  Node: Root Finding Examples,  Next: Root Finding References and Further Reading,  Prev: Root Finding Algorithms using Derivatives,  Up: One dimensional Root-Finding

32.10 Examples
==============

For any root finding algorithm we need to prepare the function to be
solved.  For this example we will use the general quadratic equation
described earlier.  We first need a header file (`demo_fn.h') to define
the function parameters,

     struct quadratic_params
       {
         double a, b, c;
       };

     double quadratic (double x, void *params);
     double quadratic_deriv (double x, void *params);
     void quadratic_fdf (double x, void *params,
                         double *y, double *dy);

We place the function definitions in a separate file (`demo_fn.c'),

     double
     quadratic (double x, void *params)
     {
       struct quadratic_params *p
         = (struct quadratic_params *) params;

       double a = p->a;
       double b = p->b;
       double c = p->c;

       return (a * x + b) * x + c;
     }

     double
     quadratic_deriv (double x, void *params)
     {
       struct quadratic_params *p
         = (struct quadratic_params *) params;

       double a = p->a;
       double b = p->b;
       double c = p->c;

       return 2.0 * a * x + b;
     }

     void
     quadratic_fdf (double x, void *params,
                    double *y, double *dy)
     {
       struct quadratic_params *p
         = (struct quadratic_params *) params;

       double a = p->a;
       double b = p->b;
       double c = p->c;

       *y = (a * x + b) * x + c;
       *dy = 2.0 * a * x + b;
     }

The first program uses the function solver `gsl_root_fsolver_brent' for
Brent's method and the general quadratic defined above to solve the
following equation,

     x^2 - 5 = 0

with solution x = \sqrt 5 = 2.236068...

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_roots.h>

     #include "demo_fn.h"
     #include "demo_fn.c"

     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_root_fsolver_type *T;
       gsl_root_fsolver *s;
       double r = 0, r_expected = sqrt (5.0);
       double x_lo = 0.0, x_hi = 5.0;
       gsl_function F;
       struct quadratic_params params = {1.0, 0.0, -5.0};

       F.function = &quadratic;
       F.params = &params;

       T = gsl_root_fsolver_brent;
       s = gsl_root_fsolver_alloc (T);
       gsl_root_fsolver_set (s, &F, x_lo, x_hi);

       printf ("using %s method\n",
               gsl_root_fsolver_name (s));

       printf ("%5s [%9s, %9s] %9s %10s %9s\n",
               "iter", "lower", "upper", "root",
               "err", "err(est)");

       do
         {
           iter++;
           status = gsl_root_fsolver_iterate (s);
           r = gsl_root_fsolver_root (s);
           x_lo = gsl_root_fsolver_x_lower (s);
           x_hi = gsl_root_fsolver_x_upper (s);
           status = gsl_root_test_interval (x_lo, x_hi,
                                            0, 0.001);

           if (status == GSL_SUCCESS)
             printf ("Converged:\n");

           printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
                   iter, x_lo, x_hi,
                   r, r - r_expected,
                   x_hi - x_lo);
         }
       while (status == GSL_CONTINUE && iter < max_iter);
       return status;
     }

Here are the results of the iterations,

     $ ./a.out
     using brent method
      iter [    lower,     upper]      root        err  err(est)
         1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
         2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
         3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
         4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
         5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
     Converged:
         6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666

If the program is modified to use the bisection solver instead of
Brent's method, by changing `gsl_root_fsolver_brent' to
`gsl_root_fsolver_bisection' the slower convergence of the Bisection
method can be observed,

     $ ./a.out
     using bisection method
      iter [    lower,     upper]      root        err  err(est)
         1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
         2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
         3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
         4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
         5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
         6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
         7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
         8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
         9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
        10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
        11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
     Converged:
        12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207

   The next program solves the same function using a derivative solver
instead.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_roots.h>

     #include "demo_fn.h"
     #include "demo_fn.c"

     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_root_fdfsolver_type *T;
       gsl_root_fdfsolver *s;
       double x0, x = 5.0, r_expected = sqrt (5.0);
       gsl_function_fdf FDF;
       struct quadratic_params params = {1.0, 0.0, -5.0};

       FDF.f = &quadratic;
       FDF.df = &quadratic_deriv;
       FDF.fdf = &quadratic_fdf;
       FDF.params = &params;

       T = gsl_root_fdfsolver_newton;
       s = gsl_root_fdfsolver_alloc (T);
       gsl_root_fdfsolver_set (s, &FDF, x);

       printf ("using %s method\n",
               gsl_root_fdfsolver_name (s));

       printf ("%-5s %10s %10s %10s\n",
               "iter", "root", "err", "err(est)");
       do
         {
           iter++;
           status = gsl_root_fdfsolver_iterate (s);
           x0 = x;
           x = gsl_root_fdfsolver_root (s);
           status = gsl_root_test_delta (x, x0, 0, 1e-3);

           if (status == GSL_SUCCESS)
             printf ("Converged:\n");

           printf ("%5d %10.7f %+10.7f %10.7f\n",
                   iter, x, x - r_expected, x - x0);
         }
       while (status == GSL_CONTINUE && iter < max_iter);
       return status;
     }

Here are the results for Newton's method,

     $ ./a.out
     using newton method
     iter        root        err   err(est)
         1  3.0000000 +0.7639320 -2.0000000
         2  2.3333333 +0.0972654 -0.6666667
         3  2.2380952 +0.0020273 -0.0952381
     Converged:
         4  2.2360689 +0.0000009 -0.0020263

Note that the error can be estimated more accurately by taking the
difference between the current iterate and next iterate rather than the
previous iterate.  The other derivative solvers can be investigated by
changing `gsl_root_fdfsolver_newton' to `gsl_root_fdfsolver_secant' or
`gsl_root_fdfsolver_steffenson'.


File: gsl-ref.info,  Node: Root Finding References and Further Reading,  Prev: Root Finding Examples,  Up: One dimensional Root-Finding

32.11 References and Further Reading
====================================

For information on the Brent-Dekker algorithm see the following two
papers,

     R. P. Brent, "An algorithm with guaranteed convergence for finding
     a zero of a function", `Computer Journal', 14 (1971) 422-425

     J. C. P. Bus and T. J. Dekker, "Two Efficient Algorithms with
     Guaranteed Convergence for Finding a Zero of a Function", `ACM
     Transactions of Mathematical Software', Vol. 1 No. 4 (1975) 330-345


File: gsl-ref.info,  Node: One dimensional Minimization,  Next: Multidimensional Root-Finding,  Prev: One dimensional Root-Finding,  Up: Top

33 One dimensional Minimization
*******************************

This chapter describes routines for finding minima of arbitrary
one-dimensional functions.  The library provides low level components
for a variety of iterative minimizers and convergence tests.  These can
be combined by the user to achieve the desired solution, with full
access to the intermediate steps of the algorithms.  Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program.  Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs.

   The header file `gsl_min.h' contains prototypes for the minimization
functions and related declarations.  To use the minimization algorithms
to find the maximum of a function simply invert its sign.

* Menu:

* Minimization Overview::
* Minimization Caveats::
* Initializing the Minimizer::
* Providing the function to minimize::
* Minimization Iteration::
* Minimization Stopping Parameters::
* Minimization Algorithms::
* Minimization Examples::
* Minimization References and Further Reading::


File: gsl-ref.info,  Node: Minimization Overview,  Next: Minimization Caveats,  Up: One dimensional Minimization

33.1 Overview
=============

The minimization algorithms begin with a bounded region known to contain
a minimum.  The region is described by a lower bound a and an upper
bound b, with an estimate of the location of the minimum x.

The value of the function at x must be less than the value of the
function at the ends of the interval,

     f(a) > f(x) < f(b)

This condition guarantees that a minimum is contained somewhere within
the interval.  On each iteration a new point x' is selected using one
of the available algorithms.  If the new point is a better estimate of
the minimum, i.e. where f(x') < f(x), then the current estimate of the
minimum x is updated.  The new point also allows the size of the
bounded interval to be reduced, by choosing the most compact set of
points which satisfies the constraint f(a) > f(x) < f(b).  The interval
is reduced until it encloses the true minimum to a desired tolerance.
This provides a best estimate of the location of the minimum and a
rigorous error estimate.

   Several bracketing algorithms are available within a single
framework.  The user provides a high-level driver for the algorithm,
and the library provides the individual functions necessary for each of
the steps.  There are three main phases of the iteration.  The steps
are,

   * initialize minimizer state, S, for algorithm T

   * update S using the iteration T

   * test S for convergence, and repeat iteration if necessary

The state for the minimizers is held in a `gsl_min_fminimizer' struct.
The updating procedure uses only function evaluations (not derivatives).


File: gsl-ref.info,  Node: Minimization Caveats,  Next: Initializing the Minimizer,  Prev: Minimization Overview,  Up: One dimensional Minimization

33.2 Caveats
============

Note that minimization functions can only search for one minimum at a
time.  When there are several minima in the search area, the first
minimum to be found will be returned; however it is difficult to predict
which of the minima this will be. _In most cases, no error will be
reported if you try to find a minimum in an area where there is more
than one._

   With all minimization algorithms it can be difficult to determine the
location of the minimum to full numerical precision.  The behavior of
the function in the region of the minimum x^* can be approximated by a
Taylor expansion,

     y = f(x^*) + (1/2) f''(x^*) (x - x^*)^2

and the second term of this expansion can be lost when added to the
first term at finite precision.  This magnifies the error in locating
x^*, making it proportional to \sqrt \epsilon (where \epsilon is the
relative accuracy of the floating point numbers).  For functions with
higher order minima, such as x^4, the magnification of the error is
correspondingly worse.  The best that can be achieved is to converge to
the limit of numerical accuracy in the function values, rather than the
location of the minimum itself.


File: gsl-ref.info,  Node: Initializing the Minimizer,  Next: Providing the function to minimize,  Prev: Minimization Caveats,  Up: One dimensional Minimization

33.3 Initializing the Minimizer
===============================

 -- Function: gsl_min_fminimizer * gsl_min_fminimizer_alloc (const
          gsl_min_fminimizer_type * T)
     This function returns a pointer to a newly allocated instance of a
     minimizer of type T.  For example, the following code creates an
     instance of a golden section minimizer,

          const gsl_min_fminimizer_type * T
            = gsl_min_fminimizer_goldensection;
          gsl_min_fminimizer * s
            = gsl_min_fminimizer_alloc (T);

     If there is insufficient memory to create the minimizer then the
     function returns a null pointer and the error handler is invoked
     with an error code of `GSL_ENOMEM'.

 -- Function: int gsl_min_fminimizer_set (gsl_min_fminimizer * S,
          gsl_function * F, double X_MINIMUM, double X_LOWER, double
          X_UPPER)
     This function sets, or resets, an existing minimizer S to use the
     function F and the initial search interval [X_LOWER, X_UPPER],
     with a guess for the location of the minimum X_MINIMUM.

     If the interval given does not contain a minimum, then the function
     returns an error code of `GSL_EINVAL'.

 -- Function: int gsl_min_fminimizer_set_with_values
          (gsl_min_fminimizer * S, gsl_function * F, double X_MINIMUM,
          double F_MINIMUM, double X_LOWER, double F_LOWER, double
          X_UPPER, double F_UPPER)
     This function is equivalent to `gsl_min_fminimizer_set' but uses
     the values F_MINIMUM, F_LOWER and F_UPPER instead of computing
     `f(x_minimum)', `f(x_lower)' and `f(x_upper)'.

 -- Function: void gsl_min_fminimizer_free (gsl_min_fminimizer * S)
     This function frees all the memory associated with the minimizer S.

 -- Function: const char * gsl_min_fminimizer_name (const
          gsl_min_fminimizer * S)
     This function returns a pointer to the name of the minimizer.  For
     example,

          printf ("s is a '%s' minimizer\n",
                  gsl_min_fminimizer_name (s));

     would print something like `s is a 'brent' minimizer'.


File: gsl-ref.info,  Node: Providing the function to minimize,  Next: Minimization Iteration,  Prev: Initializing the Minimizer,  Up: One dimensional Minimization

33.4 Providing the function to minimize
=======================================

You must provide a continuous function of one variable for the
minimizers to operate on.  In order to allow for general parameters the
functions are defined by a `gsl_function' data type (*note Providing
the function to solve::).


File: gsl-ref.info,  Node: Minimization Iteration,  Next: Minimization Stopping Parameters,  Prev: Providing the function to minimize,  Up: One dimensional Minimization

33.5 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any minimizer of
the corresponding type.  The same functions work for all minimizers so
that different methods can be substituted at runtime without
modifications to the code.

 -- Function: int gsl_min_fminimizer_iterate (gsl_min_fminimizer * S)
     This function performs a single iteration of the minimizer S.  If
     the iteration encounters an unexpected problem then an error code
     will be returned,

    `GSL_EBADFUNC'
          the iteration encountered a singular point where the function
          evaluated to `Inf' or `NaN'.

    `GSL_FAILURE'
          the algorithm could not improve the current best
          approximation or bounding interval.

   The minimizer maintains a current best estimate of the position of
the minimum at all times, and the current interval bounding the minimum.
This information can be accessed with the following auxiliary functions,

 -- Function: double gsl_min_fminimizer_x_minimum (const
          gsl_min_fminimizer * S)
     This function returns the current estimate of the position of the
     minimum for the minimizer S.

 -- Function: double gsl_min_fminimizer_x_upper (const
          gsl_min_fminimizer * S)
 -- Function: double gsl_min_fminimizer_x_lower (const
          gsl_min_fminimizer * S)
     These functions return the current upper and lower bound of the
     interval for the minimizer S.

 -- Function: double gsl_min_fminimizer_f_minimum (const
          gsl_min_fminimizer * S)
 -- Function: double gsl_min_fminimizer_f_upper (const
          gsl_min_fminimizer * S)
 -- Function: double gsl_min_fminimizer_f_lower (const
          gsl_min_fminimizer * S)
     These functions return the value of the function at the current
     estimate of the minimum and at the upper and lower bounds of the
     interval for the minimizer S.


File: gsl-ref.info,  Node: Minimization Stopping Parameters,  Next: Minimization Algorithms,  Prev: Minimization Iteration,  Up: One dimensional Minimization

33.6 Stopping Parameters
========================

A minimization procedure should stop when one of the following
conditions is true:

   * A minimum has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The function
below allows the user to test the precision of the current result.

 -- Function: int gsl_min_test_interval (double X_LOWER, double
          X_UPPER, double EPSABS, double EPSREL)
     This function tests for the convergence of the interval [X_LOWER,
     X_UPPER] with absolute error EPSABS and relative error EPSREL.
     The test returns `GSL_SUCCESS' if the following condition is
     achieved,

          |a - b| < epsabs + epsrel min(|a|,|b|)

     when the interval x = [a,b] does not include the origin.  If the
     interval includes the origin then \min(|a|,|b|) is replaced by
     zero (which is the minimum value of |x| over the interval).  This
     ensures that the relative error is accurately estimated for minima
     close to the origin.

     This condition on the interval also implies that any estimate of
     the minimum x_m in the interval satisfies the same condition with
     respect to the true minimum x_m^*,

          |x_m - x_m^*| < epsabs + epsrel x_m^*

     assuming that the true minimum x_m^* is contained within the
     interval.


File: gsl-ref.info,  Node: Minimization Algorithms,  Next: Minimization Examples,  Prev: Minimization Stopping Parameters,  Up: One dimensional Minimization

33.7 Minimization Algorithms
============================

The minimization algorithms described in this section require an initial
interval which is guaranteed to contain a minimum--if a and b are the
endpoints of the interval and x is an estimate of the minimum then f(a)
> f(x) < f(b).  This ensures that the function has at least one minimum
somewhere in the interval.  If a valid initial interval is used then
these algorithm cannot fail, provided the function is well-behaved.

 -- Minimizer: gsl_min_fminimizer_goldensection
     The "golden section algorithm" is the simplest method of bracketing
     the minimum of a function.  It is the slowest algorithm provided
     by the library, with linear convergence.

     On each iteration, the algorithm first compares the subintervals
     from the endpoints to the current minimum.  The larger subinterval
     is divided in a golden section (using the famous ratio (3-\sqrt
     5)/2 = 0.3189660...) and the value of the function at this new
     point is calculated.  The new value is used with the constraint
     f(a') > f(x') < f(b') to a select new interval containing the
     minimum, by discarding the least useful point.  This procedure can
     be continued indefinitely until the interval is sufficiently
     small.  Choosing the golden section as the bisection ratio can be
     shown to provide the fastest convergence for this type of
     algorithm.


 -- Minimizer: gsl_min_fminimizer_brent
     The "Brent minimization algorithm" combines a parabolic
     interpolation with the golden section algorithm.  This produces a
     fast algorithm which is still robust.

     The outline of the algorithm can be summarized as follows: on each
     iteration Brent's method approximates the function using an
     interpolating parabola through three existing points.  The minimum
     of the parabola is taken as a guess for the minimum.  If it lies
     within the bounds of the current interval then the interpolating
     point is accepted, and used to generate a smaller interval.  If
     the interpolating point is not accepted then the algorithm falls
     back to an ordinary golden section step.  The full details of
     Brent's method include some additional checks to improve
     convergence.


File: gsl-ref.info,  Node: Minimization Examples,  Next: Minimization References and Further Reading,  Prev: Minimization Algorithms,  Up: One dimensional Minimization

33.8 Examples
=============

The following program uses the Brent algorithm to find the minimum of
the function f(x) = \cos(x) + 1, which occurs at x = \pi.  The starting
interval is (0,6), with an initial guess for the minimum of 2.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_min.h>

     double fn1 (double x, void * params)
     {
       return cos(x) + 1.0;
     }

     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_min_fminimizer_type *T;
       gsl_min_fminimizer *s;
       double m = 2.0, m_expected = M_PI;
       double a = 0.0, b = 6.0;
       gsl_function F;

       F.function = &fn1;
       F.params = 0;

       T = gsl_min_fminimizer_brent;
       s = gsl_min_fminimizer_alloc (T);
       gsl_min_fminimizer_set (s, &F, m, a, b);

       printf ("using %s method\n",
               gsl_min_fminimizer_name (s));

       printf ("%5s [%9s, %9s] %9s %10s %9s\n",
               "iter", "lower", "upper", "min",
               "err", "err(est)");

       printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
               iter, a, b,
               m, m - m_expected, b - a);

       do
         {
           iter++;
           status = gsl_min_fminimizer_iterate (s);

           m = gsl_min_fminimizer_x_minimum (s);
           a = gsl_min_fminimizer_x_lower (s);
           b = gsl_min_fminimizer_x_upper (s);

           status
             = gsl_min_test_interval (a, b, 0.001, 0.0);

           if (status == GSL_SUCCESS)
             printf ("Converged:\n");

           printf ("%5d [%.7f, %.7f] "
                   "%.7f %.7f %+.7f %.7f\n",
                   iter, a, b,
                   m, m_expected, m - m_expected, b - a);
         }
       while (status == GSL_CONTINUE && iter < max_iter);

       return status;
     }

Here are the results of the minimization procedure.

     $ ./a.out
         0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
         1 [2.0000000, 6.0000000] 3.2758640 +0.1342713 4.0000000
         2 [2.0000000, 3.2831929] 3.2758640 +0.1342713 1.2831929
         3 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
         4 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
         5 [2.8689068, 3.2758640] 3.1460585 +0.0044658 0.4069572
         6 [3.1346075, 3.2758640] 3.1460585 +0.0044658 0.1412565
         7 [3.1346075, 3.1874620] 3.1460585 +0.0044658 0.0528545
         8 [3.1346075, 3.1460585] 3.1460585 +0.0044658 0.0114510
         9 [3.1346075, 3.1460585] 3.1424060 +0.0008133 0.0114510
        10 [3.1346075, 3.1424060] 3.1415885 -0.0000041 0.0077985
     Converged:
        11 [3.1415885, 3.1424060] 3.1415927 -0.0000000 0.0008175


File: gsl-ref.info,  Node: Minimization References and Further Reading,  Prev: Minimization Examples,  Up: One dimensional Minimization

33.9 References and Further Reading
===================================

Further information on Brent's algorithm is available in the following
book,

     Richard Brent, `Algorithms for minimization without derivatives',
     Prentice-Hall (1973), republished by Dover in paperback (2002),
     ISBN 0-486-41998-3.


File: gsl-ref.info,  Node: Multidimensional Root-Finding,  Next: Multidimensional Minimization,  Prev: One dimensional Minimization,  Up: Top

34 Multidimensional Root-Finding
********************************

This chapter describes functions for multidimensional root-finding
(solving nonlinear systems with n equations in n unknowns).  The
library provides low level components for a variety of iterative
solvers and convergence tests.  These can be combined by the user to
achieve the desired solution, with full access to the intermediate
steps of the iteration.  Each class of methods uses the same framework,
so that you can switch between solvers at runtime without needing to
recompile your program.  Each instance of a solver keeps track of its
own state, allowing the solvers to be used in multi-threaded programs.
The solvers are based on the original Fortran library MINPACK.

   The header file `gsl_multiroots.h' contains prototypes for the
multidimensional root finding functions and related declarations.

* Menu:

* Overview of Multidimensional Root Finding::
* Initializing the Multidimensional Solver::
* Providing the multidimensional system of equations to solve::
* Iteration of the multidimensional solver::
* Search Stopping Parameters for the multidimensional solver::
* Algorithms using Derivatives::
* Algorithms without Derivatives::
* Example programs for Multidimensional Root finding::
* References and Further Reading for Multidimensional Root Finding::


File: gsl-ref.info,  Node: Overview of Multidimensional Root Finding,  Next: Initializing the Multidimensional Solver,  Up: Multidimensional Root-Finding

34.1 Overview
=============

The problem of multidimensional root finding requires the simultaneous
solution of n equations, f_i, in n variables, x_i,

     f_i (x_1, ..., x_n) = 0    for i = 1 ... n.

In general there are no bracketing methods available for n dimensional
systems, and no way of knowing whether any solutions exist.  All
algorithms proceed from an initial guess using a variant of the Newton
iteration,

     x -> x' = x - J^{-1} f(x)

where x, f are vector quantities and J is the Jacobian matrix J_{ij} =
d f_i / d x_j.  Additional strategies can be used to enlarge the region
of convergence.  These include requiring a decrease in the norm |f| on
each step proposed by Newton's method, or taking steepest-descent steps
in the direction of the negative gradient of |f|.

   Several root-finding algorithms are available within a single
framework.  The user provides a high-level driver for the algorithms,
and the library provides the individual functions necessary for each of
the steps.  There are three main phases of the iteration.  The steps
are,

   * initialize solver state, S, for algorithm T

   * update S using the iteration T

   * test S for convergence, and repeat iteration if necessary

The evaluation of the Jacobian matrix can be problematic, either because
programming the derivatives is intractable or because computation of the
n^2 terms of the matrix becomes too expensive.  For these reasons the
algorithms provided by the library are divided into two classes
according to whether the derivatives are available or not.

   The state for solvers with an analytic Jacobian matrix is held in a
`gsl_multiroot_fdfsolver' struct.  The updating procedure requires both
the function and its derivatives to be supplied by the user.

   The state for solvers which do not use an analytic Jacobian matrix is
held in a `gsl_multiroot_fsolver' struct.  The updating procedure uses
only function evaluations (not derivatives).  The algorithms estimate
the matrix J or J^{-1} by approximate methods.


File: gsl-ref.info,  Node: Initializing the Multidimensional Solver,  Next: Providing the multidimensional system of equations to solve,  Prev: Overview of Multidimensional Root Finding,  Up: Multidimensional Root-Finding

34.2 Initializing the Solver
============================

The following functions initialize a multidimensional solver, either
with or without derivatives.  The solver itself depends only on the
dimension of the problem and the algorithm and can be reused for
different problems.

 -- Function: gsl_multiroot_fsolver * gsl_multiroot_fsolver_alloc
          (const gsl_multiroot_fsolver_type * T, size_t N)
     This function returns a pointer to a newly allocated instance of a
     solver of type T for a system of N dimensions.  For example, the
     following code creates an instance of a hybrid solver, to solve a
     3-dimensional system of equations.

          const gsl_multiroot_fsolver_type * T
              = gsl_multiroot_fsolver_hybrid;
          gsl_multiroot_fsolver * s
              = gsl_multiroot_fsolver_alloc (T, 3);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of `GSL_ENOMEM'.

 -- Function: gsl_multiroot_fdfsolver * gsl_multiroot_fdfsolver_alloc
          (const gsl_multiroot_fdfsolver_type * T, size_t N)
     This function returns a pointer to a newly allocated instance of a
     derivative solver of type T for a system of N dimensions.  For
     example, the following code creates an instance of a
     Newton-Raphson solver, for a 2-dimensional system of equations.

          const gsl_multiroot_fdfsolver_type * T
              = gsl_multiroot_fdfsolver_newton;
          gsl_multiroot_fdfsolver * s =
              gsl_multiroot_fdfsolver_alloc (T, 2);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of `GSL_ENOMEM'.

 -- Function: int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * S,
          gsl_multiroot_function * F, gsl_vector * X)
     This function sets, or resets, an existing solver S to use the
     function F and the initial guess X.

 -- Function: int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver
          * S, gsl_multiroot_function_fdf * FDF, gsl_vector * X)
     This function sets, or resets, an existing solver S to use the
     function and derivative FDF and the initial guess X.

 -- Function: void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver *
          S)
 -- Function: void gsl_multiroot_fdfsolver_free
          (gsl_multiroot_fdfsolver * S)
     These functions free all the memory associated with the solver S.

 -- Function: const char * gsl_multiroot_fsolver_name (const
          gsl_multiroot_fsolver * S)
 -- Function: const char * gsl_multiroot_fdfsolver_name (const
          gsl_multiroot_fdfsolver * S)
     These functions return a pointer to the name of the solver.  For
     example,

          printf ("s is a '%s' solver\n",
                  gsl_multiroot_fdfsolver_name (s));

     would print something like `s is a 'newton' solver'.


File: gsl-ref.info,  Node: Providing the multidimensional system of equations to solve,  Next: Iteration of the multidimensional solver,  Prev: Initializing the Multidimensional Solver,  Up: Multidimensional Root-Finding

34.3 Providing the function to solve
====================================

You must provide n functions of n variables for the root finders to
operate on.  In order to allow for general parameters the functions are
defined by the following data types:

 -- Data Type: gsl_multiroot_function
     This data type defines a general system of functions with
     parameters.

    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
          this function should store the vector result f(x,params) in F
          for argument X and parameters PARAMS, returning an
          appropriate error code if the function cannot be computed.

    `size_t n'
          the dimension of the system, i.e. the number of components of
          the vectors X and F.

    `void * params'
          a pointer to the parameters of the function.

Here is an example using Powell's test function,

     f_1(x) = A x_0 x_1 - 1,
     f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A)

with A = 10^4.  The following code defines a `gsl_multiroot_function'
system `F' which you could pass to a solver:

     struct powell_params { double A; };

     int
     powell (gsl_vector * x, void * p, gsl_vector * f) {
        struct powell_params * params
          = *(struct powell_params *)p;
        const double A = (params->A);
        const double x0 = gsl_vector_get(x,0);
        const double x1 = gsl_vector_get(x,1);

        gsl_vector_set (f, 0, A * x0 * x1 - 1);
        gsl_vector_set (f, 1, (exp(-x0) + exp(-x1)
                               - (1.0 + 1.0/A)));
        return GSL_SUCCESS
     }

     gsl_multiroot_function F;
     struct powell_params params = { 10000.0 };

     F.f = &powell;
     F.n = 2;
     F.params = &params;

 -- Data Type: gsl_multiroot_function_fdf
     This data type defines a general system of functions with
     parameters and the corresponding Jacobian matrix of derivatives,

    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
          this function should store the vector result f(x,params) in F
          for argument X and parameters PARAMS, returning an
          appropriate error code if the function cannot be computed.

    `int (* df) (const gsl_vector * X, void * PARAMS, gsl_matrix * J)'
          this function should store the N-by-N matrix result J_ij = d
          f_i(x,params) / d x_j in J for argument X and parameters
          PARAMS, returning an appropriate error code if the function
          cannot be computed.

    `int (* fdf) (const gsl_vector * X, void * PARAMS, gsl_vector * F, gsl_matrix * J)'
          This function should set the values of the F and J as above,
          for arguments X and parameters PARAMS.  This function
          provides an optimization of the separate functions for f(x)
          and J(x)--it is always faster to compute the function and its
          derivative at the same time.

    `size_t n'
          the dimension of the system, i.e. the number of components of
          the vectors X and F.

    `void * params'
          a pointer to the parameters of the function.

The example of Powell's test function defined above can be extended to
include analytic derivatives using the following code,

     int
     powell_df (gsl_vector * x, void * p, gsl_matrix * J)
     {
        struct powell_params * params
          = *(struct powell_params *)p;
        const double A = (params->A);
        const double x0 = gsl_vector_get(x,0);
        const double x1 = gsl_vector_get(x,1);
        gsl_matrix_set (J, 0, 0, A * x1);
        gsl_matrix_set (J, 0, 1, A * x0);
        gsl_matrix_set (J, 1, 0, -exp(-x0));
        gsl_matrix_set (J, 1, 1, -exp(-x1));
        return GSL_SUCCESS
     }

     int
     powell_fdf (gsl_vector * x, void * p,
                 gsl_matrix * f, gsl_matrix * J) {
        struct powell_params * params
          = *(struct powell_params *)p;
        const double A = (params->A);
        const double x0 = gsl_vector_get(x,0);
        const double x1 = gsl_vector_get(x,1);

        const double u0 = exp(-x0);
        const double u1 = exp(-x1);

        gsl_vector_set (f, 0, A * x0 * x1 - 1);
        gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));

        gsl_matrix_set (J, 0, 0, A * x1);
        gsl_matrix_set (J, 0, 1, A * x0);
        gsl_matrix_set (J, 1, 0, -u0);
        gsl_matrix_set (J, 1, 1, -u1);
        return GSL_SUCCESS
     }

     gsl_multiroot_function_fdf FDF;

     FDF.f = &powell_f;
     FDF.df = &powell_df;
     FDF.fdf = &powell_fdf;
     FDF.n = 2;
     FDF.params = 0;

Note that the function `powell_fdf' is able to reuse existing terms
from the function when calculating the Jacobian, thus saving time.


File: gsl-ref.info,  Node: Iteration of the multidimensional solver,  Next: Search Stopping Parameters for the multidimensional solver,  Prev: Providing the multidimensional system of equations to solve,  Up: Multidimensional Root-Finding

34.4 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any solver of the
corresponding type.  The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.

 -- Function: int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver
          * S)
 -- Function: int gsl_multiroot_fdfsolver_iterate
          (gsl_multiroot_fdfsolver * S)
     These functions perform a single iteration of the solver S.  If the
     iteration encounters an unexpected problem then an error code will
     be returned,

    `GSL_EBADFUNC'
          the iteration encountered a singular point where the function
          or its derivative evaluated to `Inf' or `NaN'.

    `GSL_ENOPROG'
          the iteration is not making any progress, preventing the
          algorithm from continuing.

   The solver maintains a current best estimate of the root at all
times.  This information can be accessed with the following auxiliary
functions,

 -- Function: gsl_vector * gsl_multiroot_fsolver_root (const
          gsl_multiroot_fsolver * S)
 -- Function: gsl_vector * gsl_multiroot_fdfsolver_root (const
          gsl_multiroot_fdfsolver * S)
     These functions return the current estimate of the root for the
     solver S.

 -- Function: gsl_vector * gsl_multiroot_fsolver_f (const
          gsl_multiroot_fsolver * S)
 -- Function: gsl_vector * gsl_multiroot_fdfsolver_f (const
          gsl_multiroot_fdfsolver * S)
     These functions return the function value f(x) at the current
     estimate of the root for the solver S.

 -- Function: gsl_vector * gsl_multiroot_fsolver_dx (const
          gsl_multiroot_fsolver * S)
 -- Function: gsl_vector * gsl_multiroot_fdfsolver_dx (const
          gsl_multiroot_fdfsolver * S)
     These functions return the last step dx taken by the solver S.


File: gsl-ref.info,  Node: Search Stopping Parameters for the multidimensional solver,  Next: Algorithms using Derivatives,  Prev: Iteration of the multidimensional solver,  Up: Multidimensional Root-Finding

34.5 Search Stopping Parameters
===============================

A root finding procedure should stop when one of the following
conditions is true:

   * A multidimensional root has been found to within the
     user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result in
several standard ways.

 -- Function: int gsl_multiroot_test_delta (const gsl_vector * DX,
          const gsl_vector * X, double EPSABS, double EPSREL)
     This function tests for the convergence of the sequence by
     comparing the last step DX with the absolute error EPSABS and
     relative error EPSREL to the current position X.  The test returns
     `GSL_SUCCESS' if the following condition is achieved,

          |dx_i| < epsabs + epsrel |x_i|

     for each component of X and returns `GSL_CONTINUE' otherwise.

 -- Function: int gsl_multiroot_test_residual (const gsl_vector * F,
          double EPSABS)
     This function tests the residual value F against the absolute
     error bound EPSABS.  The test returns `GSL_SUCCESS' if the
     following condition is achieved,

          \sum_i |f_i| < epsabs

     and returns `GSL_CONTINUE' otherwise.  This criterion is suitable
     for situations where the precise location of the root, x, is
     unimportant provided a value can be found where the residual is
     small enough.


File: gsl-ref.info,  Node: Algorithms using Derivatives,  Next: Algorithms without Derivatives,  Prev: Search Stopping Parameters for the multidimensional solver,  Up: Multidimensional Root-Finding

34.6 Algorithms using Derivatives
=================================

The root finding algorithms described in this section make use of both
the function and its derivative.  They require an initial guess for the
location of the root, but there is no absolute guarantee of
convergence--the function must be suitable for this technique and the
initial guess must be sufficiently close to the root for it to work.
When the conditions are satisfied then convergence is quadratic.

 -- Derivative Solver: gsl_multiroot_fdfsolver_hybridsj
     This is a modified version of Powell's Hybrid method as
     implemented in the HYBRJ algorithm in MINPACK.  Minpack was
     written by Jorge J. More', Burton S. Garbow and Kenneth E.
     Hillstrom.  The Hybrid algorithm retains the fast convergence of
     Newton's method but will also reduce the residual when Newton's
     method is unreliable.

     The algorithm uses a generalized trust region to keep each step
     under control.  In order to be accepted a proposed new position x'
     must satisfy the condition |D (x' - x)| < \delta, where D is a
     diagonal scaling matrix and \delta is the size of the trust
     region.  The components of D are computed internally, using the
     column norms of the Jacobian to estimate the sensitivity of the
     residual to each component of x.  This improves the behavior of the
     algorithm for badly scaled functions.

     On each iteration the algorithm first determines the standard
     Newton step by solving the system J dx = - f.  If this step falls
     inside the trust region it is used as a trial step in the next
     stage.  If not, the algorithm uses the linear combination of the
     Newton and gradient directions which is predicted to minimize the
     norm of the function while staying inside the trust region,

          dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2.

     This combination of Newton and gradient directions is referred to
     as a "dogleg step".

     The proposed step is now tested by evaluating the function at the
     resulting point, x'.  If the step reduces the norm of the function
     sufficiently then it is accepted and size of the trust region is
     increased.  If the proposed step fails to improve the solution
     then the size of the trust region is decreased and another trial
     step is computed.

     The speed of the algorithm is increased by computing the changes
     to the Jacobian approximately, using a rank-1 update.  If two
     successive attempts fail to reduce the residual then the full
     Jacobian is recomputed.  The algorithm also monitors the progress
     of the solution and returns an error if several steps fail to make
     any improvement,

    `GSL_ENOPROG'
          the iteration is not making any progress, preventing the
          algorithm from continuing.

    `GSL_ENOPROGJ'
          re-evaluations of the Jacobian indicate that the iteration is
          not making any progress, preventing the algorithm from
          continuing.


 -- Derivative Solver: gsl_multiroot_fdfsolver_hybridj
     This algorithm is an unscaled version of `hybridsj'.  The steps are
     controlled by a spherical trust region |x' - x| < \delta, instead
     of a generalized region.  This can be useful if the generalized
     region estimated by `hybridsj' is inappropriate.

 -- Derivative Solver: gsl_multiroot_fdfsolver_newton
     Newton's Method is the standard root-polishing algorithm.  The
     algorithm begins with an initial guess for the location of the
     solution.  On each iteration a linear approximation to the
     function F is used to estimate the step which will zero all the
     components of the residual.  The iteration is defined by the
     following sequence,

          x -> x' = x - J^{-1} f(x)

     where the Jacobian matrix J is computed from the derivative
     functions provided by F.  The step dx is obtained by solving the
     linear system,

          J dx = - f(x)

     using LU decomposition.

 -- Derivative Solver: gsl_multiroot_fdfsolver_gnewton
     This is a modified version of Newton's method which attempts to
     improve global convergence by requiring every step to reduce the
     Euclidean norm of the residual, |f(x)|.  If the Newton step leads
     to an increase in the norm then a reduced step of relative size,

          t = (\sqrt(1 + 6 r) - 1) / (3 r)

     is proposed, with r being the ratio of norms |f(x')|^2/|f(x)|^2.
     This procedure is repeated until a suitable step size is found.


File: gsl-ref.info,  Node: Algorithms without Derivatives,  Next: Example programs for Multidimensional Root finding,  Prev: Algorithms using Derivatives,  Up: Multidimensional Root-Finding

34.7 Algorithms without Derivatives
===================================

The algorithms described in this section do not require any derivative
information to be supplied by the user.  Any derivatives needed are
approximated by finite differences.  Note that if the
finite-differencing step size chosen by these routines is inappropriate,
an explicit user-supplied numerical derivative can always be used with
the algorithms described in the previous section.

 -- Solver: gsl_multiroot_fsolver_hybrids
     This is a version of the Hybrid algorithm which replaces calls to
     the Jacobian function by its finite difference approximation.  The
     finite difference approximation is computed using
     `gsl_multiroots_fdjac' with a relative step size of
     `GSL_SQRT_DBL_EPSILON'.

 -- Solver: gsl_multiroot_fsolver_hybrid
     This is a finite difference version of the Hybrid algorithm without
     internal scaling.

 -- Solver: gsl_multiroot_fsolver_dnewton
     The "discrete Newton algorithm" is the simplest method of solving a
     multidimensional system.  It uses the Newton iteration

          x -> x - J^{-1} f(x)

     where the Jacobian matrix J is approximated by taking finite
     differences of the function F.  The approximation scheme used by
     this implementation is,

          J_{ij} = (f_i(x + \delta_j) - f_i(x)) /  \delta_j

     where \delta_j is a step of size \sqrt\epsilon |x_j| with \epsilon
     being the machine precision (\epsilon \approx 2.22 \times 10^-16).
     The order of convergence of Newton's algorithm is quadratic, but
     the finite differences require n^2 function evaluations on each
     iteration.  The algorithm may become unstable if the finite
     differences are not a good approximation to the true derivatives.

 -- Solver: gsl_multiroot_fsolver_broyden
     The "Broyden algorithm" is a version of the discrete Newton
     algorithm which attempts to avoids the expensive update of the
     Jacobian matrix on each iteration.  The changes to the Jacobian
     are also approximated, using a rank-1 update,

          J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df

     where the vectors dx and df are the changes in x and f.  On the
     first iteration the inverse Jacobian is estimated using finite
     differences, as in the discrete Newton algorithm.

     This approximation gives a fast update but is unreliable if the
     changes are not small, and the estimate of the inverse Jacobian
     becomes worse as time passes.  The algorithm has a tendency to
     become unstable unless it starts close to the root.  The Jacobian
     is refreshed if this instability is detected (consult the source
     for details).

     This algorithm is included only for demonstration purposes, and is
     not recommended for serious use.


File: gsl-ref.info,  Node: Example programs for Multidimensional Root finding,  Next: References and Further Reading for Multidimensional Root Finding,  Prev: Algorithms without Derivatives,  Up: Multidimensional Root-Finding

34.8 Examples
=============

The multidimensional solvers are used in a similar way to the
one-dimensional root finding algorithms.  This first example
demonstrates the `hybrids' scaled-hybrid algorithm, which does not
require derivatives. The program solves the Rosenbrock system of
equations,

     f_1 (x, y) = a (1 - x)
     f_2 (x, y) = b (y - x^2)

with a = 1, b = 10. The solution of this system lies at (x,y) = (1,1)
in a narrow valley.

   The first stage of the program is to define the system of equations,

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

     struct rparams
       {
         double a;
         double b;
       };

     int
     rosenbrock_f (const gsl_vector * x, void *params,
                   gsl_vector * f)
     {
       double a = ((struct rparams *) params)->a;
       double b = ((struct rparams *) params)->b;

       const double x0 = gsl_vector_get (x, 0);
       const double x1 = gsl_vector_get (x, 1);

       const double y0 = a * (1 - x0);
       const double y1 = b * (x1 - x0 * x0);

       gsl_vector_set (f, 0, y0);
       gsl_vector_set (f, 1, y1);

       return GSL_SUCCESS;
     }

The main program begins by creating the function object `f', with the
arguments `(x,y)' and parameters `(a,b)'. The solver `s' is initialized
to use this function, with the `hybrids' method.

     int
     main (void)
     {
       const gsl_multiroot_fsolver_type *T;
       gsl_multiroot_fsolver *s;

       int status;
       size_t i, iter = 0;

       const size_t n = 2;
       struct rparams p = {1.0, 10.0};
       gsl_multiroot_function f = {&rosenbrock_f, n, &p};

       double x_init[2] = {-10.0, -5.0};
       gsl_vector *x = gsl_vector_alloc (n);

       gsl_vector_set (x, 0, x_init[0]);
       gsl_vector_set (x, 1, x_init[1]);

       T = gsl_multiroot_fsolver_hybrids;
       s = gsl_multiroot_fsolver_alloc (T, 2);
       gsl_multiroot_fsolver_set (s, &f, x);

       print_state (iter, s);

       do
         {
           iter++;
           status = gsl_multiroot_fsolver_iterate (s);

           print_state (iter, s);

           if (status)   /* check if solver is stuck */
             break;

           status =
             gsl_multiroot_test_residual (s->f, 1e-7);
         }
       while (status == GSL_CONTINUE && iter < 1000);

       printf ("status = %s\n", gsl_strerror (status));

       gsl_multiroot_fsolver_free (s);
       gsl_vector_free (x);
       return 0;
     }

Note that it is important to check the return status of each solver
step, in case the algorithm becomes stuck.  If an error condition is
detected, indicating that the algorithm cannot proceed, then the error
can be reported to the user, a new starting point chosen or a different
algorithm used.

   The intermediate state of the solution is displayed by the following
function.  The solver state contains the vector `s->x' which is the
current position, and the vector `s->f' with corresponding function
values.

     int
     print_state (size_t iter, gsl_multiroot_fsolver * s)
     {
       printf ("iter = %3u x = % .3f % .3f "
               "f(x) = % .3e % .3e\n",
               iter,
               gsl_vector_get (s->x, 0),
               gsl_vector_get (s->x, 1),
               gsl_vector_get (s->f, 0),
               gsl_vector_get (s->f, 1));
     }

Here are the results of running the program. The algorithm is started at
(-10,-5) far from the solution.  Since the solution is hidden in a
narrow valley the earliest steps follow the gradient of the function
downhill, in an attempt to reduce the large value of the residual. Once
the root has been approximately located, on iteration 8, the Newton
behavior takes over and convergence is very rapid.

     iter =  0 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
     iter =  1 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
     iter =  2 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
     iter =  3 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
     iter =  4 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
     iter =  5 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
     iter =  6 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
     iter =  7 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
     iter =  8 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
     iter =  9 x =   1.000   0.878  f(x) = 1.268e-10 -1.218e+00
     iter = 10 x =   1.000   0.989  f(x) = 1.124e-11 -1.080e-01
     iter = 11 x =   1.000   1.000  f(x) = 0.000e+00  0.000e+00
     status = success

Note that the algorithm does not update the location on every
iteration. Some iterations are used to adjust the trust-region
parameter, after trying a step which was found to be divergent, or to
recompute the Jacobian, when poor convergence behavior is detected.

   The next example program adds derivative information, in order to
accelerate the solution. There are two derivative functions
`rosenbrock_df' and `rosenbrock_fdf'. The latter computes both the
function and its derivative simultaneously. This allows the
optimization of any common terms.  For simplicity we substitute calls to
the separate `f' and `df' functions at this point in the code below.

     int
     rosenbrock_df (const gsl_vector * x, void *params,
                    gsl_matrix * J)
     {
       const double a = ((struct rparams *) params)->a;
       const double b = ((struct rparams *) params)->b;

       const double x0 = gsl_vector_get (x, 0);

       const double df00 = -a;
       const double df01 = 0;
       const double df10 = -2 * b  * x0;
       const double df11 = b;

       gsl_matrix_set (J, 0, 0, df00);
       gsl_matrix_set (J, 0, 1, df01);
       gsl_matrix_set (J, 1, 0, df10);
       gsl_matrix_set (J, 1, 1, df11);

       return GSL_SUCCESS;
     }

     int
     rosenbrock_fdf (const gsl_vector * x, void *params,
                     gsl_vector * f, gsl_matrix * J)
     {
       rosenbrock_f (x, params, f);
       rosenbrock_df (x, params, J);

       return GSL_SUCCESS;
     }

The main program now makes calls to the corresponding `fdfsolver'
versions of the functions,

     int
     main (void)
     {
       const gsl_multiroot_fdfsolver_type *T;
       gsl_multiroot_fdfsolver *s;

       int status;
       size_t i, iter = 0;

       const size_t n = 2;
       struct rparams p = {1.0, 10.0};
       gsl_multiroot_function_fdf f = {&rosenbrock_f,
                                       &rosenbrock_df,
                                       &rosenbrock_fdf,
                                       n, &p};

       double x_init[2] = {-10.0, -5.0};
       gsl_vector *x = gsl_vector_alloc (n);

       gsl_vector_set (x, 0, x_init[0]);
       gsl_vector_set (x, 1, x_init[1]);

       T = gsl_multiroot_fdfsolver_gnewton;
       s = gsl_multiroot_fdfsolver_alloc (T, n);
       gsl_multiroot_fdfsolver_set (s, &f, x);

       print_state (iter, s);

       do
         {
           iter++;

           status = gsl_multiroot_fdfsolver_iterate (s);

           print_state (iter, s);

           if (status)
             break;

           status = gsl_multiroot_test_residual (s->f, 1e-7);
         }
       while (status == GSL_CONTINUE && iter < 1000);

       printf ("status = %s\n", gsl_strerror (status));

       gsl_multiroot_fdfsolver_free (s);
       gsl_vector_free (x);
       return 0;
     }

The addition of derivative information to the `hybrids' solver does not
make any significant difference to its behavior, since it able to
approximate the Jacobian numerically with sufficient accuracy.  To
illustrate the behavior of a different derivative solver we switch to
`gnewton'. This is a traditional Newton solver with the constraint that
it scales back its step if the full step would lead "uphill". Here is
the output for the `gnewton' algorithm,

     iter = 0 x = -10.000  -5.000 f(x) =  1.100e+01 -1.050e+03
     iter = 1 x =  -4.231 -65.317 f(x) =  5.231e+00 -8.321e+02
     iter = 2 x =   1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
     iter = 3 x =   1.000   1.000 f(x) = -2.220e-16 -4.441e-15
     status = success

The convergence is much more rapid, but takes a wide excursion out to
the point (-4.23,-65.3). This could cause the algorithm to go astray in
a realistic application.  The hybrid algorithm follows the downhill
path to the solution more reliably.


File: gsl-ref.info,  Node: References and Further Reading for Multidimensional Root Finding,  Prev: Example programs for Multidimensional Root finding,  Up: Multidimensional Root-Finding

34.9 References and Further Reading
===================================

The original version of the Hybrid method is described in the following
articles by Powell,

     M.J.D. Powell, "A Hybrid Method for Nonlinear Equations" (Chap 6, p
     87-114) and "A Fortran Subroutine for Solving systems of Nonlinear
     Algebraic Equations" (Chap 7, p 115-161), in `Numerical Methods for
     Nonlinear Algebraic Equations', P. Rabinowitz, editor.  Gordon and
     Breach, 1970.

The following papers are also relevant to the algorithms described in
this section,

     J.J. More', M.Y. Cosnard, "Numerical Solution of Nonlinear
     Equations", `ACM Transactions on Mathematical Software', Vol 5, No
     1, (1979), p 64-85

     C.G. Broyden, "A Class of Methods for Solving Nonlinear
     Simultaneous Equations", `Mathematics of Computation', Vol 19
     (1965), p 577-593

     J.J. More', B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained
     Optimization Software", ACM Transactions on Mathematical Software,
     Vol 7, No 1 (1981), p 17-41


File: gsl-ref.info,  Node: Multidimensional Minimization,  Next: Least-Squares Fitting,  Prev: Multidimensional Root-Finding,  Up: Top

35 Multidimensional Minimization
********************************

This chapter describes routines for finding minima of arbitrary
multidimensional functions.  The library provides low level components
for a variety of iterative minimizers and convergence tests.  These can
be combined by the user to achieve the desired solution, while providing
full access to the intermediate steps of the algorithms.  Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program.  Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs. The minimization
algorithms can be used to maximize a function by inverting its sign.

   The header file `gsl_multimin.h' contains prototypes for the
minimization functions and related declarations.

* Menu:

* Multimin Overview::
* Multimin Caveats::
* Initializing the Multidimensional Minimizer::
* Providing a function to minimize::
* Multimin Iteration::
* Multimin Stopping Criteria::
* Multimin Algorithms::
* Multimin Examples::
* Multimin References and Further Reading::


File: gsl-ref.info,  Node: Multimin Overview,  Next: Multimin Caveats,  Up: Multidimensional Minimization

35.1 Overview
=============

The problem of multidimensional minimization requires finding a point x
such that the scalar function,

     f(x_1, ..., x_n)

takes a value which is lower than at any neighboring point. For smooth
functions the gradient g = \nabla f vanishes at the minimum. In general
there are no bracketing methods available for the minimization of
n-dimensional functions.  The algorithms proceed from an initial guess
using a search algorithm which attempts to move in a downhill direction.

   Algorithms making use of the gradient of the function perform a
one-dimensional line minimisation along this direction until the lowest
point is found to a suitable tolerance.  The search direction is then
updated with local information from the function and its derivatives,
and the whole process repeated until the true n-dimensional minimum is
found.

   The Nelder-Mead Simplex algorithm applies a different strategy.  It
maintains n+1 trial parameter vectors as the vertices of a
n-dimensional simplex.  In each iteration step it tries to improve the
worst vertex by a simple geometrical transformation until the size of
the simplex falls below a given tolerance.

   Both types of algorithms use a standard framework. The user provides
a high-level driver for the algorithms, and the library provides the
individual functions necessary for each of the steps.  There are three
main phases of the iteration.  The steps are,

   * initialize minimizer state, S, for algorithm T

   * update S using the iteration T

   * test S for convergence, and repeat iteration if necessary

Each iteration step consists either of an improvement to the
line-minimisation in the current direction or an update to the search
direction itself.  The state for the minimizers is held in a
`gsl_multimin_fdfminimizer' struct or a `gsl_multimin_fminimizer'
struct.


File: gsl-ref.info,  Node: Multimin Caveats,  Next: Initializing the Multidimensional Minimizer,  Prev: Multimin Overview,  Up: Multidimensional Minimization

35.2 Caveats
============

Note that the minimization algorithms can only search for one local
minimum at a time.  When there are several local minima in the search
area, the first minimum to be found will be returned; however it is
difficult to predict which of the minima this will be.  In most cases,
no error will be reported if you try to find a local minimum in an area
where there is more than one.

   It is also important to note that the minimization algorithms find
local minima; there is no way to determine whether a minimum is a global
minimum of the function in question.


File: gsl-ref.info,  Node: Initializing the Multidimensional Minimizer,  Next: Providing a function to minimize,  Prev: Multimin Caveats,  Up: Multidimensional Minimization

35.3 Initializing the Multidimensional Minimizer
================================================

The following function initializes a multidimensional minimizer.  The
minimizer itself depends only on the dimension of the problem and the
algorithm and can be reused for different problems.

 -- Function: gsl_multimin_fdfminimizer *
gsl_multimin_fdfminimizer_alloc (const gsl_multimin_fdfminimizer_type *
          T, size_t N)
 -- Function: gsl_multimin_fminimizer * gsl_multimin_fminimizer_alloc
          (const gsl_multimin_fminimizer_type * T, size_t N)
     This function returns a pointer to a newly allocated instance of a
     minimizer of type T for an N-dimension function.  If there is
     insufficient memory to create the minimizer then the function
     returns a null pointer and the error handler is invoked with an
     error code of `GSL_ENOMEM'.

 -- Function: int gsl_multimin_fdfminimizer_set
          (gsl_multimin_fdfminimizer * S, gsl_multimin_function_fdf *
          FDF, const gsl_vector * X, double STEP_SIZE, double TOL)
     This function initializes the minimizer S to minimize the function
     FDF starting from the initial point X.  The size of the first
     trial step is given by STEP_SIZE.  The accuracy of the line
     minimization is specified by TOL.  The precise meaning of this
     parameter depends on the method used.  Typically the line
     minimization is considered successful if the gradient of the
     function g is orthogonal to the current search direction p to a
     relative accuracy of TOL, where dot(p,g) < tol |p| |g|.

 -- Function: int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer
          * S, gsl_multimin_function * F, const gsl_vector * X, const
          gsl_vector * STEP_SIZE)
     This function initializes the minimizer S to minimize the function
     F, starting from the initial point X. The size of the initial
     trial steps is given in vector STEP_SIZE. The precise meaning of
     this parameter depends on the method used.

 -- Function: void gsl_multimin_fdfminimizer_free
          (gsl_multimin_fdfminimizer * S)
 -- Function: void gsl_multimin_fminimizer_free
          (gsl_multimin_fminimizer * S)
     This function frees all the memory associated with the minimizer S.

 -- Function: const char * gsl_multimin_fdfminimizer_name (const
          gsl_multimin_fdfminimizer * S)
 -- Function: const char * gsl_multimin_fminimizer_name (const
          gsl_multimin_fminimizer * S)
     This function returns a pointer to the name of the minimizer.  For
     example,

          printf ("s is a '%s' minimizer\n",
                  gsl_multimin_fdfminimizer_name (s));

     would print something like `s is a 'conjugate_pr' minimizer'.


File: gsl-ref.info,  Node: Providing a function to minimize,  Next: Multimin Iteration,  Prev: Initializing the Multidimensional Minimizer,  Up: Multidimensional Minimization

35.4 Providing a function to minimize
=====================================

You must provide a parametric function of n variables for the
minimizers to operate on.  You may also need to provide a routine which
calculates the gradient of the function and a third routine which
calculates both the function value and the gradient together.  In order
to allow for general parameters the functions are defined by the
following data types:

 -- Data Type: gsl_multimin_function_fdf
     This data type defines a general function of n variables with
     parameters and the corresponding gradient vector of derivatives,

    `double (* f) (const gsl_vector * X, void * PARAMS)'
          this function should return the result f(x,params) for
          argument X and parameters PARAMS.

    `void (* df) (const gsl_vector * X, void * PARAMS, gsl_vector * G)'
          this function should store the N-dimensional gradient g_i = d
          f(x,params) / d x_i in the vector G for argument X and
          parameters PARAMS, returning an appropriate error code if the
          function cannot be computed.

    `void (* fdf) (const gsl_vector * X, void * PARAMS, double * f, gsl_vector * G)'
          This function should set the values of the F and G as above,
          for arguments X and parameters PARAMS.  This function
          provides an optimization of the separate functions for f(x)
          and g(x)--it is always faster to compute the function and its
          derivative at the same time.

    `size_t n'
          the dimension of the system, i.e. the number of components of
          the vectors X.

    `void * params'
          a pointer to the parameters of the function.

 -- Data Type: gsl_multimin_function
     This data type defines a general function of n variables with
     parameters,

    `double (* f) (const gsl_vector * X, void * PARAMS)'
          this function should return the result f(x,params) for
          argument X and parameters PARAMS.

    `size_t n'
          the dimension of the system, i.e. the number of components of
          the vectors X.

    `void * params'
          a pointer to the parameters of the function.

The following example function defines a simple paraboloid with two
parameters,

     /* Paraboloid centered on (dp[0],dp[1]) */

     double
     my_f (const gsl_vector *v, void *params)
     {
       double x, y;
       double *dp = (double *)params;

       x = gsl_vector_get(v, 0);
       y = gsl_vector_get(v, 1);

       return 10.0 * (x - dp[0]) * (x - dp[0]) +
                20.0 * (y - dp[1]) * (y - dp[1]) + 30.0;
     }

     /* The gradient of f, df = (df/dx, df/dy). */
     void
     my_df (const gsl_vector *v, void *params,
            gsl_vector *df)
     {
       double x, y;
       double *dp = (double *)params;

       x = gsl_vector_get(v, 0);
       y = gsl_vector_get(v, 1);

       gsl_vector_set(df, 0, 20.0 * (x - dp[0]));
       gsl_vector_set(df, 1, 40.0 * (y - dp[1]));
     }

     /* Compute both f and df together. */
     void
     my_fdf (const gsl_vector *x, void *params,
             double *f, gsl_vector *df)
     {
       *f = my_f(x, params);
       my_df(x, params, df);
     }

The function can be initialized using the following code,

     gsl_multimin_function_fdf my_func;

     double p[2] = { 1.0, 2.0 }; /* center at (1,2) */

     my_func.f = &my_f;
     my_func.df = &my_df;
     my_func.fdf = &my_fdf;
     my_func.n = 2;
     my_func.params = (void *)p;


File: gsl-ref.info,  Node: Multimin Iteration,  Next: Multimin Stopping Criteria,  Prev: Providing a function to minimize,  Up: Multidimensional Minimization

35.5 Iteration
==============

The following function drives the iteration of each algorithm.  The
function performs one iteration to update the state of the minimizer.
The same function works for all minimizers so that different methods can
be substituted at runtime without modifications to the code.

 -- Function: int gsl_multimin_fdfminimizer_iterate
          (gsl_multimin_fdfminimizer * S)
 -- Function: int gsl_multimin_fminimizer_iterate
          (gsl_multimin_fminimizer * S)
     These functions perform a single iteration of the minimizer S.  If
     the iteration encounters an unexpected problem then an error code
     will be returned.

The minimizer maintains a current best estimate of the minimum at all
times.  This information can be accessed with the following auxiliary
functions,

 -- Function: gsl_vector * gsl_multimin_fdfminimizer_x (const
          gsl_multimin_fdfminimizer * S)
 -- Function: gsl_vector * gsl_multimin_fminimizer_x (const
          gsl_multimin_fminimizer * S)
 -- Function: double gsl_multimin_fdfminimizer_minimum (const
          gsl_multimin_fdfminimizer * S)
 -- Function: double gsl_multimin_fminimizer_minimum (const
          gsl_multimin_fminimizer * S)
 -- Function: gsl_vector * gsl_multimin_fdfminimizer_gradient (const
          gsl_multimin_fdfminimizer * S)
 -- Function: double gsl_multimin_fminimizer_size (const
          gsl_multimin_fminimizer * S)
     These functions return the current best estimate of the location
     of the minimum, the value of the function at that point, its
     gradient, and minimizer specific characteristic size for the
     minimizer S.

 -- Function: int gsl_multimin_fdfminimizer_restart
          (gsl_multimin_fdfminimizer * S)
     This function resets the minimizer S to use the current point as a
     new starting point.


File: gsl-ref.info,  Node: Multimin Stopping Criteria,  Next: Multimin Algorithms,  Prev: Multimin Iteration,  Up: Multidimensional Minimization

35.6 Stopping Criteria
======================

A minimization procedure should stop when one of the following
conditions is true:

   * A minimum has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result.

 -- Function: int gsl_multimin_test_gradient (const gsl_vector * G,
          double EPSABS)
     This function tests the norm of the gradient G against the
     absolute tolerance EPSABS. The gradient of a multidimensional
     function goes to zero at a minimum. The test returns `GSL_SUCCESS'
     if the following condition is achieved,

          |g| < epsabs

     and returns `GSL_CONTINUE' otherwise.  A suitable choice of EPSABS
     can be made from the desired accuracy in the function for small
     variations in x.  The relationship between these quantities is
     given by \delta f = g \delta x.

 -- Function: int gsl_multimin_test_size (const double SIZE, double
          EPSABS)
     This function tests the minimizer specific characteristic size (if
     applicable to the used minimizer) against absolute tolerance
     EPSABS.  The test returns `GSL_SUCCESS' if the size is smaller
     than tolerance, otherwise `GSL_CONTINUE' is returned.


File: gsl-ref.info,  Node: Multimin Algorithms,  Next: Multimin Examples,  Prev: Multimin Stopping Criteria,  Up: Multidimensional Minimization

35.7 Algorithms
===============

There are several minimization methods available. The best choice of
algorithm depends on the problem.  All of the algorithms use the value
of the function and its gradient at each evaluation point, except for
the Simplex algorithm which uses function values only.

 -- Minimizer: gsl_multimin_fdfminimizer_conjugate_fr
     This is the Fletcher-Reeves conjugate gradient algorithm. The
     conjugate gradient algorithm proceeds as a succession of line
     minimizations. The sequence of search directions is used to build
     up an approximation to the curvature of the function in the
     neighborhood of the minimum.

     An initial search direction P is chosen using the gradient, and
     line minimization is carried out in that direction.  The accuracy
     of the line minimization is specified by the parameter TOL.  The
     minimum along this line occurs when the function gradient G and
     the search direction P are orthogonal.  The line minimization
     terminates when dot(p,g) < tol |p| |g|.  The search direction is
     updated  using the Fletcher-Reeves formula p' = g' - \beta g where
     \beta=-|g'|^2/|g|^2, and the line minimization is then repeated
     for the new search direction.

 -- Minimizer: gsl_multimin_fdfminimizer_conjugate_pr
     This is the Polak-Ribiere conjugate gradient algorithm.  It is
     similar to the Fletcher-Reeves method, differing only in the
     choice of the coefficient \beta. Both methods work well when the
     evaluation point is close enough to the minimum of the objective
     function that it is well approximated by a quadratic hypersurface.

 -- Minimizer: gsl_multimin_fdfminimizer_vector_bfgs
     This is the vector Broyden-Fletcher-Goldfarb-Shanno (BFGS)
     conjugate gradient algorithm.  It is a quasi-Newton method which
     builds up an approximation to the second derivatives of the
     function f using the difference between successive gradient
     vectors.  By combining the first and second derivatives the
     algorithm is able to take Newton-type steps towards the function
     minimum, assuming quadratic behavior in that region.

 -- Minimizer: gsl_multimin_fdfminimizer_steepest_descent
     The steepest descent algorithm follows the downhill gradient of the
     function at each step. When a downhill step is successful the
     step-size is increased by a factor of two.  If the downhill step
     leads to a higher function value then the algorithm backtracks and
     the step size is decreased using the parameter TOL.  A suitable
     value of TOL for most applications is 0.1.  The steepest descent
     method is inefficient and is included only for demonstration
     purposes.

 -- Minimizer: gsl_multimin_fminimizer_nmsimplex
     This is the Simplex algorithm of Nelder and Mead. It constructs n
     vectors p_i from the starting vector X and the vector STEP_SIZE as
     follows:

          p_0 = (x_0, x_1, ... , x_n)
          p_1 = (x_0 + step_size_0, x_1, ... , x_n)
          p_2 = (x_0, x_1 + step_size_1, ... , x_n)
          ... = ...
          p_n = (x_0, x_1, ... , x_n+step_size_n)

     These vectors form the n+1 vertices of a simplex in n dimensions.
     On each iteration the algorithm tries to improve the parameter
     vector p_i corresponding to the highest function value by simple
     geometrical transformations.  These are reflection, reflection
     followed by expansion, contraction and multiple contraction. Using
     these transformations the simplex moves through the parameter
     space towards the minimum, where it contracts itself.

     After each iteration, the best vertex is returned.  Note, that due
     to the nature of the algorithm not every step improves the current
     best parameter vector.  Usually several iterations are required.

     The routine calculates the minimizer specific characteristic size
     as the average distance from the geometrical center of the simplex
     to all its vertices.  This size can be used as a stopping
     criteria, as the simplex contracts itself near the minimum. The
     size is returned by the function `gsl_multimin_fminimizer_size'.


File: gsl-ref.info,  Node: Multimin Examples,  Next: Multimin References and Further Reading,  Prev: Multimin Algorithms,  Up: Multidimensional Minimization

35.8 Examples
=============

This example program finds the minimum of the paraboloid function
defined earlier.  The location of the minimum is offset from the origin
in x and y, and the function value at the minimum is non-zero. The main
program is given below, it requires the example function given earlier
in this chapter.

     int
     main (void)
     {
       size_t iter = 0;
       int status;

       const gsl_multimin_fdfminimizer_type *T;
       gsl_multimin_fdfminimizer *s;

       /* Position of the minimum (1,2). */
       double par[2] = { 1.0, 2.0 };

       gsl_vector *x;
       gsl_multimin_function_fdf my_func;

       my_func.f = &my_f;
       my_func.df = &my_df;
       my_func.fdf = &my_fdf;
       my_func.n = 2;
       my_func.params = &par;

       /* Starting point, x = (5,7) */
       x = gsl_vector_alloc (2);
       gsl_vector_set (x, 0, 5.0);
       gsl_vector_set (x, 1, 7.0);

       T = gsl_multimin_fdfminimizer_conjugate_fr;
       s = gsl_multimin_fdfminimizer_alloc (T, 2);

       gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);

       do
         {
           iter++;
           status = gsl_multimin_fdfminimizer_iterate (s);

           if (status)
             break;

           status = gsl_multimin_test_gradient (s->gradient, 1e-3);

           if (status == GSL_SUCCESS)
             printf ("Minimum found at:\n");

           printf ("%5d %.5f %.5f %10.5f\n", iter,
                   gsl_vector_get (s->x, 0),
                   gsl_vector_get (s->x, 1),
                   s->f);

         }
       while (status == GSL_CONTINUE && iter < 100);

       gsl_multimin_fdfminimizer_free (s);
       gsl_vector_free (x);

       return 0;
     }

The initial step-size is chosen as 0.01, a conservative estimate in this
case, and the line minimization parameter is set at 0.0001.  The program
terminates when the norm of the gradient has been reduced below 0.001.
The output of the program is shown below,

              x       y         f
         1 4.99629 6.99072  687.84780
         2 4.98886 6.97215  683.55456
         3 4.97400 6.93501  675.01278
         4 4.94429 6.86073  658.10798
         5 4.88487 6.71217  625.01340
         6 4.76602 6.41506  561.68440
         7 4.52833 5.82083  446.46694
         8 4.05295 4.63238  261.79422
         9 3.10219 2.25548   75.49762
        10 2.85185 1.62963   67.03704
        11 2.19088 1.76182   45.31640
        12 0.86892 2.02622   30.18555
     Minimum found at:
        13 1.00000 2.00000   30.00000

Note that the algorithm gradually increases the step size as it
successfully moves downhill, as can be seen by plotting the successive
points.

The conjugate gradient algorithm finds the minimum on its second
direction because the function is purely quadratic. Additional
iterations would be needed for a more complicated function.

   Here is another example using the Nelder-Mead Simplex algorithm to
minimize the same example object function, as above.

     int
     main(void)
     {
       size_t np = 2;
       double par[2] = {1.0, 2.0};

       const gsl_multimin_fminimizer_type *T =
         gsl_multimin_fminimizer_nmsimplex;
       gsl_multimin_fminimizer *s = NULL;
       gsl_vector *ss, *x;
       gsl_multimin_function minex_func;

       size_t iter = 0, i;
       int status;
       double size;

       /* Initial vertex size vector */
       ss = gsl_vector_alloc (np);

       /* Set all step sizes to 1 */
       gsl_vector_set_all (ss, 1.0);

       /* Starting point */
       x = gsl_vector_alloc (np);

       gsl_vector_set (x, 0, 5.0);
       gsl_vector_set (x, 1, 7.0);

       /* Initialize method and iterate */
       minex_func.f = &my_f;
       minex_func.n = np;
       minex_func.params = (void *)&par;

       s = gsl_multimin_fminimizer_alloc (T, np);
       gsl_multimin_fminimizer_set (s, &minex_func, x, ss);

       do
         {
           iter++;
           status = gsl_multimin_fminimizer_iterate(s);

           if (status)
             break;

           size = gsl_multimin_fminimizer_size (s);
           status = gsl_multimin_test_size (size, 1e-2);

           if (status == GSL_SUCCESS)
             {
               printf ("converged to minimum at\n");
             }

           printf ("%5d ", iter);
           for (i = 0; i < np; i++)
             {
               printf ("%10.3e ", gsl_vector_get (s->x, i));
             }
           printf ("f() = %7.3f size = %.3f\n", s->fval, size);
         }
       while (status == GSL_CONTINUE && iter < 100);

       gsl_vector_free(x);
       gsl_vector_free(ss);
       gsl_multimin_fminimizer_free (s);

       return status;
     }

The minimum search stops when the Simplex size drops to 0.01. The
output is shown below.

         1  6.500e+00  5.000e+00 f() = 512.500 size = 1.082
         2  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
         3  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
         4  5.500e+00  1.000e+00 f() = 252.500 size = 1.372
         5  2.625e+00  3.500e+00 f() = 101.406 size = 1.823
         6  3.469e+00  1.375e+00 f() = 98.760  size = 1.526
         7  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
         8  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
         9  1.016e+00  2.812e+00 f() = 43.206  size = 1.105
        10  2.041e+00  2.008e+00 f() = 40.838  size = 0.645
        11  1.236e+00  1.664e+00 f() = 32.816  size = 0.645
        12  1.236e+00  1.664e+00 f() = 32.816  size = 0.447
        13  5.225e-01  1.980e+00 f() = 32.288  size = 0.447
        14  1.103e+00  2.073e+00 f() = 30.214  size = 0.345
        15  1.103e+00  2.073e+00 f() = 30.214  size = 0.264
        16  1.103e+00  2.073e+00 f() = 30.214  size = 0.160
        17  9.864e-01  1.934e+00 f() = 30.090  size = 0.132
        18  9.190e-01  1.987e+00 f() = 30.069  size = 0.092
        19  1.028e+00  2.017e+00 f() = 30.013  size = 0.056
        20  1.028e+00  2.017e+00 f() = 30.013  size = 0.046
        21  1.028e+00  2.017e+00 f() = 30.013  size = 0.033
        22  9.874e-01  1.985e+00 f() = 30.006  size = 0.028
        23  9.846e-01  1.995e+00 f() = 30.003  size = 0.023
        24  1.007e+00  2.003e+00 f() = 30.001  size = 0.012
     converged to minimum at
        25  1.007e+00  2.003e+00 f() = 30.001  size = 0.010

The simplex size first increases, while the simplex moves towards the
minimum. After a while the size begins to decrease as the simplex
contracts around the minimum.


File: gsl-ref.info,  Node: Multimin References and Further Reading,  Prev: Multimin Examples,  Up: Multidimensional Minimization

35.9 References and Further Reading
===================================

A brief description of multidimensional minimization algorithms and
further references can be found in the following book,

     C.W. Ueberhuber, `Numerical Computation (Volume 2)', Chapter 14,
     Section 4.4 "Minimization Methods", p. 325-335, Springer (1997),
     ISBN 3-540-62057-5.

The simplex algorithm is described in the following paper,

     J.A. Nelder and R. Mead, `A simplex method for function
     minimization', Computer Journal vol. 7 (1965), 308-315.



File: gsl-ref.info,  Node: Least-Squares Fitting,  Next: Nonlinear Least-Squares Fitting,  Prev: Multidimensional Minimization,  Up: Top

36 Least-Squares Fitting
************************

This chapter describes routines for performing least squares fits to
experimental data using linear combinations of functions.  The data may
be weighted or unweighted, i.e. with known or unknown errors.  For
weighted data the functions compute the best fit parameters and their
associated covariance matrix.  For unweighted data the covariance
matrix is estimated from the scatter of the points, giving a
variance-covariance matrix.

   The functions are divided into separate versions for simple one- or
two-parameter regression and multiple-parameter fits.  The functions
are declared in the header file `gsl_fit.h'.

* Menu:

* Fitting Overview::
* Linear regression::
* Linear fitting without a constant term::
* Multi-parameter fitting::
* Fitting Examples::
* Fitting References and Further Reading::


File: gsl-ref.info,  Node: Fitting Overview,  Next: Linear regression,  Up: Least-Squares Fitting

36.1 Overview
=============

Least-squares fits are found by minimizing \chi^2 (chi-squared), the
weighted sum of squared residuals over n experimental datapoints (x_i,
y_i) for the model Y(c,x),

     \chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2

The p parameters of the model are c = {c_0, c_1, ...}.  The weight
factors w_i are given by w_i = 1/\sigma_i^2, where \sigma_i is the
experimental error on the data-point y_i.  The errors are assumed to be
gaussian and uncorrelated.  For unweighted data the chi-squared sum is
computed without any weight factors.

   The fitting routines return the best-fit parameters c and their p
\times p covariance matrix.  The covariance matrix measures the
statistical errors on the best-fit parameters resulting from the errors
on the data \sigma_i, and is defined as C_{ab} = <\delta c_a \delta
c_b> where \langle \, \rangle denotes an average over the gaussian
error distributions of the underlying datapoints.

   The covariance matrix is calculated by error propagation from the
data errors \sigma_i.  The change in a fitted parameter \delta c_a
caused by a small change in the data \delta y_i is given by

allowing the covariance matrix to be written in terms of the errors on
the data,

For uncorrelated data the fluctuations of the underlying datapoints
satisfy <\delta y_i \delta y_j> = \sigma_i^2 \delta_{ij}, giving a
corresponding parameter covariance matrix of

When computing the covariance matrix for unweighted data, i.e. data
with unknown errors, the weight factors w_i in this sum are replaced by
the single estimate w = 1/\sigma^2, where \sigma^2 is the computed
variance of the residuals about the best-fit model, \sigma^2 = \sum
(y_i - Y(c,x_i))^2 / (n-p).  This is referred to as the
"variance-covariance matrix".  

   The standard deviations of the best-fit parameters are given by the
square root of the corresponding diagonal elements of the covariance
matrix, \sigma_{c_a} = \sqrt{C_{aa}}.


File: gsl-ref.info,  Node: Linear regression,  Next: Linear fitting without a constant term,  Prev: Fitting Overview,  Up: Least-Squares Fitting

36.2 Linear regression
======================

The functions described in this section can be used to perform
least-squares fits to a straight line model, Y(c,x) = c_0 + c_1 x.

 -- Function: int gsl_fit_linear (const double * X, const size_t
          XSTRIDE, const double * Y, const size_t YSTRIDE, size_t N,
          double * C0, double * C1, double * COV00, double * COV01,
          double * COV11, double * SUMSQ)
     This function computes the best-fit linear regression coefficients
     (C0,C1) of the model Y = c_0 + c_1 X for the dataset (X, Y), two
     vectors of length N with strides XSTRIDE and YSTRIDE.  The errors
     on Y are assumed unknown so the variance-covariance matrix for the
     parameters (C0, C1) is estimated from the scatter of the points
     around the best-fit line and returned via the parameters (COV00,
     COV01, COV11).  The sum of squares of the residuals from the
     best-fit line is returned in SUMSQ.

 -- Function: int gsl_fit_wlinear (const double * X, const size_t
          XSTRIDE, const double * W, const size_t WSTRIDE, const double
          * Y, const size_t YSTRIDE, size_t N, double * C0, double *
          C1, double * COV00, double * COV01, double * COV11, double *
          CHISQ)
     This function computes the best-fit linear regression coefficients
     (C0,C1) of the model Y = c_0 + c_1 X for the weighted dataset (X,
     Y), two vectors of length N with strides XSTRIDE and YSTRIDE.  The
     vector W, of length N and stride WSTRIDE, specifies the weight of
     each datapoint. The weight is the reciprocal of the variance for
     each datapoint in Y.

     The covariance matrix for the parameters (C0, C1) is computed
     using the weights and returned via the parameters (COV00, COV01,
     COV11).  The weighted sum of squares of the residuals from the
     best-fit line, \chi^2, is returned in CHISQ.

 -- Function: int gsl_fit_linear_est (double X, double C0, double C1,
          double C00, double C01, double C11, double * Y, double *
          Y_ERR)
     This function uses the best-fit linear regression coefficients
     C0,C1 and their covariance COV00,COV01,COV11 to compute the fitted
     function Y and its standard deviation Y_ERR for the model Y = c_0
     + c_1 X at the point X.


File: gsl-ref.info,  Node: Linear fitting without a constant term,  Next: Multi-parameter fitting,  Prev: Linear regression,  Up: Least-Squares Fitting

36.3 Linear fitting without a constant term
===========================================

The functions described in this section can be used to perform
least-squares fits to a straight line model without a constant term, Y
= c_1 X.

 -- Function: int gsl_fit_mul (const double * X, const size_t XSTRIDE,
          const double * Y, const size_t YSTRIDE, size_t N, double *
          C1, double * COV11, double * SUMSQ)
     This function computes the best-fit linear regression coefficient
     C1 of the model Y = c_1 X for the datasets (X, Y), two vectors of
     length N with strides XSTRIDE and YSTRIDE.  The errors on Y are
     assumed unknown so the variance of the parameter C1 is estimated
     from the scatter of the points around the best-fit line and
     returned via the parameter COV11.  The sum of squares of the
     residuals from the best-fit line is returned in SUMSQ.

 -- Function: int gsl_fit_wmul (const double * X, const size_t XSTRIDE,
          const double * W, const size_t WSTRIDE, const double * Y,
          const size_t YSTRIDE, size_t N, double * C1, double * COV11,
          double * SUMSQ)
     This function computes the best-fit linear regression coefficient
     C1 of the model Y = c_1 X for the weighted datasets (X, Y), two
     vectors of length N with strides XSTRIDE and YSTRIDE.  The vector
     W, of length N and stride WSTRIDE, specifies the weight of each
     datapoint. The weight is the reciprocal of the variance for each
     datapoint in Y.

     The variance of the parameter C1 is computed using the weights and
     returned via the parameter COV11.  The weighted sum of squares of
     the residuals from the best-fit line, \chi^2, is returned in CHISQ.

 -- Function: int gsl_fit_mul_est (double X, double C1, double C11,
          double * Y, double * Y_ERR)
     This function uses the best-fit linear regression coefficient C1
     and its covariance COV11 to compute the fitted function Y and its
     standard deviation Y_ERR for the model Y = c_1 X at the point X.


File: gsl-ref.info,  Node: Multi-parameter fitting,  Next: Fitting Examples,  Prev: Linear fitting without a constant term,  Up: Least-Squares Fitting

36.4 Multi-parameter fitting
============================

The functions described in this section perform least-squares fits to a
general linear model, y = X c where y is a vector of n observations, X
is an n by p matrix of predictor variables, and the elements of the
vector c are the p unknown best-fit parameters which are to be
estimated.

   This formulation can be used for fits to any number of functions
and/or variables by preparing the n-by-p matrix X appropriately.  For
example, to fit to a p-th order polynomial in X, use the following
matrix,

     X_{ij} = x_i^j

where the index i runs over the observations and the index j runs from
0 to p-1.

   To fit to a set of p sinusoidal functions with fixed frequencies
\omega_1, \omega_2, ..., \omega_p, use,

     X_{ij} = sin(\omega_j x_i)

To fit to p independent variables x_1, x_2, ..., x_p, use,

     X_{ij} = x_j(i)

where x_j(i) is the i-th value of the predictor variable x_j.

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

   The solution of the general linear least-squares system requires an
additional working space for intermediate results, such as the singular
value decomposition of the matrix X.

 -- Function: gsl_multifit_linear_workspace * gsl_multifit_linear_alloc
          (size_t N, size_t P)
     This function allocates a workspace for fitting a model to N
     observations using P parameters.

 -- Function: void gsl_multifit_linear_free
          (gsl_multifit_linear_workspace * WORK)
     This function frees the memory associated with the workspace W.

 -- Function: int gsl_multifit_linear (const gsl_matrix * X, const
          gsl_vector * Y, gsl_vector * C, gsl_matrix * COV, double *
          CHISQ, gsl_multifit_linear_workspace * WORK)
 -- Function: int gsl_multifit_linear_svd (const gsl_matrix * X, const
          gsl_vector * Y, double TOL, size_t * RANK, gsl_vector * C,
          gsl_matrix * COV, double * CHISQ,
          gsl_multifit_linear_workspace * WORK)
     These functions compute the best-fit parameters C of the model y =
     X c for the observations Y and the matrix of predictor variables
     X.  The variance-covariance matrix of the model parameters COV is
     estimated from the scatter of the observations about the best-fit.
     The sum of squares of the residuals from the best-fit, \chi^2, is
     returned in CHISQ.

     The best-fit is found by singular value decomposition of the matrix
     X using the preallocated workspace provided in WORK. The modified
     Golub-Reinsch SVD algorithm is used, with column scaling to
     improve the accuracy of the singular values. Any components which
     have zero singular value (to machine precision) are discarded from
     the fit.  In the second form of the function the components are
     discarded if the ratio of singular values s_i/s_0 falls below the
     user-specified tolerance TOL, and the effective rank is returned
     in RANK.

 -- Function: int gsl_multifit_wlinear (const gsl_matrix * X, const
          gsl_vector * W, const gsl_vector * Y, gsl_vector * C,
          gsl_matrix * COV, double * CHISQ,
          gsl_multifit_linear_workspace * WORK)
 -- Function: int gsl_multifit_wlinear_svd (const gsl_matrix * X, const
          gsl_vector * W, const gsl_vector * Y, double TOL, size_t *
          RANK, gsl_vector * C, gsl_matrix * COV, double * CHISQ,
          gsl_multifit_linear_workspace * WORK)
     This function computes the best-fit parameters C of the weighted
     model y = X c for the observations Y with weights W and the matrix
     of predictor variables X.  The covariance matrix of the model
     parameters COV is computed with the given weights.  The weighted
     sum of squares of the residuals from the best-fit, \chi^2, is
     returned in CHISQ.

     The best-fit is found by singular value decomposition of the matrix
     X using the preallocated workspace provided in WORK. Any
     components which have zero singular value (to machine precision)
     are discarded from the fit.  In the second form of the function the
     components are discarded if the ratio of singular values s_i/s_0
     falls below the user-specified tolerance TOL, and the effective
     rank is returned in RANK.

 -- Function: int gsl_multifit_linear_est (const gsl_vector * X, const
          gsl_vector * C, const gsl_matrix * COV, double * Y, double *
          Y_ERR)
     This function uses the best-fit multilinear regression coefficients
     C and their covariance matrix COV to compute the fitted function
     value Y and its standard deviation Y_ERR for the model y = x.c at
     the point X.


File: gsl-ref.info,  Node: Fitting Examples,  Next: Fitting References and Further Reading,  Prev: Multi-parameter fitting,  Up: Least-Squares Fitting

36.5 Examples
=============

The following program computes a least squares straight-line fit to a
simple dataset, and outputs the best-fit line and its associated one
standard-deviation error bars.

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

     int
     main (void)
     {
       int i, n = 4;
       double x[4] = { 1970, 1980, 1990, 2000 };
       double y[4] = {   12,   11,   14,   13 };
       double w[4] = {  0.1,  0.2,  0.3,  0.4 };

       double c0, c1, cov00, cov01, cov11, chisq;

       gsl_fit_wlinear (x, 1, w, 1, y, 1, n,
                        &c0, &c1, &cov00, &cov01, &cov11,
                        &chisq);

       printf ("# best fit: Y = %g + %g X\n", c0, c1);
       printf ("# covariance matrix:\n");
       printf ("# [ %g, %g\n#   %g, %g]\n",
               cov00, cov01, cov01, cov11);
       printf ("# chisq = %g\n", chisq);

       for (i = 0; i < n; i++)
         printf ("data: %g %g %g\n",
                        x[i], y[i], 1/sqrt(w[i]));

       printf ("\n");

       for (i = -30; i < 130; i++)
         {
           double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);
           double yf, yf_err;

           gsl_fit_linear_est (xf,
                               c0, c1,
                               cov00, cov01, cov11,
                               &yf, &yf_err);

           printf ("fit: %g %g\n", xf, yf);
           printf ("hi : %g %g\n", xf, yf + yf_err);
           printf ("lo : %g %g\n", xf, yf - yf_err);
         }
       return 0;
     }

The following commands extract the data from the output of the program
and display it using the GNU plotutils `graph' utility,

     $ ./demo > tmp
     $ more tmp
     # best fit: Y = -106.6 + 0.06 X
     # covariance matrix:
     # [ 39602, -19.9
     #   -19.9, 0.01]
     # chisq = 0.8

     $ for n in data fit hi lo ;
        do
          grep "^$n" tmp | cut -d: -f2 > $n ;
        done
     $ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data
          -S 0 -I a -m 1 fit -m 2 hi -m 2 lo

   The next program performs a quadratic fit y = c_0 + c_1 x + c_2 x^2
to a weighted dataset using the generalised linear fitting function
`gsl_multifit_wlinear'.  The model matrix X for a quadratic fit is
given by,

     X = [ 1   , x_0  , x_0^2 ;
           1   , x_1  , x_1^2 ;
           1   , x_2  , x_2^2 ;
           ... , ...  , ...   ]

where the column of ones corresponds to the constant term c_0.  The two
remaining columns corresponds to the terms c_1 x and c_2 x^2.

   The program reads N lines of data in the format (X, Y, ERR) where
ERR is the error (standard deviation) in the value Y.

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

     int
     main (int argc, char **argv)
     {
       int i, n;
       double xi, yi, ei, chisq;
       gsl_matrix *X, *cov;
       gsl_vector *y, *w, *c;

       if (argc != 2)
         {
           fprintf (stderr,"usage: fit n < data\n");
           exit (-1);
         }

       n = atoi (argv[1]);

       X = gsl_matrix_alloc (n, 3);
       y = gsl_vector_alloc (n);
       w = gsl_vector_alloc (n);

       c = gsl_vector_alloc (3);
       cov = gsl_matrix_alloc (3, 3);

       for (i = 0; i < n; i++)
         {
           int count = fscanf (stdin, "%lg %lg %lg",
                               &xi, &yi, &ei);

           if (count != 3)
             {
               fprintf (stderr, "error reading file\n");
               exit (-1);
             }

           printf ("%g %g +/- %g\n", xi, yi, ei);

           gsl_matrix_set (X, i, 0, 1.0);
           gsl_matrix_set (X, i, 1, xi);
           gsl_matrix_set (X, i, 2, xi*xi);

           gsl_vector_set (y, i, yi);
           gsl_vector_set (w, i, 1.0/(ei*ei));
         }

       {
         gsl_multifit_linear_workspace * work
           = gsl_multifit_linear_alloc (n, 3);
         gsl_multifit_wlinear (X, w, y, c, cov,
                               &chisq, work);
         gsl_multifit_linear_free (work);
       }

     #define C(i) (gsl_vector_get(c,(i)))
     #define COV(i,j) (gsl_matrix_get(cov,(i),(j)))

       {
         printf ("# best fit: Y = %g + %g X + %g X^2\n",
                 C(0), C(1), C(2));

         printf ("# covariance matrix:\n");
         printf ("[ %+.5e, %+.5e, %+.5e  \n",
                    COV(0,0), COV(0,1), COV(0,2));
         printf ("  %+.5e, %+.5e, %+.5e  \n",
                    COV(1,0), COV(1,1), COV(1,2));
         printf ("  %+.5e, %+.5e, %+.5e ]\n",
                    COV(2,0), COV(2,1), COV(2,2));
         printf ("# chisq = %g\n", chisq);
       }
       return 0;
     }

A suitable set of data for fitting can be generated using the following
program.  It outputs a set of points with gaussian errors from the curve
y = e^x in the region 0 < x < 2.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_randist.h>

     int
     main (void)
     {
       double x;
       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (x = 0.1; x < 2; x+= 0.1)
         {
           double y0 = exp (x);
           double sigma = 0.1 * y0;
           double dy = gsl_ran_gaussian (r, sigma);

           printf ("%g %g %g\n", x, y0 + dy, sigma);
         }
       return 0;
     }

The data can be prepared by running the resulting executable program,

     $ ./generate > exp.dat
     $ more exp.dat
     0.1 0.97935 0.110517
     0.2 1.3359 0.12214
     0.3 1.52573 0.134986
     0.4 1.60318 0.149182
     0.5 1.81731 0.164872
     0.6 1.92475 0.182212
     ....

To fit the data use the previous program, with the number of data points
given as the first argument.  In this case there are 19 data points.

     $ ./fit 19 < exp.dat
     0.1 0.97935 +/- 0.110517
     0.2 1.3359 +/- 0.12214
     ...
     # best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
     # covariance matrix:
     [ +1.25612e-02, -3.64387e-02, +1.94389e-02
       -3.64387e-02, +1.42339e-01, -8.48761e-02
       +1.94389e-02, -8.48761e-02, +5.60243e-02 ]
     # chisq = 23.0987

The parameters of the quadratic fit match the coefficients of the
expansion of e^x, taking into account the errors on the parameters and
the O(x^3) difference between the exponential and quadratic functions
for the larger values of x.  The errors on the parameters are given by
the square-root of the corresponding diagonal elements of the
covariance matrix.  The chi-squared per degree of freedom is 1.4,
indicating a reasonable fit to the data.


File: gsl-ref.info,  Node: Fitting References and Further Reading,  Prev: Fitting Examples,  Up: Least-Squares Fitting

36.6 References and Further Reading
===================================

A summary of formulas and techniques for least squares fitting can be
found in the "Statistics" chapter of the Annual Review of Particle
Physics prepared by the Particle Data Group,

     `Review of Particle Properties', R.M. Barnett et al., Physical
     Review D54, 1 (1996) `http://pdg.lbl.gov/'

The Review of Particle Physics is available online at the website given
above.

   The tests used to prepare these routines are based on the NIST
Statistical Reference Datasets. The datasets and their documentation are
available from NIST at the following website,

           `http://www.nist.gov/itl/div898/strd/index.html'.


File: gsl-ref.info,  Node: Nonlinear Least-Squares Fitting,  Next: Physical Constants,  Prev: Least-Squares Fitting,  Up: Top

37 Nonlinear Least-Squares Fitting
**********************************

This chapter describes functions for multidimensional nonlinear
least-squares fitting.  The library provides low level components for a
variety of iterative solvers and convergence tests.  These can be
combined by the user to achieve the desired solution, with full access
to the intermediate steps of the iteration.  Each class of methods uses
the same framework, so that you can switch between solvers at runtime
without needing to recompile your program.  Each instance of a solver
keeps track of its own state, allowing the solvers to be used in
multi-threaded programs.

   The header file `gsl_multifit_nlin.h' contains prototypes for the
multidimensional nonlinear fitting functions and related declarations.

* Menu:

* Overview of Nonlinear Least-Squares Fitting::
* Initializing the Nonlinear Least-Squares Solver::
* Providing the Function to be Minimized::
* Iteration of the Minimization Algorithm::
* Search Stopping Parameters for Minimization Algorithms::
* Minimization Algorithms using Derivatives::
* Minimization Algorithms without Derivatives::
* Computing the covariance matrix of best fit parameters::
* Example programs for Nonlinear Least-Squares Fitting::
* References and Further Reading for Nonlinear Least-Squares Fitting::


File: gsl-ref.info,  Node: Overview of Nonlinear Least-Squares Fitting,  Next: Initializing the Nonlinear Least-Squares Solver,  Up: Nonlinear Least-Squares Fitting

37.1 Overview
=============

The problem of multidimensional nonlinear least-squares fitting requires
the minimization of the squared residuals of n functions, f_i, in p
parameters, x_i,

     \Phi(x) = (1/2) || F(x) ||^2
             = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2

All algorithms proceed from an initial guess using the linearization,

     \psi(p) = || F(x+p) || ~=~ || F(x) + J p ||

where x is the initial point, p is the proposed step and J is the
Jacobian matrix J_{ij} = d f_i / d x_j.  Additional strategies are used
to enlarge the region of convergence.  These include requiring a
decrease in the norm ||F|| on each step or using a trust region to
avoid steps which fall outside the linear regime.

   To perform a weighted least-squares fit of a nonlinear model Y(x,t)
to data (t_i, y_i) with independent gaussian errors \sigma_i, use
function components of the following form,

     f_i = (Y(x, t_i) - y_i) / \sigma_i

Note that the model parameters are denoted by x in this chapter since
the non-linear least-squares algorithms are described geometrically
(i.e. finding the minimum of a surface).  The independent variable of
any data to be fitted is denoted by t.

   With the definition above the Jacobian is J_{ij} =(1 / \sigma_i)  d
Y_i / d x_j, where Y_i = Y(x,t_i).


File: gsl-ref.info,  Node: Initializing the Nonlinear Least-Squares Solver,  Next: Providing the Function to be Minimized,  Prev: Overview of Nonlinear Least-Squares Fitting,  Up: Nonlinear Least-Squares Fitting

37.2 Initializing the Solver
============================

 -- Function: gsl_multifit_fsolver * gsl_multifit_fsolver_alloc (const
          gsl_multifit_fsolver_type * T, size_t N, size_t P)
     This function returns a pointer to a newly allocated instance of a
     solver of type T for N observations and P parameters.  The number
     of observations N must be greater than or equal to parameters P.

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of `GSL_ENOMEM'.

 -- Function: gsl_multifit_fdfsolver * gsl_multifit_fdfsolver_alloc
          (const gsl_multifit_fdfsolver_type * T, size_t N, size_t P)
     This function returns a pointer to a newly allocated instance of a
     derivative solver of type T for N observations and P parameters.
     For example, the following code creates an instance of a
     Levenberg-Marquardt solver for 100 data points and 3 parameters,

          const gsl_multifit_fdfsolver_type * T
              = gsl_multifit_fdfsolver_lmder;
          gsl_multifit_fdfsolver * s
              = gsl_multifit_fdfsolver_alloc (T, 100, 3);

     The number of observations N must be greater than or equal to
     parameters P.

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of `GSL_ENOMEM'.

 -- Function: int gsl_multifit_fsolver_set (gsl_multifit_fsolver * S,
          gsl_multifit_function * F, gsl_vector * X)
     This function initializes, or reinitializes, an existing solver S
     to use the function F and the initial guess X.

 -- Function: int gsl_multifit_fdfsolver_set (gsl_multifit_fdfsolver *
          S, gsl_function_fdf * FDF, gsl_vector * X)
     This function initializes, or reinitializes, an existing solver S
     to use the function and derivative FDF and the initial guess X.

 -- Function: void gsl_multifit_fsolver_free (gsl_multifit_fsolver * S)
 -- Function: void gsl_multifit_fdfsolver_free (gsl_multifit_fdfsolver
          * S)
     These functions free all the memory associated with the solver S.

 -- Function: const char * gsl_multifit_fsolver_name (const
          gsl_multifit_fdfsolver * S)
 -- Function: const char * gsl_multifit_fdfsolver_name (const
          gsl_multifit_fdfsolver * S)
     These functions return a pointer to the name of the solver.  For
     example,

          printf ("s is a '%s' solver\n",
                  gsl_multifit_fdfsolver_name (s));

     would print something like `s is a 'lmder' solver'.


File: gsl-ref.info,  Node: Providing the Function to be Minimized,  Next: Iteration of the Minimization Algorithm,  Prev: Initializing the Nonlinear Least-Squares Solver,  Up: Nonlinear Least-Squares Fitting

37.3 Providing the Function to be Minimized
===========================================

You must provide n functions of p variables for the minimization
algorithms to operate on.  In order to allow for arbitrary parameters
the functions are defined by the following data types:

 -- Data Type: gsl_multifit_function
     This data type defines a general system of functions with
     arbitrary parameters.

    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
          this function should store the vector result f(x,params) in F
          for argument X and arbitrary parameters PARAMS, returning an
          appropriate error code if the function cannot be computed.

    `size_t n'
          the number of functions, i.e. the number of components of the
          vector F.

    `size_t p'
          the number of independent variables, i.e. the number of
          components of the vector X.

    `void * params'
          a pointer to the arbitrary parameters of the function.

 -- Data Type: gsl_multifit_function_fdf
     This data type defines a general system of functions with
     arbitrary parameters and the corresponding Jacobian matrix of
     derivatives,

    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
          this function should store the vector result f(x,params) in F
          for argument X and arbitrary parameters PARAMS, returning an
          appropriate error code if the function cannot be computed.

    `int (* df) (const gsl_vector * X, void * PARAMS, gsl_matrix * J)'
          this function should store the N-by-P matrix result J_ij = d
          f_i(x,params) / d x_j in J for argument X and arbitrary
          parameters PARAMS, returning an appropriate error code if the
          function cannot be computed.

    `int (* fdf) (const gsl_vector * X, void * PARAMS, gsl_vector * F, gsl_matrix * J)'
          This function should set the values of the F and J as above,
          for arguments X and arbitrary parameters PARAMS.  This
          function provides an optimization of the separate functions
          for f(x) and J(x)--it is always faster to compute the
          function and its derivative at the same time.

    `size_t n'
          the number of functions, i.e. the number of components of the
          vector F.

    `size_t p'
          the number of independent variables, i.e. the number of
          components of the vector X.

    `void * params'
          a pointer to the arbitrary parameters of the function.

   Note that when fitting a non-linear model against experimental data,
the data is passed to the functions above using the PARAMS argument and
the trial best-fit parameters through the X argument.


File: gsl-ref.info,  Node: Iteration of the Minimization Algorithm,  Next: Search Stopping Parameters for Minimization Algorithms,  Prev: Providing the Function to be Minimized,  Up: Nonlinear Least-Squares Fitting

37.4 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any solver of the
corresponding type.  The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.

 -- Function: int gsl_multifit_fsolver_iterate (gsl_multifit_fsolver *
          S)
 -- Function: int gsl_multifit_fdfsolver_iterate
          (gsl_multifit_fdfsolver * S)
     These functions perform a single iteration of the solver S.  If
     the iteration encounters an unexpected problem then an error code
     will be returned.  The solver maintains a current estimate of the
     best-fit parameters at all times.

   The solver struct S contains the following entries, which can be
used to track the progress of the solution:

`gsl_vector * x'
     The current position.

`gsl_vector * f'
     The function value at the current position.

`gsl_vector * dx'
     The difference between the current position and the previous
     position, i.e. the last step, taken as a vector.

`gsl_matrix * J'
     The Jacobian matrix at the current position (for the
     `gsl_multifit_fdfsolver' struct only)

   The best-fit information also can be accessed with the following
auxiliary functions,

 -- Function: gsl_vector * gsl_multifit_fsolver_position (const
          gsl_multifit_fsolver * S)
 -- Function: gsl_vector * gsl_multifit_fdfsolver_position (const
          gsl_multifit_fdfsolver * S)
     These functions return the current position (i.e. best-fit
     parameters) `s->x' of the solver S.


File: gsl-ref.info,  Node: Search Stopping Parameters for Minimization Algorithms,  Next: Minimization Algorithms using Derivatives,  Prev: Iteration of the Minimization Algorithm,  Up: Nonlinear Least-Squares Fitting

37.5 Search Stopping Parameters
===============================

A minimization procedure should stop when one of the following
conditions is true:

   * A minimum has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the current estimate of the best-fit
parameters in several standard ways.

 -- Function: int gsl_multifit_test_delta (const gsl_vector * DX, const
          gsl_vector * X, double EPSABS, double EPSREL)
     This function tests for the convergence of the sequence by
     comparing the last step DX with the absolute error EPSABS and
     relative error EPSREL to the current position X.  The test returns
     `GSL_SUCCESS' if the following condition is achieved,

          |dx_i| < epsabs + epsrel |x_i|

     for each component of X and returns `GSL_CONTINUE' otherwise.

 -- Function: int gsl_multifit_test_gradient (const gsl_vector * G,
          double EPSABS)
     This function tests the residual gradient G against the absolute
     error bound EPSABS.  Mathematically, the gradient should be
     exactly zero at the minimum. The test returns `GSL_SUCCESS' if the
     following condition is achieved,

          \sum_i |g_i| < epsabs

     and returns `GSL_CONTINUE' otherwise.  This criterion is suitable
     for situations where the precise location of the minimum, x, is
     unimportant provided a value can be found where the gradient is
     small enough.

 -- Function: int gsl_multifit_gradient (const gsl_matrix * J, const
          gsl_vector * F, gsl_vector * G)
     This function computes the gradient G of \Phi(x) = (1/2)
     ||F(x)||^2 from the Jacobian matrix J and the function values F,
     using the formula g = J^T f.


File: gsl-ref.info,  Node: Minimization Algorithms using Derivatives,  Next: Minimization Algorithms without Derivatives,  Prev: Search Stopping Parameters for Minimization Algorithms,  Up: Nonlinear Least-Squares Fitting

37.6 Minimization Algorithms using Derivatives
==============================================

The minimization algorithms described in this section make use of both
the function and its derivative.  They require an initial guess for the
location of the minimum. There is no absolute guarantee of
convergence--the function must be suitable for this technique and the
initial guess must be sufficiently close to the minimum for it to work.

 -- Derivative Solver: gsl_multifit_fdfsolver_lmsder
     This is a robust and efficient version of the Levenberg-Marquardt
     algorithm as implemented in the scaled LMDER routine in MINPACK.
     Minpack was written by Jorge J. More', Burton S. Garbow and
     Kenneth E. Hillstrom.

     The algorithm uses a generalized trust region to keep each step
     under control.  In order to be accepted a proposed new position x'
     must satisfy the condition |D (x' - x)| < \delta, where D is a
     diagonal scaling matrix and \delta is the size of the trust
     region.  The components of D are computed internally, using the
     column norms of the Jacobian to estimate the sensitivity of the
     residual to each component of x.  This improves the behavior of the
     algorithm for badly scaled functions.

     On each iteration the algorithm attempts to minimize the linear
     system |F + J p| subject to the constraint |D p| < \Delta.  The
     solution to this constrained linear system is found using the
     Levenberg-Marquardt method.

     The proposed step is now tested by evaluating the function at the
     resulting point, x'.  If the step reduces the norm of the function
     sufficiently, and follows the predicted behavior of the function
     within the trust region, then it is accepted and the size of the
     trust region is increased.  If the proposed step fails to improve
     the solution, or differs significantly from the expected behavior
     within the trust region, then the size of the trust region is
     decreased and another trial step is computed.

     The algorithm also monitors the progress of the solution and
     returns an error if the changes in the solution are smaller than
     the machine precision.  The possible error codes are,

    `GSL_ETOLF'
          the decrease in the function falls below machine precision

    `GSL_ETOLX'
          the change in the position vector falls below machine
          precision

    `GSL_ETOLG'
          the norm of the gradient, relative to the norm of the
          function, falls below machine precision

     These error codes indicate that further iterations will be
     unlikely to change the solution from its current value.


 -- Derivative Solver: gsl_multifit_fdfsolver_lmder
     This is an unscaled version of the LMDER algorithm.  The elements
     of the diagonal scaling matrix D are set to 1.  This algorithm may
     be useful in circumstances where the scaled version of LMDER
     converges too slowly, or the function is already scaled
     appropriately.


File: gsl-ref.info,  Node: Minimization Algorithms without Derivatives,  Next: Computing the covariance matrix of best fit parameters,  Prev: Minimization Algorithms using Derivatives,  Up: Nonlinear Least-Squares Fitting

37.7 Minimization Algorithms without Derivatives
================================================

There are no algorithms implemented in this section at the moment.


File: gsl-ref.info,  Node: Computing the covariance matrix of best fit parameters,  Next: Example programs for Nonlinear Least-Squares Fitting,  Prev: Minimization Algorithms without Derivatives,  Up: Nonlinear Least-Squares Fitting

37.8 Computing the covariance matrix of best fit parameters
===========================================================

 -- Function: int gsl_multifit_covar (const gsl_matrix * J, double
          EPSREL, gsl_matrix * COVAR)
     This function uses the Jacobian matrix J to compute the covariance
     matrix of the best-fit parameters, COVAR.  The parameter EPSREL is
     used to remove linear-dependent columns when J is rank deficient.

     The covariance matrix is given by,

          covar = (J^T J)^{-1}

     and is computed by QR decomposition of J with column-pivoting.  Any
     columns of R which satisfy

          |R_{kk}| <= epsrel |R_{11}|

     are considered linearly-dependent and are excluded from the
     covariance matrix (the corresponding rows and columns of the
     covariance matrix are set to zero).

     If the minimisation uses the weighted least-squares function f_i =
     (Y(x, t_i) - y_i) / \sigma_i then the covariance matrix above
     gives the statistical error on the best-fit parameters resulting
     from the gaussian errors \sigma_i on the underlying data y_i.
     This can be verified from the relation \delta f = J \delta c and
     the fact that the fluctuations in f from the data y_i are
     normalised by \sigma_i and so satisfy <\delta f \delta f^T> = I.

     For an unweighted least-squares function f_i = (Y(x, t_i) - y_i)
     the covariance matrix above should be multiplied by the variance
     of the residuals about the best-fit \sigma^2 = \sum (y_i -
     Y(x,t_i))^2 / (n-p) to give the variance-covariance matrix
     \sigma^2 C.  This estimates the statistical error on the best-fit
     parameters from the scatter of the underlying data.

     For more information about covariance matrices see *Note Fitting
     Overview::.


File: gsl-ref.info,  Node: Example programs for Nonlinear Least-Squares Fitting,  Next: References and Further Reading for Nonlinear Least-Squares Fitting,  Prev: Computing the covariance matrix of best fit parameters,  Up: Nonlinear Least-Squares Fitting

37.9 Examples
=============

The following example program fits a weighted exponential model with
background to experimental data, Y = A \exp(-\lambda t) + b. The first
part of the program sets up the functions `expb_f' and `expb_df' to
calculate the model and its Jacobian.  The appropriate fitting function
is given by,

     f_i = ((A \exp(-\lambda t_i) + b) - y_i)/\sigma_i

where we have chosen t_i = i.  The Jacobian matrix J is the derivative
of these functions with respect to the three parameters (A, \lambda,
b).  It is given by,

     J_{ij} = d f_i / d x_j

where x_0 = A, x_1 = \lambda and x_2 = b.

     /* expfit.c -- model functions for exponential + background */

     struct data {
       size_t n;
       double * y;
       double * sigma;
     };

     int
     expb_f (const gsl_vector * x, void *data,
             gsl_vector * f)
     {
       size_t n = ((struct data *)data)->n;
       double *y = ((struct data *)data)->y;
       double *sigma = ((struct data *) data)->sigma;

       double A = gsl_vector_get (x, 0);
       double lambda = gsl_vector_get (x, 1);
       double b = gsl_vector_get (x, 2);

       size_t i;

       for (i = 0; i < n; i++)
         {
           /* Model Yi = A * exp(-lambda * i) + b */
           double t = i;
           double Yi = A * exp (-lambda * t) + b;
           gsl_vector_set (f, i, (Yi - y[i])/sigma[i]);
         }

       return GSL_SUCCESS;
     }

     int
     expb_df (const gsl_vector * x, void *data,
              gsl_matrix * J)
     {
       size_t n = ((struct data *)data)->n;
       double *sigma = ((struct data *) data)->sigma;

       double A = gsl_vector_get (x, 0);
       double lambda = gsl_vector_get (x, 1);

       size_t i;

       for (i = 0; i < n; i++)
         {
           /* Jacobian matrix J(i,j) = dfi / dxj, */
           /* where fi = (Yi - yi)/sigma[i],      */
           /*       Yi = A * exp(-lambda * i) + b  */
           /* and the xj are the parameters (A,lambda,b) */
           double t = i;
           double s = sigma[i];
           double e = exp(-lambda * t);
           gsl_matrix_set (J, i, 0, e/s);
           gsl_matrix_set (J, i, 1, -t * A * e/s);
           gsl_matrix_set (J, i, 2, 1/s);
         }
       return GSL_SUCCESS;
     }

     int
     expb_fdf (const gsl_vector * x, void *data,
               gsl_vector * f, gsl_matrix * J)
     {
       expb_f (x, data, f);
       expb_df (x, data, J);

       return GSL_SUCCESS;
     }

The main part of the program sets up a Levenberg-Marquardt solver and
some simulated random data. The data uses the known parameters
(1.0,5.0,0.1) combined with gaussian noise (standard deviation = 0.1)
over a range of 40 timesteps. The initial guess for the parameters is
chosen as (0.0, 1.0, 0.0).

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlin.h>

     #include "expfit.c"

     #define N 40

     void print_state (size_t iter, gsl_multifit_fdfsolver * s);

     int
     main (void)
     {
       const gsl_multifit_fdfsolver_type *T;
       gsl_multifit_fdfsolver *s;

       int status;
       size_t i, iter = 0;

       const size_t n = N;
       const size_t p = 3;

       gsl_matrix *covar = gsl_matrix_alloc (p, p);
       double y[N], sigma[N];
       struct data d = { n, y, sigma};
       gsl_multifit_function_fdf f;
       double x_init[3] = { 1.0, 0.0, 0.0 };
       gsl_vector_view x = gsl_vector_view_array (x_init, p);
       const gsl_rng_type * type;
       gsl_rng * r;

       gsl_rng_env_setup();

       type = gsl_rng_default;
       r = gsl_rng_alloc (type);

       f.f = &expb_f;
       f.df = &expb_df;
       f.fdf = &expb_fdf;
       f.n = n;
       f.p = p;
       f.params = &d;

       /* This is the data to be fitted */

       for (i = 0; i < n; i++)
         {
           double t = i;
           y[i] = 1.0 + 5 * exp (-0.1 * t)
                      + gsl_ran_gaussian (r, 0.1);
           sigma[i] = 0.1;
           printf ("data: %d %g %g\n", i, y[i], sigma[i]);
         };

       T = gsl_multifit_fdfsolver_lmsder;
       s = gsl_multifit_fdfsolver_alloc (T, n, p);
       gsl_multifit_fdfsolver_set (s, &f, &x.vector);

       print_state (iter, s);

       do
         {
           iter++;
           status = gsl_multifit_fdfsolver_iterate (s);

           printf ("status = %s\n", gsl_strerror (status));

           print_state (iter, s);

           if (status)
             break;

           status = gsl_multifit_test_delta (s->dx, s->x,
                                             1e-4, 1e-4);
         }
       while (status == GSL_CONTINUE && iter < 500);

       gsl_multifit_covar (s->J, 0.0, covar);

     #define FIT(i) gsl_vector_get(s->x, i)
     #define ERR(i) sqrt(gsl_matrix_get(covar,i,i))

       {
         double chi = gsl_blas_dnrm2(s->f);
         double dof = n - p;
         double c = GSL_MAX_DBL(1, chi / sqrt(dof));

         printf("chisq/dof = %g\n",  pow(chi, 2.0) / dof);

         printf ("A      = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
         printf ("lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
         printf ("b      = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
       }

       printf ("status = %s\n", gsl_strerror (status));

       gsl_multifit_fdfsolver_free (s);
       return 0;
     }

     void
     print_state (size_t iter, gsl_multifit_fdfsolver * s)
     {
       printf ("iter: %3u x = % 15.8f % 15.8f % 15.8f "
               "|f(x)| = %g\n",
               iter,
               gsl_vector_get (s->x, 0),
               gsl_vector_get (s->x, 1),
               gsl_vector_get (s->x, 2),
               gsl_blas_dnrm2 (s->f));
     }

The iteration terminates when the change in x is smaller than 0.0001, as
both an absolute and relative change.  Here are the results of running
the program:

     iter: 0 x=1.00000000 0.00000000 0.00000000 |f(x)|=117.349
     status=success
     iter: 1 x=1.64659312 0.01814772 0.64659312 |f(x)|=76.4578
     status=success
     iter: 2 x=2.85876037 0.08092095 1.44796363 |f(x)|=37.6838
     status=success
     iter: 3 x=4.94899512 0.11942928 1.09457665 |f(x)|=9.58079
     status=success
     iter: 4 x=5.02175572 0.10287787 1.03388354 |f(x)|=5.63049
     status=success
     iter: 5 x=5.04520433 0.10405523 1.01941607 |f(x)|=5.44398
     status=success
     iter: 6 x=5.04535782 0.10404906 1.01924871 |f(x)|=5.44397
     chisq/dof = 0.800996
     A      = 5.04536 +/- 0.06028
     lambda = 0.10405 +/- 0.00316
     b      = 1.01925 +/- 0.03782
     status = success

The approximate values of the parameters are found correctly, and the
chi-squared value indicates a good fit (the chi-squared per degree of
freedom is approximately 1).  In this case the errors on the parameters
can be estimated from the square roots of the diagonal elements of the
covariance matrix.

   If the chi-squared value shows a poor fit (i.e. chi^2/dof >> 1) then
the error estimates obtained from the covariance matrix will be too
small.  In the example program the error estimates are multiplied by
\sqrt{\chi^2/dof} in this case, a common way of increasing the errors
for a poor fit.  Note that a poor fit will result from the use an
inappropriate model, and the scaled error estimates may then be outside
the range of validity for gaussian errors.


File: gsl-ref.info,  Node: References and Further Reading for Nonlinear Least-Squares Fitting,  Prev: Example programs for Nonlinear Least-Squares Fitting,  Up: Nonlinear Least-Squares Fitting

37.10 References and Further Reading
====================================

The MINPACK algorithm is described in the following article,

     J.J. More', `The Levenberg-Marquardt Algorithm: Implementation and
     Theory', Lecture Notes in Mathematics, v630 (1978), ed G. Watson.

The following paper is also relevant to the algorithms described in this
section,

     J.J. More', B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained
     Optimization Software", ACM Transactions on Mathematical Software,
     Vol 7, No 1 (1981), p 17-41.


File: gsl-ref.info,  Node: Physical Constants,  Next: IEEE floating-point arithmetic,  Prev: Nonlinear Least-Squares Fitting,  Up: Top

38 Physical Constants
*********************

This chapter describes macros for the values of physical constants, such
as the speed of light, c, and gravitational constant, G.  The values
are available in different unit systems, including the standard MKSA
system (meters, kilograms, seconds, amperes) and the CGSM system
(centimeters, grams, seconds, gauss), which is commonly used in
Astronomy.

   The definitions of constants in the MKSA system are available in the
file `gsl_const_mksa.h'.  The constants in the CGSM system are defined
in `gsl_const_cgsm.h'.  Dimensionless constants, such as the fine
structure constant, which are pure numbers are defined in
`gsl_const_num.h'.

* Menu:

* Fundamental Constants::
* Astronomy and Astrophysics::
* Atomic and Nuclear Physics::
* Measurement of Time::
* Imperial Units ::
* Speed and Nautical Units::
* Printers Units::
* Volume Area and Length::
* Mass and Weight ::
* Thermal Energy and Power::
* Pressure::
* Viscosity::
* Light and Illumination::
* Radioactivity::
* Force and Energy::
* Prefixes::
* Physical Constant Examples::
* Physical Constant References and Further Reading::

   The full list of constants is described briefly below.  Consult the
header files themselves for the values of the constants used in the
library.


File: gsl-ref.info,  Node: Fundamental Constants,  Next: Astronomy and Astrophysics,  Up: Physical Constants

38.1 Fundamental Constants
==========================

`GSL_CONST_MKSA_SPEED_OF_LIGHT'
     The speed of light in vacuum, c.

`GSL_CONST_MKSA_VACUUM_PERMEABILITY'
     The permeability of free space, \mu_0. This constant is defined in
     the MKSA system only.

`GSL_CONST_MKSA_VACUUM_PERMITTIVITY'
     The permittivity of free space, \epsilon_0.  This constant is
     defined in the MKSA system only.

`GSL_CONST_MKSA_PLANCKS_CONSTANT_H'
     Planck's constant, h.

`GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR'
     Planck's constant divided by 2\pi, \hbar.

`GSL_CONST_NUM_AVOGADRO'
     Avogadro's number, N_a.

`GSL_CONST_MKSA_FARADAY'
     The molar charge of 1 Faraday.

`GSL_CONST_MKSA_BOLTZMANN'
     The Boltzmann constant, k.

`GSL_CONST_MKSA_MOLAR_GAS'
     The molar gas constant, R_0.

`GSL_CONST_MKSA_STANDARD_GAS_VOLUME'
     The standard gas volume, V_0.

`GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT'
     The Stefan-Boltzmann radiation constant, \sigma.

`GSL_CONST_MKSA_GAUSS'
     The magnetic field of 1 Gauss.


File: gsl-ref.info,  Node: Astronomy and Astrophysics,  Next: Atomic and Nuclear Physics,  Prev: Fundamental Constants,  Up: Physical Constants

38.2 Astronomy and Astrophysics
===============================

`GSL_CONST_MKSA_ASTRONOMICAL_UNIT'
     The length of 1 astronomical unit (mean earth-sun distance), au.

`GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT'
     The gravitational constant, G.

`GSL_CONST_MKSA_LIGHT_YEAR'
     The distance of 1 light-year, ly.

`GSL_CONST_MKSA_PARSEC'
     The distance of 1 parsec, pc.

`GSL_CONST_MKSA_GRAV_ACCEL'
     The standard gravitational acceleration on Earth, g.

`GSL_CONST_MKSA_SOLAR_MASS'
     The mass of the Sun.


File: gsl-ref.info,  Node: Atomic and Nuclear Physics,  Next: Measurement of Time,  Prev: Astronomy and Astrophysics,  Up: Physical Constants

38.3 Atomic and Nuclear Physics
===============================

`GSL_CONST_MKSA_ELECTRON_CHARGE'
     The charge of the electron, e.

`GSL_CONST_MKSA_ELECTRON_VOLT'
     The energy of 1 electron volt, eV.

`GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS'
     The unified atomic mass, amu.

`GSL_CONST_MKSA_MASS_ELECTRON'
     The mass of the electron, m_e.

`GSL_CONST_MKSA_MASS_MUON'
     The mass of the muon, m_\mu.

`GSL_CONST_MKSA_MASS_PROTON'
     The mass of the proton, m_p.

`GSL_CONST_MKSA_MASS_NEUTRON'
     The mass of the neutron, m_n.

`GSL_CONST_NUM_FINE_STRUCTURE'
     The electromagnetic fine structure constant \alpha.

`GSL_CONST_MKSA_RYDBERG'
     The Rydberg constant, Ry, in units of energy.  This is related to
     the Rydberg inverse wavelength R by Ry = h c R.

`GSL_CONST_MKSA_BOHR_RADIUS'
     The Bohr radius, a_0.

`GSL_CONST_MKSA_ANGSTROM'
     The length of 1 angstrom.

`GSL_CONST_MKSA_BARN'
     The area of 1 barn.

`GSL_CONST_MKSA_BOHR_MAGNETON'
     The Bohr Magneton, \mu_B.

`GSL_CONST_MKSA_NUCLEAR_MAGNETON'
     The Nuclear Magneton, \mu_N.

`GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT'
     The absolute value of the magnetic moment of the electron, \mu_e.
     The physical magnetic moment of the electron is negative.

`GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT'
     The magnetic moment of the proton, \mu_p.

`GSL_CONST_MKSA_THOMSON_CROSS_SECTION'
     The Thomson cross section, \sigma_T.

`GSL_CONST_MKSA_DEBYE'
     The electric dipole moment of 1 Debye, D.


File: gsl-ref.info,  Node: Measurement of Time,  Next: Imperial Units,  Prev: Atomic and Nuclear Physics,  Up: Physical Constants

38.4 Measurement of Time
========================

`GSL_CONST_MKSA_MINUTE'
     The number of seconds in 1 minute.

`GSL_CONST_MKSA_HOUR'
     The number of seconds in 1 hour.

`GSL_CONST_MKSA_DAY'
     The number of seconds in 1 day.

`GSL_CONST_MKSA_WEEK'
     The number of seconds in 1 week.


File: gsl-ref.info,  Node: Imperial Units,  Next: Speed and Nautical Units,  Prev: Measurement of Time,  Up: Physical Constants

38.5 Imperial Units
===================

`GSL_CONST_MKSA_INCH'
     The length of 1 inch.

`GSL_CONST_MKSA_FOOT'
     The length of 1 foot.

`GSL_CONST_MKSA_YARD'
     The length of 1 yard.

`GSL_CONST_MKSA_MILE'
     The length of 1 mile.

`GSL_CONST_MKSA_MIL'
     The length of 1 mil (1/1000th of an inch).


File: gsl-ref.info,  Node: Speed and Nautical Units,  Next: Printers Units,  Prev: Imperial Units,  Up: Physical Constants

38.6 Speed and Nautical Units
=============================

`GSL_CONST_MKSA_KILOMETERS_PER_HOUR'
     The speed of 1 kilometer per hour.

`GSL_CONST_MKSA_MILES_PER_HOUR'
     The speed of 1 mile per hour.

`GSL_CONST_MKSA_NAUTICAL_MILE'
     The length of 1 nautical mile.

`GSL_CONST_MKSA_FATHOM'
     The length of 1 fathom.

`GSL_CONST_MKSA_KNOT'
     The speed of 1 knot.


File: gsl-ref.info,  Node: Printers Units,  Next: Volume Area and Length,  Prev: Speed and Nautical Units,  Up: Physical Constants

38.7 Printers Units
===================

`GSL_CONST_MKSA_POINT'
     The length of 1 printer's point (1/72 inch).

`GSL_CONST_MKSA_TEXPOINT'
     The length of 1 TeX point (1/72.27 inch).


File: gsl-ref.info,  Node: Volume Area and Length,  Next: Mass and Weight,  Prev: Printers Units,  Up: Physical Constants

38.8 Volume, Area and Length
============================

`GSL_CONST_MKSA_MICRON'
     The length of 1 micron.

`GSL_CONST_MKSA_HECTARE'
     The area of 1 hectare.

`GSL_CONST_MKSA_ACRE'
     The area of 1 acre.

`GSL_CONST_MKSA_LITER'
     The volume of 1 liter.

`GSL_CONST_MKSA_US_GALLON'
     The volume of 1 US gallon.

`GSL_CONST_MKSA_CANADIAN_GALLON'
     The volume of 1 Canadian gallon.

`GSL_CONST_MKSA_UK_GALLON'
     The volume of 1 UK gallon.

`GSL_CONST_MKSA_QUART'
     The volume of 1 quart.

`GSL_CONST_MKSA_PINT'
     The volume of 1 pint.


File: gsl-ref.info,  Node: Mass and Weight,  Next: Thermal Energy and Power,  Prev: Volume Area and Length,  Up: Physical Constants

38.9 Mass and Weight
====================

`GSL_CONST_MKSA_POUND_MASS'
     The mass of 1 pound.

`GSL_CONST_MKSA_OUNCE_MASS'
     The mass of 1 ounce.

`GSL_CONST_MKSA_TON'
     The mass of 1 ton.

`GSL_CONST_MKSA_METRIC_TON'
     The mass of 1 metric ton (1000 kg).

`GSL_CONST_MKSA_UK_TON'
     The mass of 1 UK ton.

`GSL_CONST_MKSA_TROY_OUNCE'
     The mass of 1 troy ounce.

`GSL_CONST_MKSA_CARAT'
     The mass of 1 carat.

`GSL_CONST_MKSA_GRAM_FORCE'
     The force of 1 gram weight.

`GSL_CONST_MKSA_POUND_FORCE'
     The force of 1 pound weight.

`GSL_CONST_MKSA_KILOPOUND_FORCE'
     The force of 1 kilopound weight.

`GSL_CONST_MKSA_POUNDAL'
     The force of 1 poundal.


File: gsl-ref.info,  Node: Thermal Energy and Power,  Next: Pressure,  Prev: Mass and Weight,  Up: Physical Constants

38.10 Thermal Energy and Power
==============================

`GSL_CONST_MKSA_CALORIE'
     The energy of 1 calorie.

`GSL_CONST_MKSA_BTU'
     The energy of 1 British Thermal Unit, btu.

`GSL_CONST_MKSA_THERM'
     The energy of 1 Therm.

`GSL_CONST_MKSA_HORSEPOWER'
     The power of 1 horsepower.


File: gsl-ref.info,  Node: Pressure,  Next: Viscosity,  Prev: Thermal Energy and Power,  Up: Physical Constants

38.11 Pressure
==============

`GSL_CONST_MKSA_BAR'
     The pressure of 1 bar.

`GSL_CONST_MKSA_STD_ATMOSPHERE'
     The pressure of 1 standard atmosphere.

`GSL_CONST_MKSA_TORR'
     The pressure of 1 torr.

`GSL_CONST_MKSA_METER_OF_MERCURY'
     The pressure of 1 meter of mercury.

`GSL_CONST_MKSA_INCH_OF_MERCURY'
     The pressure of 1 inch of mercury.

`GSL_CONST_MKSA_INCH_OF_WATER'
     The pressure of 1 inch of water.

`GSL_CONST_MKSA_PSI'
     The pressure of 1 pound per square inch.


File: gsl-ref.info,  Node: Viscosity,  Next: Light and Illumination,  Prev: Pressure,  Up: Physical Constants

38.12 Viscosity
===============

`GSL_CONST_MKSA_POISE'
     The dynamic viscosity of 1 poise.

`GSL_CONST_MKSA_STOKES'
     The kinematic viscosity of 1 stokes.


File: gsl-ref.info,  Node: Light and Illumination,  Next: Radioactivity,  Prev: Viscosity,  Up: Physical Constants

38.13 Light and Illumination
============================

`GSL_CONST_MKSA_STILB'
     The luminance of 1 stilb.

`GSL_CONST_MKSA_LUMEN'
     The luminous flux of 1 lumen.

`GSL_CONST_MKSA_LUX'
     The illuminance of 1 lux.

`GSL_CONST_MKSA_PHOT'
     The illuminance of 1 phot.

`GSL_CONST_MKSA_FOOTCANDLE'
     The illuminance of 1 footcandle.

`GSL_CONST_MKSA_LAMBERT'
     The luminance of 1 lambert.

`GSL_CONST_MKSA_FOOTLAMBERT'
     The luminance of 1 footlambert.


File: gsl-ref.info,  Node: Radioactivity,  Next: Force and Energy,  Prev: Light and Illumination,  Up: Physical Constants

38.14 Radioactivity
===================

`GSL_CONST_MKSA_CURIE'
     The activity of 1 curie.

`GSL_CONST_MKSA_ROENTGEN'
     The exposure of 1 roentgen.

`GSL_CONST_MKSA_RAD'
     The absorbed dose of 1 rad.


File: gsl-ref.info,  Node: Force and Energy,  Next: Prefixes,  Prev: Radioactivity,  Up: Physical Constants

38.15 Force and Energy
======================

`GSL_CONST_MKSA_NEWTON'
     The SI unit of force, 1 Newton.

`GSL_CONST_MKSA_DYNE'
     The force of 1 Dyne = 10^-5 Newton.

`GSL_CONST_MKSA_JOULE'
     The SI unit of energy, 1 Joule.

`GSL_CONST_MKSA_ERG'
     The energy 1 erg = 10^-7 Joule.


File: gsl-ref.info,  Node: Prefixes,  Next: Physical Constant Examples,  Prev: Force and Energy,  Up: Physical Constants

38.16 Prefixes
==============

These constants are dimensionless scaling factors.

`GSL_CONST_NUM_YOTTA'
     10^24

`GSL_CONST_NUM_ZETTA'
     10^21

`GSL_CONST_NUM_EXA'
     10^18

`GSL_CONST_NUM_PETA'
     10^15

`GSL_CONST_NUM_TERA'
     10^12

`GSL_CONST_NUM_GIGA'
     10^9

`GSL_CONST_NUM_MEGA'
     10^6

`GSL_CONST_NUM_KILO'
     10^3

`GSL_CONST_NUM_MILLI'
     10^-3

`GSL_CONST_NUM_MICRO'
     10^-6

`GSL_CONST_NUM_NANO'
     10^-9

`GSL_CONST_NUM_PICO'
     10^-12

`GSL_CONST_NUM_FEMTO'
     10^-15

`GSL_CONST_NUM_ATTO'
     10^-18

`GSL_CONST_NUM_ZEPTO'
     10^-21

`GSL_CONST_NUM_YOCTO'
     10^-24


File: gsl-ref.info,  Node: Physical Constant Examples,  Next: Physical Constant References and Further Reading,  Prev: Prefixes,  Up: Physical Constants

38.17 Examples
==============

The following program demonstrates the use of the physical constants in
a calculation.  In this case, the goal is to calculate the range of
light-travel times from Earth to Mars.

   The required data is the average distance of each planet from the
Sun in astronomical units (the eccentricities and inclinations of the
orbits will be neglected for the purposes of this calculation).  The
average radius of the orbit of Mars is 1.52 astronomical units, and for
the orbit of Earth it is 1 astronomical unit (by definition).  These
values are combined with the MKSA values of the constants for the speed
of light and the length of an astronomical unit to produce a result for
the shortest and longest light-travel times in seconds.  The figures are
converted into minutes before being displayed.

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

     int
     main (void)
     {
       double c  = GSL_CONST_MKSA_SPEED_OF_LIGHT;
       double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT;
       double minutes = GSL_CONST_MKSA_MINUTE;

       /* distance stored in meters */
       double r_earth = 1.00 * au;
       double r_mars  = 1.52 * au;

       double t_min, t_max;

       t_min = (r_mars - r_earth) / c;
       t_max = (r_mars + r_earth) / c;

       printf ("light travel time from Earth to Mars:\n");
       printf ("minimum = %.1f minutes\n", t_min / minutes);
       printf ("maximum = %.1f minutes\n", t_max / minutes);

       return 0;
     }

Here is the output from the program,

     light travel time from Earth to Mars:
     minimum = 4.3 minutes
     maximum = 21.0 minutes


File: gsl-ref.info,  Node: Physical Constant References and Further Reading,  Prev: Physical Constant Examples,  Up: Physical Constants

38.18 References and Further Reading
====================================

The authoritative sources for physical constanst are the 2002 CODATA
recommended values, published in the articles below. Further information
on the values of physical constants is also available from the cited
articles and the NIST website.

     Journal of Physical and Chemical Reference Data, 28(6), 1713-1852,
     1999

     Reviews of Modern Physics, 72(2), 351-495, 2000

     `http://www.physics.nist.gov/cuu/Constants/index.html'

     `http://physics.nist.gov/Pubs/SP811/appenB9.html'


File: gsl-ref.info,  Node: IEEE floating-point arithmetic,  Next: Debugging Numerical Programs,  Prev: Physical Constants,  Up: Top

39 IEEE floating-point arithmetic
*********************************

This chapter describes functions for examining the representation of
floating point numbers and controlling the floating point environment of
your program.  The functions described in this chapter are declared in
the header file `gsl_ieee_utils.h'.

* Menu:

* Representation of floating point numbers::
* Setting up your IEEE environment::
* IEEE References and Further Reading::


File: gsl-ref.info,  Node: Representation of floating point numbers,  Next: Setting up your IEEE environment,  Up: IEEE floating-point arithmetic

39.1 Representation of floating point numbers
=============================================

The IEEE Standard for Binary Floating-Point Arithmetic defines binary
formats for single and double precision numbers.  Each number is
composed of three parts: a "sign bit" (s), an "exponent" (E) and a
"fraction" (f).  The numerical value of the combination (s,E,f) is
given by the following formula,

     (-1)^s (1.fffff...) 2^E

The sign bit is either zero or one.  The exponent ranges from a minimum
value E_min to a maximum value E_max depending on the precision.  The
exponent is converted to an unsigned number e, known as the "biased
exponent", for storage by adding a "bias" parameter, e = E + bias.  The
sequence fffff... represents the digits of the binary fraction f.  The
binary digits are stored in "normalized form", by adjusting the
exponent to give a leading digit of 1.  Since the leading digit is
always 1 for normalized numbers it is assumed implicitly and does not
have to be stored.  Numbers smaller than 2^(E_min) are be stored in
"denormalized form" with a leading zero,

     (-1)^s (0.fffff...) 2^(E_min)

This allows gradual underflow down to 2^(E_min - p) for p bits of
precision.  A zero is encoded with the special exponent of 2^(E_min -
1) and infinities with the exponent of 2^(E_max + 1).

The format for single precision numbers uses 32 bits divided in the
following way,

     seeeeeeeefffffffffffffffffffffff

     s = sign bit, 1 bit
     e = exponent, 8 bits  (E_min=-126, E_max=127, bias=127)
     f = fraction, 23 bits

The format for double precision numbers uses 64 bits divided in the
following way,

     seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff

     s = sign bit, 1 bit
     e = exponent, 11 bits  (E_min=-1022, E_max=1023, bias=1023)
     f = fraction, 52 bits

It is often useful to be able to investigate the behavior of a
calculation at the bit-level and the library provides functions for
printing the IEEE representations in a human-readable form.

 -- Function: void gsl_ieee_fprintf_float (FILE * STREAM, const float *
          X)
 -- Function: void gsl_ieee_fprintf_double (FILE * STREAM, const double
          * X)
     These functions output a formatted version of the IEEE
     floating-point number pointed to by X to the stream STREAM. A
     pointer is used to pass the number indirectly, to avoid any
     undesired promotion from `float' to `double'.  The output takes
     one of the following forms,

    `NaN'
          the Not-a-Number symbol

    `Inf, -Inf'
          positive or negative infinity

    `1.fffff...*2^E, -1.fffff...*2^E'
          a normalized floating point number

    `0.fffff...*2^E, -0.fffff...*2^E'
          a denormalized floating point number

    `0, -0'
          positive or negative zero


     The output can be used directly in GNU Emacs Calc mode by
     preceding it with `2#' to indicate binary.

 -- Function: void gsl_ieee_printf_float (const float * X)
 -- Function: void gsl_ieee_printf_double (const double * X)
     These functions output a formatted version of the IEEE
     floating-point number pointed to by X to the stream `stdout'.

The following program demonstrates the use of the functions by printing
the single and double precision representations of the fraction 1/3.
For comparison the representation of the value promoted from single to
double precision is also printed.

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

     int
     main (void)
     {
       float f = 1.0/3.0;
       double d = 1.0/3.0;

       double fd = f; /* promote from float to double */

       printf (" f="); gsl_ieee_printf_float(&f);
       printf ("\n");

       printf ("fd="); gsl_ieee_printf_double(&fd);
       printf ("\n");

       printf (" d="); gsl_ieee_printf_double(&d);
       printf ("\n");

       return 0;
     }

The binary representation of 1/3 is 0.01010101... .  The output below
shows that the IEEE format normalizes this fraction to give a leading
digit of 1,

      f= 1.01010101010101010101011*2^-2
     fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
      d= 1.0101010101010101010101010101010101010101010101010101*2^-2

The output also shows that a single-precision number is promoted to
double-precision by adding zeros in the binary representation.


File: gsl-ref.info,  Node: Setting up your IEEE environment,  Next: IEEE References and Further Reading,  Prev: Representation of floating point numbers,  Up: IEEE floating-point arithmetic

39.2 Setting up your IEEE environment
=====================================

The IEEE standard defines several "modes" for controlling the behavior
of floating point operations.  These modes specify the important
properties of computer arithmetic: the direction used for rounding (e.g.
whether numbers should be rounded up, down or to the nearest number),
the rounding precision and how the program should handle arithmetic
exceptions, such as division by zero.

   Many of these features can now be controlled via standard functions
such as `fpsetround', which should be used whenever they are available.
Unfortunately in the past there has been no universal API for
controlling their behavior--each system has had its own low-level way
of accessing them.  To help you write portable programs GSL allows you
to specify modes in a platform-independent way using the environment
variable `GSL_IEEE_MODE'.  The library then takes care of all the
necessary machine-specific initializations for you when you call the
function `gsl_ieee_env_setup'.

 -- Function: void gsl_ieee_env_setup ()
     This function reads the environment variable `GSL_IEEE_MODE' and
     attempts to set up the corresponding specified IEEE modes.  The
     environment variable should be a list of keywords, separated by
     commas, like this,

          `GSL_IEEE_MODE' = "KEYWORD,KEYWORD,..."

     where KEYWORD is one of the following mode-names,

          `single-precision'

          `double-precision'

          `extended-precision'

          `round-to-nearest'

          `round-down'

          `round-up'

          `round-to-zero'

          `mask-all'

          `mask-invalid'

          `mask-denormalized'

          `mask-division-by-zero'

          `mask-overflow'

          `mask-underflow'

          `trap-inexact'

          `trap-common'

     If `GSL_IEEE_MODE' is empty or undefined then the function returns
     immediately and no attempt is made to change the system's IEEE
     mode.  When the modes from `GSL_IEEE_MODE' are turned on the
     function prints a short message showing the new settings to remind
     you that the results of the program will be affected.

     If the requested modes are not supported by the platform being
     used then the function calls the error handler and returns an
     error code of `GSL_EUNSUP'.

     When options are specified using this method, the resulting mode is
     based on a default setting of the highest available precision
     (double precision or extended precision, depending on the
     platform) in round-to-nearest mode, with all exceptions enabled
     apart from the INEXACT exception.  The INEXACT exception is
     generated whenever rounding occurs, so it must generally be
     disabled in typical scientific calculations.  All other
     floating-point exceptions are enabled by default, including
     underflows and the use of denormalized numbers, for safety.  They
     can be disabled with the individual `mask-' settings or together
     using `mask-all'.

     The following adjusted combination of modes is convenient for many
     purposes,

          GSL_IEEE_MODE="double-precision,"\
                          "mask-underflow,"\
                            "mask-denormalized"

     This choice ignores any errors relating to small numbers (either
     denormalized, or underflowing to zero) but traps overflows,
     division by zero and invalid operations.

To demonstrate the effects of different rounding modes consider the
following program which computes e, the base of natural logarithms, by
summing a rapidly-decreasing series,

     e = 1 + 1/2! + 1/3! + 1/4! + ...
       = 2.71828182846...

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

     int
     main (void)
     {
       double x = 1, oldsum = 0, sum = 0;
       int i = 0;

       gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */

       do
         {
           i++;

           oldsum = sum;
           sum += x;
           x = x / i;

           printf ("i=%2d sum=%.18f error=%g\n",
                   i, sum, sum - M_E);

           if (i > 30)
              break;
         }
       while (sum != oldsum);

       return 0;
     }

Here are the results of running the program in `round-to-nearest' mode.
This is the IEEE default so it isn't really necessary to specify it
here,

     $ GSL_IEEE_MODE="round-to-nearest" ./a.out
     i= 1 sum=1.000000000000000000 error=-1.71828
     i= 2 sum=2.000000000000000000 error=-0.718282
     ....
     i=18 sum=2.718281828459045535 error=4.44089e-16
     i=19 sum=2.718281828459045535 error=4.44089e-16

After nineteen terms the sum converges to within 4 \times 10^-16 of the
correct value.  If we now change the rounding mode to `round-down' the
final result is less accurate,

     $ GSL_IEEE_MODE="round-down" ./a.out
     i= 1 sum=1.000000000000000000 error=-1.71828
     ....
     i=19 sum=2.718281828459041094 error=-3.9968e-15

The result is about 4 \times 10^-15 below the correct value, an order
of magnitude worse than the result obtained in the `round-to-nearest'
mode.

   If we change to rounding mode to `round-up' then the series no
longer converges (the reason is that when we add each term to the sum
the final result is always rounded up.  This is guaranteed to increase
the sum by at least one tick on each iteration).  To avoid this problem
we would need to use a safer converge criterion, such as `while
(fabs(sum - oldsum) > epsilon)', with a suitably chosen value of
epsilon.

   Finally we can see the effect of computing the sum using
single-precision rounding, in the default `round-to-nearest' mode.  In
this case the program thinks it is still using double precision numbers
but the CPU rounds the result of each floating point operation to
single-precision accuracy.  This simulates the effect of writing the
program using single-precision `float' variables instead of `double'
variables.  The iteration stops after about half the number of
iterations and the final result is much less accurate,

     $ GSL_IEEE_MODE="single-precision" ./a.out
     ....
     i=12 sum=2.718281984329223633 error=1.5587e-07

with an error of O(10^-7), which corresponds to single precision
accuracy (about 1 part in 10^7).  Continuing the iterations further
does not decrease the error because all the subsequent results are
rounded to the same value.


File: gsl-ref.info,  Node: IEEE References and Further Reading,  Prev: Setting up your IEEE environment,  Up: IEEE floating-point arithmetic

39.3 References and Further Reading
===================================

The reference for the IEEE standard is,

     ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point
     Arithmetic.

A more pedagogical introduction to the standard can be found in the
following paper,

     David Goldberg: What Every Computer Scientist Should Know About
     Floating-Point Arithmetic. `ACM Computing Surveys', Vol. 23, No. 1
     (March 1991), pages 5-48.

     Corrigendum: `ACM Computing Surveys', Vol. 23, No. 3 (September
     1991), page 413. and see also the sections by B. A. Wichmann and
     Charles B. Dunham in Surveyor's Forum: "What Every Computer
     Scientist Should Know About Floating-Point Arithmetic". `ACM
     Computing Surveys', Vol. 24, No. 3 (September 1992), page 319.

A detailed textbook on IEEE arithmetic and its practical use is
available from SIAM Press,

     Michael L. Overton, `Numerical Computing with IEEE Floating Point
     Arithmetic', SIAM Press, ISBN 0898715717.



File: gsl-ref.info,  Node: Debugging Numerical Programs,  Next: Contributors to GSL,  Prev: IEEE floating-point arithmetic,  Up: Top

Appendix A Debugging Numerical Programs
***************************************

This chapter describes some tips and tricks for debugging numerical
programs which use GSL.

* Menu:

* Using gdb::
* Examining floating point registers::
* Handling floating point exceptions::
* GCC warning options for numerical programs::
* Debugging References::


File: gsl-ref.info,  Node: Using gdb,  Next: Examining floating point registers,  Up: Debugging Numerical Programs

A.1 Using gdb
=============

Any errors reported by the library are passed to the function
`gsl_error'.  By running your programs under gdb and setting a
breakpoint in this function you can automatically catch any library
errors.  You can add a breakpoint for every session by putting

     break gsl_error

into your `.gdbinit' file in the directory where your program is
started.

   If the breakpoint catches an error then you can use a backtrace
(`bt') to see the call-tree, and the arguments which possibly caused
the error.  By moving up into the calling function you can investigate
the values of variables at that point.  Here is an example from the
program `fft/test_trap', which contains the following line,

     status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);

The function `gsl_fft_complex_wavetable_alloc' takes the length of an
FFT as its first argument.  When this line is executed an error will be
generated because the length of an FFT is not allowed to be zero.

   To debug this problem we start `gdb', using the file `.gdbinit' to
define a breakpoint in `gsl_error',

     $ gdb test_trap

     GDB is free software and you are welcome to distribute copies
     of it under certain conditions; type "show copying" to see
     the conditions.  There is absolutely no warranty for GDB;
     type "show warranty" for details.  GDB 4.16 (i586-debian-linux),
     Copyright 1996 Free Software Foundation, Inc.

     Breakpoint 1 at 0x8050b1e: file error.c, line 14.

When we run the program this breakpoint catches the error and shows the
reason for it.

     (gdb) run
     Starting program: test_trap

     Breakpoint 1, gsl_error (reason=0x8052b0d
         "length n must be positive integer",
         file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
         at error.c:14
     14        if (gsl_error_handler)

The first argument of `gsl_error' is always a string describing the
error.  Now we can look at the backtrace to see what caused the problem,

     (gdb) bt
     #0  gsl_error (reason=0x8052b0d
         "length n must be positive integer",
         file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
         at error.c:14
     #1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
         wavetable=0xbffff778) at c_init.c:108
     #2  0x8048a00 in main (argc=1, argv=0xbffff9bc)
         at test_trap.c:94
     #3  0x80488be in ___crt_dummy__ ()

We can see that the error was generated in the function
`gsl_fft_complex_wavetable_alloc' when it was called with an argument
of N=0.  The original call came from line 94 in the file `test_trap.c'.

   By moving up to the level of the original call we can find the line
that caused the error,

     (gdb) up
     #1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
         wavetable=0xbffff778) at c_init.c:108
     108   GSL_ERROR ("length n must be positive integer", GSL_EDOM);
     (gdb) up
     #2  0x8048a00 in main (argc=1, argv=0xbffff9bc)
         at test_trap.c:94
     94    status = gsl_fft_complex_wavetable_alloc (0,
             &complex_wavetable);

Thus we have found the line that caused the problem.  From this point we
could also print out the values of other variables such as
`complex_wavetable'.


File: gsl-ref.info,  Node: Examining floating point registers,  Next: Handling floating point exceptions,  Prev: Using gdb,  Up: Debugging Numerical Programs

A.2 Examining floating point registers
======================================

The contents of floating point registers can be examined using the
command `info float' (on supported platforms).

     (gdb) info float
          st0: 0xc4018b895aa17a945000  Valid Normal -7.838871e+308
          st1: 0x3ff9ea3f50e4d7275000  Valid Normal 0.0285946
          st2: 0x3fe790c64ce27dad4800  Valid Normal 6.7415931e-08
          st3: 0x3ffaa3ef0df6607d7800  Spec  Normal 0.0400229
          st4: 0x3c028000000000000000  Valid Normal 4.4501477e-308
          st5: 0x3ffef5412c22219d9000  Zero  Normal 0.9580257
          st6: 0x3fff8000000000000000  Valid Normal 1
          st7: 0xc4028b65a1f6d243c800  Valid Normal -1.566206e+309
        fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
        fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS
         ftag: 0x3fff
          fip: 0x08048b5c
          fcs: 0x051a0023
       fopoff: 0x08086820
       fopsel: 0x002b

Individual registers can be examined using the variables $REG, where
REG is the register name.

     (gdb) p $st1
     $1 = 0.02859464454261210347719


File: gsl-ref.info,  Node: Handling floating point exceptions,  Next: GCC warning options for numerical programs,  Prev: Examining floating point registers,  Up: Debugging Numerical Programs

A.3 Handling floating point exceptions
======================================

It is possible to stop the program whenever a `SIGFPE' floating point
exception occurs.  This can be useful for finding the cause of an
unexpected infinity or `NaN'.  The current handler settings can be
shown with the command `info signal SIGFPE'.

     (gdb) info signal SIGFPE
     Signal  Stop  Print  Pass to program Description
     SIGFPE  Yes   Yes    Yes             Arithmetic exception

Unless the program uses a signal handler the default setting should be
changed so that SIGFPE is not passed to the program, as this would cause
it to exit.  The command `handle SIGFPE stop nopass' prevents this.

     (gdb) handle SIGFPE stop nopass
     Signal  Stop  Print  Pass to program Description
     SIGFPE  Yes   Yes    No              Arithmetic exception

Depending on the platform it may be necessary to instruct the kernel to
generate signals for floating point exceptions.  For programs using GSL
this can be achieved using the `GSL_IEEE_MODE' environment variable in
conjunction with the function `gsl_ieee_env_setup()' as described in
*note IEEE floating-point arithmetic::.

     (gdb) set env GSL_IEEE_MODE=double-precision


File: gsl-ref.info,  Node: GCC warning options for numerical programs,  Next: Debugging References,  Prev: Handling floating point exceptions,  Up: Debugging Numerical Programs

A.4 GCC warning options for numerical programs
==============================================

Writing reliable numerical programs in C requires great care.  The
following GCC warning options are recommended when compiling numerical
programs:

     gcc -ansi -pedantic -Werror -Wall -W
       -Wmissing-prototypes -Wstrict-prototypes
       -Wtraditional -Wconversion -Wshadow
       -Wpointer-arith -Wcast-qual -Wcast-align
       -Wwrite-strings -Wnested-externs
       -fshort-enums -fno-common -Dinline= -g -O4

For details of each option consult the manual `Using and Porting GCC'.
The following table gives a brief explanation of what types of errors
these options catch.

`-ansi -pedantic'
     Use ANSI C, and reject any non-ANSI extensions.  These flags help
     in writing portable programs that will compile on other systems.

`-Werror'
     Consider warnings to be errors, so that compilation stops.  This
     prevents warnings from scrolling off the top of the screen and
     being lost.  You won't be able to compile the program until it is
     completely warning-free.

`-Wall'
     This turns on a set of warnings for common programming problems.
     You need `-Wall', but it is not enough on its own.

`-O2'
     Turn on optimization.  The warnings for uninitialized variables in
     `-Wall' rely on the optimizer to analyze the code.  If there is no
     optimization then these warnings aren't generated.

`-W'
     This turns on some extra warnings not included in `-Wall', such as
     missing return values and comparisons between signed and unsigned
     integers.

`-Wmissing-prototypes -Wstrict-prototypes'
     Warn if there are any missing or inconsistent prototypes.  Without
     prototypes it is harder to detect problems with incorrect
     arguments.

`-Wtraditional'
     This warns about certain constructs that behave differently in
     traditional and ANSI C. Whether the traditional or ANSI
     interpretation is used might be unpredictable on other compilers.

`-Wconversion'
     The main use of this option is to warn about conversions from
     signed to unsigned integers.  For example, `unsigned int x = -1'.
     If you need to perform such a conversion you can use an explicit
     cast.

`-Wshadow'
     This warns whenever a local variable shadows another local
     variable.  If two variables have the same name then it is a
     potential source of confusion.

`-Wpointer-arith -Wcast-qual -Wcast-align'
     These options warn if you try to do pointer arithmetic for types
     which don't have a size, such as `void', if you remove a `const'
     cast from a pointer, or if you cast a pointer to a type which has a
     different size, causing an invalid alignment.

`-Wwrite-strings'
     This option gives string constants a `const' qualifier so that it
     will be a compile-time error to attempt to overwrite them.

`-fshort-enums'
     This option makes the type of `enum' as short as possible.
     Normally this makes an `enum' different from an `int'.
     Consequently any attempts to assign a pointer-to-int to a
     pointer-to-enum will generate a cast-alignment warning.

`-fno-common'
     This option prevents global variables being simultaneously defined
     in different object files (you get an error at link time).  Such a
     variable should be defined in one file and referred to in other
     files with an `extern' declaration.

`-Wnested-externs'
     This warns if an `extern' declaration is encountered within a
     function.

`-Dinline='
     The `inline' keyword is not part of ANSI C. Thus if you want to use
     `-ansi' with a program which uses inline functions you can use this
     preprocessor definition to remove the `inline' keywords.

`-g'
     It always makes sense to put debugging symbols in the executable
     so that you can debug it using `gdb'.  The only effect of
     debugging symbols is to increase the size of the file, and you can
     use the `strip' command to remove them later if necessary.


File: gsl-ref.info,  Node: Debugging References,  Prev: GCC warning options for numerical programs,  Up: Debugging Numerical Programs

A.5 References and Further Reading
==================================

The following books are essential reading for anyone writing and
debugging numerical programs with GCC and GDB.

     R.M. Stallman, `Using and Porting GNU CC', Free Software
     Foundation, ISBN 1882114388

     R.M. Stallman, R.H. Pesch, `Debugging with GDB: The GNU
     Source-Level Debugger', Free Software Foundation, ISBN 1882114779

For a tutorial introduction to the GNU C Compiler and related programs,
see

     B.J. Gough, `An Introduction to GCC', Network Theory Ltd, ISBN
     0954161793


File: gsl-ref.info,  Node: Contributors to GSL,  Next: Autoconf Macros,  Prev: Debugging Numerical Programs,  Up: Top

Appendix B Contributors to GSL
******************************

(See the AUTHORS file in the distribution for up-to-date information.)

*Mark Galassi*
     Conceived GSL (with James Theiler) and wrote the design document.
     Wrote the simulated annealing package and the relevant chapter in
     the manual.

*James Theiler*
     Conceived GSL (with Mark Galassi).  Wrote the random number
     generators and the relevant chapter in this manual.

*Jim Davies*
     Wrote the statistical routines and the relevant chapter in this
     manual.

*Brian Gough*
     FFTs, numerical integration, random number generators and
     distributions, root finding, minimization and fitting, polynomial
     solvers, complex numbers, physical constants, permutations, vector
     and matrix functions, histograms, statistics, ieee-utils, revised
     CBLAS Level 2 & 3, matrix decompositions, eigensystems, cumulative
     distribution functions, testing, documentation and releases.

*Reid Priedhorsky*
     Wrote and documented the initial version of the root finding
     routines while at Los Alamos National Laboratory, Mathematical
     Modeling and Analysis Group.

*Gerard Jungman*
     Special Functions, Series acceleration, ODEs, BLAS, Linear Algebra,
     Eigensystems, Hankel Transforms.

*Mike Booth*
     Wrote the Monte Carlo library.

*Jorma Olavi Ta"htinen*
     Wrote the initial complex arithmetic functions.

*Thomas Walter*
     Wrote the initial heapsort routines and cholesky decomposition.

*Fabrice Rossi*
     Multidimensional minimization.

*Carlo Perassi*
     Implementation of the random number generators in Knuth's
     `Seminumerical Algorithms', 3rd Ed.

*Szymon Jaroszewicz*
     Wrote the routines for generating combinations.

*Nicolas Darnis*
     Wrote the initial routines for canonical permutations.

*Jason H. Stover*
     Wrote the major cumulative distribution functions.

*Ivo Alxneit*
     Wrote the routines for wavelet transforms.

*Tuomo Keskitalo*
     Improved the implementation of the ODE solvers.


   Thanks to Nigel Lowry for help in proofreading the manual.


File: gsl-ref.info,  Node: Autoconf Macros,  Next: GSL CBLAS Library,  Prev: Contributors to GSL,  Up: Top

Appendix C Autoconf Macros
**************************

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

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

It is important to check for `libm' and `libgslcblas' before `libgsl',
otherwise the tests will fail.  Assuming the libraries are found the
output during the configure stage looks like this,

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

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

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

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

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

     AC_MSG_ERROR(could not find required version of GSL)

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

     libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas

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

   To test for `inline' the following test should be placed in your
`configure.in' file,

     AC_C_INLINE

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

and the macro will then be defined in the compilation flags or by
including the file `config.h' before any library headers.

   The following autoconf test will check for `extern inline',

     dnl Check for "extern inline", using a modified version
     dnl of the test for AC_C_INLINE from acspecific.mt
     dnl
     AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,
     [ac_cv_c_extern_inline=no
     AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
     extern $ac_cv_c_inline double foo(double x) { return x+1.0; };
     double foo (double x) { return x + 1.0; };],
     [  foo(1.0)  ],
     [ac_cv_c_extern_inline="yes"])
     ])

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

   The substitution of portability functions can be made automatically
if you use `autoconf'. For example, to test whether the BSD function
`hypot' is available you can include the following line in the
configure file `configure.in' for your application,

     AC_CHECK_FUNCS(hypot)

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

     /* Substitute gsl_hypot for missing system hypot */

     #ifndef HAVE_HYPOT
     #define hypot gsl_hypot
     #endif

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


File: gsl-ref.info,  Node: GSL CBLAS Library,  Next: Free Software Needs Free Documentation,  Prev: Autoconf Macros,  Up: Top

Appendix D GSL CBLAS Library
****************************

The prototypes for the low-level CBLAS functions are declared in the
file `gsl_cblas.h'.  For the definition of the functions consult the
documentation available from Netlib (*note BLAS References and Further
Reading::).

* Menu:

* Level 1 CBLAS Functions::
* Level 2 CBLAS Functions::
* Level 3 CBLAS Functions::
* GSL CBLAS Examples::


File: gsl-ref.info,  Node: Level 1 CBLAS Functions,  Next: Level 2 CBLAS Functions,  Up: GSL CBLAS Library

D.1 Level 1
===========

 -- Function: float cblas_sdsdot (const int N, const float ALPHA, const
          float * X, const int INCX, const float * Y, const int INCY)

 -- Function: double cblas_dsdot (const int N, const float * X, const
          int INCX, const float * Y, const int INCY)

 -- Function: float cblas_sdot (const int N, const float * X, const int
          INCX, const float * Y, const int INCY)

 -- Function: double cblas_ddot (const int N, const double * X, const
          int INCX, const double * Y, const int INCY)

 -- Function: void cblas_cdotu_sub (const int N, const void * X, const
          int INCX, const void * Y, const int INCY, void * DOTU)

 -- Function: void cblas_cdotc_sub (const int N, const void * X, const
          int INCX, const void * Y, const int INCY, void * DOTC)

 -- Function: void cblas_zdotu_sub (const int N, const void * X, const
          int INCX, const void * Y, const int INCY, void * DOTU)

 -- Function: void cblas_zdotc_sub (const int N, const void * X, const
          int INCX, const void * Y, const int INCY, void * DOTC)

 -- Function: float cblas_snrm2 (const int N, const float * X, const
          int INCX)

 -- Function: float cblas_sasum (const int N, const float * X, const
          int INCX)

 -- Function: double cblas_dnrm2 (const int N, const double * X, const
          int INCX)

 -- Function: double cblas_dasum (const int N, const double * X, const
          int INCX)

 -- Function: float cblas_scnrm2 (const int N, const void * X, const
          int INCX)

 -- Function: float cblas_scasum (const int N, const void * X, const
          int INCX)

 -- Function: double cblas_dznrm2 (const int N, const void * X, const
          int INCX)

 -- Function: double cblas_dzasum (const int N, const void * X, const
          int INCX)

 -- Function: CBLAS_INDEX cblas_isamax (const int N, const float * X,
          const int INCX)

 -- Function: CBLAS_INDEX cblas_idamax (const int N, const double * X,
          const int INCX)

 -- Function: CBLAS_INDEX cblas_icamax (const int N, const void * X,
          const int INCX)

 -- Function: CBLAS_INDEX cblas_izamax (const int N, const void * X,
          const int INCX)

 -- Function: void cblas_sswap (const int N, float * X, const int INCX,
          float * Y, const int INCY)

 -- Function: void cblas_scopy (const int N, const float * X, const int
          INCX, float * Y, const int INCY)

 -- Function: void cblas_saxpy (const int N, const float ALPHA, const
          float * X, const int INCX, float * Y, const int INCY)

 -- Function: void cblas_dswap (const int N, double * X, const int
          INCX, double * Y, const int INCY)

 -- Function: void cblas_dcopy (const int N, const double * X, const
          int INCX, double * Y, const int INCY)

 -- Function: void cblas_daxpy (const int N, const double ALPHA, const
          double * X, const int INCX, double * Y, const int INCY)

 -- Function: void cblas_cswap (const int N, void * X, const int INCX,
          void * Y, const int INCY)

 -- Function: void cblas_ccopy (const int N, const void * X, const int
          INCX, void * Y, const int INCY)

 -- Function: void cblas_caxpy (const int N, const void * ALPHA, const
          void * X, const int INCX, void * Y, const int INCY)

 -- Function: void cblas_zswap (const int N, void * X, const int INCX,
          void * Y, const int INCY)

 -- Function: void cblas_zcopy (const int N, const void * X, const int
          INCX, void * Y, const int INCY)

 -- Function: void cblas_zaxpy (const int N, const void * ALPHA, const
          void * X, const int INCX, void * Y, const int INCY)

 -- Function: void cblas_srotg (float * A, float * B, float * C, float
          * S)

 -- Function: void cblas_srotmg (float * D1, float * D2, float * B1,
          const float B2, float * P)

 -- Function: void cblas_srot (const int N, float * X, const int INCX,
          float * Y, const int INCY, const float C, const float S)

 -- Function: void cblas_srotm (const int N, float * X, const int INCX,
          float * Y, const int INCY, const float * P)

 -- Function: void cblas_drotg (double * A, double * B, double * C,
          double * S)

 -- Function: void cblas_drotmg (double * D1, double * D2, double * B1,
          const double B2, double * P)

 -- Function: void cblas_drot (const int N, double * X, const int INCX,
          double * Y, const int INCY, const double C, const double S)

 -- Function: void cblas_drotm (const int N, double * X, const int
          INCX, double * Y, const int INCY, const double * P)

 -- Function: void cblas_sscal (const int N, const float ALPHA, float *
          X, const int INCX)

 -- Function: void cblas_dscal (const int N, const double ALPHA, double
          * X, const int INCX)

 -- Function: void cblas_cscal (const int N, const void * ALPHA, void *
          X, const int INCX)

 -- Function: void cblas_zscal (const int N, const void * ALPHA, void *
          X, const int INCX)

 -- Function: void cblas_csscal (const int N, const float ALPHA, void *
          X, const int INCX)

 -- Function: void cblas_zdscal (const int N, const double ALPHA, void
          * X, const int INCX)


File: gsl-ref.info,  Node: Level 2 CBLAS Functions,  Next: Level 3 CBLAS Functions,  Prev: Level 1 CBLAS Functions,  Up: GSL CBLAS Library

D.2 Level 2
===========

 -- Function: void cblas_sgemv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
          float ALPHA, const float * A, const int LDA, const float * X,
          const int INCX, const float BETA, float * Y, const int INCY)

 -- Function: void cblas_sgbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
          int KL, const int KU, const float ALPHA, const float * A,
          const int LDA, const float * X, const int INCX, const float
          BETA, float * Y, const int INCY)

 -- Function: void cblas_strmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const float * A,
          const int LDA, float * X, const int INCX)

 -- Function: void cblas_stbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const int K, const
          float * A, const int LDA, float * X, const int INCX)

 -- Function: void cblas_stpmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const float * AP,
          float * X, const int INCX)

 -- Function: void cblas_strsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const float * A,
          const int LDA, float * X, const int INCX)

 -- Function: void cblas_stbsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const int K, const
          float * A, const int LDA, float * X, const int INCX)

 -- Function: void cblas_stpsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const float * AP,
          float * X, const int INCX)

 -- Function: void cblas_dgemv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
          double ALPHA, const double * A, const int LDA, const double *
          X, const int INCX, const double BETA, double * Y, const int
          INCY)

 -- Function: void cblas_dgbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
          int KL, const int KU, const double ALPHA, const double * A,
          const int LDA, const double * X, const int INCX, const double
          BETA, double * Y, const int INCY)

 -- Function: void cblas_dtrmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const double * A,
          const int LDA, double * X, const int INCX)

 -- Function: void cblas_dtbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const int K, const
          double * A, const int LDA, double * X, const int INCX)

 -- Function: void cblas_dtpmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const double * AP,
          double * X, const int INCX)

 -- Function: void cblas_dtrsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const double * A,
          const int LDA, double * X, const int INCX)

 -- Function: void cblas_dtbsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const int K, const
          double * A, const int LDA, double * X, const int INCX)

 -- Function: void cblas_dtpsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const double * AP,
          double * X, const int INCX)

 -- Function: void cblas_cgemv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
          void * ALPHA, const void * A, const int LDA, const void * X,
          const int INCX, const void * BETA, void * Y, const int INCY)

 -- Function: void cblas_cgbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
          int KL, const int KU, const void * ALPHA, const void * A,
          const int LDA, const void * X, const int INCX, const void *
          BETA, void * Y, const int INCY)

 -- Function: void cblas_ctrmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const void * A,
          const int LDA, void * X, const int INCX)

 -- Function: void cblas_ctbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const int K, const
          void * A, const int LDA, void * X, const int INCX)

 -- Function: void cblas_ctpmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const void * AP,
          void * X, const int INCX)

 -- Function: void cblas_ctrsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const void * A,
          const int LDA, void * X, const int INCX)

 -- Function: void cblas_ctbsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const int K, const
          void * A, const int LDA, void * X, const int INCX)

 -- Function: void cblas_ctpsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const void * AP,
          void * X, const int INCX)

 -- Function: void cblas_zgemv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
          void * ALPHA, const void * A, const int LDA, const void * X,
          const int INCX, const void * BETA, void * Y, const int INCY)

 -- Function: void cblas_zgbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
          int KL, const int KU, const void * ALPHA, const void * A,
          const int LDA, const void * X, const int INCX, const void *
          BETA, void * Y, const int INCY)

 -- Function: void cblas_ztrmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const void * A,
          const int LDA, void * X, const int INCX)

 -- Function: void cblas_ztbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const int K, const
          void * A, const int LDA, void * X, const int INCX)

 -- Function: void cblas_ztpmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const void * AP,
          void * X, const int INCX)

 -- Function: void cblas_ztrsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const void * A,
          const int LDA, void * X, const int INCX)

 -- Function: void cblas_ztbsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const int K, const
          void * A, const int LDA, void * X, const int INCX)

 -- Function: void cblas_ztpsv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
          const enum CBLAS_DIAG DIAG, const int N, const void * AP,
          void * X, const int INCX)

 -- Function: void cblas_ssymv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const
          float * A, const int LDA, const float * X, const int INCX,
          const float BETA, float * Y, const int INCY)

 -- Function: void cblas_ssbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const int K, const float
          ALPHA, const float * A, const int LDA, const float * X, const
          int INCX, const float BETA, float * Y, const int INCY)

 -- Function: void cblas_sspmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const
          float * AP, const float * X, const int INCX, const float
          BETA, float * Y, const int INCY)

 -- Function: void cblas_sger (const enum CBLAS_ORDER ORDER, const int
          M, const int N, const float ALPHA, const float * X, const int
          INCX, const float * Y, const int INCY, float * A, const int
          LDA)

 -- Function: void cblas_ssyr (const enum CBLAS_ORDER ORDER, const enum
          CBLAS_UPLO UPLO, const int N, const float ALPHA, const float
          * X, const int INCX, float * A, const int LDA)

 -- Function: void cblas_sspr (const enum CBLAS_ORDER ORDER, const enum
          CBLAS_UPLO UPLO, const int N, const float ALPHA, const float
          * X, const int INCX, float * AP)

 -- Function: void cblas_ssyr2 (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const
          float * X, const int INCX, const float * Y, const int INCY,
          float * A, const int LDA)

 -- Function: void cblas_sspr2 (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const
          float * X, const int INCX, const float * Y, const int INCY,
          float * A)

 -- Function: void cblas_dsymv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const
          double * A, const int LDA, const double * X, const int INCX,
          const double BETA, double * Y, const int INCY)

 -- Function: void cblas_dsbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const int K, const double
          ALPHA, const double * A, const int LDA, const double * X,
          const int INCX, const double BETA, double * Y, const int INCY)

 -- Function: void cblas_dspmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const
          double * AP, const double * X, const int INCX, const double
          BETA, double * Y, const int INCY)

 -- Function: void cblas_dger (const enum CBLAS_ORDER ORDER, const int
          M, const int N, const double ALPHA, const double * X, const
          int INCX, const double * Y, const int INCY, double * A, const
          int LDA)

 -- Function: void cblas_dsyr (const enum CBLAS_ORDER ORDER, const enum
          CBLAS_UPLO UPLO, const int N, const double ALPHA, const
          double * X, const int INCX, double * A, const int LDA)

 -- Function: void cblas_dspr (const enum CBLAS_ORDER ORDER, const enum
          CBLAS_UPLO UPLO, const int N, const double ALPHA, const
          double * X, const int INCX, double * AP)

 -- Function: void cblas_dsyr2 (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const
          double * X, const int INCX, const double * Y, const int INCY,
          double * A, const int LDA)

 -- Function: void cblas_dspr2 (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const
          double * X, const int INCX, const double * Y, const int INCY,
          double * A)

 -- Function: void cblas_chemv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
          void * A, const int LDA, const void * X, const int INCX,
          const void * BETA, void * Y, const int INCY)

 -- Function: void cblas_chbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const int K, const void *
          ALPHA, const void * A, const int LDA, const void * X, const
          int INCX, const void * BETA, void * Y, const int INCY)

 -- Function: void cblas_chpmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
          void * AP, const void * X, const int INCX, const void * BETA,
          void * Y, const int INCY)

 -- Function: void cblas_cgeru (const enum CBLAS_ORDER ORDER, const int
          M, const int N, const void * ALPHA, const void * X, const int
          INCX, const void * Y, const int INCY, void * A, const int LDA)

 -- Function: void cblas_cgerc (const enum CBLAS_ORDER ORDER, const int
          M, const int N, const void * ALPHA, const void * X, const int
          INCX, const void * Y, const int INCY, void * A, const int LDA)

 -- Function: void cblas_cher (const enum CBLAS_ORDER ORDER, const enum
          CBLAS_UPLO UPLO, const int N, const float ALPHA, const void *
          X, const int INCX, void * A, const int LDA)

 -- Function: void cblas_chpr (const enum CBLAS_ORDER ORDER, const enum
          CBLAS_UPLO UPLO, const int N, const float ALPHA, const void *
          X, const int INCX, void * A)

 -- Function: void cblas_cher2 (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
          void * X, const int INCX, const void * Y, const int INCY,
          void * A, const int LDA)

 -- Function: void cblas_chpr2 (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
          void * X, const int INCX, const void * Y, const int INCY,
          void * AP)

 -- Function: void cblas_zhemv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
          void * A, const int LDA, const void * X, const int INCX,
          const void * BETA, void * Y, const int INCY)

 -- Function: void cblas_zhbmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const int K, const void *
          ALPHA, const void * A, const int LDA, const void * X, const
          int INCX, const void * BETA, void * Y, const int INCY)

 -- Function: void cblas_zhpmv (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
          void * AP, const void * X, const int INCX, const void * BETA,
          void * Y, const int INCY)

 -- Function: void cblas_zgeru (const enum CBLAS_ORDER ORDER, const int
          M, const int N, const void * ALPHA, const void * X, const int
          INCX, const void * Y, const int INCY, void * A, const int LDA)

 -- Function: void cblas_zgerc (const enum CBLAS_ORDER ORDER, const int
          M, const int N, const void * ALPHA, const void * X, const int
          INCX, const void * Y, const int INCY, void * A, const int LDA)

 -- Function: void cblas_zher (const enum CBLAS_ORDER ORDER, const enum
          CBLAS_UPLO UPLO, const int N, const double ALPHA, const void
          * X, const int INCX, void * A, const int LDA)

 -- Function: void cblas_zhpr (const enum CBLAS_ORDER ORDER, const enum
          CBLAS_UPLO UPLO, const int N, const double ALPHA, const void
          * X, const int INCX, void * A)

 -- Function: void cblas_zher2 (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
          void * X, const int INCX, const void * Y, const int INCY,
          void * A, const int LDA)

 -- Function: void cblas_zhpr2 (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
          void * X, const int INCX, const void * Y, const int INCY,
          void * AP)


File: gsl-ref.info,  Node: Level 3 CBLAS Functions,  Next: GSL CBLAS Examples,  Prev: Level 2 CBLAS Functions,  Up: GSL CBLAS Library

D.3 Level 3
===========

 -- Function: void cblas_sgemm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE
          TRANSB, const int M, const int N, const int K, const float
          ALPHA, const float * A, const int LDA, const float * B, const
          int LDB, const float BETA, float * C, const int LDC)

 -- Function: void cblas_ssymm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
          M, const int N, const float ALPHA, const float * A, const int
          LDA, const float * B, const int LDB, const float BETA, float
          * C, const int LDC)

 -- Function: void cblas_ssyrk (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const float ALPHA, const float * A, const
          int LDA, const float BETA, float * C, const int LDC)

 -- Function: void cblas_ssyr2k (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const float ALPHA, const float * A, const
          int LDA, const float * B, const int LDB, const float BETA,
          float * C, const int LDC)

 -- Function: void cblas_strmm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
          M, const int N, const float ALPHA, const float * A, const int
          LDA, float * B, const int LDB)

 -- Function: void cblas_strsm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
          M, const int N, const float ALPHA, const float * A, const int
          LDA, float * B, const int LDB)

 -- Function: void cblas_dgemm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE
          TRANSB, const int M, const int N, const int K, const double
          ALPHA, const double * A, const int LDA, const double * B,
          const int LDB, const double BETA, double * C, const int LDC)

 -- Function: void cblas_dsymm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
          M, const int N, const double ALPHA, const double * A, const
          int LDA, const double * B, const int LDB, const double BETA,
          double * C, const int LDC)

 -- Function: void cblas_dsyrk (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const double ALPHA, const double * A,
          const int LDA, const double BETA, double * C, const int LDC)

 -- Function: void cblas_dsyr2k (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const double ALPHA, const double * A,
          const int LDA, const double * B, const int LDB, const double
          BETA, double * C, const int LDC)

 -- Function: void cblas_dtrmm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
          M, const int N, const double ALPHA, const double * A, const
          int LDA, double * B, const int LDB)

 -- Function: void cblas_dtrsm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
          M, const int N, const double ALPHA, const double * A, const
          int LDA, double * B, const int LDB)

 -- Function: void cblas_cgemm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE
          TRANSB, const int M, const int N, const int K, const void *
          ALPHA, const void * A, const int LDA, const void * B, const
          int LDB, const void * BETA, void * C, const int LDC)

 -- Function: void cblas_csymm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
          M, const int N, const void * ALPHA, const void * A, const int
          LDA, const void * B, const int LDB, const void * BETA, void *
          C, const int LDC)

 -- Function: void cblas_csyrk (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const void * ALPHA, const void * A, const
          int LDA, const void * BETA, void * C, const int LDC)

 -- Function: void cblas_csyr2k (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const void * ALPHA, const void * A, const
          int LDA, const void * B, const int LDB, const void * BETA,
          void * C, const int LDC)

 -- Function: void cblas_ctrmm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
          M, const int N, const void * ALPHA, const void * A, const int
          LDA, void * B, const int LDB)

 -- Function: void cblas_ctrsm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
          M, const int N, const void * ALPHA, const void * A, const int
          LDA, void * B, const int LDB)

 -- Function: void cblas_zgemm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE
          TRANSB, const int M, const int N, const int K, const void *
          ALPHA, const void * A, const int LDA, const void * B, const
          int LDB, const void * BETA, void * C, const int LDC)

 -- Function: void cblas_zsymm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
          M, const int N, const void * ALPHA, const void * A, const int
          LDA, const void * B, const int LDB, const void * BETA, void *
          C, const int LDC)

 -- Function: void cblas_zsyrk (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const void * ALPHA, const void * A, const
          int LDA, const void * BETA, void * C, const int LDC)

 -- Function: void cblas_zsyr2k (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const void * ALPHA, const void * A, const
          int LDA, const void * B, const int LDB, const void * BETA,
          void * C, const int LDC)

 -- Function: void cblas_ztrmm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
          M, const int N, const void * ALPHA, const void * A, const int
          LDA, void * B, const int LDB)

 -- Function: void cblas_ztrsm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
          M, const int N, const void * ALPHA, const void * A, const int
          LDA, void * B, const int LDB)

 -- Function: void cblas_chemm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
          M, const int N, const void * ALPHA, const void * A, const int
          LDA, const void * B, const int LDB, const void * BETA, void *
          C, const int LDC)

 -- Function: void cblas_cherk (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const float ALPHA, const void * A, const
          int LDA, const float BETA, void * C, const int LDC)

 -- Function: void cblas_cher2k (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const void * ALPHA, const void * A, const
          int LDA, const void * B, const int LDB, const float BETA,
          void * C, const int LDC)

 -- Function: void cblas_zhemm (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
          M, const int N, const void * ALPHA, const void * A, const int
          LDA, const void * B, const int LDB, const void * BETA, void *
          C, const int LDC)

 -- Function: void cblas_zherk (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const double ALPHA, const void * A, const
          int LDA, const double BETA, void * C, const int LDC)

 -- Function: void cblas_zher2k (const enum CBLAS_ORDER ORDER, const
          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
          int N, const int K, const void * ALPHA, const void * A, const
          int LDA, const void * B, const int LDB, const double BETA,
          void * C, const int LDC)

 -- Function: void cblas_xerbla (int P, const char * ROUT, const char *
          FORM, ...)


File: gsl-ref.info,  Node: GSL CBLAS Examples,  Prev: Level 3 CBLAS Functions,  Up: GSL CBLAS Library

D.4 Examples
============

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

     [ 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 but could be stored in column
major order if the first argument of the call to `cblas_sgemm' was
changed to `CblasColMajor'.

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

     int
     main (void)
     {
       int lda = 3;

       float A[] = { 0.11, 0.12, 0.13,
                     0.21, 0.22, 0.23 };

       int ldb = 2;

       float B[] = { 1011, 1012,
                     1021, 1022,
                     1031, 1032 };

       int ldc = 2;

       float C[] = { 0.00, 0.00,
                     0.00, 0.00 };

       /* Compute C = A B */

       cblas_sgemm (CblasRowMajor,
                    CblasNoTrans, CblasNoTrans, 2, 2, 3,
                    1.0, A, lda, B, ldb, 0.0, C, ldc);

       printf ("[ %g, %g\n", C[0], C[1]);
       printf ("  %g, %g ]\n", C[2], C[3]);

       return 0;
     }

To compile the program use the following command line,

     $ gcc -Wall demo.c -lgslcblas

There is no need to link with the main library `-lgsl' in this case as
the CBLAS library is an independent unit. Here is the output from the
program,

     $ ./a.out
     [ 367.76, 368.12
       674.06, 674.72 ]


File: gsl-ref.info,  Node: Free Software Needs Free Documentation,  Next: GNU General Public License,  Prev: GSL CBLAS Library,  Up: Top

Free Software Needs Free Documentation
**************************************

     The following article was written by Richard Stallman, founder of
     the GNU Project.

   The biggest deficiency in the free software community today is not in
the software--it is the lack of good free documentation that we can
include with the free software.  Many of our most important programs do
not come with free reference manuals and free introductory texts.
Documentation is an essential part of any software package; when an
important free software package does not come with a free manual and a
free tutorial, that is a major gap.  We have many such gaps today.

   Consider Perl, for instance.  The tutorial manuals that people
normally use are non-free.  How did this come about?  Because the
authors of those manuals published them with restrictive terms--no
copying, no modification, source files not available--which exclude
them from the free software world.

   That wasn't the first time this sort of thing happened, and it was
far from the last.  Many times we have heard a GNU user eagerly
describe a manual that he is writing, his intended contribution to the
community, only to learn that he had ruined everything by signing a
publication contract to make it non-free.

   Free documentation, like free software, is a matter of freedom, not
price.  The problem with the non-free manual is not that publishers
charge a price for printed copies--that in itself is fine.  (The Free
Software Foundation sells printed copies of manuals, too.)  The problem
is the restrictions on the use of the manual.  Free manuals are
available in source code form, and give you permission to copy and
modify.  Non-free manuals do not allow this.

   The criteria of freedom for a free manual are roughly the same as for
free software.  Redistribution (including the normal kinds of
commercial redistribution) must be permitted, so that the manual can
accompany every copy of the program, both on-line and on paper.

   Permission for modification of the technical content is crucial too.
When people modify the software, adding or changing features, if they
are conscientious they will change the manual too--so they can provide
accurate and clear documentation for the modified program.  A manual
that leaves you no choice but to write a new manual to document a
changed version of the program is not really available to our community.

   Some kinds of limits on the way modification is handled are
acceptable.  For example, requirements to preserve the original
author's copyright notice, the distribution terms, or the list of
authors, are ok.  It is also no problem to require modified versions to
include notice that they were modified.  Even entire sections that may
not be deleted or changed are acceptable, as long as they deal with
nontechnical topics (like this one).  These kinds of restrictions are
acceptable because they don't obstruct the community's normal use of
the manual.

   However, it must be possible to modify all the _technical_ content
of the manual, and then distribute the result in all the usual media,
through all the usual channels.  Otherwise, the restrictions obstruct
the use of the manual, it is not free, and we need another manual to
replace it.

   Please spread the word about this issue.  Our community continues to
lose manuals to proprietary publishing.  If we spread the word that
free software needs free reference manuals and free tutorials, perhaps
the next person who wants to contribute by writing documentation will
realize, before it is too late, that only free manuals contribute to
the free software community.

   If you are writing documentation, please insist on publishing it
under the GNU Free Documentation License or another free documentation
license.  Remember that this decision requires your approval--you don't
have to let the publisher decide.  Some commercial publishers will use
a free license if you insist, but they will not propose the option; it
is up to you to raise the issue and say firmly that this is what you
want.  If the publisher you are dealing with refuses, please try other
publishers.  If you're not sure whether a proposed license is free,
write to <licensing@gnu.org>.

   You can encourage commercial publishers to sell more free, copylefted
manuals and tutorials by buying them, and particularly by buying copies
from the publishers that paid for their writing or for major
improvements.  Meanwhile, try to avoid buying non-free documentation at
all.  Check the distribution terms of a manual before you buy it, and
insist that whoever seeks your business must respect your freedom.
Check the history of the book, and try reward the publishers that have
paid or pay the authors to work on it.

   The Free Software Foundation maintains a list of free documentation
published by other publishers:

     `http://www.fsf.org/doc/other-free-books.html'


File: gsl-ref.info,  Node: GNU General Public License,  Next: GNU Free Documentation License,  Prev: Free Software Needs Free Documentation,  Up: Top

GNU General Public License
**************************

                         Version 2, June 1991
     Copyright (C) 1989, 1991 Free Software Foundation, Inc.
     59 Temple Place - Suite 330, Boston, MA  02111-1307, USA

     Everyone is permitted to copy and distribute verbatim copies of this
     license document, but changing it is not allowed.

Preamble
========

The licenses for most software are designed to take away your freedom
to share and change it.  By contrast, the GNU General Public License is
intended to guarantee your freedom to share and change free
software--to make sure the software is free for all its users.  This
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                      END OF TERMS AND CONDITIONS
Appendix: How to Apply These Terms to Your New Programs
=======================================================

If you develop a new program, and you want it to be of the greatest
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free software which everyone can redistribute and change under these
terms.

   To do so, attach the following notices to the program.  It is safest
to attach them to the start of each source file to most effectively
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the "copyright" line and a pointer to where the full notice is found.

     one line to give the program's name and a brief idea
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     This program is free software; you can redistribute it
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     details.

     You should have received a copy of the GNU General Public
     License along with this program; if not, write to the Free
     Software Foundation, Inc., 59 Temple Place - Suite 330,
     Boston, MA 02111-1307, USA.

   Also add information on how to contact you by electronic and paper
mail.

   If the program is interactive, make it output a short notice like
this when it starts in an interactive mode:

     Gnomovision version 69, Copyright (C) 19YY NAME OF AUTHOR
     Gnomovision comes with ABSOLUTELY NO WARRANTY; for details
     type `show w'.  This is free software, and you are welcome
     to redistribute it under certain conditions; type `show c'
     for details.

   The hypothetical commands `show w' and `show c' should show the
appropriate parts of the General Public License.  Of course, the
commands you use may be called something other than `show w' and `show
c'; they could even be mouse-clicks or menu items--whatever suits your
program.

   You should also get your employer (if you work as a programmer) or
your school, if any, to sign a "copyright disclaimer" for the program,
if necessary.  Here is a sample; alter the names:

     Yoyodyne, Inc., hereby disclaims all copyright interest in
     the program `Gnomovision' (which makes passes at compilers)
     written by James Hacker.

     SIGNATURE OF TY COON, 1 April 1989
     Ty Coon, President of Vice

   This General Public License does not permit incorporating your
program into proprietary programs.  If your program is a subroutine
library, you may consider it more useful to permit linking proprietary
applications with the library.  If this is what you want to do, use the
GNU Library General Public License instead of this License.


File: gsl-ref.info,  Node: GNU Free Documentation License,  Next: Function Index,  Prev: GNU General Public License,  Up: Top

GNU Free Documentation License
******************************

                      Version 1.2, November 2002
     Copyright (C) 2000,2001,2002 Free Software Foundation, Inc.
     51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA

     Everyone is permitted to copy and distribute verbatim copies of this license
     document, but changing it is not allowed.

  0. PREAMBLE

     The purpose of this License is to make a manual, textbook, or other
     functional and useful document "free" in the sense of freedom: to
     assure everyone the effective freedom to copy and redistribute it,
     with or without modifying it, either commercially or
     noncommercially.  Secondarily, this License preserves for the
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     This License is a kind of "copyleft", which means that derivative
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     It complements the GNU General Public License, which is a copyleft
     license designed for free software.

     We have designed this License in order to use it for manuals for
     free software, because free software needs free documentation: a
     free program should come with manuals providing the same freedoms
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     software manuals; it can be used for any textual work, regardless
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     We recommend this License principally for works whose purpose is
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  1. APPLICABILITY AND DEFINITIONS

     This License applies to any manual or other work, in any medium,
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     A "Modified Version" of the Document means any work containing the
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     The "Invariant Sections" are certain Secondary Sections whose
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     The Document may include Warranty Disclaimers next to the notice
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  2. VERBATIM COPYING

     You may copy and distribute the Document in any medium, either
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  3. COPYING IN QUANTITY

     If you publish printed copies (or copies in media that commonly
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     If the required texts for either cover are too voluminous to fit
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     If you publish or distribute Opaque copies of the Document
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     state in or with each Opaque copy a computer-network location from
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     It is requested, but not required, that you contact the authors of
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  4. MODIFICATIONS

     You may copy and distribute a Modified Version of the Document
     under the conditions of sections 2 and 3 above, provided that you
     release the Modified Version under precisely this License, with
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     licensing distribution and modification of the Modified Version to
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     things in the Modified Version:

       A. Use in the Title Page (and on the covers, if any) a title
          distinct from that of the Document, and from those of
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       B. List on the Title Page, as authors, one or more persons or
          entities responsible for authorship of the modifications in
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          principal authors of the Document (all of its principal
          authors, if it has fewer than five), unless they release you
          from this requirement.

       C. State on the Title page the name of the publisher of the
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       D. Preserve all the copyright notices of the Document.

       E. Add an appropriate copyright notice for your modifications
          adjacent to the other copyright notices.

       F. Include, immediately after the copyright notices, a license
          notice giving the public permission to use the Modified
          Version under the terms of this License, in the form shown in
          the Addendum below.

       G. Preserve in that license notice the full lists of Invariant
          Sections and required Cover Texts given in the Document's
          license notice.

       H. Include an unaltered copy of this License.

       I. Preserve the section Entitled "History", Preserve its Title,
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          authors, and publisher of the Modified Version as given on
          the Title Page.  If there is no section Entitled "History" in
          the Document, create one stating the title, year, authors,
          and publisher of the Document as given on its Title Page,
          then add an item describing the Modified Version as stated in
          the previous sentence.

       J. Preserve the network location, if any, given in the Document
          for public access to a Transparent copy of the Document, and
          likewise the network locations given in the Document for
          previous versions it was based on.  These may be placed in
          the "History" section.  You may omit a network location for a
          work that was published at least four years before the
          Document itself, or if the original publisher of the version
          it refers to gives permission.

       K. For any section Entitled "Acknowledgements" or "Dedications",
          Preserve the Title of the section, and preserve in the
          section all the substance and tone of each of the contributor
          acknowledgements and/or dedications given therein.

       L. Preserve all the Invariant Sections of the Document,
          unaltered in their text and in their titles.  Section numbers
          or the equivalent are not considered part of the section
          titles.

       M. Delete any section Entitled "Endorsements".  Such a section
          may not be included in the Modified Version.

       N. Do not retitle any existing section to be Entitled
          "Endorsements" or to conflict in title with any Invariant
          Section.

       O. Preserve any Warranty Disclaimers.

     If the Modified Version includes new front-matter sections or
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     You may add a section Entitled "Endorsements", provided it contains
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     You may add a passage of up to five words as a Front-Cover Text,
     and a passage of up to 25 words as a Back-Cover Text, to the end
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     License give permission to use their names for publicity for or to
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  5. COMBINING DOCUMENTS

     You may combine the Document with other documents released under
     this License, under the terms defined in section 4 above for
     modified versions, provided that you include in the combination
     all of the Invariant Sections of all of the original documents,
     unmodified, and list them all as Invariant Sections of your
     combined work in its license notice, and that you preserve all
     their Warranty Disclaimers.

     The combined work need only contain one copy of this License, and
     multiple identical Invariant Sections may be replaced with a single
     copy.  If there are multiple Invariant Sections with the same name
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     by adding at the end of it, in parentheses, the name of the
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     In the combination, you must combine any sections Entitled
     "History" in the various original documents, forming one section
     Entitled "History"; likewise combine any sections Entitled
     "Acknowledgements", and any sections Entitled "Dedications".  You
     must delete all sections Entitled "Endorsements."

  6. COLLECTIONS OF DOCUMENTS

     You may make a collection consisting of the Document and other
     documents released under this License, and replace the individual
     copies of this License in the various documents with a single copy
     that is included in the collection, provided that you follow the
     rules of this License for verbatim copying of each of the
     documents in all other respects.

     You may extract a single document from such a collection, and
     distribute it individually under this License, provided you insert
     a copy of this License into the extracted document, and follow
     this License in all other respects regarding verbatim copying of
     that document.

  7. AGGREGATION WITH INDEPENDENT WORKS

     A compilation of the Document or its derivatives with other
     separate and independent documents or works, in or on a volume of
     a storage or distribution medium, is called an "aggregate" if the
     copyright resulting from the compilation is not used to limit the
     legal rights of the compilation's users beyond what the individual
     works permit.  When the Document is included in an aggregate, this
     License does not apply to the other works in the aggregate which
     are not themselves derivative works of the Document.

     If the Cover Text requirement of section 3 is applicable to these
     copies of the Document, then if the Document is less than one half
     of the entire aggregate, the Document's Cover Texts may be placed
     on covers that bracket the Document within the aggregate, or the
     electronic equivalent of covers if the Document is in electronic
     form.  Otherwise they must appear on printed covers that bracket
     the whole aggregate.

  8. TRANSLATION

     Translation is considered a kind of modification, so you may
     distribute translations of the Document under the terms of section
     4.  Replacing Invariant Sections with translations requires special
     permission from their copyright holders, but you may include
     translations of some or all Invariant Sections in addition to the
     original versions of these Invariant Sections.  You may include a
     translation of this License, and all the license notices in the
     Document, and any Warranty Disclaimers, provided that you also
     include the original English version of this License and the
     original versions of those notices and disclaimers.  In case of a
     disagreement between the translation and the original version of
     this License or a notice or disclaimer, the original version will
     prevail.

     If a section in the Document is Entitled "Acknowledgements",
     "Dedications", or "History", the requirement (section 4) to
     Preserve its Title (section 1) will typically require changing the
     actual title.

  9. TERMINATION

     You may not copy, modify, sublicense, or distribute the Document
     except as expressly provided for under this License.  Any other
     attempt to copy, modify, sublicense or distribute the Document is
     void, and will automatically terminate your rights under this
     License.  However, parties who have received copies, or rights,
     from you under this License will not have their licenses
     terminated so long as such parties remain in full compliance.

 10. FUTURE REVISIONS OF THIS LICENSE

     The Free Software Foundation may publish new, revised versions of
     the GNU Free Documentation License from time to time.  Such new
     versions will be similar in spirit to the present version, but may
     differ in detail to address new problems or concerns.  See
     `http://www.gnu.org/copyleft/'.

     Each version of the License is given a distinguishing version
     number.  If the Document specifies that a particular numbered
     version of this License "or any later version" applies to it, you
     have the option of following the terms and conditions either of
     that specified version or of any later version that has been
     published (not as a draft) by the Free Software Foundation.  If
     the Document does not specify a version number of this License,
     you may choose any version ever published (not as a draft) by the
     Free Software Foundation.

ADDENDUM: How to use this License for your documents
====================================================

To use this License in a document you have written, include a copy of
the License in the document and put the following copyright and license
notices just after the title page:

       Copyright (C) YEAR YOUR NAME.
       Permission is granted to copy, distribute and/or modify
       this document under the terms of the GNU Free
       Documentation License, Version 1.2 or any later version
       published by the Free Software Foundation; with no
       Invariant Sections, no Front-Cover Texts, and no
       Back-Cover Texts.  A copy of the license is included in
       the section entitled ``GNU Free Documentation License''.

   If you have Invariant Sections, Front-Cover Texts and Back-Cover
Texts, replace the "with...Texts." line with this:

       with the Invariant Sections being LIST THEIR
       TITLES, with the Front-Cover Texts being LIST, and
       with the Back-Cover Texts being LIST.

   If you have Invariant Sections without Cover Texts, or some other
combination of the three, merge those two alternatives to suit the
situation.

   If your document contains nontrivial examples of program code, we
recommend releasing these examples in parallel under your choice of
free software license, such as the GNU General Public License, to
permit their use in free software.


File: gsl-ref.info,  Node: Function Index,  Next: Variable Index,  Prev: GNU Free Documentation License,  Up: Top

Function Index
**************

[index]
* Menu:

* cblas_caxpy:                           Level 1 CBLAS Functions.
                                                              (line  92)
* cblas_ccopy:                           Level 1 CBLAS Functions.
                                                              (line  89)
* cblas_cdotc_sub:                       Level 1 CBLAS Functions.
                                                              (line  23)
* cblas_cdotu_sub:                       Level 1 CBLAS Functions.
                                                              (line  20)
* cblas_cgbmv:                           Level 2 CBLAS Functions.
                                                              (line  99)
* cblas_cgemm:                           Level 3 CBLAS Functions.
                                                              (line  81)
* cblas_cgemv:                           Level 2 CBLAS Functions.
                                                              (line  93)
* cblas_cgerc:                           Level 2 CBLAS Functions.
                                                              (line 269)
* cblas_cgeru:                           Level 2 CBLAS Functions.
                                                              (line 265)
* cblas_chbmv:                           Level 2 CBLAS Functions.
                                                              (line 256)
* cblas_chemm:                           Level 3 CBLAS Functions.
                                                              (line 151)
* cblas_chemv:                           Level 2 CBLAS Functions.
                                                              (line 251)
* cblas_cher:                            Level 2 CBLAS Functions.
                                                              (line 273)
* cblas_cher2:                           Level 2 CBLAS Functions.
                                                              (line 282)
* cblas_cher2k:                          Level 3 CBLAS Functions.
                                                              (line 162)
* cblas_cherk:                           Level 3 CBLAS Functions.
                                                              (line 156)
* cblas_chpmv:                           Level 2 CBLAS Functions.
                                                              (line 261)
* cblas_chpr:                            Level 2 CBLAS Functions.
                                                              (line 277)
* cblas_chpr2:                           Level 2 CBLAS Functions.
                                                              (line 287)
* cblas_cscal:                           Level 1 CBLAS Functions.
                                                              (line 134)
* cblas_csscal:                          Level 1 CBLAS Functions.
                                                              (line 140)
* cblas_cswap:                           Level 1 CBLAS Functions.
                                                              (line  86)
* cblas_csymm:                           Level 3 CBLAS Functions.
                                                              (line  87)
* cblas_csyr2k:                          Level 3 CBLAS Functions.
                                                              (line  98)
* cblas_csyrk:                           Level 3 CBLAS Functions.
                                                              (line  92)
* cblas_ctbmv:                           Level 2 CBLAS Functions.
                                                              (line 109)
* cblas_ctbsv:                           Level 2 CBLAS Functions.
                                                              (line 124)
* cblas_ctpmv:                           Level 2 CBLAS Functions.
                                                              (line 114)
* cblas_ctpsv:                           Level 2 CBLAS Functions.
                                                              (line 129)
* cblas_ctrmm:                           Level 3 CBLAS Functions.
                                                              (line 104)
* cblas_ctrmv:                           Level 2 CBLAS Functions.
                                                              (line 104)
* cblas_ctrsm:                           Level 3 CBLAS Functions.
                                                              (line 110)
* cblas_ctrsv:                           Level 2 CBLAS Functions.
                                                              (line 119)
* cblas_dasum:                           Level 1 CBLAS Functions.
                                                              (line  41)
* cblas_daxpy:                           Level 1 CBLAS Functions.
                                                              (line  83)
* cblas_dcopy:                           Level 1 CBLAS Functions.
                                                              (line  80)
* cblas_ddot:                            Level 1 CBLAS Functions.
                                                              (line  17)
* cblas_dgbmv:                           Level 2 CBLAS Functions.
                                                              (line  58)
* cblas_dgemm:                           Level 3 CBLAS Functions.
                                                              (line  46)
* cblas_dgemv:                           Level 2 CBLAS Functions.
                                                              (line  52)
* cblas_dger:                            Level 2 CBLAS Functions.
                                                              (line 228)
* cblas_dnrm2:                           Level 1 CBLAS Functions.
                                                              (line  38)
* cblas_drot:                            Level 1 CBLAS Functions.
                                                              (line 122)
* cblas_drotg:                           Level 1 CBLAS Functions.
                                                              (line 116)
* cblas_drotm:                           Level 1 CBLAS Functions.
                                                              (line 125)
* cblas_drotmg:                          Level 1 CBLAS Functions.
                                                              (line 119)
* cblas_dsbmv:                           Level 2 CBLAS Functions.
                                                              (line 218)
* cblas_dscal:                           Level 1 CBLAS Functions.
                                                              (line 131)
* cblas_dsdot:                           Level 1 CBLAS Functions.
                                                              (line  11)
* cblas_dspmv:                           Level 2 CBLAS Functions.
                                                              (line 223)
* cblas_dspr:                            Level 2 CBLAS Functions.
                                                              (line 236)
* cblas_dspr2:                           Level 2 CBLAS Functions.
                                                              (line 246)
* cblas_dswap:                           Level 1 CBLAS Functions.
                                                              (line  77)
* cblas_dsymm:                           Level 3 CBLAS Functions.
                                                              (line  52)
* cblas_dsymv:                           Level 2 CBLAS Functions.
                                                              (line 213)
* cblas_dsyr:                            Level 2 CBLAS Functions.
                                                              (line 232)
* cblas_dsyr2:                           Level 2 CBLAS Functions.
                                                              (line 241)
* cblas_dsyr2k:                          Level 3 CBLAS Functions.
                                                              (line  63)
* cblas_dsyrk:                           Level 3 CBLAS Functions.
                                                              (line  57)
* cblas_dtbmv:                           Level 2 CBLAS Functions.
                                                              (line  68)
* cblas_dtbsv:                           Level 2 CBLAS Functions.
                                                              (line  83)
* cblas_dtpmv:                           Level 2 CBLAS Functions.
                                                              (line  73)
* cblas_dtpsv:                           Level 2 CBLAS Functions.
                                                              (line  88)
* cblas_dtrmm:                           Level 3 CBLAS Functions.
                                                              (line  69)
* cblas_dtrmv:                           Level 2 CBLAS Functions.
                                                              (line  63)
* cblas_dtrsm:                           Level 3 CBLAS Functions.
                                                              (line  75)
* cblas_dtrsv:                           Level 2 CBLAS Functions.
                                                              (line  78)
* cblas_dzasum:                          Level 1 CBLAS Functions.
                                                              (line  53)
* cblas_dznrm2:                          Level 1 CBLAS Functions.
                                                              (line  50)
* cblas_icamax:                          Level 1 CBLAS Functions.
                                                              (line  62)
* cblas_idamax:                          Level 1 CBLAS Functions.
                                                              (line  59)
* cblas_isamax:                          Level 1 CBLAS Functions.
                                                              (line  56)
* cblas_izamax:                          Level 1 CBLAS Functions.
                                                              (line  65)
* cblas_sasum:                           Level 1 CBLAS Functions.
                                                              (line  35)
* cblas_saxpy:                           Level 1 CBLAS Functions.
                                                              (line  74)
* cblas_scasum:                          Level 1 CBLAS Functions.
                                                              (line  47)
* cblas_scnrm2:                          Level 1 CBLAS Functions.
                                                              (line  44)
* cblas_scopy:                           Level 1 CBLAS Functions.
                                                              (line  71)
* cblas_sdot:                            Level 1 CBLAS Functions.
                                                              (line  14)
* cblas_sdsdot:                          Level 1 CBLAS Functions.
                                                              (line   8)
* cblas_sgbmv:                           Level 2 CBLAS Functions.
                                                              (line  16)
* cblas_sgemm:                           Level 3 CBLAS Functions.
                                                              (line  11)
* cblas_sgemv:                           Level 2 CBLAS Functions.
                                                              (line  10)
* cblas_sger:                            Level 2 CBLAS Functions.
                                                              (line 190)
* cblas_snrm2:                           Level 1 CBLAS Functions.
                                                              (line  32)
* cblas_srot:                            Level 1 CBLAS Functions.
                                                              (line 110)
* cblas_srotg:                           Level 1 CBLAS Functions.
                                                              (line 104)
* cblas_srotm:                           Level 1 CBLAS Functions.
                                                              (line 113)
* cblas_srotmg:                          Level 1 CBLAS Functions.
                                                              (line 107)
* cblas_ssbmv:                           Level 2 CBLAS Functions.
                                                              (line 180)
* cblas_sscal:                           Level 1 CBLAS Functions.
                                                              (line 128)
* cblas_sspmv:                           Level 2 CBLAS Functions.
                                                              (line 185)
* cblas_sspr:                            Level 2 CBLAS Functions.
                                                              (line 198)
* cblas_sspr2:                           Level 2 CBLAS Functions.
                                                              (line 208)
* cblas_sswap:                           Level 1 CBLAS Functions.
                                                              (line  68)
* cblas_ssymm:                           Level 3 CBLAS Functions.
                                                              (line  17)
* cblas_ssymv:                           Level 2 CBLAS Functions.
                                                              (line 175)
* cblas_ssyr:                            Level 2 CBLAS Functions.
                                                              (line 194)
* cblas_ssyr2:                           Level 2 CBLAS Functions.
                                                              (line 203)
* cblas_ssyr2k:                          Level 3 CBLAS Functions.
                                                              (line  28)
* cblas_ssyrk:                           Level 3 CBLAS Functions.
                                                              (line  22)
* cblas_stbmv:                           Level 2 CBLAS Functions.
                                                              (line  26)
* cblas_stbsv:                           Level 2 CBLAS Functions.
                                                              (line  41)
* cblas_stpmv:                           Level 2 CBLAS Functions.
                                                              (line  31)
* cblas_stpsv:                           Level 2 CBLAS Functions.
                                                              (line  46)
* cblas_strmm:                           Level 3 CBLAS Functions.
                                                              (line  34)
* cblas_strmv:                           Level 2 CBLAS Functions.
                                                              (line  21)
* cblas_strsm:                           Level 3 CBLAS Functions.
                                                              (line  40)
* cblas_strsv:                           Level 2 CBLAS Functions.
                                                              (line  36)
* cblas_xerbla:                          Level 3 CBLAS Functions.
                                                              (line 182)
* cblas_zaxpy:                           Level 1 CBLAS Functions.
                                                              (line 101)
* cblas_zcopy:                           Level 1 CBLAS Functions.
                                                              (line  98)
* cblas_zdotc_sub:                       Level 1 CBLAS Functions.
                                                              (line  29)
* cblas_zdotu_sub:                       Level 1 CBLAS Functions.
                                                              (line  26)
* cblas_zdscal:                          Level 1 CBLAS Functions.
                                                              (line 143)
* cblas_zgbmv:                           Level 2 CBLAS Functions.
                                                              (line 140)
* cblas_zgemm:                           Level 3 CBLAS Functions.
                                                              (line 116)
* cblas_zgemv:                           Level 2 CBLAS Functions.
                                                              (line 134)
* cblas_zgerc:                           Level 2 CBLAS Functions.
                                                              (line 310)
* cblas_zgeru:                           Level 2 CBLAS Functions.
                                                              (line 306)
* cblas_zhbmv:                           Level 2 CBLAS Functions.
                                                              (line 297)
* cblas_zhemm:                           Level 3 CBLAS Functions.
                                                              (line 168)
* cblas_zhemv:                           Level 2 CBLAS Functions.
                                                              (line 292)
* cblas_zher:                            Level 2 CBLAS Functions.
                                                              (line 314)
* cblas_zher2:                           Level 2 CBLAS Functions.
                                                              (line 323)
* cblas_zher2k:                          Level 3 CBLAS Functions.
                                                              (line 179)
* cblas_zherk:                           Level 3 CBLAS Functions.
                                                              (line 173)
* cblas_zhpmv:                           Level 2 CBLAS Functions.
                                                              (line 302)
* cblas_zhpr:                            Level 2 CBLAS Functions.
                                                              (line 318)
* cblas_zhpr2:                           Level 2 CBLAS Functions.
                                                              (line 328)
* cblas_zscal:                           Level 1 CBLAS Functions.
                                                              (line 137)
* cblas_zswap:                           Level 1 CBLAS Functions.
                                                              (line  95)
* cblas_zsymm:                           Level 3 CBLAS Functions.
                                                              (line 122)
* cblas_zsyr2k:                          Level 3 CBLAS Functions.
                                                              (line 133)
* cblas_zsyrk:                           Level 3 CBLAS Functions.
                                                              (line 127)
* cblas_ztbmv:                           Level 2 CBLAS Functions.
                                                              (line 150)
* cblas_ztbsv:                           Level 2 CBLAS Functions.
                                                              (line 165)
* cblas_ztpmv:                           Level 2 CBLAS Functions.
                                                              (line 155)
* cblas_ztpsv:                           Level 2 CBLAS Functions.
                                                              (line 170)
* cblas_ztrmm:                           Level 3 CBLAS Functions.
                                                              (line 139)
* cblas_ztrmv:                           Level 2 CBLAS Functions.
                                                              (line 145)
* cblas_ztrsm:                           Level 3 CBLAS Functions.
                                                              (line 145)
* cblas_ztrsv:                           Level 2 CBLAS Functions.
                                                              (line 160)
* gsl_acosh:                             Elementary Functions.
                                                              (line  28)
* gsl_asinh:                             Elementary Functions.
                                                              (line  32)
* gsl_atanh:                             Elementary Functions.
                                                              (line  36)
* gsl_blas_caxpy:                        Level 1 GSL BLAS Interface.
                                                              (line  94)
* gsl_blas_ccopy:                        Level 1 GSL BLAS Interface.
                                                              (line  83)
* gsl_blas_cdotc:                        Level 1 GSL BLAS Interface.
                                                              (line  29)
* gsl_blas_cdotu:                        Level 1 GSL BLAS Interface.
                                                              (line  22)
* gsl_blas_cgemm:                        Level 3 GSL BLAS Interface.
                                                              (line  18)
* gsl_blas_cgemv:                        Level 2 GSL BLAS Interface.
                                                              (line  16)
* gsl_blas_cgerc:                        Level 2 GSL BLAS Interface.
                                                              (line 110)
* gsl_blas_cgeru:                        Level 2 GSL BLAS Interface.
                                                              (line 101)
* gsl_blas_chemm:                        Level 3 GSL BLAS Interface.
                                                              (line  52)
* gsl_blas_chemv:                        Level 2 GSL BLAS Interface.
                                                              (line  82)
* gsl_blas_cher:                         Level 2 GSL BLAS Interface.
                                                              (line 129)
* gsl_blas_cher2:                        Level 2 GSL BLAS Interface.
                                                              (line 155)
* gsl_blas_cher2k:                       Level 3 GSL BLAS Interface.
                                                              (line 176)
* gsl_blas_cherk:                        Level 3 GSL BLAS Interface.
                                                              (line 136)
* gsl_blas_cscal:                        Level 1 GSL BLAS Interface.
                                                              (line 103)
* gsl_blas_csscal:                       Level 1 GSL BLAS Interface.
                                                              (line 107)
* gsl_blas_cswap:                        Level 1 GSL BLAS Interface.
                                                              (line  74)
* gsl_blas_csymm:                        Level 3 GSL BLAS Interface.
                                                              (line  37)
* gsl_blas_csyr2k:                       Level 3 GSL BLAS Interface.
                                                              (line 159)
* gsl_blas_csyrk:                        Level 3 GSL BLAS Interface.
                                                              (line 122)
* gsl_blas_ctrmm:                        Level 3 GSL BLAS Interface.
                                                              (line  74)
* gsl_blas_ctrmv:                        Level 2 GSL BLAS Interface.
                                                              (line  33)
* gsl_blas_ctrsm:                        Level 3 GSL BLAS Interface.
                                                              (line  98)
* gsl_blas_ctrsv:                        Level 2 GSL BLAS Interface.
                                                              (line  54)
* gsl_blas_dasum:                        Level 1 GSL BLAS Interface.
                                                              (line  47)
* gsl_blas_daxpy:                        Level 1 GSL BLAS Interface.
                                                              (line  92)
* gsl_blas_dcopy:                        Level 1 GSL BLAS Interface.
                                                              (line  81)
* gsl_blas_ddot:                         Level 1 GSL BLAS Interface.
                                                              (line  17)
* gsl_blas_dgemm:                        Level 3 GSL BLAS Interface.
                                                              (line  13)
* gsl_blas_dgemv:                        Level 2 GSL BLAS Interface.
                                                              (line  12)
* gsl_blas_dger:                         Level 2 GSL BLAS Interface.
                                                              (line  98)
* gsl_blas_dnrm2:                        Level 1 GSL BLAS Interface.
                                                              (line  36)
* gsl_blas_drot:                         Level 1 GSL BLAS Interface.
                                                              (line 128)
* gsl_blas_drotg:                        Level 1 GSL BLAS Interface.
                                                              (line 116)
* gsl_blas_drotm:                        Level 1 GSL BLAS Interface.
                                                              (line 143)
* gsl_blas_drotmg:                       Level 1 GSL BLAS Interface.
                                                              (line 135)
* gsl_blas_dscal:                        Level 1 GSL BLAS Interface.
                                                              (line 101)
* gsl_blas_dsdot:                        Level 1 GSL BLAS Interface.
                                                              (line  15)
* gsl_blas_dswap:                        Level 1 GSL BLAS Interface.
                                                              (line  72)
* gsl_blas_dsymm:                        Level 3 GSL BLAS Interface.
                                                              (line  33)
* gsl_blas_dsymv:                        Level 2 GSL BLAS Interface.
                                                              (line  71)
* gsl_blas_dsyr:                         Level 2 GSL BLAS Interface.
                                                              (line 120)
* gsl_blas_dsyr2:                        Level 2 GSL BLAS Interface.
                                                              (line 144)
* gsl_blas_dsyr2k:                       Level 3 GSL BLAS Interface.
                                                              (line 154)
* gsl_blas_dsyrk:                        Level 3 GSL BLAS Interface.
                                                              (line 118)
* gsl_blas_dtrmm:                        Level 3 GSL BLAS Interface.
                                                              (line  70)
* gsl_blas_dtrmv:                        Level 2 GSL BLAS Interface.
                                                              (line  30)
* gsl_blas_dtrsm:                        Level 3 GSL BLAS Interface.
                                                              (line  94)
* gsl_blas_dtrsv:                        Level 2 GSL BLAS Interface.
                                                              (line  51)
* gsl_blas_dzasum:                       Level 1 GSL BLAS Interface.
                                                              (line  52)
* gsl_blas_dznrm2:                       Level 1 GSL BLAS Interface.
                                                              (line  41)
* gsl_blas_icamax:                       Level 1 GSL BLAS Interface.
                                                              (line  60)
* gsl_blas_idamax:                       Level 1 GSL BLAS Interface.
                                                              (line  58)
* gsl_blas_isamax:                       Level 1 GSL BLAS Interface.
                                                              (line  57)
* gsl_blas_izamax:                       Level 1 GSL BLAS Interface.
                                                              (line  62)
* gsl_blas_sasum:                        Level 1 GSL BLAS Interface.
                                                              (line  46)
* gsl_blas_saxpy:                        Level 1 GSL BLAS Interface.
                                                              (line  90)
* gsl_blas_scasum:                       Level 1 GSL BLAS Interface.
                                                              (line  51)
* gsl_blas_scnrm2:                       Level 1 GSL BLAS Interface.
                                                              (line  40)
* gsl_blas_scopy:                        Level 1 GSL BLAS Interface.
                                                              (line  80)
* gsl_blas_sdot:                         Level 1 GSL BLAS Interface.
                                                              (line  13)
* gsl_blas_sdsdot:                       Level 1 GSL BLAS Interface.
                                                              (line   8)
* gsl_blas_sgemm:                        Level 3 GSL BLAS Interface.
                                                              (line  10)
* gsl_blas_sgemv:                        Level 2 GSL BLAS Interface.
                                                              (line   9)
* gsl_blas_sger:                         Level 2 GSL BLAS Interface.
                                                              (line  96)
* gsl_blas_snrm2:                        Level 1 GSL BLAS Interface.
                                                              (line  35)
* gsl_blas_srot:                         Level 1 GSL BLAS Interface.
                                                              (line 126)
* gsl_blas_srotg:                        Level 1 GSL BLAS Interface.
                                                              (line 114)
* gsl_blas_srotm:                        Level 1 GSL BLAS Interface.
                                                              (line 141)
* gsl_blas_srotmg:                       Level 1 GSL BLAS Interface.
                                                              (line 133)
* gsl_blas_sscal:                        Level 1 GSL BLAS Interface.
                                                              (line 100)
* gsl_blas_sswap:                        Level 1 GSL BLAS Interface.
                                                              (line  71)
* gsl_blas_ssymm:                        Level 3 GSL BLAS Interface.
                                                              (line  30)
* gsl_blas_ssymv:                        Level 2 GSL BLAS Interface.
                                                              (line  68)
* gsl_blas_ssyr:                         Level 2 GSL BLAS Interface.
                                                              (line 118)
* gsl_blas_ssyr2:                        Level 2 GSL BLAS Interface.
                                                              (line 142)
* gsl_blas_ssyr2k:                       Level 3 GSL BLAS Interface.
                                                              (line 151)
* gsl_blas_ssyrk:                        Level 3 GSL BLAS Interface.
                                                              (line 115)
* gsl_blas_strmm:                        Level 3 GSL BLAS Interface.
                                                              (line  67)
* gsl_blas_strmv:                        Level 2 GSL BLAS Interface.
                                                              (line  27)
* gsl_blas_strsm:                        Level 3 GSL BLAS Interface.
                                                              (line  91)
* gsl_blas_strsv:                        Level 2 GSL BLAS Interface.
                                                              (line  48)
* gsl_blas_zaxpy:                        Level 1 GSL BLAS Interface.
                                                              (line  96)
* gsl_blas_zcopy:                        Level 1 GSL BLAS Interface.
                                                              (line  85)
* gsl_blas_zdotc:                        Level 1 GSL BLAS Interface.
                                                              (line  31)
* gsl_blas_zdotu:                        Level 1 GSL BLAS Interface.
                                                              (line  24)
* gsl_blas_zdscal:                       Level 1 GSL BLAS Interface.
                                                              (line 109)
* gsl_blas_zgemm:                        Level 3 GSL BLAS Interface.
                                                              (line  22)
* gsl_blas_zgemv:                        Level 2 GSL BLAS Interface.
                                                              (line  20)
* gsl_blas_zgerc:                        Level 2 GSL BLAS Interface.
                                                              (line 113)
* gsl_blas_zgeru:                        Level 2 GSL BLAS Interface.
                                                              (line 104)
* gsl_blas_zhemm:                        Level 3 GSL BLAS Interface.
                                                              (line  56)
* gsl_blas_zhemv:                        Level 2 GSL BLAS Interface.
                                                              (line  85)
* gsl_blas_zher:                         Level 2 GSL BLAS Interface.
                                                              (line 131)
* gsl_blas_zher2:                        Level 2 GSL BLAS Interface.
                                                              (line 158)
* gsl_blas_zher2k:                       Level 3 GSL BLAS Interface.
                                                              (line 180)
* gsl_blas_zherk:                        Level 3 GSL BLAS Interface.
                                                              (line 139)
* gsl_blas_zscal:                        Level 1 GSL BLAS Interface.
                                                              (line 105)
* gsl_blas_zswap:                        Level 1 GSL BLAS Interface.
                                                              (line  76)
* gsl_blas_zsymm:                        Level 3 GSL BLAS Interface.
                                                              (line  41)
* gsl_blas_zsyr2k:                       Level 3 GSL BLAS Interface.
                                                              (line 163)
* gsl_blas_zsyrk:                        Level 3 GSL BLAS Interface.
                                                              (line 125)
* gsl_blas_ztrmm:                        Level 3 GSL BLAS Interface.
                                                              (line  78)
* gsl_blas_ztrmv:                        Level 2 GSL BLAS Interface.
                                                              (line  36)
* gsl_blas_ztrsm:                        Level 3 GSL BLAS Interface.
                                                              (line 102)
* gsl_blas_ztrsv:                        Level 2 GSL BLAS Interface.
                                                              (line  57)
* gsl_block_alloc:                       Block allocation.    (line  15)
* gsl_block_calloc:                      Block allocation.    (line  25)
* gsl_block_fprintf:                     Reading and writing blocks.
                                                              (line  27)
* gsl_block_fread:                       Reading and writing blocks.
                                                              (line  17)
* gsl_block_free:                        Block allocation.    (line  29)
* gsl_block_fscanf:                      Reading and writing blocks.
                                                              (line  34)
* gsl_block_fwrite:                      Reading and writing blocks.
                                                              (line  10)
* gsl_cdf_beta_P:                        The Beta Distribution.
                                                              (line  22)
* gsl_cdf_beta_Pinv:                     The Beta Distribution.
                                                              (line  24)
* gsl_cdf_beta_Q:                        The Beta Distribution.
                                                              (line  23)
* gsl_cdf_beta_Qinv:                     The Beta Distribution.
                                                              (line  25)
* gsl_cdf_binomial_P:                    The Binomial Distribution.
                                                              (line  27)
* gsl_cdf_binomial_Q:                    The Binomial Distribution.
                                                              (line  29)
* gsl_cdf_cauchy_P:                      The Cauchy Distribution.
                                                              (line  23)
* gsl_cdf_cauchy_Pinv:                   The Cauchy Distribution.
                                                              (line  25)
* gsl_cdf_cauchy_Q:                      The Cauchy Distribution.
                                                              (line  24)
* gsl_cdf_cauchy_Qinv:                   The Cauchy Distribution.
                                                              (line  26)
* gsl_cdf_chisq_P:                       The Chi-squared Distribution.
                                                              (line  30)
* gsl_cdf_chisq_Pinv:                    The Chi-squared Distribution.
                                                              (line  32)
* gsl_cdf_chisq_Q:                       The Chi-squared Distribution.
                                                              (line  31)
* gsl_cdf_chisq_Qinv:                    The Chi-squared Distribution.
                                                              (line  33)
* gsl_cdf_exponential_P:                 The Exponential Distribution.
                                                              (line  22)
* gsl_cdf_exponential_Pinv:              The Exponential Distribution.
                                                              (line  24)
* gsl_cdf_exponential_Q:                 The Exponential Distribution.
                                                              (line  23)
* gsl_cdf_exponential_Qinv:              The Exponential Distribution.
                                                              (line  25)
* gsl_cdf_exppow_P:                      The Exponential Power Distribution.
                                                              (line  25)
* gsl_cdf_exppow_Q:                      The Exponential Power Distribution.
                                                              (line  26)
* gsl_cdf_fdist_P:                       The F-distribution.  (line  35)
* gsl_cdf_fdist_Pinv:                    The F-distribution.  (line  38)
* gsl_cdf_fdist_Q:                       The F-distribution.  (line  36)
* gsl_cdf_fdist_Qinv:                    The F-distribution.  (line  40)
* gsl_cdf_flat_P:                        The Flat (Uniform) Distribution.
                                                              (line  21)
* gsl_cdf_flat_Pinv:                     The Flat (Uniform) Distribution.
                                                              (line  23)
* gsl_cdf_flat_Q:                        The Flat (Uniform) Distribution.
                                                              (line  22)
* gsl_cdf_flat_Qinv:                     The Flat (Uniform) Distribution.
                                                              (line  24)
* gsl_cdf_gamma_P:                       The Gamma Distribution.
                                                              (line  32)
* gsl_cdf_gamma_Pinv:                    The Gamma Distribution.
                                                              (line  34)
* gsl_cdf_gamma_Q:                       The Gamma Distribution.
                                                              (line  33)
* gsl_cdf_gamma_Qinv:                    The Gamma Distribution.
                                                              (line  35)
* gsl_cdf_gaussian_P:                    The Gaussian Distribution.
                                                              (line  42)
* gsl_cdf_gaussian_Pinv:                 The Gaussian Distribution.
                                                              (line  44)
* gsl_cdf_gaussian_Q:                    The Gaussian Distribution.
                                                              (line  43)
* gsl_cdf_gaussian_Qinv:                 The Gaussian Distribution.
                                                              (line  45)
* gsl_cdf_geometric_P:                   The Geometric Distribution.
                                                              (line  26)
* gsl_cdf_geometric_Q:                   The Geometric Distribution.
                                                              (line  27)
* gsl_cdf_gumbel1_P:                     The Type-1 Gumbel Distribution.
                                                              (line  22)
* gsl_cdf_gumbel1_Pinv:                  The Type-1 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_gumbel1_Q:                     The Type-1 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_gumbel1_Qinv:                  The Type-1 Gumbel Distribution.
                                                              (line  25)
* gsl_cdf_gumbel2_P:                     The Type-2 Gumbel Distribution.
                                                              (line  22)
* gsl_cdf_gumbel2_Pinv:                  The Type-2 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_gumbel2_Q:                     The Type-2 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_gumbel2_Qinv:                  The Type-2 Gumbel Distribution.
                                                              (line  25)
* gsl_cdf_hypergeometric_P:              The Hypergeometric Distribution.
                                                              (line  31)
* gsl_cdf_hypergeometric_Q:              The Hypergeometric Distribution.
                                                              (line  33)
* gsl_cdf_laplace_P:                     The Laplace Distribution.
                                                              (line  20)
* gsl_cdf_laplace_Pinv:                  The Laplace Distribution.
                                                              (line  22)
* gsl_cdf_laplace_Q:                     The Laplace Distribution.
                                                              (line  21)
* gsl_cdf_laplace_Qinv:                  The Laplace Distribution.
                                                              (line  23)
* gsl_cdf_logistic_P:                    The Logistic Distribution.
                                                              (line  21)
* gsl_cdf_logistic_Pinv:                 The Logistic Distribution.
                                                              (line  23)
* gsl_cdf_logistic_Q:                    The Logistic Distribution.
                                                              (line  22)
* gsl_cdf_logistic_Qinv:                 The Logistic Distribution.
                                                              (line  24)
* gsl_cdf_lognormal_P:                   The Lognormal Distribution.
                                                              (line  24)
* gsl_cdf_lognormal_Pinv:                The Lognormal Distribution.
                                                              (line  28)
* gsl_cdf_lognormal_Q:                   The Lognormal Distribution.
                                                              (line  26)
* gsl_cdf_lognormal_Qinv:                The Lognormal Distribution.
                                                              (line  30)
* gsl_cdf_negative_binomial_P:           The Negative Binomial Distribution.
                                                              (line  26)
* gsl_cdf_negative_binomial_Q:           The Negative Binomial Distribution.
                                                              (line  28)
* gsl_cdf_pareto_P:                      The Pareto Distribution.
                                                              (line  22)
* gsl_cdf_pareto_Pinv:                   The Pareto Distribution.
                                                              (line  24)
* gsl_cdf_pareto_Q:                      The Pareto Distribution.
                                                              (line  23)
* gsl_cdf_pareto_Qinv:                   The Pareto Distribution.
                                                              (line  25)
* gsl_cdf_pascal_P:                      The Pascal Distribution.
                                                              (line  25)
* gsl_cdf_pascal_Q:                      The Pascal Distribution.
                                                              (line  27)
* gsl_cdf_poisson_P:                     The Poisson Distribution.
                                                              (line  22)
* gsl_cdf_poisson_Q:                     The Poisson Distribution.
                                                              (line  23)
* gsl_cdf_rayleigh_P:                    The Rayleigh Distribution.
                                                              (line  21)
* gsl_cdf_rayleigh_Pinv:                 The Rayleigh Distribution.
                                                              (line  23)
* gsl_cdf_rayleigh_Q:                    The Rayleigh Distribution.
                                                              (line  22)
* gsl_cdf_rayleigh_Qinv:                 The Rayleigh Distribution.
                                                              (line  24)
* gsl_cdf_tdist_P:                       The t-distribution.  (line  30)
* gsl_cdf_tdist_Pinv:                    The t-distribution.  (line  32)
* gsl_cdf_tdist_Q:                       The t-distribution.  (line  31)
* gsl_cdf_tdist_Qinv:                    The t-distribution.  (line  33)
* gsl_cdf_ugaussian_P:                   The Gaussian Distribution.
                                                              (line  50)
* gsl_cdf_ugaussian_Pinv:                The Gaussian Distribution.
                                                              (line  52)
* gsl_cdf_ugaussian_Q:                   The Gaussian Distribution.
                                                              (line  51)
* gsl_cdf_ugaussian_Qinv:                The Gaussian Distribution.
                                                              (line  53)
* gsl_cdf_weibull_P:                     The Weibull Distribution.
                                                              (line  22)
* gsl_cdf_weibull_Pinv:                  The Weibull Distribution.
                                                              (line  24)
* gsl_cdf_weibull_Q:                     The Weibull Distribution.
                                                              (line  23)
* gsl_cdf_weibull_Qinv:                  The Weibull Distribution.
                                                              (line  25)
* gsl_cheb_alloc:                        Creation and Calculation of Chebyshev Series.
                                                              (line   7)
* gsl_cheb_calc_deriv:                   Derivatives and Integrals.
                                                              (line  14)
* gsl_cheb_calc_integ:                   Derivatives and Integrals.
                                                              (line  21)
* gsl_cheb_eval:                         Chebyshev Series Evaluation.
                                                              (line   8)
* gsl_cheb_eval_err:                     Chebyshev Series Evaluation.
                                                              (line  12)
* gsl_cheb_eval_n:                       Chebyshev Series Evaluation.
                                                              (line  19)
* gsl_cheb_eval_n_err:                   Chebyshev Series Evaluation.
                                                              (line  25)
* gsl_cheb_free:                         Creation and Calculation of Chebyshev Series.
                                                              (line  11)
* gsl_cheb_init:                         Creation and Calculation of Chebyshev Series.
                                                              (line  15)
* gsl_combination_alloc:                 Combination allocation.
                                                              (line   8)
* gsl_combination_calloc:                Combination allocation.
                                                              (line  17)
* gsl_combination_data:                  Combination properties.
                                                              (line  14)
* gsl_combination_fprintf:               Reading and writing combinations.
                                                              (line  29)
* gsl_combination_fread:                 Reading and writing combinations.
                                                              (line  19)
* gsl_combination_free:                  Combination allocation.
                                                              (line  31)
* gsl_combination_fscanf:                Reading and writing combinations.
                                                              (line  38)
* gsl_combination_fwrite:                Reading and writing combinations.
                                                              (line  11)
* gsl_combination_get:                   Accessing combination elements.
                                                              (line  11)
* gsl_combination_init_first:            Combination allocation.
                                                              (line  23)
* gsl_combination_init_last:             Combination allocation.
                                                              (line  27)
* gsl_combination_k:                     Combination properties.
                                                              (line  10)
* gsl_combination_memcpy:                Combination allocation.
                                                              (line  35)
* gsl_combination_n:                     Combination properties.
                                                              (line   7)
* gsl_combination_next:                  Combination functions.
                                                              (line   7)
* gsl_combination_prev:                  Combination functions.
                                                              (line  15)
* gsl_combination_valid:                 Combination properties.
                                                              (line  18)
* gsl_complex_abs:                       Properties of complex numbers.
                                                              (line  11)
* gsl_complex_abs2:                      Properties of complex numbers.
                                                              (line  14)
* gsl_complex_add:                       Complex arithmetic operators.
                                                              (line   7)
* gsl_complex_add_imag:                  Complex arithmetic operators.
                                                              (line  39)
* gsl_complex_add_real:                  Complex arithmetic operators.
                                                              (line  23)
* gsl_complex_arccos:                    Inverse Complex Trigonometric Functions.
                                                              (line  20)
* gsl_complex_arccos_real:               Inverse Complex Trigonometric Functions.
                                                              (line  25)
* gsl_complex_arccosh:                   Inverse Complex Hyperbolic Functions.
                                                              (line  12)
* gsl_complex_arccosh_real:              Inverse Complex Hyperbolic Functions.
                                                              (line  17)
* gsl_complex_arccot:                    Inverse Complex Trigonometric Functions.
                                                              (line  53)
* gsl_complex_arccoth:                   Inverse Complex Hyperbolic Functions.
                                                              (line  38)
* gsl_complex_arccsc:                    Inverse Complex Trigonometric Functions.
                                                              (line  45)
* gsl_complex_arccsc_real:               Inverse Complex Trigonometric Functions.
                                                              (line  49)
* gsl_complex_arccsch:                   Inverse Complex Hyperbolic Functions.
                                                              (line  34)
* gsl_complex_arcsec:                    Inverse Complex Trigonometric Functions.
                                                              (line  37)
* gsl_complex_arcsec_real:               Inverse Complex Trigonometric Functions.
                                                              (line  41)
* gsl_complex_arcsech:                   Inverse Complex Hyperbolic Functions.
                                                              (line  30)
* gsl_complex_arcsin:                    Inverse Complex Trigonometric Functions.
                                                              (line   7)
* gsl_complex_arcsin_real:               Inverse Complex Trigonometric Functions.
                                                              (line  12)
* gsl_complex_arcsinh:                   Inverse Complex Hyperbolic Functions.
                                                              (line   7)
* gsl_complex_arctan:                    Inverse Complex Trigonometric Functions.
                                                              (line  32)
* gsl_complex_arctanh:                   Inverse Complex Hyperbolic Functions.
                                                              (line  21)
* gsl_complex_arctanh_real:              Inverse Complex Hyperbolic Functions.
                                                              (line  26)
* gsl_complex_arg:                       Properties of complex numbers.
                                                              (line   7)
* gsl_complex_conjugate:                 Complex arithmetic operators.
                                                              (line  55)
* gsl_complex_cos:                       Complex Trigonometric Functions.
                                                              (line  11)
* gsl_complex_cosh:                      Complex Hyperbolic Functions.
                                                              (line  11)
* gsl_complex_cot:                       Complex Trigonometric Functions.
                                                              (line  27)
* gsl_complex_coth:                      Complex Hyperbolic Functions.
                                                              (line  27)
* gsl_complex_csc:                       Complex Trigonometric Functions.
                                                              (line  23)
* gsl_complex_csch:                      Complex Hyperbolic Functions.
                                                              (line  23)
* gsl_complex_div:                       Complex arithmetic operators.
                                                              (line  19)
* gsl_complex_div_imag:                  Complex arithmetic operators.
                                                              (line  51)
* gsl_complex_div_real:                  Complex arithmetic operators.
                                                              (line  35)
* gsl_complex_exp:                       Elementary Complex Functions.
                                                              (line  25)
* gsl_complex_inverse:                   Complex arithmetic operators.
                                                              (line  59)
* gsl_complex_log:                       Elementary Complex Functions.
                                                              (line  29)
* gsl_complex_log10:                     Elementary Complex Functions.
                                                              (line  34)
* gsl_complex_log_b:                     Elementary Complex Functions.
                                                              (line  39)
* gsl_complex_logabs:                    Properties of complex numbers.
                                                              (line  18)
* gsl_complex_mul:                       Complex arithmetic operators.
                                                              (line  15)
* gsl_complex_mul_imag:                  Complex arithmetic operators.
                                                              (line  47)
* gsl_complex_mul_real:                  Complex arithmetic operators.
                                                              (line  31)
* gsl_complex_negative:                  Complex arithmetic operators.
                                                              (line  63)
* gsl_complex_polar:                     Complex numbers.     (line  29)
* gsl_complex_pow:                       Elementary Complex Functions.
                                                              (line  16)
* gsl_complex_pow_real:                  Elementary Complex Functions.
                                                              (line  21)
* gsl_complex_rect:                      Complex numbers.     (line  25)
* gsl_complex_sec:                       Complex Trigonometric Functions.
                                                              (line  19)
* gsl_complex_sech:                      Complex Hyperbolic Functions.
                                                              (line  19)
* gsl_complex_sin:                       Complex Trigonometric Functions.
                                                              (line   7)
* gsl_complex_sinh:                      Complex Hyperbolic Functions.
                                                              (line   7)
* gsl_complex_sqrt:                      Elementary Complex Functions.
                                                              (line   7)
* gsl_complex_sqrt_real:                 Elementary Complex Functions.
                                                              (line  12)
* gsl_complex_sub:                       Complex arithmetic operators.
                                                              (line  11)
* gsl_complex_sub_imag:                  Complex arithmetic operators.
                                                              (line  43)
* gsl_complex_sub_real:                  Complex arithmetic operators.
                                                              (line  27)
* gsl_complex_tan:                       Complex Trigonometric Functions.
                                                              (line  15)
* gsl_complex_tanh:                      Complex Hyperbolic Functions.
                                                              (line  15)
* gsl_deriv_backward:                    Numerical Differentiation functions.
                                                              (line  42)
* gsl_deriv_central:                     Numerical Differentiation functions.
                                                              (line   8)
* gsl_deriv_forward:                     Numerical Differentiation functions.
                                                              (line  24)
* gsl_dht_alloc:                         Discrete Hankel Transform Functions.
                                                              (line   7)
* gsl_dht_apply:                         Discrete Hankel Transform Functions.
                                                              (line  24)
* gsl_dht_free:                          Discrete Hankel Transform Functions.
                                                              (line  20)
* gsl_dht_init:                          Discrete Hankel Transform Functions.
                                                              (line  11)
* gsl_dht_k_sample:                      Discrete Hankel Transform Functions.
                                                              (line  34)
* gsl_dht_new:                           Discrete Hankel Transform Functions.
                                                              (line  16)
* gsl_dht_x_sample:                      Discrete Hankel Transform Functions.
                                                              (line  29)
* GSL_EDOM:                              Error Codes.         (line  13)
* gsl_eigen_herm:                        Complex Hermitian Matrices.
                                                              (line  17)
* gsl_eigen_herm_alloc:                  Complex Hermitian Matrices.
                                                              (line   8)
* gsl_eigen_herm_free:                   Complex Hermitian Matrices.
                                                              (line  13)
* gsl_eigen_hermv:                       Complex Hermitian Matrices.
                                                              (line  37)
* gsl_eigen_hermv_alloc:                 Complex Hermitian Matrices.
                                                              (line  27)
* gsl_eigen_hermv_free:                  Complex Hermitian Matrices.
                                                              (line  32)
* gsl_eigen_hermv_sort:                  Sorting Eigenvalues and Eigenvectors.
                                                              (line  28)
* gsl_eigen_symm:                        Real Symmetric Matrices.
                                                              (line  17)
* gsl_eigen_symm_alloc:                  Real Symmetric Matrices.
                                                              (line   8)
* gsl_eigen_symm_free:                   Real Symmetric Matrices.
                                                              (line  13)
* gsl_eigen_symmv:                       Real Symmetric Matrices.
                                                              (line  35)
* gsl_eigen_symmv_alloc:                 Real Symmetric Matrices.
                                                              (line  26)
* gsl_eigen_symmv_free:                  Real Symmetric Matrices.
                                                              (line  31)
* gsl_eigen_symmv_sort:                  Sorting Eigenvalues and Eigenvectors.
                                                              (line   8)
* GSL_EINVAL:                            Error Codes.         (line  29)
* GSL_ENOMEM:                            Error Codes.         (line  23)
* GSL_ERANGE:                            Error Codes.         (line  18)
* GSL_ERROR:                             Using GSL error reporting in your own functions.
                                                              (line  16)
* GSL_ERROR_VAL:                         Using GSL error reporting in your own functions.
                                                              (line  36)
* gsl_expm1:                             Elementary Functions.
                                                              (line  18)
* gsl_fcmp:                              Approximate Comparison of Floating Point Numbers.
                                                              (line  13)
* gsl_fft_complex_backward:              Mixed-radix FFT routines for complex data.
                                                              (line 130)
* gsl_fft_complex_forward:               Mixed-radix FFT routines for complex data.
                                                              (line 122)
* gsl_fft_complex_inverse:               Mixed-radix FFT routines for complex data.
                                                              (line 134)
* gsl_fft_complex_radix2_backward:       Radix-2 FFT routines for complex data.
                                                              (line  23)
* gsl_fft_complex_radix2_dif_backward:   Radix-2 FFT routines for complex data.
                                                              (line  43)
* gsl_fft_complex_radix2_dif_forward:    Radix-2 FFT routines for complex data.
                                                              (line  38)
* gsl_fft_complex_radix2_dif_inverse:    Radix-2 FFT routines for complex data.
                                                              (line  45)
* gsl_fft_complex_radix2_dif_transform:  Radix-2 FFT routines for complex data.
                                                              (line  41)
* gsl_fft_complex_radix2_forward:        Radix-2 FFT routines for complex data.
                                                              (line  18)
* gsl_fft_complex_radix2_inverse:        Radix-2 FFT routines for complex data.
                                                              (line  25)
* gsl_fft_complex_radix2_transform:      Radix-2 FFT routines for complex data.
                                                              (line  21)
* gsl_fft_complex_transform:             Mixed-radix FFT routines for complex data.
                                                              (line 126)
* gsl_fft_complex_wavetable_alloc:       Mixed-radix FFT routines for complex data.
                                                              (line  47)
* gsl_fft_complex_wavetable_free:        Mixed-radix FFT routines for complex data.
                                                              (line  66)
* gsl_fft_complex_workspace_alloc:       Mixed-radix FFT routines for complex data.
                                                              (line 107)
* gsl_fft_complex_workspace_free:        Mixed-radix FFT routines for complex data.
                                                              (line 112)
* gsl_fft_halfcomplex_radix2_backward:   Radix-2 FFT routines for real data.
                                                              (line  55)
* gsl_fft_halfcomplex_radix2_inverse:    Radix-2 FFT routines for real data.
                                                              (line  53)
* gsl_fft_halfcomplex_transform:         Mixed-radix FFT routines for real data.
                                                              (line 125)
* gsl_fft_halfcomplex_unpack:            Mixed-radix FFT routines for real data.
                                                              (line 155)
* gsl_fft_halfcomplex_wavetable_alloc:   Mixed-radix FFT routines for real data.
                                                              (line  76)
* gsl_fft_halfcomplex_wavetable_free:    Mixed-radix FFT routines for real data.
                                                              (line  97)
* gsl_fft_real_radix2_transform:         Radix-2 FFT routines for real data.
                                                              (line  15)
* gsl_fft_real_transform:                Mixed-radix FFT routines for real data.
                                                              (line 122)
* gsl_fft_real_unpack:                   Mixed-radix FFT routines for real data.
                                                              (line 139)
* gsl_fft_real_wavetable_alloc:          Mixed-radix FFT routines for real data.
                                                              (line  74)
* gsl_fft_real_wavetable_free:           Mixed-radix FFT routines for real data.
                                                              (line  95)
* gsl_fft_real_workspace_alloc:          Mixed-radix FFT routines for real data.
                                                              (line 106)
* gsl_fft_real_workspace_free:           Mixed-radix FFT routines for real data.
                                                              (line 112)
* gsl_finite:                            Infinities and Not-a-number.
                                                              (line  26)
* gsl_fit_linear:                        Linear regression.   (line  13)
* gsl_fit_linear_est:                    Linear regression.   (line  42)
* gsl_fit_mul:                           Linear fitting without a constant term.
                                                              (line  13)
* gsl_fit_mul_est:                       Linear fitting without a constant term.
                                                              (line  38)
* gsl_fit_wlinear:                       Linear regression.   (line  27)
* gsl_fit_wmul:                          Linear fitting without a constant term.
                                                              (line  25)
* gsl_frexp:                             Elementary Functions.
                                                              (line  44)
* gsl_heapsort:                          Sorting objects.     (line  18)
* gsl_heapsort_index:                    Sorting objects.     (line  59)
* gsl_histogram2d_accumulate:            Updating and accessing 2D histogram elements.
                                                              (line  27)
* gsl_histogram2d_add:                   2D Histogram Operations.
                                                              (line  13)
* gsl_histogram2d_alloc:                 2D Histogram allocation.
                                                              (line  16)
* gsl_histogram2d_clone:                 Copying 2D Histograms.
                                                              (line  14)
* gsl_histogram2d_cov:                   2D Histogram Statistics.
                                                              (line  53)
* gsl_histogram2d_div:                   2D Histogram Operations.
                                                              (line  33)
* gsl_histogram2d_equal_bins_p:          2D Histogram Operations.
                                                              (line   8)
* gsl_histogram2d_find:                  Searching 2D histogram ranges.
                                                              (line  11)
* gsl_histogram2d_fprintf:               Reading and writing 2D histograms.
                                                              (line  30)
* gsl_histogram2d_fread:                 Reading and writing 2D histograms.
                                                              (line  19)
* gsl_histogram2d_free:                  2D Histogram allocation.
                                                              (line  38)
* gsl_histogram2d_fscanf:                Reading and writing 2D histograms.
                                                              (line  66)
* gsl_histogram2d_fwrite:                Reading and writing 2D histograms.
                                                              (line  11)
* gsl_histogram2d_get:                   Updating and accessing 2D histogram elements.
                                                              (line  33)
* gsl_histogram2d_get_xrange:            Updating and accessing 2D histogram elements.
                                                              (line  40)
* gsl_histogram2d_get_yrange:            Updating and accessing 2D histogram elements.
                                                              (line  42)
* gsl_histogram2d_increment:             Updating and accessing 2D histogram elements.
                                                              (line  14)
* gsl_histogram2d_max_bin:               2D Histogram Statistics.
                                                              (line  12)
* gsl_histogram2d_max_val:               2D Histogram Statistics.
                                                              (line   7)
* gsl_histogram2d_memcpy:                Copying 2D Histograms.
                                                              (line   8)
* gsl_histogram2d_min_bin:               2D Histogram Statistics.
                                                              (line  23)
* gsl_histogram2d_min_val:               2D Histogram Statistics.
                                                              (line  18)
* gsl_histogram2d_mul:                   2D Histogram Operations.
                                                              (line  26)
* gsl_histogram2d_nx:                    Updating and accessing 2D histogram elements.
                                                              (line  56)
* gsl_histogram2d_ny:                    Updating and accessing 2D histogram elements.
                                                              (line  59)
* gsl_histogram2d_pdf_alloc:             Resampling from 2D histograms.
                                                              (line  43)
* gsl_histogram2d_pdf_free:              Resampling from 2D histograms.
                                                              (line  58)
* gsl_histogram2d_pdf_init:              Resampling from 2D histograms.
                                                              (line  51)
* gsl_histogram2d_pdf_sample:            Resampling from 2D histograms.
                                                              (line  63)
* gsl_histogram2d_reset:                 Updating and accessing 2D histogram elements.
                                                              (line  65)
* gsl_histogram2d_scale:                 2D Histogram Operations.
                                                              (line  40)
* gsl_histogram2d_set_ranges:            2D Histogram allocation.
                                                              (line  27)
* gsl_histogram2d_set_ranges_uniform:    2D Histogram allocation.
                                                              (line  33)
* gsl_histogram2d_shift:                 2D Histogram Operations.
                                                              (line  45)
* gsl_histogram2d_sub:                   2D Histogram Operations.
                                                              (line  19)
* gsl_histogram2d_sum:                   2D Histogram Statistics.
                                                              (line  59)
* gsl_histogram2d_xmax:                  Updating and accessing 2D histogram elements.
                                                              (line  54)
* gsl_histogram2d_xmean:                 2D Histogram Statistics.
                                                              (line  29)
* gsl_histogram2d_xmin:                  Updating and accessing 2D histogram elements.
                                                              (line  55)
* gsl_histogram2d_xsigma:                2D Histogram Statistics.
                                                              (line  41)
* gsl_histogram2d_ymax:                  Updating and accessing 2D histogram elements.
                                                              (line  57)
* gsl_histogram2d_ymean:                 2D Histogram Statistics.
                                                              (line  35)
* gsl_histogram2d_ymin:                  Updating and accessing 2D histogram elements.
                                                              (line  58)
* gsl_histogram2d_ysigma:                2D Histogram Statistics.
                                                              (line  47)
* gsl_histogram_accumulate:              Updating and accessing histogram elements.
                                                              (line  27)
* gsl_histogram_add:                     Histogram Operations.
                                                              (line  13)
* gsl_histogram_alloc:                   Histogram allocation.
                                                              (line  15)
* gsl_histogram_bins:                    Updating and accessing histogram elements.
                                                              (line  54)
* gsl_histogram_clone:                   Copying Histograms.  (line  14)
* gsl_histogram_div:                     Histogram Operations.
                                                              (line  32)
* gsl_histogram_equal_bins_p:            Histogram Operations.
                                                              (line   8)
* gsl_histogram_find:                    Searching histogram ranges.
                                                              (line  11)
* gsl_histogram_fprintf:                 Reading and writing histograms.
                                                              (line  29)
* gsl_histogram_fread:                   Reading and writing histograms.
                                                              (line  18)
* gsl_histogram_free:                    Histogram allocation.
                                                              (line  62)
* gsl_histogram_fscanf:                  Reading and writing histograms.
                                                              (line  53)
* gsl_histogram_fwrite:                  Reading and writing histograms.
                                                              (line  11)
* gsl_histogram_get:                     Updating and accessing histogram elements.
                                                              (line  33)
* gsl_histogram_get_range:               Updating and accessing histogram elements.
                                                              (line  40)
* gsl_histogram_increment:               Updating and accessing histogram elements.
                                                              (line  12)
* gsl_histogram_max:                     Updating and accessing histogram elements.
                                                              (line  52)
* gsl_histogram_max_bin:                 Histogram Statistics.
                                                              (line  11)
* gsl_histogram_max_val:                 Histogram Statistics.
                                                              (line   7)
* gsl_histogram_mean:                    Histogram Statistics.
                                                              (line  25)
* gsl_histogram_memcpy:                  Copying Histograms.  (line   8)
* gsl_histogram_min:                     Updating and accessing histogram elements.
                                                              (line  53)
* gsl_histogram_min_bin:                 Histogram Statistics.
                                                              (line  20)
* gsl_histogram_min_val:                 Histogram Statistics.
                                                              (line  16)
* gsl_histogram_mul:                     Histogram Operations.
                                                              (line  25)
* gsl_histogram_pdf_alloc:               The histogram probability distribution struct.
                                                              (line  36)
* gsl_histogram_pdf_free:                The histogram probability distribution struct.
                                                              (line  50)
* gsl_histogram_pdf_init:                The histogram probability distribution struct.
                                                              (line  44)
* gsl_histogram_pdf_sample:              The histogram probability distribution struct.
                                                              (line  55)
* gsl_histogram_reset:                   Updating and accessing histogram elements.
                                                              (line  60)
* gsl_histogram_scale:                   Histogram Operations.
                                                              (line  38)
* gsl_histogram_set_ranges:              Histogram allocation.
                                                              (line  25)
* gsl_histogram_set_ranges_uniform:      Histogram allocation.
                                                              (line  50)
* gsl_histogram_shift:                   Histogram Operations.
                                                              (line  42)
* gsl_histogram_sigma:                   Histogram Statistics.
                                                              (line  31)
* gsl_histogram_sub:                     Histogram Operations.
                                                              (line  19)
* gsl_histogram_sum:                     Histogram Statistics.
                                                              (line  38)
* gsl_hypot:                             Elementary Functions.
                                                              (line  23)
* gsl_ieee_env_setup:                    Setting up your IEEE environment.
                                                              (line  24)
* gsl_ieee_fprintf_double:               Representation of floating point numbers.
                                                              (line  57)
* gsl_ieee_fprintf_float:                Representation of floating point numbers.
                                                              (line  55)
* gsl_ieee_printf_double:                Representation of floating point numbers.
                                                              (line  84)
* gsl_ieee_printf_float:                 Representation of floating point numbers.
                                                              (line  83)
* GSL_IMAG:                              Complex numbers.     (line  35)
* gsl_integration_qag:                   QAG adaptive integration.
                                                              (line  28)
* gsl_integration_qagi:                  QAGI adaptive integration on infinite intervals.
                                                              (line  10)
* gsl_integration_qagil:                 QAGI adaptive integration on infinite intervals.
                                                              (line  41)
* gsl_integration_qagiu:                 QAGI adaptive integration on infinite intervals.
                                                              (line  27)
* gsl_integration_qagp:                  QAGP adaptive integration with known singular points.
                                                              (line  10)
* gsl_integration_qags:                  QAGS adaptive integration with singularities.
                                                              (line  19)
* gsl_integration_qawc:                  QAWC adaptive integration for Cauchy principal values.
                                                              (line  10)
* gsl_integration_qawf:                  QAWF adaptive integration for Fourier integrals.
                                                              (line  12)
* gsl_integration_qawo:                  QAWO adaptive integration for oscillatory functions.
                                                              (line  59)
* gsl_integration_qawo_table_alloc:      QAWO adaptive integration for oscillatory functions.
                                                              (line  14)
* gsl_integration_qawo_table_free:       QAWO adaptive integration for oscillatory functions.
                                                              (line  52)
* gsl_integration_qawo_table_set:        QAWO adaptive integration for oscillatory functions.
                                                              (line  42)
* gsl_integration_qawo_table_set_length: QAWO adaptive integration for oscillatory functions.
                                                              (line  47)
* gsl_integration_qaws:                  QAWS adaptive integration for singular functions.
                                                              (line  53)
* gsl_integration_qaws_table_alloc:      QAWS adaptive integration for singular functions.
                                                              (line  14)
* gsl_integration_qaws_table_free:       QAWS adaptive integration for singular functions.
                                                              (line  45)
* gsl_integration_qaws_table_set:        QAWS adaptive integration for singular functions.
                                                              (line  40)
* gsl_integration_qng:                   QNG non-adaptive Gauss-Kronrod integration.
                                                              (line  13)
* gsl_integration_workspace_alloc:       QAG adaptive integration.
                                                              (line  17)
* gsl_integration_workspace_free:        QAG adaptive integration.
                                                              (line  22)
* gsl_interp_accel_alloc:                Index Look-up and Acceleration.
                                                              (line  19)
* gsl_interp_accel_find:                 Index Look-up and Acceleration.
                                                              (line  26)
* gsl_interp_accel_free:                 Index Look-up and Acceleration.
                                                              (line  32)
* gsl_interp_akima:                      Interpolation Types. (line  36)
* gsl_interp_akima_periodic:             Interpolation Types. (line  40)
* gsl_interp_alloc:                      Interpolation Functions.
                                                              (line  11)
* gsl_interp_bsearch:                    Index Look-up and Acceleration.
                                                              (line  14)
* gsl_interp_cspline:                    Interpolation Types. (line  20)
* gsl_interp_cspline_periodic:           Interpolation Types. (line  26)
* gsl_interp_eval:                       Evaluation of Interpolating Functions.
                                                              (line   9)
* gsl_interp_eval_deriv:                 Evaluation of Interpolating Functions.
                                                              (line  19)
* gsl_interp_eval_deriv2:                Evaluation of Interpolating Functions.
                                                              (line  29)
* gsl_interp_eval_deriv2_e:              Evaluation of Interpolating Functions.
                                                              (line  32)
* gsl_interp_eval_deriv_e:               Evaluation of Interpolating Functions.
                                                              (line  22)
* gsl_interp_eval_e:                     Evaluation of Interpolating Functions.
                                                              (line  12)
* gsl_interp_eval_integ:                 Evaluation of Interpolating Functions.
                                                              (line  39)
* gsl_interp_eval_integ_e:               Evaluation of Interpolating Functions.
                                                              (line  42)
* gsl_interp_free:                       Interpolation Functions.
                                                              (line  24)
* gsl_interp_init:                       Interpolation Functions.
                                                              (line  16)
* gsl_interp_linear:                     Interpolation Types. (line   9)
* gsl_interp_min_size:                   Interpolation Types. (line  58)
* gsl_interp_name:                       Interpolation Types. (line  46)
* gsl_interp_polynomial:                 Interpolation Types. (line  13)
* GSL_IS_EVEN:                           Testing for Odd and Even Numbers.
                                                              (line  11)
* GSL_IS_ODD:                            Testing for Odd and Even Numbers.
                                                              (line   7)
* gsl_isinf:                             Infinities and Not-a-number.
                                                              (line  22)
* gsl_isnan:                             Infinities and Not-a-number.
                                                              (line  19)
* gsl_ldexp:                             Elementary Functions.
                                                              (line  40)
* gsl_linalg_bidiag_decomp:              Bidiagonalization.   (line  17)
* gsl_linalg_bidiag_unpack:              Bidiagonalization.   (line  29)
* gsl_linalg_bidiag_unpack2:             Bidiagonalization.   (line  37)
* gsl_linalg_bidiag_unpack_B:            Bidiagonalization.   (line  44)
* gsl_linalg_cholesky_decomp:            Cholesky Decomposition.
                                                              (line  19)
* gsl_linalg_cholesky_solve:             Cholesky Decomposition.
                                                              (line  29)
* gsl_linalg_cholesky_svx:               Cholesky Decomposition.
                                                              (line  35)
* gsl_linalg_complex_LU_decomp:          LU Decomposition.    (line  22)
* gsl_linalg_complex_LU_det:             LU Decomposition.    (line  85)
* gsl_linalg_complex_LU_invert:          LU Decomposition.    (line  73)
* gsl_linalg_complex_LU_lndet:           LU Decomposition.    (line  93)
* gsl_linalg_complex_LU_refine:          LU Decomposition.    (line  63)
* gsl_linalg_complex_LU_sgndet:          LU Decomposition.    (line 101)
* gsl_linalg_complex_LU_solve:           LU Decomposition.    (line  44)
* gsl_linalg_complex_LU_svx:             LU Decomposition.    (line  52)
* gsl_linalg_hermtd_decomp:              Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  16)
* gsl_linalg_hermtd_unpack:              Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  29)
* gsl_linalg_hermtd_unpack_T:            Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  36)
* gsl_linalg_HH_solve:                   Householder solver for linear systems.
                                                              (line   8)
* gsl_linalg_HH_svx:                     Householder solver for linear systems.
                                                              (line  14)
* gsl_linalg_householder_hm:             Householder Transformations.
                                                              (line  25)
* gsl_linalg_householder_hv:             Householder Transformations.
                                                              (line  37)
* gsl_linalg_householder_mh:             Householder Transformations.
                                                              (line  31)
* gsl_linalg_householder_transform:      Householder Transformations.
                                                              (line  18)
* gsl_linalg_LU_decomp:                  LU Decomposition.    (line  20)
* gsl_linalg_LU_det:                     LU Decomposition.    (line  83)
* gsl_linalg_LU_invert:                  LU Decomposition.    (line  70)
* gsl_linalg_LU_lndet:                   LU Decomposition.    (line  91)
* gsl_linalg_LU_refine:                  LU Decomposition.    (line  59)
* gsl_linalg_LU_sgndet:                  LU Decomposition.    (line  99)
* gsl_linalg_LU_solve:                   LU Decomposition.    (line  41)
* gsl_linalg_LU_svx:                     LU Decomposition.    (line  50)
* gsl_linalg_QR_decomp:                  QR Decomposition.    (line  21)
* gsl_linalg_QR_lssolve:                 QR Decomposition.    (line  53)
* gsl_linalg_QR_QRsolve:                 QR Decomposition.    (line  96)
* gsl_linalg_QR_QTvec:                   QR Decomposition.    (line  63)
* gsl_linalg_QR_Qvec:                    QR Decomposition.    (line  71)
* gsl_linalg_QR_Rsolve:                  QR Decomposition.    (line  78)
* gsl_linalg_QR_Rsvx:                    QR Decomposition.    (line  84)
* gsl_linalg_QR_solve:                   QR Decomposition.    (line  38)
* gsl_linalg_QR_svx:                     QR Decomposition.    (line  45)
* gsl_linalg_QR_unpack:                  QR Decomposition.    (line  91)
* gsl_linalg_QR_update:                  QR Decomposition.    (line 102)
* gsl_linalg_QRPT_decomp:                QR Decomposition with Column Pivoting.
                                                              (line  20)
* gsl_linalg_QRPT_decomp2:               QR Decomposition with Column Pivoting.
                                                              (line  42)
* gsl_linalg_QRPT_QRsolve:               QR Decomposition with Column Pivoting.
                                                              (line  63)
* gsl_linalg_QRPT_Rsolve:                QR Decomposition with Column Pivoting.
                                                              (line  78)
* gsl_linalg_QRPT_Rsvx:                  QR Decomposition with Column Pivoting.
                                                              (line  83)
* gsl_linalg_QRPT_solve:                 QR Decomposition with Column Pivoting.
                                                              (line  49)
* gsl_linalg_QRPT_svx:                   QR Decomposition with Column Pivoting.
                                                              (line  55)
* gsl_linalg_QRPT_update:                QR Decomposition with Column Pivoting.
                                                              (line  70)
* gsl_linalg_R_solve:                    QR Decomposition.    (line 109)
* gsl_linalg_R_svx:                      QR Decomposition.    (line 114)
* gsl_linalg_solve_cyc_tridiag:          Tridiagonal Systems. (line  35)
* gsl_linalg_solve_symm_cyc_tridiag:     Tridiagonal Systems. (line  49)
* gsl_linalg_solve_symm_tridiag:         Tridiagonal Systems. (line  22)
* gsl_linalg_solve_tridiag:              Tridiagonal Systems. (line   9)
* gsl_linalg_SV_decomp:                  Singular Value Decomposition.
                                                              (line  28)
* gsl_linalg_SV_decomp_jacobi:           Singular Value Decomposition.
                                                              (line  47)
* gsl_linalg_SV_decomp_mod:              Singular Value Decomposition.
                                                              (line  41)
* gsl_linalg_SV_solve:                   Singular Value Decomposition.
                                                              (line  54)
* gsl_linalg_symmtd_decomp:              Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  15)
* gsl_linalg_symmtd_unpack:              Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  27)
* gsl_linalg_symmtd_unpack_T:            Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  34)
* gsl_log1p:                             Elementary Functions.
                                                              (line  13)
* gsl_matrix_add:                        Matrix operations.   (line   9)
* gsl_matrix_add_constant:               Matrix operations.   (line  36)
* gsl_matrix_alloc:                      Matrix allocation.   (line  15)
* gsl_matrix_calloc:                     Matrix allocation.   (line  22)
* gsl_matrix_column:                     Creating row and column views.
                                                              (line  26)
* gsl_matrix_const_column:               Creating row and column views.
                                                              (line  28)
* gsl_matrix_const_diagonal:             Creating row and column views.
                                                              (line  39)
* gsl_matrix_const_ptr:                  Accessing matrix elements.
                                                              (line  36)
* gsl_matrix_const_row:                  Creating row and column views.
                                                              (line  16)
* gsl_matrix_const_subdiagonal:          Creating row and column views.
                                                              (line  52)
* gsl_matrix_const_submatrix:            Matrix views.        (line  22)
* gsl_matrix_const_superdiagonal:        Creating row and column views.
                                                              (line  64)
* gsl_matrix_const_view_array:           Matrix views.        (line  54)
* gsl_matrix_const_view_array_with_tda:  Matrix views.        (line  79)
* gsl_matrix_const_view_vector:          Matrix views.        (line 105)
* gsl_matrix_const_view_vector_with_tda: Matrix views.        (line 130)
* gsl_matrix_diagonal:                   Creating row and column views.
                                                              (line  37)
* gsl_matrix_div_elements:               Matrix operations.   (line  26)
* gsl_matrix_fprintf:                    Reading and writing matrices.
                                                              (line  28)
* gsl_matrix_fread:                      Reading and writing matrices.
                                                              (line  18)
* gsl_matrix_free:                       Matrix allocation.   (line  26)
* gsl_matrix_fscanf:                     Reading and writing matrices.
                                                              (line  35)
* gsl_matrix_fwrite:                     Reading and writing matrices.
                                                              (line  11)
* gsl_matrix_get:                        Accessing matrix elements.
                                                              (line  22)
* gsl_matrix_get_col:                    Copying rows and columns.
                                                              (line  21)
* gsl_matrix_get_row:                    Copying rows and columns.
                                                              (line  15)
* gsl_matrix_isnull:                     Matrix properties.   (line   7)
* gsl_matrix_max:                        Finding maximum and minimum elements of matrices.
                                                              (line   9)
* gsl_matrix_max_index:                  Finding maximum and minimum elements of matrices.
                                                              (line  21)
* gsl_matrix_memcpy:                     Copying matrices.    (line   8)
* gsl_matrix_min:                        Finding maximum and minimum elements of matrices.
                                                              (line  12)
* gsl_matrix_min_index:                  Finding maximum and minimum elements of matrices.
                                                              (line  28)
* gsl_matrix_minmax:                     Finding maximum and minimum elements of matrices.
                                                              (line  16)
* gsl_matrix_minmax_index:               Finding maximum and minimum elements of matrices.
                                                              (line  35)
* gsl_matrix_mul_elements:               Matrix operations.   (line  20)
* gsl_matrix_ptr:                        Accessing matrix elements.
                                                              (line  34)
* gsl_matrix_row:                        Creating row and column views.
                                                              (line  14)
* gsl_matrix_scale:                      Matrix operations.   (line  31)
* gsl_matrix_set:                        Accessing matrix elements.
                                                              (line  28)
* gsl_matrix_set_all:                    Initializing matrix elements.
                                                              (line   7)
* gsl_matrix_set_col:                    Copying rows and columns.
                                                              (line  33)
* gsl_matrix_set_identity:               Initializing matrix elements.
                                                              (line  13)
* gsl_matrix_set_row:                    Copying rows and columns.
                                                              (line  27)
* gsl_matrix_set_zero:                   Initializing matrix elements.
                                                              (line  10)
* gsl_matrix_sub:                        Matrix operations.   (line  14)
* gsl_matrix_subdiagonal:                Creating row and column views.
                                                              (line  50)
* gsl_matrix_submatrix:                  Matrix views.        (line  20)
* gsl_matrix_superdiagonal:              Creating row and column views.
                                                              (line  62)
* gsl_matrix_swap:                       Copying matrices.    (line  12)
* gsl_matrix_swap_columns:               Exchanging rows and columns.
                                                              (line  16)
* gsl_matrix_swap_rowcol:                Exchanging rows and columns.
                                                              (line  21)
* gsl_matrix_swap_rows:                  Exchanging rows and columns.
                                                              (line  11)
* gsl_matrix_transpose:                  Exchanging rows and columns.
                                                              (line  33)
* gsl_matrix_transpose_memcpy:           Exchanging rows and columns.
                                                              (line  27)
* gsl_matrix_view_array:                 Matrix views.        (line  52)
* gsl_matrix_view_array_with_tda:        Matrix views.        (line  76)
* gsl_matrix_view_vector:                Matrix views.        (line 103)
* gsl_matrix_view_vector_with_tda:       Matrix views.        (line 127)
* GSL_MAX:                               Maximum and Minimum functions.
                                                              (line   7)
* GSL_MAX_DBL:                           Maximum and Minimum functions.
                                                              (line  15)
* GSL_MAX_INT:                           Maximum and Minimum functions.
                                                              (line  29)
* GSL_MAX_LDBL:                          Maximum and Minimum functions.
                                                              (line  37)
* GSL_MIN:                               Maximum and Minimum functions.
                                                              (line  11)
* GSL_MIN_DBL:                           Maximum and Minimum functions.
                                                              (line  22)
* gsl_min_fminimizer_alloc:              Initializing the Minimizer.
                                                              (line   8)
* gsl_min_fminimizer_brent:              Minimization Algorithms.
                                                              (line  32)
* gsl_min_fminimizer_f_lower:            Minimization Iteration.
                                                              (line  47)
* gsl_min_fminimizer_f_minimum:          Minimization Iteration.
                                                              (line  43)
* gsl_min_fminimizer_f_upper:            Minimization Iteration.
                                                              (line  45)
* gsl_min_fminimizer_free:               Initializing the Minimizer.
                                                              (line  40)
* gsl_min_fminimizer_goldensection:      Minimization Algorithms.
                                                              (line  14)
* gsl_min_fminimizer_iterate:            Minimization Iteration.
                                                              (line  13)
* gsl_min_fminimizer_name:               Initializing the Minimizer.
                                                              (line  44)
* gsl_min_fminimizer_set:                Initializing the Minimizer.
                                                              (line  24)
* gsl_min_fminimizer_set_with_values:    Initializing the Minimizer.
                                                              (line  35)
* gsl_min_fminimizer_x_lower:            Minimization Iteration.
                                                              (line  38)
* gsl_min_fminimizer_x_minimum:          Minimization Iteration.
                                                              (line  31)
* gsl_min_fminimizer_x_upper:            Minimization Iteration.
                                                              (line  36)
* GSL_MIN_INT:                           Maximum and Minimum functions.
                                                              (line  30)
* GSL_MIN_LDBL:                          Maximum and Minimum functions.
                                                              (line  39)
* gsl_min_test_interval:                 Minimization Stopping Parameters.
                                                              (line  20)
* gsl_monte_miser_alloc:                 MISER.               (line  47)
* gsl_monte_miser_free:                  MISER.               (line  70)
* gsl_monte_miser_init:                  MISER.               (line  52)
* gsl_monte_miser_integrate:             MISER.               (line  60)
* gsl_monte_plain_alloc:                 PLAIN Monte Carlo.   (line  30)
* gsl_monte_plain_free:                  PLAIN Monte Carlo.   (line  52)
* gsl_monte_plain_init:                  PLAIN Monte Carlo.   (line  34)
* gsl_monte_plain_integrate:             PLAIN Monte Carlo.   (line  42)
* gsl_monte_vegas_alloc:                 VEGAS.               (line  51)
* gsl_monte_vegas_free:                  VEGAS.               (line  78)
* gsl_monte_vegas_init:                  VEGAS.               (line  56)
* gsl_monte_vegas_integrate:             VEGAS.               (line  64)
* gsl_multifit_covar:                    Computing the covariance matrix of best fit parameters.
                                                              (line   8)
* gsl_multifit_fdfsolver_alloc:          Initializing the Nonlinear Least-Squares Solver.
                                                              (line  18)
* gsl_multifit_fdfsolver_free:           Initializing the Nonlinear Least-Squares Solver.
                                                              (line  48)
* gsl_multifit_fdfsolver_iterate:        Iteration of the Minimization Algorithm.
                                                              (line  16)
* gsl_multifit_fdfsolver_lmder:          Minimization Algorithms using Derivatives.
                                                              (line  61)
* gsl_multifit_fdfsolver_lmsder:         Minimization Algorithms using Derivatives.
                                                              (line  13)
* gsl_multifit_fdfsolver_name:           Initializing the Nonlinear Least-Squares Solver.
                                                              (line  54)
* gsl_multifit_fdfsolver_position:       Iteration of the Minimization Algorithm.
                                                              (line  45)
* gsl_multifit_fdfsolver_set:            Initializing the Nonlinear Least-Squares Solver.
                                                              (line  42)
* gsl_multifit_fsolver_alloc:            Initializing the Nonlinear Least-Squares Solver.
                                                              (line   8)
* gsl_multifit_fsolver_free:             Initializing the Nonlinear Least-Squares Solver.
                                                              (line  46)
* gsl_multifit_fsolver_iterate:          Iteration of the Minimization Algorithm.
                                                              (line  14)
* gsl_multifit_fsolver_name:             Initializing the Nonlinear Least-Squares Solver.
                                                              (line  52)
* gsl_multifit_fsolver_position:         Iteration of the Minimization Algorithm.
                                                              (line  43)
* gsl_multifit_fsolver_set:              Initializing the Nonlinear Least-Squares Solver.
                                                              (line  37)
* gsl_multifit_gradient:                 Search Stopping Parameters for Minimization Algorithms.
                                                              (line  46)
* gsl_multifit_linear:                   Multi-parameter fitting.
                                                              (line  52)
* gsl_multifit_linear_alloc:             Multi-parameter fitting.
                                                              (line  42)
* gsl_multifit_linear_est:               Multi-parameter fitting.
                                                              (line  99)
* gsl_multifit_linear_free:              Multi-parameter fitting.
                                                              (line  47)
* gsl_multifit_linear_svd:               Multi-parameter fitting.
                                                              (line  56)
* gsl_multifit_test_delta:               Search Stopping Parameters for Minimization Algorithms.
                                                              (line  21)
* gsl_multifit_test_gradient:            Search Stopping Parameters for Minimization Algorithms.
                                                              (line  32)
* gsl_multifit_wlinear:                  Multi-parameter fitting.
                                                              (line  77)
* gsl_multifit_wlinear_svd:              Multi-parameter fitting.
                                                              (line  81)
* gsl_multimin_fdfminimizer_alloc:       Initializing the Multidimensional Minimizer.
                                                              (line  13)
* gsl_multimin_fdfminimizer_conjugate_fr: Multimin Algorithms.
                                                              (line  12)
* gsl_multimin_fdfminimizer_conjugate_pr: Multimin Algorithms.
                                                              (line  29)
* gsl_multimin_fdfminimizer_free:        Initializing the Multidimensional Minimizer.
                                                              (line  43)
* gsl_multimin_fdfminimizer_gradient:    Multimin Iteration.  (line  33)
* gsl_multimin_fdfminimizer_iterate:     Multimin Iteration.  (line  13)
* gsl_multimin_fdfminimizer_minimum:     Multimin Iteration.  (line  29)
* gsl_multimin_fdfminimizer_name:        Initializing the Multidimensional Minimizer.
                                                              (line  49)
* gsl_multimin_fdfminimizer_restart:     Multimin Iteration.  (line  42)
* gsl_multimin_fdfminimizer_set:         Initializing the Multidimensional Minimizer.
                                                              (line  24)
* gsl_multimin_fdfminimizer_steepest_descent: Multimin Algorithms.
                                                              (line  45)
* gsl_multimin_fdfminimizer_vector_bfgs: Multimin Algorithms. (line  36)
* gsl_multimin_fdfminimizer_x:           Multimin Iteration.  (line  25)
* gsl_multimin_fminimizer_alloc:         Initializing the Multidimensional Minimizer.
                                                              (line  15)
* gsl_multimin_fminimizer_free:          Initializing the Multidimensional Minimizer.
                                                              (line  45)
* gsl_multimin_fminimizer_iterate:       Multimin Iteration.  (line  15)
* gsl_multimin_fminimizer_minimum:       Multimin Iteration.  (line  31)
* gsl_multimin_fminimizer_name:          Initializing the Multidimensional Minimizer.
                                                              (line  51)
* gsl_multimin_fminimizer_nmsimplex:     Multimin Algorithms. (line  55)
* gsl_multimin_fminimizer_set:           Initializing the Multidimensional Minimizer.
                                                              (line  36)
* gsl_multimin_fminimizer_size:          Multimin Iteration.  (line  35)
* gsl_multimin_fminimizer_x:             Multimin Iteration.  (line  27)
* gsl_multimin_test_gradient:            Multimin Stopping Criteria.
                                                              (line  20)
* gsl_multimin_test_size:                Multimin Stopping Criteria.
                                                              (line  34)
* gsl_multiroot_fdfsolver_alloc:         Initializing the Multidimensional Solver.
                                                              (line  29)
* gsl_multiroot_fdfsolver_dx:            Iteration of the multidimensional solver.
                                                              (line  50)
* gsl_multiroot_fdfsolver_f:             Iteration of the multidimensional solver.
                                                              (line  43)
* gsl_multiroot_fdfsolver_free:          Initializing the Multidimensional Solver.
                                                              (line  57)
* gsl_multiroot_fdfsolver_gnewton:       Algorithms using Derivatives.
                                                              (line  91)
* gsl_multiroot_fdfsolver_hybridj:       Algorithms using Derivatives.
                                                              (line  67)
* gsl_multiroot_fdfsolver_hybridsj:      Algorithms using Derivatives.
                                                              (line  14)
* gsl_multiroot_fdfsolver_iterate:       Iteration of the multidimensional solver.
                                                              (line  16)
* gsl_multiroot_fdfsolver_name:          Initializing the Multidimensional Solver.
                                                              (line  63)
* gsl_multiroot_fdfsolver_newton:        Algorithms using Derivatives.
                                                              (line  73)
* gsl_multiroot_fdfsolver_root:          Iteration of the multidimensional solver.
                                                              (line  36)
* gsl_multiroot_fdfsolver_set:           Initializing the Multidimensional Solver.
                                                              (line  50)
* gsl_multiroot_fsolver_alloc:           Initializing the Multidimensional Solver.
                                                              (line  13)
* gsl_multiroot_fsolver_broyden:         Algorithms without Derivatives.
                                                              (line  44)
* gsl_multiroot_fsolver_dnewton:         Algorithms without Derivatives.
                                                              (line  25)
* gsl_multiroot_fsolver_dx:              Iteration of the multidimensional solver.
                                                              (line  48)
* gsl_multiroot_fsolver_f:               Iteration of the multidimensional solver.
                                                              (line  41)
* gsl_multiroot_fsolver_free:            Initializing the Multidimensional Solver.
                                                              (line  55)
* gsl_multiroot_fsolver_hybrid:          Algorithms without Derivatives.
                                                              (line  21)
* gsl_multiroot_fsolver_hybrids:         Algorithms without Derivatives.
                                                              (line  14)
* gsl_multiroot_fsolver_iterate:         Iteration of the multidimensional solver.
                                                              (line  14)
* gsl_multiroot_fsolver_name:            Initializing the Multidimensional Solver.
                                                              (line  61)
* gsl_multiroot_fsolver_root:            Iteration of the multidimensional solver.
                                                              (line  34)
* gsl_multiroot_fsolver_set:             Initializing the Multidimensional Solver.
                                                              (line  45)
* gsl_multiroot_test_delta:              Search Stopping Parameters for the multidimensional solver.
                                                              (line  22)
* gsl_multiroot_test_residual:           Search Stopping Parameters for the multidimensional solver.
                                                              (line  33)
* GSL_NAN:                               Infinities and Not-a-number.
                                                              (line  15)
* GSL_NEGINF:                            Infinities and Not-a-number.
                                                              (line  11)
* gsl_ntuple_bookdata:                   Writing ntuples.     (line  11)
* gsl_ntuple_close:                      Closing an ntuple file.
                                                              (line   7)
* gsl_ntuple_create:                     Creating ntuples.    (line   8)
* gsl_ntuple_open:                       Opening an existing ntuple file.
                                                              (line   8)
* gsl_ntuple_project:                    Histogramming ntuple values.
                                                              (line  36)
* gsl_ntuple_read:                       Reading ntuples.     (line   7)
* gsl_ntuple_write:                      Writing ntuples.     (line   7)
* gsl_odeiv_control_alloc:               Adaptive Step-size Control.
                                                              (line  75)
* gsl_odeiv_control_free:                Adaptive Step-size Control.
                                                              (line  87)
* gsl_odeiv_control_hadjust:             Adaptive Step-size Control.
                                                              (line  93)
* gsl_odeiv_control_init:                Adaptive Step-size Control.
                                                              (line  82)
* gsl_odeiv_control_name:                Adaptive Step-size Control.
                                                              (line 107)
* gsl_odeiv_control_scaled_new:          Adaptive Step-size Control.
                                                              (line  63)
* gsl_odeiv_control_standard_new:        Adaptive Step-size Control.
                                                              (line  12)
* gsl_odeiv_control_y_new:               Adaptive Step-size Control.
                                                              (line  46)
* gsl_odeiv_control_yp_new:              Adaptive Step-size Control.
                                                              (line  54)
* gsl_odeiv_evolve_alloc:                Evolution.           (line  15)
* gsl_odeiv_evolve_apply:                Evolution.           (line  22)
* gsl_odeiv_evolve_free:                 Evolution.           (line  47)
* gsl_odeiv_evolve_reset:                Evolution.           (line  42)
* gsl_odeiv_step_alloc:                  Stepping Functions.  (line  12)
* gsl_odeiv_step_apply:                  Stepping Functions.  (line  43)
* gsl_odeiv_step_bsimp:                  Stepping Functions.  (line  89)
* gsl_odeiv_step_free:                   Stepping Functions.  (line  21)
* gsl_odeiv_step_gear1:                  Stepping Functions.  (line  93)
* gsl_odeiv_step_gear2:                  Stepping Functions.  (line  96)
* gsl_odeiv_step_name:                   Stepping Functions.  (line  26)
* gsl_odeiv_step_order:                  Stepping Functions.  (line  36)
* gsl_odeiv_step_reset:                  Stepping Functions.  (line  16)
* gsl_odeiv_step_rk2:                    Stepping Functions.  (line  67)
* gsl_odeiv_step_rk2imp:                 Stepping Functions.  (line  83)
* gsl_odeiv_step_rk4:                    Stepping Functions.  (line  70)
* gsl_odeiv_step_rk4imp:                 Stepping Functions.  (line  86)
* gsl_odeiv_step_rk8pd:                  Stepping Functions.  (line  80)
* gsl_odeiv_step_rkck:                   Stepping Functions.  (line  77)
* gsl_odeiv_step_rkf45:                  Stepping Functions.  (line  73)
* gsl_permutation_alloc:                 Permutation allocation.
                                                              (line   7)
* gsl_permutation_calloc:                Permutation allocation.
                                                              (line  15)
* gsl_permutation_canonical_cycles:      Permutations in cyclic form.
                                                              (line  71)
* gsl_permutation_canonical_to_linear:   Permutations in cyclic form.
                                                              (line  53)
* gsl_permutation_data:                  Permutation properties.
                                                              (line  10)
* gsl_permutation_fprintf:               Reading and writing permutations.
                                                              (line  29)
* gsl_permutation_fread:                 Reading and writing permutations.
                                                              (line  19)
* gsl_permutation_free:                  Permutation allocation.
                                                              (line  24)
* gsl_permutation_fscanf:                Reading and writing permutations.
                                                              (line  38)
* gsl_permutation_fwrite:                Reading and writing permutations.
                                                              (line  11)
* gsl_permutation_get:                   Accessing permutation elements.
                                                              (line  11)
* gsl_permutation_init:                  Permutation allocation.
                                                              (line  20)
* gsl_permutation_inverse:               Permutation functions.
                                                              (line  11)
* gsl_permutation_inversions:            Permutations in cyclic form.
                                                              (line  58)
* gsl_permutation_linear_cycles:         Permutations in cyclic form.
                                                              (line  66)
* gsl_permutation_linear_to_canonical:   Permutations in cyclic form.
                                                              (line  48)
* gsl_permutation_memcpy:                Permutation allocation.
                                                              (line  28)
* gsl_permutation_mul:                   Applying Permutations.
                                                              (line  37)
* gsl_permutation_next:                  Permutation functions.
                                                              (line  15)
* gsl_permutation_prev:                  Permutation functions.
                                                              (line  23)
* gsl_permutation_reverse:               Permutation functions.
                                                              (line   7)
* gsl_permutation_size:                  Permutation properties.
                                                              (line   7)
* gsl_permutation_swap:                  Accessing permutation elements.
                                                              (line  17)
* gsl_permutation_valid:                 Permutation properties.
                                                              (line  14)
* gsl_permute:                           Applying Permutations.
                                                              (line   8)
* gsl_permute_inverse:                   Applying Permutations.
                                                              (line  13)
* gsl_permute_vector:                    Applying Permutations.
                                                              (line  18)
* gsl_permute_vector_inverse:            Applying Permutations.
                                                              (line  27)
* gsl_poly_complex_solve:                General Polynomial Equations.
                                                              (line  27)
* gsl_poly_complex_solve_cubic:          Cubic Equations.     (line  24)
* gsl_poly_complex_solve_quadratic:      Quadratic Equations. (line  32)
* gsl_poly_complex_workspace_alloc:      General Polynomial Equations.
                                                              (line  13)
* gsl_poly_complex_workspace_free:       General Polynomial Equations.
                                                              (line  23)
* gsl_poly_dd_eval:                      Divided Difference Representation of Polynomials.
                                                              (line  20)
* gsl_poly_dd_init:                      Divided Difference Representation of Polynomials.
                                                              (line  12)
* gsl_poly_dd_taylor:                    Divided Difference Representation of Polynomials.
                                                              (line  26)
* gsl_poly_eval:                         Polynomial Evaluation.
                                                              (line   8)
* gsl_poly_solve_cubic:                  Cubic Equations.     (line   8)
* gsl_poly_solve_quadratic:              Quadratic Equations. (line   8)
* GSL_POSINF:                            Infinities and Not-a-number.
                                                              (line   7)
* gsl_pow_2:                             Small integer powers.
                                                              (line  19)
* gsl_pow_3:                             Small integer powers.
                                                              (line  20)
* gsl_pow_4:                             Small integer powers.
                                                              (line  21)
* gsl_pow_5:                             Small integer powers.
                                                              (line  22)
* gsl_pow_6:                             Small integer powers.
                                                              (line  23)
* gsl_pow_7:                             Small integer powers.
                                                              (line  24)
* gsl_pow_8:                             Small integer powers.
                                                              (line  25)
* gsl_pow_9:                             Small integer powers.
                                                              (line  26)
* gsl_pow_int:                           Small integer powers.
                                                              (line  12)
* gsl_qrng_alloc:                        Quasi-random number generator initialization.
                                                              (line   8)
* gsl_qrng_clone:                        Saving and resorting quasi-random number generator state.
                                                              (line  13)
* gsl_qrng_free:                         Quasi-random number generator initialization.
                                                              (line  15)
* gsl_qrng_get:                          Sampling from a quasi-random number generator.
                                                              (line   7)
* gsl_qrng_init:                         Quasi-random number generator initialization.
                                                              (line  18)
* gsl_qrng_memcpy:                       Saving and resorting quasi-random number generator state.
                                                              (line   8)
* gsl_qrng_name:                         Auxiliary quasi-random number generator functions.
                                                              (line   7)
* gsl_qrng_niederreiter_2:               Quasi-random number generator algorithms.
                                                              (line   9)
* gsl_qrng_size:                         Auxiliary quasi-random number generator functions.
                                                              (line  10)
* gsl_qrng_sobol:                        Quasi-random number generator algorithms.
                                                              (line  14)
* gsl_qrng_state:                        Auxiliary quasi-random number generator functions.
                                                              (line  11)
* gsl_ran_bernoulli:                     The Bernoulli Distribution.
                                                              (line   8)
* gsl_ran_bernoulli_pdf:                 The Bernoulli Distribution.
                                                              (line  17)
* gsl_ran_beta:                          The Beta Distribution.
                                                              (line   8)
* gsl_ran_beta_pdf:                      The Beta Distribution.
                                                              (line  16)
* gsl_ran_binomial:                      The Binomial Distribution.
                                                              (line   9)
* gsl_ran_binomial_pdf:                  The Binomial Distribution.
                                                              (line  20)
* gsl_ran_bivariate_gaussian:            The Bivariate Gaussian Distribution.
                                                              (line   9)
* gsl_ran_bivariate_gaussian_pdf:        The Bivariate Gaussian Distribution.
                                                              (line  21)
* gsl_ran_cauchy:                        The Cauchy Distribution.
                                                              (line   7)
* gsl_ran_cauchy_pdf:                    The Cauchy Distribution.
                                                              (line  17)
* gsl_ran_chisq:                         The Chi-squared Distribution.
                                                              (line  15)
* gsl_ran_chisq_pdf:                     The Chi-squared Distribution.
                                                              (line  24)
* gsl_ran_choose:                        Shuffling and Sampling.
                                                              (line  36)
* gsl_ran_dir_2d:                        Spherical Vector Distributions.
                                                              (line  12)
* gsl_ran_dir_2d_trig_method:            Spherical Vector Distributions.
                                                              (line  14)
* gsl_ran_dir_3d:                        Spherical Vector Distributions.
                                                              (line  33)
* gsl_ran_dir_nd:                        Spherical Vector Distributions.
                                                              (line  42)
* gsl_ran_dirichlet:                     The Dirichlet Distribution.
                                                              (line   8)
* gsl_ran_dirichlet_lnpdf:               The Dirichlet Distribution.
                                                              (line  32)
* gsl_ran_dirichlet_pdf:                 The Dirichlet Distribution.
                                                              (line  26)
* gsl_ran_discrete:                      General Discrete Distributions.
                                                              (line  62)
* gsl_ran_discrete_free:                 General Discrete Distributions.
                                                              (line  74)
* gsl_ran_discrete_pdf:                  General Discrete Distributions.
                                                              (line  67)
* gsl_ran_discrete_preproc:              General Discrete Distributions.
                                                              (line  52)
* gsl_ran_exponential:                   The Exponential Distribution.
                                                              (line   8)
* gsl_ran_exponential_pdf:               The Exponential Distribution.
                                                              (line  16)
* gsl_ran_exppow:                        The Exponential Power Distribution.
                                                              (line   8)
* gsl_ran_exppow_pdf:                    The Exponential Power Distribution.
                                                              (line  19)
* gsl_ran_fdist:                         The F-distribution.  (line  16)
* gsl_ran_fdist_pdf:                     The F-distribution.  (line  29)
* gsl_ran_flat:                          The Flat (Uniform) Distribution.
                                                              (line   8)
* gsl_ran_flat_pdf:                      The Flat (Uniform) Distribution.
                                                              (line  16)
* gsl_ran_gamma:                         The Gamma Distribution.
                                                              (line   9)
* gsl_ran_gamma_mt:                      The Gamma Distribution.
                                                              (line  22)
* gsl_ran_gamma_pdf:                     The Gamma Distribution.
                                                              (line  26)
* gsl_ran_gaussian:                      The Gaussian Distribution.
                                                              (line   7)
* gsl_ran_gaussian_pdf:                  The Gaussian Distribution.
                                                              (line  20)
* gsl_ran_gaussian_ratio_method:         The Gaussian Distribution.
                                                              (line  29)
* gsl_ran_gaussian_tail:                 The Gaussian Tail Distribution.
                                                              (line   8)
* gsl_ran_gaussian_tail_pdf:             The Gaussian Tail Distribution.
                                                              (line  27)
* gsl_ran_gaussian_ziggurat:             The Gaussian Distribution.
                                                              (line  27)
* gsl_ran_geometric:                     The Geometric Distribution.
                                                              (line   8)
* gsl_ran_geometric_pdf:                 The Geometric Distribution.
                                                              (line  20)
* gsl_ran_gumbel1:                       The Type-1 Gumbel Distribution.
                                                              (line   8)
* gsl_ran_gumbel1_pdf:                   The Type-1 Gumbel Distribution.
                                                              (line  16)
* gsl_ran_gumbel2:                       The Type-2 Gumbel Distribution.
                                                              (line   8)
* gsl_ran_gumbel2_pdf:                   The Type-2 Gumbel Distribution.
                                                              (line  16)
* gsl_ran_hypergeometric:                The Hypergeometric Distribution.
                                                              (line   8)
* gsl_ran_hypergeometric_pdf:            The Hypergeometric Distribution.
                                                              (line  24)
* gsl_ran_landau:                        The Landau Distribution.
                                                              (line   8)
* gsl_ran_landau_pdf:                    The Landau Distribution.
                                                              (line  19)
* gsl_ran_laplace:                       The Laplace Distribution.
                                                              (line   7)
* gsl_ran_laplace_pdf:                   The Laplace Distribution.
                                                              (line  15)
* gsl_ran_levy:                          The Levy alpha-Stable Distributions.
                                                              (line   9)
* gsl_ran_levy_skew:                     The Levy skew alpha-Stable Distribution.
                                                              (line   9)
* gsl_ran_logarithmic:                   The Logarithmic Distribution.
                                                              (line   8)
* gsl_ran_logarithmic_pdf:               The Logarithmic Distribution.
                                                              (line  17)
* gsl_ran_logistic:                      The Logistic Distribution.
                                                              (line   7)
* gsl_ran_logistic_pdf:                  The Logistic Distribution.
                                                              (line  15)
* gsl_ran_lognormal:                     The Lognormal Distribution.
                                                              (line   8)
* gsl_ran_lognormal_pdf:                 The Lognormal Distribution.
                                                              (line  17)
* gsl_ran_multinomial:                   The Multinomial Distribution.
                                                              (line   8)
* gsl_ran_multinomial_lnpdf:             The Multinomial Distribution.
                                                              (line  33)
* gsl_ran_multinomial_pdf:               The Multinomial Distribution.
                                                              (line  27)
* gsl_ran_negative_binomial:             The Negative Binomial Distribution.
                                                              (line   8)
* gsl_ran_negative_binomial_pdf:         The Negative Binomial Distribution.
                                                              (line  19)
* gsl_ran_pareto:                        The Pareto Distribution.
                                                              (line   8)
* gsl_ran_pareto_pdf:                    The Pareto Distribution.
                                                              (line  16)
* gsl_ran_pascal:                        The Pascal Distribution.
                                                              (line   8)
* gsl_ran_pascal_pdf:                    The Pascal Distribution.
                                                              (line  18)
* gsl_ran_poisson:                       The Poisson Distribution.
                                                              (line   8)
* gsl_ran_poisson_pdf:                   The Poisson Distribution.
                                                              (line  17)
* gsl_ran_rayleigh:                      The Rayleigh Distribution.
                                                              (line   7)
* gsl_ran_rayleigh_pdf:                  The Rayleigh Distribution.
                                                              (line  15)
* gsl_ran_rayleigh_tail:                 The Rayleigh Tail Distribution.
                                                              (line   8)
* gsl_ran_rayleigh_tail_pdf:             The Rayleigh Tail Distribution.
                                                              (line  18)
* gsl_ran_sample:                        Shuffling and Sampling.
                                                              (line  64)
* gsl_ran_shuffle:                       Shuffling and Sampling.
                                                              (line  16)
* gsl_ran_tdist:                         The t-distribution.  (line  15)
* gsl_ran_tdist_pdf:                     The t-distribution.  (line  24)
* gsl_ran_ugaussian:                     The Gaussian Distribution.
                                                              (line  35)
* gsl_ran_ugaussian_pdf:                 The Gaussian Distribution.
                                                              (line  36)
* gsl_ran_ugaussian_ratio_method:        The Gaussian Distribution.
                                                              (line  37)
* gsl_ran_ugaussian_tail:                The Gaussian Tail Distribution.
                                                              (line  34)
* gsl_ran_ugaussian_tail_pdf:            The Gaussian Tail Distribution.
                                                              (line  35)
* gsl_ran_weibull:                       The Weibull Distribution.
                                                              (line   8)
* gsl_ran_weibull_pdf:                   The Weibull Distribution.
                                                              (line  16)
* GSL_REAL:                              Complex numbers.     (line  34)
* gsl_rng_alloc:                         Random number generator initialization.
                                                              (line   7)
* gsl_rng_borosh13:                      Other random number generators.
                                                              (line 158)
* gsl_rng_clone:                         Copying random number generator state.
                                                              (line  17)
* gsl_rng_cmrg:                          Random number generator algorithms.
                                                              (line  89)
* gsl_rng_coveyou:                       Other random number generators.
                                                              (line 168)
* gsl_rng_env_setup:                     Random number environment variables.
                                                              (line  12)
* gsl_rng_fishman18:                     Other random number generators.
                                                              (line 177)
* gsl_rng_fishman20:                     Other random number generators.
                                                              (line 187)
* gsl_rng_fishman2x:                     Other random number generators.
                                                              (line 197)
* gsl_rng_fread:                         Reading and writing random number generator state.
                                                              (line  18)
* gsl_rng_free:                          Random number generator initialization.
                                                              (line  38)
* gsl_rng_fwrite:                        Reading and writing random number generator state.
                                                              (line  10)
* gsl_rng_get:                           Sampling from a random number generator.
                                                              (line  11)
* gsl_rng_gfsr4:                         Random number generator algorithms.
                                                              (line 164)
* gsl_rng_knuthran:                      Other random number generators.
                                                              (line 217)
* gsl_rng_knuthran2:                     Other random number generators.
                                                              (line 208)
* gsl_rng_lecuyer21:                     Other random number generators.
                                                              (line 222)
* gsl_rng_max:                           Auxiliary random number generator functions.
                                                              (line  20)
* gsl_rng_memcpy:                        Copying random number generator state.
                                                              (line  12)
* gsl_rng_min:                           Auxiliary random number generator functions.
                                                              (line  24)
* gsl_rng_minstd:                        Other random number generators.
                                                              (line 114)
* gsl_rng_mrg:                           Random number generator algorithms.
                                                              (line 111)
* gsl_rng_mt19937:                       Random number generator algorithms.
                                                              (line  18)
* gsl_rng_name:                          Auxiliary random number generator functions.
                                                              (line  11)
* gsl_rng_r250:                          Other random number generators.
                                                              (line  60)
* gsl_rng_rand:                          Unix random number generators.
                                                              (line  18)
* gsl_rng_rand48:                        Unix random number generators.
                                                              (line  59)
* gsl_rng_random_bsd:                    Unix random number generators.
                                                              (line  27)
* gsl_rng_random_glibc2:                 Unix random number generators.
                                                              (line  29)
* gsl_rng_random_libc5:                  Unix random number generators.
                                                              (line  28)
* gsl_rng_randu:                         Other random number generators.
                                                              (line 105)
* gsl_rng_ranf:                          Other random number generators.
                                                              (line  23)
* gsl_rng_ranlux:                        Random number generator algorithms.
                                                              (line  62)
* gsl_rng_ranlux389:                     Random number generator algorithms.
                                                              (line  63)
* gsl_rng_ranlxd1:                       Random number generator algorithms.
                                                              (line  56)
* gsl_rng_ranlxd2:                       Random number generator algorithms.
                                                              (line  57)
* gsl_rng_ranlxs0:                       Random number generator algorithms.
                                                              (line  41)
* gsl_rng_ranlxs1:                       Random number generator algorithms.
                                                              (line  42)
* gsl_rng_ranlxs2:                       Random number generator algorithms.
                                                              (line  43)
* gsl_rng_ranmar:                        Other random number generators.
                                                              (line  54)
* gsl_rng_set:                           Random number generator initialization.
                                                              (line  26)
* gsl_rng_size:                          Auxiliary random number generator functions.
                                                              (line  31)
* gsl_rng_slatec:                        Other random number generators.
                                                              (line 141)
* gsl_rng_state:                         Auxiliary random number generator functions.
                                                              (line  30)
* gsl_rng_taus:                          Random number generator algorithms.
                                                              (line 128)
* gsl_rng_taus2:                         Random number generator algorithms.
                                                              (line 129)
* gsl_rng_transputer:                    Other random number generators.
                                                              (line  96)
* gsl_rng_tt800:                         Other random number generators.
                                                              (line  75)
* gsl_rng_types_setup:                   Auxiliary random number generator functions.
                                                              (line  41)
* gsl_rng_uni:                           Other random number generators.
                                                              (line 134)
* gsl_rng_uni32:                         Other random number generators.
                                                              (line 135)
* gsl_rng_uniform:                       Sampling from a random number generator.
                                                              (line  18)
* gsl_rng_uniform_int:                   Sampling from a random number generator.
                                                              (line  37)
* gsl_rng_uniform_pos:                   Sampling from a random number generator.
                                                              (line  28)
* gsl_rng_vax:                           Other random number generators.
                                                              (line  87)
* gsl_rng_waterman14:                    Other random number generators.
                                                              (line 232)
* gsl_rng_zuf:                           Other random number generators.
                                                              (line 145)
* gsl_root_fdfsolver_alloc:              Initializing the Solver.
                                                              (line  23)
* gsl_root_fdfsolver_free:               Initializing the Solver.
                                                              (line  49)
* gsl_root_fdfsolver_iterate:            Root Finding Iteration.
                                                              (line  14)
* gsl_root_fdfsolver_name:               Initializing the Solver.
                                                              (line  55)
* gsl_root_fdfsolver_newton:             Root Finding Algorithms using Derivatives.
                                                              (line  16)
* gsl_root_fdfsolver_root:               Root Finding Iteration.
                                                              (line  35)
* gsl_root_fdfsolver_secant:             Root Finding Algorithms using Derivatives.
                                                              (line  30)
* gsl_root_fdfsolver_set:                Initializing the Solver.
                                                              (line  44)
* gsl_root_fdfsolver_steffenson:         Root Finding Algorithms using Derivatives.
                                                              (line  61)
* gsl_root_fsolver_alloc:                Initializing the Solver.
                                                              (line   8)
* gsl_root_fsolver_bisection:            Root Bracketing Algorithms.
                                                              (line  17)
* gsl_root_fsolver_brent:                Root Bracketing Algorithms.
                                                              (line  50)
* gsl_root_fsolver_falsepos:             Root Bracketing Algorithms.
                                                              (line  33)
* gsl_root_fsolver_free:                 Initializing the Solver.
                                                              (line  48)
* gsl_root_fsolver_iterate:              Root Finding Iteration.
                                                              (line  13)
* gsl_root_fsolver_name:                 Initializing the Solver.
                                                              (line  53)
* gsl_root_fsolver_root:                 Root Finding Iteration.
                                                              (line  33)
* gsl_root_fsolver_set:                  Initializing the Solver.
                                                              (line  38)
* gsl_root_fsolver_x_lower:              Root Finding Iteration.
                                                              (line  40)
* gsl_root_fsolver_x_upper:              Root Finding Iteration.
                                                              (line  42)
* gsl_root_test_delta:                   Search Stopping Parameters.
                                                              (line  44)
* gsl_root_test_interval:                Search Stopping Parameters.
                                                              (line  21)
* gsl_root_test_residual:                Search Stopping Parameters.
                                                              (line  53)
* GSL_SET_COMPLEX:                       Complex numbers.     (line  39)
* gsl_set_error_handler:                 Error Handlers.      (line  44)
* gsl_set_error_handler_off:             Error Handlers.      (line  68)
* GSL_SET_IMAG:                          Complex numbers.     (line  49)
* GSL_SET_REAL:                          Complex numbers.     (line  48)
* gsl_sf_airy_Ai:                        Airy Functions.      (line   7)
* gsl_sf_airy_Ai_deriv:                  Derivatives of Airy Functions.
                                                              (line   7)
* gsl_sf_airy_Ai_deriv_e:                Derivatives of Airy Functions.
                                                              (line   9)
* gsl_sf_airy_Ai_deriv_scaled:           Derivatives of Airy Functions.
                                                              (line  20)
* gsl_sf_airy_Ai_deriv_scaled_e:         Derivatives of Airy Functions.
                                                              (line  22)
* gsl_sf_airy_Ai_e:                      Airy Functions.      (line   9)
* gsl_sf_airy_Ai_scaled:                 Airy Functions.      (line  19)
* gsl_sf_airy_Ai_scaled_e:               Airy Functions.      (line  21)
* gsl_sf_airy_Bi:                        Airy Functions.      (line  13)
* gsl_sf_airy_Bi_deriv:                  Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_Bi_deriv_e:                Derivatives of Airy Functions.
                                                              (line  15)
* gsl_sf_airy_Bi_deriv_scaled:           Derivatives of Airy Functions.
                                                              (line  28)
* gsl_sf_airy_Bi_deriv_scaled_e:         Derivatives of Airy Functions.
                                                              (line  30)
* gsl_sf_airy_Bi_e:                      Airy Functions.      (line  15)
* gsl_sf_airy_Bi_scaled:                 Airy Functions.      (line  26)
* gsl_sf_airy_Bi_scaled_e:               Airy Functions.      (line  28)
* gsl_sf_airy_zero_Ai:                   Zeros of Airy Functions.
                                                              (line   7)
* gsl_sf_airy_zero_Ai_deriv:             Zeros of Derivatives of Airy Functions.
                                                              (line   7)
* gsl_sf_airy_zero_Ai_deriv_e:           Zeros of Derivatives of Airy Functions.
                                                              (line   9)
* gsl_sf_airy_zero_Ai_e:                 Zeros of Airy Functions.
                                                              (line   9)
* gsl_sf_airy_zero_Bi:                   Zeros of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_zero_Bi_deriv:             Zeros of Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_zero_Bi_deriv_e:           Zeros of Derivatives of Airy Functions.
                                                              (line  15)
* gsl_sf_airy_zero_Bi_e:                 Zeros of Airy Functions.
                                                              (line  15)
* gsl_sf_angle_restrict_pos:             Restriction Functions.
                                                              (line  12)
* gsl_sf_angle_restrict_pos_e:           Restriction Functions.
                                                              (line  13)
* gsl_sf_angle_restrict_symm:            Restriction Functions.
                                                              (line   7)
* gsl_sf_angle_restrict_symm_e:          Restriction Functions.
                                                              (line   8)
* gsl_sf_atanint:                        Arctangent Integral. (line   7)
* gsl_sf_atanint_e:                      Arctangent Integral. (line   8)
* gsl_sf_bessel_I0:                      Regular Modified Cylindrical Bessel Functions.
                                                              (line   7)
* gsl_sf_bessel_I0_e:                    Regular Modified Cylindrical Bessel Functions.
                                                              (line   8)
* gsl_sf_bessel_I0_scaled:               Regular Modified Cylindrical Bessel Functions.
                                                              (line  32)
* gsl_sf_bessel_i0_scaled:               Regular Modified Spherical Bessel Functions.
                                                              (line  11)
* gsl_sf_bessel_i0_scaled_e:             Regular Modified Spherical Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_I0_scaled_e:             Regular Modified Cylindrical Bessel Functions.
                                                              (line  34)
* gsl_sf_bessel_I1:                      Regular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_I1_e:                    Regular Modified Cylindrical Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_I1_scaled:               Regular Modified Cylindrical Bessel Functions.
                                                              (line  38)
* gsl_sf_bessel_i1_scaled:               Regular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_I1_scaled_e:             Regular Modified Cylindrical Bessel Functions.
                                                              (line  40)
* gsl_sf_bessel_i1_scaled_e:             Regular Modified Spherical Bessel Functions.
                                                              (line  19)
* gsl_sf_bessel_i2_scaled:               Regular Modified Spherical Bessel Functions.
                                                              (line  23)
* gsl_sf_bessel_i2_scaled_e:             Regular Modified Spherical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_il_scaled:               Regular Modified Spherical Bessel Functions.
                                                              (line  29)
* gsl_sf_bessel_il_scaled_array:         Regular Modified Spherical Bessel Functions.
                                                              (line  36)
* gsl_sf_bessel_il_scaled_e:             Regular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_In:                      Regular Modified Cylindrical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_In_array:                Regular Modified Cylindrical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_In_e:                    Regular Modified Cylindrical Bessel Functions.
                                                              (line  19)
* gsl_sf_bessel_In_scaled:               Regular Modified Cylindrical Bessel Functions.
                                                              (line  44)
* gsl_sf_bessel_In_scaled_array:         Regular Modified Cylindrical Bessel Functions.
                                                              (line  51)
* gsl_sf_bessel_In_scaled_e:             Regular Modified Cylindrical Bessel Functions.
                                                              (line  46)
* gsl_sf_bessel_Inu:                     Regular Modified Bessel Functions - Fractional Order.
                                                              (line   7)
* gsl_sf_bessel_Inu_e:                   Regular Modified Bessel Functions - Fractional Order.
                                                              (line   9)
* gsl_sf_bessel_Inu_scaled:              Regular Modified Bessel Functions - Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_Inu_scaled_e:            Regular Modified Bessel Functions - Fractional Order.
                                                              (line  15)
* gsl_sf_bessel_J0:                      Regular Cylindrical Bessel Functions.
                                                              (line   7)
* gsl_sf_bessel_j0:                      Regular Spherical Bessel Functions.
                                                              (line   7)
* gsl_sf_bessel_j0_e:                    Regular Spherical Bessel Functions.
                                                              (line   8)
* gsl_sf_bessel_J0_e:                    Regular Cylindrical Bessel Functions.
                                                              (line   8)
* gsl_sf_bessel_J1:                      Regular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_j1:                      Regular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_J1_e:                    Regular Cylindrical Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_j1_e:                    Regular Spherical Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_j2:                      Regular Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_j2_e:                    Regular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_jl:                      Regular Spherical Bessel Functions.
                                                              (line  22)
* gsl_sf_bessel_jl_array:                Regular Spherical Bessel Functions.
                                                              (line  29)
* gsl_sf_bessel_jl_e:                    Regular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_jl_steed_array:          Regular Spherical Bessel Functions.
                                                              (line  37)
* gsl_sf_bessel_Jn:                      Regular Cylindrical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_Jn_array:                Regular Cylindrical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_Jn_e:                    Regular Cylindrical Bessel Functions.
                                                              (line  19)
* gsl_sf_bessel_Jnu:                     Regular Bessel Function - Fractional Order.
                                                              (line   7)
* gsl_sf_bessel_Jnu_e:                   Regular Bessel Function - Fractional Order.
                                                              (line   9)
* gsl_sf_bessel_K0:                      Irregular Modified Cylindrical Bessel Functions.
                                                              (line   7)
* gsl_sf_bessel_K0_e:                    Irregular Modified Cylindrical Bessel Functions.
                                                              (line   8)
* gsl_sf_bessel_K0_scaled:               Irregular Modified Cylindrical Bessel Functions.
                                                              (line  33)
* gsl_sf_bessel_k0_scaled:               Irregular Modified Spherical Bessel Functions.
                                                              (line  11)
* gsl_sf_bessel_k0_scaled_e:             Irregular Modified Spherical Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_K0_scaled_e:             Irregular Modified Cylindrical Bessel Functions.
                                                              (line  35)
* gsl_sf_bessel_K1:                      Irregular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_K1_e:                    Irregular Modified Cylindrical Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_k1_scaled:               Irregular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_K1_scaled:               Irregular Modified Cylindrical Bessel Functions.
                                                              (line  39)
* gsl_sf_bessel_K1_scaled_e:             Irregular Modified Cylindrical Bessel Functions.
                                                              (line  41)
* gsl_sf_bessel_k1_scaled_e:             Irregular Modified Spherical Bessel Functions.
                                                              (line  19)
* gsl_sf_bessel_k2_scaled:               Irregular Modified Spherical Bessel Functions.
                                                              (line  23)
* gsl_sf_bessel_k2_scaled_e:             Irregular Modified Spherical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_kl_scaled:               Irregular Modified Spherical Bessel Functions.
                                                              (line  29)
* gsl_sf_bessel_kl_scaled_array:         Irregular Modified Spherical Bessel Functions.
                                                              (line  36)
* gsl_sf_bessel_kl_scaled_e:             Irregular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_Kn:                      Irregular Modified Cylindrical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_Kn_array:                Irregular Modified Cylindrical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_Kn_e:                    Irregular Modified Cylindrical Bessel Functions.
                                                              (line  19)
* gsl_sf_bessel_Kn_scaled:               Irregular Modified Cylindrical Bessel Functions.
                                                              (line  45)
* gsl_sf_bessel_Kn_scaled_array:         Irregular Modified Cylindrical Bessel Functions.
                                                              (line  52)
* gsl_sf_bessel_Kn_scaled_e:             Irregular Modified Cylindrical Bessel Functions.
                                                              (line  47)
* gsl_sf_bessel_Knu:                     Irregular Modified Bessel Functions - Fractional Order.
                                                              (line   7)
* gsl_sf_bessel_Knu_e:                   Irregular Modified Bessel Functions - Fractional Order.
                                                              (line   9)
* gsl_sf_bessel_Knu_scaled:              Irregular Modified Bessel Functions - Fractional Order.
                                                              (line  20)
* gsl_sf_bessel_Knu_scaled_e:            Irregular Modified Bessel Functions - Fractional Order.
                                                              (line  22)
* gsl_sf_bessel_lnKnu:                   Irregular Modified Bessel Functions - Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_lnKnu_e:                 Irregular Modified Bessel Functions - Fractional Order.
                                                              (line  15)
* gsl_sf_bessel_sequence_Jnu_e:          Regular Bessel Function - Fractional Order.
                                                              (line  14)
* gsl_sf_bessel_y0:                      Irregular Spherical Bessel Functions.
                                                              (line   7)
* gsl_sf_bessel_Y0:                      Irregular Cylindrical Bessel Functions.
                                                              (line   7)
* gsl_sf_bessel_y0_e:                    Irregular Spherical Bessel Functions.
                                                              (line   8)
* gsl_sf_bessel_Y0_e:                    Irregular Cylindrical Bessel Functions.
                                                              (line   8)
* gsl_sf_bessel_Y1:                      Irregular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_y1:                      Irregular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_Y1_e:                    Irregular Cylindrical Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_y1_e:                    Irregular Spherical Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_y2:                      Irregular Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_y2_e:                    Irregular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_yl:                      Irregular Spherical Bessel Functions.
                                                              (line  22)
* gsl_sf_bessel_yl_array:                Irregular Spherical Bessel Functions.
                                                              (line  29)
* gsl_sf_bessel_yl_e:                    Irregular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_Yn:                      Irregular Cylindrical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_Yn_array:                Irregular Cylindrical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_Yn_e:                    Irregular Cylindrical Bessel Functions.
                                                              (line  19)
* gsl_sf_bessel_Ynu:                     Irregular Bessel Functions - Fractional Order.
                                                              (line   7)
* gsl_sf_bessel_Ynu_e:                   Irregular Bessel Functions - Fractional Order.
                                                              (line   9)
* gsl_sf_bessel_zero_J0:                 Zeros of Regular Bessel Functions.
                                                              (line   7)
* gsl_sf_bessel_zero_J0_e:               Zeros of Regular Bessel Functions.
                                                              (line   9)
* gsl_sf_bessel_zero_J1:                 Zeros of Regular Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_zero_J1_e:               Zeros of Regular Bessel Functions.
                                                              (line  15)
* gsl_sf_bessel_zero_Jnu:                Zeros of Regular Bessel Functions.
                                                              (line  19)
* gsl_sf_bessel_zero_Jnu_e:              Zeros of Regular Bessel Functions.
                                                              (line  21)
* gsl_sf_beta:                           Beta Functions.      (line   7)
* gsl_sf_beta_e:                         Beta Functions.      (line   9)
* gsl_sf_beta_inc:                       Incomplete Beta Function.
                                                              (line   7)
* gsl_sf_beta_inc_e:                     Incomplete Beta Function.
                                                              (line   9)
* gsl_sf_Chi:                            Hyperbolic Integrals.
                                                              (line  12)
* gsl_sf_Chi_e:                          Hyperbolic Integrals.
                                                              (line  13)
* gsl_sf_choose:                         Factorials.          (line  40)
* gsl_sf_choose_e:                       Factorials.          (line  42)
* gsl_sf_Ci:                             Trigonometric Integrals.
                                                              (line  12)
* gsl_sf_Ci_e:                           Trigonometric Integrals.
                                                              (line  13)
* gsl_sf_clausen:                        Clausen Functions.   (line  15)
* gsl_sf_clausen_e:                      Clausen Functions.   (line  16)
* gsl_sf_complex_cos_e:                  Trigonometric Functions for Complex Arguments.
                                                              (line  13)
* gsl_sf_complex_dilog_e:                Complex Argument.    (line   8)
* gsl_sf_complex_log_e:                  Logarithm and Related Functions.
                                                              (line  21)
* gsl_sf_complex_logsin_e:               Trigonometric Functions for Complex Arguments.
                                                              (line  18)
* gsl_sf_complex_sin_e:                  Trigonometric Functions for Complex Arguments.
                                                              (line   8)
* gsl_sf_conicalP_0:                     Conical Functions.   (line  23)
* gsl_sf_conicalP_0_e:                   Conical Functions.   (line  25)
* gsl_sf_conicalP_1:                     Conical Functions.   (line  29)
* gsl_sf_conicalP_1_e:                   Conical Functions.   (line  31)
* gsl_sf_conicalP_cyl_reg:               Conical Functions.   (line  43)
* gsl_sf_conicalP_cyl_reg_e:             Conical Functions.   (line  45)
* gsl_sf_conicalP_half:                  Conical Functions.   (line  11)
* gsl_sf_conicalP_half_e:                Conical Functions.   (line  13)
* gsl_sf_conicalP_mhalf:                 Conical Functions.   (line  17)
* gsl_sf_conicalP_mhalf_e:               Conical Functions.   (line  19)
* gsl_sf_conicalP_sph_reg:               Conical Functions.   (line  36)
* gsl_sf_conicalP_sph_reg_e:             Conical Functions.   (line  38)
* gsl_sf_cos:                            Circular Trigonometric Functions.
                                                              (line  11)
* gsl_sf_cos_e:                          Circular Trigonometric Functions.
                                                              (line  12)
* gsl_sf_cos_err_e:                      Trigonometric Functions With Error Estimates.
                                                              (line  15)
* gsl_sf_coulomb_CL_array:               Coulomb Wave Function Normalization Constant.
                                                              (line  16)
* gsl_sf_coulomb_CL_e:                   Coulomb Wave Function Normalization Constant.
                                                              (line  11)
* gsl_sf_coulomb_wave_F_array:           Coulomb Wave Functions.
                                                              (line  36)
* gsl_sf_coulomb_wave_FG_array:          Coulomb Wave Functions.
                                                              (line  43)
* gsl_sf_coulomb_wave_FG_e:              Coulomb Wave Functions.
                                                              (line  24)
* gsl_sf_coulomb_wave_FGp_array:         Coulomb Wave Functions.
                                                              (line  52)
* gsl_sf_coulomb_wave_sphF_array:        Coulomb Wave Functions.
                                                              (line  61)
* gsl_sf_coupling_3j:                    3-j Symbols.         (line   8)
* gsl_sf_coupling_3j_e:                  3-j Symbols.         (line  11)
* gsl_sf_coupling_6j:                    6-j Symbols.         (line   8)
* gsl_sf_coupling_6j_e:                  6-j Symbols.         (line  11)
* gsl_sf_coupling_9j:                    9-j Symbols.         (line   9)
* gsl_sf_coupling_9j_e:                  9-j Symbols.         (line  12)
* gsl_sf_dawson:                         Dawson Function.     (line  12)
* gsl_sf_dawson_e:                       Dawson Function.     (line  13)
* gsl_sf_debye_1:                        Debye Functions.     (line  14)
* gsl_sf_debye_1_e:                      Debye Functions.     (line  15)
* gsl_sf_debye_2:                        Debye Functions.     (line  19)
* gsl_sf_debye_2_e:                      Debye Functions.     (line  20)
* gsl_sf_debye_3:                        Debye Functions.     (line  24)
* gsl_sf_debye_3_e:                      Debye Functions.     (line  25)
* gsl_sf_debye_4:                        Debye Functions.     (line  29)
* gsl_sf_debye_4_e:                      Debye Functions.     (line  30)
* gsl_sf_debye_5:                        Debye Functions.     (line  34)
* gsl_sf_debye_5_e:                      Debye Functions.     (line  35)
* gsl_sf_debye_6:                        Debye Functions.     (line  39)
* gsl_sf_debye_6_e:                      Debye Functions.     (line  40)
* gsl_sf_dilog:                          Real Argument.       (line   7)
* gsl_sf_dilog_e:                        Real Argument.       (line   8)
* gsl_sf_doublefact:                     Factorials.          (line  20)
* gsl_sf_doublefact_e:                   Factorials.          (line  22)
* gsl_sf_ellint_D:                       Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  29)
* gsl_sf_ellint_D_e:                     Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  31)
* gsl_sf_ellint_E:                       Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  15)
* gsl_sf_ellint_E_e:                     Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  17)
* gsl_sf_ellint_Ecomp:                   Legendre Form of Complete Elliptic Integrals.
                                                              (line  13)
* gsl_sf_ellint_Ecomp_e:                 Legendre Form of Complete Elliptic Integrals.
                                                              (line  15)
* gsl_sf_ellint_F:                       Legendre Form of Incomplete Elliptic Integrals.
                                                              (line   8)
* gsl_sf_ellint_F_e:                     Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  10)
* gsl_sf_ellint_Kcomp:                   Legendre Form of Complete Elliptic Integrals.
                                                              (line   7)
* gsl_sf_ellint_Kcomp_e:                 Legendre Form of Complete Elliptic Integrals.
                                                              (line   9)
* gsl_sf_ellint_P:                       Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  22)
* gsl_sf_ellint_P_e:                     Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  24)
* gsl_sf_ellint_RC:                      Carlson Forms.       (line   8)
* gsl_sf_ellint_RC_e:                    Carlson Forms.       (line  10)
* gsl_sf_ellint_RD:                      Carlson Forms.       (line  15)
* gsl_sf_ellint_RD_e:                    Carlson Forms.       (line  17)
* gsl_sf_ellint_RF:                      Carlson Forms.       (line  22)
* gsl_sf_ellint_RF_e:                    Carlson Forms.       (line  24)
* gsl_sf_ellint_RJ:                      Carlson Forms.       (line  29)
* gsl_sf_ellint_RJ_e:                    Carlson Forms.       (line  31)
* gsl_sf_elljac_e:                       Elliptic Functions (Jacobi).
                                                              (line  12)
* gsl_sf_erf:                            Error Function.      (line   7)
* gsl_sf_erf_e:                          Error Function.      (line   8)
* gsl_sf_erf_Q:                          Probability functions.
                                                              (line  15)
* gsl_sf_erf_Q_e:                        Probability functions.
                                                              (line  16)
* gsl_sf_erf_Z:                          Probability functions.
                                                              (line  10)
* gsl_sf_erf_Z_e:                        Probability functions.
                                                              (line  11)
* gsl_sf_erfc:                           Complementary Error Function.
                                                              (line   7)
* gsl_sf_erfc_e:                         Complementary Error Function.
                                                              (line   8)
* gsl_sf_eta:                            Eta Function.        (line  13)
* gsl_sf_eta_e:                          Eta Function.        (line  14)
* gsl_sf_eta_int:                        Eta Function.        (line   9)
* gsl_sf_eta_int_e:                      Eta Function.        (line  10)
* gsl_sf_exp:                            Exponential Function.
                                                              (line   7)
* gsl_sf_exp_e:                          Exponential Function.
                                                              (line   8)
* gsl_sf_exp_e10_e:                      Exponential Function.
                                                              (line  13)
* gsl_sf_exp_err_e:                      Exponentiation With Error Estimate.
                                                              (line   8)
* gsl_sf_exp_err_e10_e:                  Exponentiation With Error Estimate.
                                                              (line  12)
* gsl_sf_exp_mult:                       Exponential Function.
                                                              (line  19)
* gsl_sf_exp_mult_e:                     Exponential Function.
                                                              (line  21)
* gsl_sf_exp_mult_e10_e:                 Exponential Function.
                                                              (line  26)
* gsl_sf_exp_mult_err_e:                 Exponentiation With Error Estimate.
                                                              (line  18)
* gsl_sf_exp_mult_err_e10_e:             Exponentiation With Error Estimate.
                                                              (line  23)
* gsl_sf_expint_3:                       Ei_3(x).             (line   7)
* gsl_sf_expint_3_e:                     Ei_3(x).             (line   8)
* gsl_sf_expint_E1:                      Exponential Integral.
                                                              (line   7)
* gsl_sf_expint_E1_e:                    Exponential Integral.
                                                              (line   8)
* gsl_sf_expint_E2:                      Exponential Integral.
                                                              (line  15)
* gsl_sf_expint_E2_e:                    Exponential Integral.
                                                              (line  16)
* gsl_sf_expint_Ei:                      Ei(x).               (line   7)
* gsl_sf_expint_Ei_e:                    Ei(x).               (line   8)
* gsl_sf_expm1:                          Relative Exponential Functions.
                                                              (line   7)
* gsl_sf_expm1_e:                        Relative Exponential Functions.
                                                              (line   8)
* gsl_sf_exprel:                         Relative Exponential Functions.
                                                              (line  12)
* gsl_sf_exprel_2:                       Relative Exponential Functions.
                                                              (line  19)
* gsl_sf_exprel_2_e:                     Relative Exponential Functions.
                                                              (line  20)
* gsl_sf_exprel_e:                       Relative Exponential Functions.
                                                              (line  13)
* gsl_sf_exprel_n:                       Relative Exponential Functions.
                                                              (line  26)
* gsl_sf_exprel_n_e:                     Relative Exponential Functions.
                                                              (line  28)
* gsl_sf_fact:                           Factorials.          (line  13)
* gsl_sf_fact_e:                         Factorials.          (line  14)
* gsl_sf_fermi_dirac_0:                  Complete Fermi-Dirac Integrals.
                                                              (line  18)
* gsl_sf_fermi_dirac_0_e:                Complete Fermi-Dirac Integrals.
                                                              (line  20)
* gsl_sf_fermi_dirac_1:                  Complete Fermi-Dirac Integrals.
                                                              (line  24)
* gsl_sf_fermi_dirac_1_e:                Complete Fermi-Dirac Integrals.
                                                              (line  26)
* gsl_sf_fermi_dirac_2:                  Complete Fermi-Dirac Integrals.
                                                              (line  30)
* gsl_sf_fermi_dirac_2_e:                Complete Fermi-Dirac Integrals.
                                                              (line  32)
* gsl_sf_fermi_dirac_3half:              Complete Fermi-Dirac Integrals.
                                                              (line  55)
* gsl_sf_fermi_dirac_3half_e:            Complete Fermi-Dirac Integrals.
                                                              (line  57)
* gsl_sf_fermi_dirac_half:               Complete Fermi-Dirac Integrals.
                                                              (line  49)
* gsl_sf_fermi_dirac_half_e:             Complete Fermi-Dirac Integrals.
                                                              (line  51)
* gsl_sf_fermi_dirac_inc_0:              Incomplete Fermi-Dirac Integrals.
                                                              (line  11)
* gsl_sf_fermi_dirac_inc_0_e:            Incomplete Fermi-Dirac Integrals.
                                                              (line  13)
* gsl_sf_fermi_dirac_int:                Complete Fermi-Dirac Integrals.
                                                              (line  36)
* gsl_sf_fermi_dirac_int_e:              Complete Fermi-Dirac Integrals.
                                                              (line  38)
* gsl_sf_fermi_dirac_m1:                 Complete Fermi-Dirac Integrals.
                                                              (line  11)
* gsl_sf_fermi_dirac_m1_e:               Complete Fermi-Dirac Integrals.
                                                              (line  13)
* gsl_sf_fermi_dirac_mhalf:              Complete Fermi-Dirac Integrals.
                                                              (line  43)
* gsl_sf_fermi_dirac_mhalf_e:            Complete Fermi-Dirac Integrals.
                                                              (line  45)
* gsl_sf_gamma:                          Gamma Functions.     (line  16)
* gsl_sf_gamma_e:                        Gamma Functions.     (line  17)
* gsl_sf_gamma_inc:                      Incomplete Gamma Functions.
                                                              (line   7)
* gsl_sf_gamma_inc_e:                    Incomplete Gamma Functions.
                                                              (line   9)
* gsl_sf_gamma_inc_P:                    Incomplete Gamma Functions.
                                                              (line  21)
* gsl_sf_gamma_inc_P_e:                  Incomplete Gamma Functions.
                                                              (line  23)
* gsl_sf_gamma_inc_Q:                    Incomplete Gamma Functions.
                                                              (line  14)
* gsl_sf_gamma_inc_Q_e:                  Incomplete Gamma Functions.
                                                              (line  16)
* gsl_sf_gammainv:                       Gamma Functions.     (line  49)
* gsl_sf_gammainv_e:                     Gamma Functions.     (line  50)
* gsl_sf_gammastar:                      Gamma Functions.     (line  40)
* gsl_sf_gammastar_e:                    Gamma Functions.     (line  41)
* gsl_sf_gegenpoly_1:                    Gegenbauer Functions.
                                                              (line  12)
* gsl_sf_gegenpoly_1_e:                  Gegenbauer Functions.
                                                              (line  16)
* gsl_sf_gegenpoly_2:                    Gegenbauer Functions.
                                                              (line  13)
* gsl_sf_gegenpoly_2_e:                  Gegenbauer Functions.
                                                              (line  18)
* gsl_sf_gegenpoly_3:                    Gegenbauer Functions.
                                                              (line  14)
* gsl_sf_gegenpoly_3_e:                  Gegenbauer Functions.
                                                              (line  20)
* gsl_sf_gegenpoly_array:                Gegenbauer Functions.
                                                              (line  32)
* gsl_sf_gegenpoly_n:                    Gegenbauer Functions.
                                                              (line  24)
* gsl_sf_gegenpoly_n_e:                  Gegenbauer Functions.
                                                              (line  26)
* gsl_sf_hazard:                         Probability functions.
                                                              (line  28)
* gsl_sf_hazard_e:                       Probability functions.
                                                              (line  29)
* gsl_sf_hydrogenicR:                    Normalized Hydrogenic Bound States.
                                                              (line  14)
* gsl_sf_hydrogenicR_1:                  Normalized Hydrogenic Bound States.
                                                              (line   7)
* gsl_sf_hydrogenicR_1_e:                Normalized Hydrogenic Bound States.
                                                              (line   9)
* gsl_sf_hydrogenicR_e:                  Normalized Hydrogenic Bound States.
                                                              (line  16)
* gsl_sf_hyperg_0F1:                     Hypergeometric Functions.
                                                              (line  11)
* gsl_sf_hyperg_0F1_e:                   Hypergeometric Functions.
                                                              (line  13)
* gsl_sf_hyperg_1F1:                     Hypergeometric Functions.
                                                              (line  22)
* gsl_sf_hyperg_1F1_e:                   Hypergeometric Functions.
                                                              (line  24)
* gsl_sf_hyperg_1F1_int:                 Hypergeometric Functions.
                                                              (line  16)
* gsl_sf_hyperg_1F1_int_e:               Hypergeometric Functions.
                                                              (line  18)
* gsl_sf_hyperg_2F0:                     Hypergeometric Functions.
                                                              (line  86)
* gsl_sf_hyperg_2F0_e:                   Hypergeometric Functions.
                                                              (line  88)
* gsl_sf_hyperg_2F1:                     Hypergeometric Functions.
                                                              (line  52)
* gsl_sf_hyperg_2F1_conj:                Hypergeometric Functions.
                                                              (line  64)
* gsl_sf_hyperg_2F1_conj_e:              Hypergeometric Functions.
                                                              (line  66)
* gsl_sf_hyperg_2F1_conj_renorm:         Hypergeometric Functions.
                                                              (line  79)
* gsl_sf_hyperg_2F1_conj_renorm_e:       Hypergeometric Functions.
                                                              (line  81)
* gsl_sf_hyperg_2F1_e:                   Hypergeometric Functions.
                                                              (line  54)
* gsl_sf_hyperg_2F1_renorm:              Hypergeometric Functions.
                                                              (line  72)
* gsl_sf_hyperg_2F1_renorm_e:            Hypergeometric Functions.
                                                              (line  74)
* gsl_sf_hyperg_U:                       Hypergeometric Functions.
                                                              (line  40)
* gsl_sf_hyperg_U_e:                     Hypergeometric Functions.
                                                              (line  41)
* gsl_sf_hyperg_U_e10_e:                 Hypergeometric Functions.
                                                              (line  46)
* gsl_sf_hyperg_U_int:                   Hypergeometric Functions.
                                                              (line  28)
* gsl_sf_hyperg_U_int_e:                 Hypergeometric Functions.
                                                              (line  30)
* gsl_sf_hyperg_U_int_e10_e:             Hypergeometric Functions.
                                                              (line  35)
* gsl_sf_hypot:                          Circular Trigonometric Functions.
                                                              (line  15)
* gsl_sf_hypot_e:                        Circular Trigonometric Functions.
                                                              (line  17)
* gsl_sf_hzeta:                          Hurwitz Zeta Function.
                                                              (line  10)
* gsl_sf_hzeta_e:                        Hurwitz Zeta Function.
                                                              (line  12)
* gsl_sf_laguerre_1:                     Laguerre Functions.  (line  17)
* gsl_sf_laguerre_1_e:                   Laguerre Functions.  (line  21)
* gsl_sf_laguerre_2:                     Laguerre Functions.  (line  18)
* gsl_sf_laguerre_2_e:                   Laguerre Functions.  (line  23)
* gsl_sf_laguerre_3:                     Laguerre Functions.  (line  19)
* gsl_sf_laguerre_3_e:                   Laguerre Functions.  (line  25)
* gsl_sf_laguerre_n:                     Laguerre Functions.  (line  30)
* gsl_sf_laguerre_n_e:                   Laguerre Functions.  (line  32)
* gsl_sf_lambert_W0:                     Lambert W Functions. (line  14)
* gsl_sf_lambert_W0_e:                   Lambert W Functions. (line  15)
* gsl_sf_lambert_Wm1:                    Lambert W Functions. (line  19)
* gsl_sf_lambert_Wm1_e:                  Lambert W Functions. (line  21)
* gsl_sf_legendre_array_size:            Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  51)
* gsl_sf_legendre_H3d:                   Radial Functions for Hyperbolic Space.
                                                              (line  32)
* gsl_sf_legendre_H3d_0:                 Radial Functions for Hyperbolic Space.
                                                              (line  12)
* gsl_sf_legendre_H3d_0_e:               Radial Functions for Hyperbolic Space.
                                                              (line  14)
* gsl_sf_legendre_H3d_1:                 Radial Functions for Hyperbolic Space.
                                                              (line  21)
* gsl_sf_legendre_H3d_1_e:               Radial Functions for Hyperbolic Space.
                                                              (line  23)
* gsl_sf_legendre_H3d_array:             Radial Functions for Hyperbolic Space.
                                                              (line  41)
* gsl_sf_legendre_H3d_e:                 Radial Functions for Hyperbolic Space.
                                                              (line  34)
* gsl_sf_legendre_P1:                    Legendre Polynomials.
                                                              (line   7)
* gsl_sf_legendre_P1_e:                  Legendre Polynomials.
                                                              (line  11)
* gsl_sf_legendre_P2:                    Legendre Polynomials.
                                                              (line   8)
* gsl_sf_legendre_P2_e:                  Legendre Polynomials.
                                                              (line  13)
* gsl_sf_legendre_P3:                    Legendre Polynomials.
                                                              (line   9)
* gsl_sf_legendre_P3_e:                  Legendre Polynomials.
                                                              (line  15)
* gsl_sf_legendre_Pl:                    Legendre Polynomials.
                                                              (line  19)
* gsl_sf_legendre_Pl_array:              Legendre Polynomials.
                                                              (line  26)
* gsl_sf_legendre_Pl_deriv_array:        Legendre Polynomials.
                                                              (line  28)
* gsl_sf_legendre_Pl_e:                  Legendre Polynomials.
                                                              (line  21)
* gsl_sf_legendre_Plm:                   Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  18)
* gsl_sf_legendre_Plm_array:             Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  25)
* gsl_sf_legendre_Plm_deriv_array:       Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  27)
* gsl_sf_legendre_Plm_e:                 Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  20)
* gsl_sf_legendre_Q0:                    Legendre Polynomials.
                                                              (line  33)
* gsl_sf_legendre_Q0_e:                  Legendre Polynomials.
                                                              (line  35)
* gsl_sf_legendre_Q1:                    Legendre Polynomials.
                                                              (line  39)
* gsl_sf_legendre_Q1_e:                  Legendre Polynomials.
                                                              (line  41)
* gsl_sf_legendre_Ql:                    Legendre Polynomials.
                                                              (line  45)
* gsl_sf_legendre_Ql_e:                  Legendre Polynomials.
                                                              (line  47)
* gsl_sf_legendre_sphPlm:                Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  32)
* gsl_sf_legendre_sphPlm_array:          Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  42)
* gsl_sf_legendre_sphPlm_deriv_array:    Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  44)
* gsl_sf_legendre_sphPlm_e:              Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  34)
* gsl_sf_lnbeta:                         Beta Functions.      (line  13)
* gsl_sf_lnbeta_e:                       Beta Functions.      (line  15)
* gsl_sf_lnchoose:                       Factorials.          (line  46)
* gsl_sf_lnchoose_e:                     Factorials.          (line  48)
* gsl_sf_lncosh:                         Hyperbolic Trigonometric Functions.
                                                              (line  11)
* gsl_sf_lncosh_e:                       Hyperbolic Trigonometric Functions.
                                                              (line  12)
* gsl_sf_lndoublefact:                   Factorials.          (line  34)
* gsl_sf_lndoublefact_e:                 Factorials.          (line  36)
* gsl_sf_lnfact:                         Factorials.          (line  27)
* gsl_sf_lnfact_e:                       Factorials.          (line  29)
* gsl_sf_lngamma:                        Gamma Functions.     (line  24)
* gsl_sf_lngamma_complex_e:              Gamma Functions.     (line  55)
* gsl_sf_lngamma_e:                      Gamma Functions.     (line  25)
* gsl_sf_lngamma_sgn_e:                  Gamma Functions.     (line  33)
* gsl_sf_lnpoch:                         Pochhammer Symbol.   (line  15)
* gsl_sf_lnpoch_e:                       Pochhammer Symbol.   (line  17)
* gsl_sf_lnpoch_sgn_e:                   Pochhammer Symbol.   (line  22)
* gsl_sf_lnsinh:                         Hyperbolic Trigonometric Functions.
                                                              (line   7)
* gsl_sf_lnsinh_e:                       Hyperbolic Trigonometric Functions.
                                                              (line   8)
* gsl_sf_log:                            Logarithm and Related Functions.
                                                              (line  11)
* gsl_sf_log_1plusx:                     Logarithm and Related Functions.
                                                              (line  26)
* gsl_sf_log_1plusx_e:                   Logarithm and Related Functions.
                                                              (line  27)
* gsl_sf_log_1plusx_mx:                  Logarithm and Related Functions.
                                                              (line  31)
* gsl_sf_log_1plusx_mx_e:                Logarithm and Related Functions.
                                                              (line  33)
* gsl_sf_log_abs:                        Logarithm and Related Functions.
                                                              (line  15)
* gsl_sf_log_abs_e:                      Logarithm and Related Functions.
                                                              (line  16)
* gsl_sf_log_e:                          Logarithm and Related Functions.
                                                              (line  12)
* gsl_sf_log_erfc:                       Log Complementary Error Function.
                                                              (line   7)
* gsl_sf_log_erfc_e:                     Log Complementary Error Function.
                                                              (line   8)
* gsl_sf_multiply_e:                     Elementary Operations.
                                                              (line  12)
* gsl_sf_multiply_err_e:                 Elementary Operations.
                                                              (line  17)
* gsl_sf_poch:                           Pochhammer Symbol.   (line   7)
* gsl_sf_poch_e:                         Pochhammer Symbol.   (line   9)
* gsl_sf_pochrel:                        Pochhammer Symbol.   (line  28)
* gsl_sf_pochrel_e:                      Pochhammer Symbol.   (line  30)
* gsl_sf_polar_to_rect:                  Conversion Functions.
                                                              (line   8)
* gsl_sf_pow_int:                        Power Function.      (line  11)
* gsl_sf_pow_int_e:                      Power Function.      (line  13)
* gsl_sf_psi:                            Digamma Function.    (line  12)
* gsl_sf_psi_1:                          Trigamma Function.   (line  12)
* gsl_sf_psi_1_e:                        Trigamma Function.   (line  13)
* gsl_sf_psi_1_int:                      Trigamma Function.   (line   7)
* gsl_sf_psi_1_int_e:                    Trigamma Function.   (line   8)
* gsl_sf_psi_1piy:                       Digamma Function.    (line  17)
* gsl_sf_psi_1piy_e:                     Digamma Function.    (line  18)
* gsl_sf_psi_e:                          Digamma Function.    (line  13)
* gsl_sf_psi_int:                        Digamma Function.    (line   7)
* gsl_sf_psi_int_e:                      Digamma Function.    (line   8)
* gsl_sf_psi_n:                          Polygamma Function.  (line   7)
* gsl_sf_psi_n_e:                        Polygamma Function.  (line   9)
* gsl_sf_rect_to_polar:                  Conversion Functions.
                                                              (line  14)
* gsl_sf_Shi:                            Hyperbolic Integrals.
                                                              (line   7)
* gsl_sf_Shi_e:                          Hyperbolic Integrals.
                                                              (line   8)
* gsl_sf_Si:                             Trigonometric Integrals.
                                                              (line   7)
* gsl_sf_Si_e:                           Trigonometric Integrals.
                                                              (line   8)
* gsl_sf_sin:                            Circular Trigonometric Functions.
                                                              (line   7)
* gsl_sf_sin_e:                          Circular Trigonometric Functions.
                                                              (line   8)
* gsl_sf_sin_err_e:                      Trigonometric Functions With Error Estimates.
                                                              (line   8)
* gsl_sf_sinc:                           Circular Trigonometric Functions.
                                                              (line  21)
* gsl_sf_sinc_e:                         Circular Trigonometric Functions.
                                                              (line  22)
* gsl_sf_synchrotron_1:                  Synchrotron Functions.
                                                              (line  10)
* gsl_sf_synchrotron_1_e:                Synchrotron Functions.
                                                              (line  12)
* gsl_sf_synchrotron_2:                  Synchrotron Functions.
                                                              (line  16)
* gsl_sf_synchrotron_2_e:                Synchrotron Functions.
                                                              (line  18)
* gsl_sf_taylorcoeff:                    Factorials.          (line  52)
* gsl_sf_taylorcoeff_e:                  Factorials.          (line  54)
* gsl_sf_transport_2:                    Transport Functions. (line  11)
* gsl_sf_transport_2_e:                  Transport Functions. (line  13)
* gsl_sf_transport_3:                    Transport Functions. (line  16)
* gsl_sf_transport_3_e:                  Transport Functions. (line  18)
* gsl_sf_transport_4:                    Transport Functions. (line  21)
* gsl_sf_transport_4_e:                  Transport Functions. (line  23)
* gsl_sf_transport_5:                    Transport Functions. (line  26)
* gsl_sf_transport_5_e:                  Transport Functions. (line  28)
* gsl_sf_zeta:                           Riemann Zeta Function.
                                                              (line  15)
* gsl_sf_zeta_e:                         Riemann Zeta Function.
                                                              (line  16)
* gsl_sf_zeta_int:                       Riemann Zeta Function.
                                                              (line  10)
* gsl_sf_zeta_int_e:                     Riemann Zeta Function.
                                                              (line  11)
* gsl_sf_zetam1:                         Riemann Zeta Function Minus One.
                                                              (line  15)
* gsl_sf_zetam1_e:                       Riemann Zeta Function Minus One.
                                                              (line  16)
* gsl_sf_zetam1_int:                     Riemann Zeta Function Minus One.
                                                              (line  11)
* gsl_sf_zetam1_int_e:                   Riemann Zeta Function Minus One.
                                                              (line  12)
* GSL_SIGN:                              Testing the Sign of Numbers.
                                                              (line   7)
* gsl_siman_solve:                       Simulated Annealing functions.
                                                              (line  13)
* gsl_sort:                              Sorting vectors.     (line  24)
* gsl_sort_index:                        Sorting vectors.     (line  33)
* gsl_sort_largest:                      Selecting the k smallest or largest elements.
                                                              (line  25)
* gsl_sort_largest_index:                Selecting the k smallest or largest elements.
                                                              (line  51)
* gsl_sort_smallest:                     Selecting the k smallest or largest elements.
                                                              (line  18)
* gsl_sort_smallest_index:               Selecting the k smallest or largest elements.
                                                              (line  43)
* gsl_sort_vector:                       Sorting vectors.     (line  28)
* gsl_sort_vector_index:                 Sorting vectors.     (line  43)
* gsl_sort_vector_largest:               Selecting the k smallest or largest elements.
                                                              (line  34)
* gsl_sort_vector_largest_index:         Selecting the k smallest or largest elements.
                                                              (line  61)
* gsl_sort_vector_smallest:              Selecting the k smallest or largest elements.
                                                              (line  32)
* gsl_sort_vector_smallest_index:        Selecting the k smallest or largest elements.
                                                              (line  59)
* gsl_spline_alloc:                      Higher-level Interface.
                                                              (line  16)
* gsl_spline_eval:                       Higher-level Interface.
                                                              (line  29)
* gsl_spline_eval_deriv:                 Higher-level Interface.
                                                              (line  34)
* gsl_spline_eval_deriv2:                Higher-level Interface.
                                                              (line  39)
* gsl_spline_eval_deriv2_e:              Higher-level Interface.
                                                              (line  41)
* gsl_spline_eval_deriv_e:               Higher-level Interface.
                                                              (line  36)
* gsl_spline_eval_e:                     Higher-level Interface.
                                                              (line  31)
* gsl_spline_eval_integ:                 Higher-level Interface.
                                                              (line  44)
* gsl_spline_eval_integ_e:               Higher-level Interface.
                                                              (line  46)
* gsl_spline_free:                       Higher-level Interface.
                                                              (line  21)
* gsl_spline_init:                       Higher-level Interface.
                                                              (line  19)
* gsl_spline_min_size:                   Higher-level Interface.
                                                              (line  26)
* gsl_spline_name:                       Higher-level Interface.
                                                              (line  23)
* gsl_stats_absdev:                      Absolute deviation.  (line   8)
* gsl_stats_absdev_m:                    Absolute deviation.  (line  21)
* gsl_stats_covariance:                  Covariance.          (line   9)
* gsl_stats_covariance_m:                Covariance.          (line  17)
* gsl_stats_kurtosis:                    Higher moments (skewness and kurtosis).
                                                              (line  31)
* gsl_stats_kurtosis_m_sd:               Higher moments (skewness and kurtosis).
                                                              (line  42)
* gsl_stats_lag1_autocorrelation:        Autocorrelation.     (line   8)
* gsl_stats_lag1_autocorrelation_m:      Autocorrelation.     (line  18)
* gsl_stats_max:                         Maximum and Minimum values.
                                                              (line  14)
* gsl_stats_max_index:                   Maximum and Minimum values.
                                                              (line  39)
* gsl_stats_mean:                        Mean and standard deviation and variance.
                                                              (line   8)
* gsl_stats_median_from_sorted_data:     Median and Percentiles.
                                                              (line  12)
* gsl_stats_min:                         Maximum and Minimum values.
                                                              (line  24)
* gsl_stats_min_index:                   Maximum and Minimum values.
                                                              (line  47)
* gsl_stats_minmax:                      Maximum and Minimum values.
                                                              (line  34)
* gsl_stats_minmax_index:                Maximum and Minimum values.
                                                              (line  55)
* gsl_stats_quantile_from_sorted_data:   Median and Percentiles.
                                                              (line  27)
* gsl_stats_sd:                          Mean and standard deviation and variance.
                                                              (line  46)
* gsl_stats_sd_m:                        Mean and standard deviation and variance.
                                                              (line  48)
* gsl_stats_sd_with_fixed_mean:          Mean and standard deviation and variance.
                                                              (line  64)
* gsl_stats_skew:                        Higher moments (skewness and kurtosis).
                                                              (line   8)
* gsl_stats_skew_m_sd:                   Higher moments (skewness and kurtosis).
                                                              (line  21)
* gsl_stats_variance:                    Mean and standard deviation and variance.
                                                              (line  20)
* gsl_stats_variance_m:                  Mean and standard deviation and variance.
                                                              (line  38)
* gsl_stats_variance_with_fixed_mean:    Mean and standard deviation and variance.
                                                              (line  54)
* gsl_stats_wabsdev:                     Weighted Samples.    (line  73)
* gsl_stats_wabsdev_m:                   Weighted Samples.    (line  82)
* gsl_stats_wkurtosis:                   Weighted Samples.    (line 100)
* gsl_stats_wkurtosis_m_sd:              Weighted Samples.    (line 107)
* gsl_stats_wmean:                       Weighted Samples.    (line  16)
* gsl_stats_wsd:                         Weighted Samples.    (line  44)
* gsl_stats_wsd_m:                       Weighted Samples.    (line  50)
* gsl_stats_wsd_with_fixed_mean:         Weighted Samples.    (line  67)
* gsl_stats_wskew:                       Weighted Samples.    (line  87)
* gsl_stats_wskew_m_sd:                  Weighted Samples.    (line  94)
* gsl_stats_wvariance:                   Weighted Samples.    (line  24)
* gsl_stats_wvariance_m:                 Weighted Samples.    (line  39)
* gsl_stats_wvariance_with_fixed_mean:   Weighted Samples.    (line  56)
* gsl_strerror:                          Error Codes.         (line  37)
* gsl_sum_levin_u_accel:                 Acceleration functions.
                                                              (line  34)
* gsl_sum_levin_u_alloc:                 Acceleration functions.
                                                              (line  25)
* gsl_sum_levin_u_free:                  Acceleration functions.
                                                              (line  29)
* gsl_sum_levin_utrunc_accel:            Acceleration functions without error estimation.
                                                              (line  35)
* gsl_sum_levin_utrunc_alloc:            Acceleration functions without error estimation.
                                                              (line  24)
* gsl_sum_levin_utrunc_free:             Acceleration functions without error estimation.
                                                              (line  30)
* gsl_vector_add:                        Vector operations.   (line   9)
* gsl_vector_add_constant:               Vector operations.   (line  34)
* gsl_vector_alloc:                      Vector allocation.   (line  15)
* gsl_vector_calloc:                     Vector allocation.   (line  22)
* gsl_vector_complex_const_imag:         Vector views.        (line 110)
* gsl_vector_complex_const_real:         Vector views.        (line  99)
* gsl_vector_complex_imag:               Vector views.        (line 108)
* gsl_vector_complex_real:               Vector views.        (line  97)
* gsl_vector_const_ptr:                  Accessing vector elements.
                                                              (line  41)
* gsl_vector_const_subvector:            Vector views.        (line  31)
* gsl_vector_const_subvector_with_stride: Vector views.       (line  62)
* gsl_vector_const_view_array:           Vector views.        (line 121)
* gsl_vector_const_view_array_with_stride: Vector views.      (line 145)
* gsl_vector_div:                        Vector operations.   (line  24)
* gsl_vector_fprintf:                    Reading and writing vectors.
                                                              (line  28)
* gsl_vector_fread:                      Reading and writing vectors.
                                                              (line  18)
* gsl_vector_free:                       Vector allocation.   (line  26)
* gsl_vector_fscanf:                     Reading and writing vectors.
                                                              (line  35)
* gsl_vector_fwrite:                     Reading and writing vectors.
                                                              (line  11)
* gsl_vector_get:                        Accessing vector elements.
                                                              (line  29)
* gsl_vector_isnull:                     Vector properties.   (line   7)
* gsl_vector_max:                        Finding maximum and minimum elements of vectors.
                                                              (line   7)
* gsl_vector_max_index:                  Finding maximum and minimum elements of vectors.
                                                              (line  18)
* gsl_vector_memcpy:                     Copying vectors.     (line  14)
* gsl_vector_min:                        Finding maximum and minimum elements of vectors.
                                                              (line  10)
* gsl_vector_min_index:                  Finding maximum and minimum elements of vectors.
                                                              (line  23)
* gsl_vector_minmax:                     Finding maximum and minimum elements of vectors.
                                                              (line  14)
* gsl_vector_minmax_index:               Finding maximum and minimum elements of vectors.
                                                              (line  29)
* gsl_vector_mul:                        Vector operations.   (line  19)
* gsl_vector_ptr:                        Accessing vector elements.
                                                              (line  39)
* gsl_vector_reverse:                    Exchanging elements. (line  15)
* gsl_vector_scale:                      Vector operations.   (line  29)
* gsl_vector_set:                        Accessing vector elements.
                                                              (line  34)
* gsl_vector_set_all:                    Initializing vector elements.
                                                              (line   7)
* gsl_vector_set_basis:                  Initializing vector elements.
                                                              (line  13)
* gsl_vector_set_zero:                   Initializing vector elements.
                                                              (line  10)
* gsl_vector_sub:                        Vector operations.   (line  14)
* gsl_vector_subvector:                  Vector views.        (line  29)
* gsl_vector_subvector_with_stride:      Vector views.        (line  59)
* gsl_vector_swap:                       Copying vectors.     (line  18)
* gsl_vector_swap_elements:              Exchanging elements. (line  11)
* gsl_vector_view_array:                 Vector views.        (line 119)
* gsl_vector_view_array_with_stride:     Vector views.        (line 142)
* gsl_wavelet2d_nstransform:             DWT in two dimension.
                                                              (line  67)
* gsl_wavelet2d_nstransform_forward:     DWT in two dimension.
                                                              (line  70)
* gsl_wavelet2d_nstransform_inverse:     DWT in two dimension.
                                                              (line  73)
* gsl_wavelet2d_nstransform_matrix:      DWT in two dimension.
                                                              (line  79)
* gsl_wavelet2d_nstransform_matrix_forward: DWT in two dimension.
                                                              (line  81)
* gsl_wavelet2d_nstransform_matrix_inverse: DWT in two dimension.
                                                              (line  83)
* gsl_wavelet2d_transform:               DWT in two dimension.
                                                              (line  32)
* gsl_wavelet2d_transform_forward:       DWT in two dimension.
                                                              (line  35)
* gsl_wavelet2d_transform_inverse:       DWT in two dimension.
                                                              (line  38)
* gsl_wavelet2d_transform_matrix:        DWT in two dimension.
                                                              (line  57)
* gsl_wavelet2d_transform_matrix_forward: DWT in two dimension.
                                                              (line  59)
* gsl_wavelet2d_transform_matrix_inverse: DWT in two dimension.
                                                              (line  61)
* gsl_wavelet_alloc:                     DWT Initialization.  (line  11)
* gsl_wavelet_bspline:                   DWT Initialization.  (line  30)
* gsl_wavelet_bspline_centered:          DWT Initialization.  (line  31)
* gsl_wavelet_daubechies:                DWT Initialization.  (line  19)
* gsl_wavelet_daubechies_centered:       DWT Initialization.  (line  20)
* gsl_wavelet_free:                      DWT Initialization.  (line  45)
* gsl_wavelet_haar:                      DWT Initialization.  (line  25)
* gsl_wavelet_haar_centered:             DWT Initialization.  (line  26)
* gsl_wavelet_name:                      DWT Initialization.  (line  41)
* gsl_wavelet_transform:                 DWT in one dimension.
                                                              (line   9)
* gsl_wavelet_transform_forward:         DWT in one dimension.
                                                              (line  12)
* gsl_wavelet_transform_inverse:         DWT in one dimension.
                                                              (line  15)
* gsl_wavelet_workspace_alloc:           DWT Initialization.  (line  53)
* gsl_wavelet_workspace_free:            DWT Initialization.  (line  62)


File: gsl-ref.info,  Node: Variable Index,  Next: Type Index,  Prev: Function Index,  Up: Top

Variable Index
**************

[index]
* Menu:

* alpha <1>:                             VEGAS.               (line 124)
* alpha:                                 MISER.               (line  99)
* chisq:                                 VEGAS.               (line 115)
* dither:                                MISER.               (line 115)
* estimate_frac:                         MISER.               (line  78)
* iterations:                            VEGAS.               (line 129)
* min_calls:                             MISER.               (line  83)
* min_calls_per_bisection:               MISER.               (line  91)
* mode:                                  VEGAS.               (line 147)
* ostream:                               VEGAS.               (line 157)
* result:                                VEGAS.               (line 110)
* sigma:                                 VEGAS.               (line 111)
* stage:                                 VEGAS.               (line 133)
* verbose:                               VEGAS.               (line 156)


File: gsl-ref.info,  Node: Type Index,  Next: Concept Index,  Prev: Variable Index,  Up: Top

Type Index
**********

[index]
* Menu:

* gsl_error_handler_t:                   Error Handlers.       (line 24)
* gsl_fft_complex_wavetable:             Mixed-radix FFT routines for complex data.
                                                               (line 79)
* gsl_function:                          Providing the function to solve.
                                                               (line 12)
* gsl_function_fdf:                      Providing the function to solve.
                                                               (line 53)
* gsl_histogram:                         The histogram struct. (line  9)
* gsl_histogram2d:                       The 2D histogram struct.
                                                               (line  9)
* gsl_histogram2d_pdf:                   Resampling from 2D histograms.
                                                               (line 21)
* gsl_histogram_pdf:                     The histogram probability distribution struct.
                                                               (line 19)
* gsl_monte_function:                    Monte Carlo Interface.
                                                               (line 28)
* gsl_multifit_function:                 Providing the Function to be Minimized.
                                                               (line 11)
* gsl_multifit_function_fdf:             Providing the Function to be Minimized.
                                                               (line 31)
* gsl_multimin_function:                 Providing a function to minimize.
                                                               (line 42)
* gsl_multimin_function_fdf:             Providing a function to minimize.
                                                               (line 14)
* gsl_multiroot_function:                Providing the multidimensional system of equations to solve.
                                                               (line 11)
* gsl_multiroot_function_fdf:            Providing the multidimensional system of equations to solve.
                                                               (line 58)
* gsl_odeiv_system:                      Defining the ODE System.
                                                               (line 15)
* gsl_siman_copy_construct_t:            Simulated Annealing functions.
                                                               (line 82)
* gsl_siman_copy_t:                      Simulated Annealing functions.
                                                               (line 77)
* gsl_siman_destroy_t:                   Simulated Annealing functions.
                                                               (line 88)
* gsl_siman_Efunc_t:                     Simulated Annealing functions.
                                                               (line 52)
* gsl_siman_metric_t:                    Simulated Annealing functions.
                                                               (line 65)
* gsl_siman_params_t:                    Simulated Annealing functions.
                                                               (line 94)
* gsl_siman_print_t:                     Simulated Annealing functions.
                                                               (line 71)
* gsl_siman_step_t:                      Simulated Annealing functions.
                                                               (line 57)


File: gsl-ref.info,  Node: Concept Index,  Prev: Type Index,  Up: Top

Concept Index
*************

[index]
* Menu:

* $, shell prompt:                       Conventions used in this manual.
                                                              (line   6)
* 2D histograms:                         Two dimensional histograms.
                                                              (line   6)
* 2D random direction vector:            Spherical Vector Distributions.
                                                              (line  14)
* 3-j symbols:                           Coupling Coefficients.
                                                              (line   6)
* 3D random direction vector:            Spherical Vector Distributions.
                                                              (line  33)
* 6-j symbols:                           Coupling Coefficients.
                                                              (line   6)
* 9-j symbols:                           Coupling Coefficients.
                                                              (line   6)
* acceleration of series:                Series Acceleration. (line   6)
* acosh:                                 Elementary Functions.
                                                              (line  28)
* Adaptive step-size control, differential equations: Adaptive Step-size Control.
                                                              (line   6)
* Ai(x):                                 Airy Functions and Derivatives.
                                                              (line   6)
* Airy functions:                        Airy Functions and Derivatives.
                                                              (line   6)
* Akima splines:                         Interpolation Types. (line  36)
* aliasing of arrays:                    Aliasing of arrays.  (line   6)
* alternative optimized functions:       Alternative optimized functions.
                                                              (line   6)
* AMAX, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line  62)
* angular reduction:                     Restriction Functions.
                                                              (line   6)
* ANSI C, use of:                        Using the library.   (line   6)
* Apell symbol, see Pochammer symbol:    Pochhammer Symbol.   (line   9)
* approximate comparison of floating point numbers: Approximate Comparison of Floating Point Numbers.
                                                              (line  13)
* arctangent integral:                   Arctangent Integral. (line   6)
* argument of complex number:            Properties of complex numbers.
                                                              (line   7)
* arithmetic exceptions:                 Setting up your IEEE environment.
                                                              (line   6)
* asinh:                                 Elementary Functions.
                                                              (line  32)
* astronomical constants:                Astronomy and Astrophysics.
                                                              (line   6)
* ASUM, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line  47)
* atanh:                                 Elementary Functions.
                                                              (line  36)
* atomic physics, constants:             Atomic and Nuclear Physics.
                                                              (line   6)
* autoconf, using with GSL:              Autoconf Macros.     (line   6)
* AXPY, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line  96)
* B-spline wavelets:                     DWT Initialization.  (line  31)
* Bader and Deuflhard, Bulirsch-Stoer method.: Stepping Functions.
                                                              (line  89)
* Basic Linear Algebra Subroutines (BLAS) <1>: GSL CBLAS Library.
                                                              (line   6)
* Basic Linear Algebra Subroutines (BLAS): BLAS Support.      (line   6)
* Bernoulli trial, random variates:      The Bernoulli Distribution.
                                                              (line   8)
* Bessel functions:                      Bessel Functions.    (line   6)
* Bessel Functions, Fractional Order:    Regular Bessel Function - Fractional Order.
                                                              (line   6)
* best-fit parameters, covariance:       Computing the covariance matrix of best fit parameters.
                                                              (line   6)
* Beta distribution:                     The Beta Distribution.
                                                              (line   8)
* Beta function:                         Beta Functions.      (line   9)
* Beta function, incomplete normalized:  Incomplete Beta Function.
                                                              (line   9)
* BFGS conjugate gradient algorithm, minimization: Multimin Algorithms.
                                                              (line  36)
* Bi(x):                                 Airy Functions and Derivatives.
                                                              (line   6)
* bias, IEEE format:                     Representation of floating point numbers.
                                                              (line   6)
* bidiagonalization of real matrices:    Bidiagonalization.   (line   6)
* binning data:                          Histograms.          (line   6)
* Binomial random variates:              The Binomial Distribution.
                                                              (line   9)
* biorthogonal wavelets:                 DWT Initialization.  (line  31)
* bisection algorithm for finding roots: Root Bracketing Algorithms.
                                                              (line  17)
* Bivariate Gaussian distribution:       The Bivariate Gaussian Distribution.
                                                              (line   9)
* BLAS:                                  BLAS Support.        (line   6)
* BLAS, Low-level C interface:           GSL CBLAS Library.   (line   6)
* blocks:                                Vectors and Matrices.
                                                              (line   6)
* bounds checking, extension to GCC:     Accessing vector elements.
                                                              (line   6)
* breakpoints:                           Using gdb.           (line   6)
* Brent's method for finding minima:     Minimization Algorithms.
                                                              (line  32)
* Brent's method for finding roots:      Root Bracketing Algorithms.
                                                              (line  50)
* Broyden algorithm for multidimensional roots: Algorithms without Derivatives.
                                                              (line  44)
* BSD random number generator:           Unix random number generators.
                                                              (line  18)
* bug-gsl mailing list:                  Reporting Bugs.      (line   6)
* bugs, how to report:                   Reporting Bugs.      (line   6)
* Bulirsch-Stoer method:                 Stepping Functions.  (line  89)
* C extensions, compatible use of:       Using the library.   (line   6)
* C++, compatibility:                    Compatibility with C++.
                                                              (line   6)
* Carlson forms of Elliptic integrals:   Definition of Carlson Forms.
                                                              (line   6)
* Cash-Karp, Runge-Kutta method:         Stepping Functions.  (line  77)
* Cauchy distribution:                   The Cauchy Distribution.
                                                              (line   7)
* Cauchy principal value, by numerical quadrature: QAWC adaptive integration for Cauchy principal values.
                                                              (line   6)
* CBLAS:                                 BLAS Support.        (line   6)
* CBLAS, Low-level interface:            GSL CBLAS Library.   (line   6)
* CDFs, cumulative distribution functions: Random Number Distributions.
                                                              (line   6)
* Chebyshev series:                      Chebyshev Approximations.
                                                              (line   6)
* checking combination for validity:     Combination properties.
                                                              (line  18)
* checking permutation for validity:     Permutation properties.
                                                              (line  14)
* Chi(x):                                Hyperbolic Integrals.
                                                              (line   6)
* Chi-squared distribution:              The Chi-squared Distribution.
                                                              (line  15)
* Cholesky decomposition:                Cholesky Decomposition.
                                                              (line   6)
* Ci(x):                                 Trigonometric Integrals.
                                                              (line   6)
* Clausen functions:                     Clausen Functions.   (line   6)
* Clenshaw-Curtis quadrature:            Integrands with weight functions.
                                                              (line   6)
* CMRG, combined multiple recursive random number generator: Random number generator algorithms.
                                                              (line  89)
* code reuse in applications:            Code Reuse.          (line   6)
* combinations:                          Combinations.        (line   6)
* combinatorial factor C(m,n):           Factorials.          (line  42)
* combinatorial optimization:            Simulated Annealing. (line   6)
* comparison functions, definition:      Sorting objects.     (line  16)
* compatibility:                         Using the library.   (line   6)
* compiling programs, include paths:     Compiling and Linking.
                                                              (line   6)
* compiling programs, library paths:     Linking programs with the library.
                                                              (line   6)
* complementary incomplete Gamma function: Incomplete Gamma Functions.
                                                              (line  23)
* complete Fermi-Dirac integrals:        Complete Fermi-Dirac Integrals.
                                                              (line   6)
* complex arithmetic:                    Complex arithmetic operators.
                                                              (line   6)
* complex cosine function, special functions: Trigonometric Functions for Complex Arguments.
                                                              (line  13)
* Complex Gamma function:                Gamma Functions.     (line  55)
* complex hermitian matrix, eigensystem: Complex Hermitian Matrices.
                                                              (line   6)
* complex log sine function, special functions: Trigonometric Functions for Complex Arguments.
                                                              (line  18)
* complex numbers:                       Complex Numbers.     (line   6)
* complex sinc function, special functions: Circular Trigonometric Functions.
                                                              (line  22)
* complex sine function, special functions: Trigonometric Functions for Complex Arguments.
                                                              (line   8)
* confluent hypergeometric function:     Laguerre Functions.  (line   6)
* confluent hypergeometric functions:    Hypergeometric Functions.
                                                              (line   6)
* conical functions:                     Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* Conjugate gradient algorithm, minimization: Multimin Algorithms.
                                                              (line  12)
* conjugate of complex number:           Complex arithmetic operators.
                                                              (line  55)
* constant matrix:                       Initializing matrix elements.
                                                              (line   6)
* constants, fundamental:                Fundamental Constants.
                                                              (line   6)
* constants, mathematical--defined as macros: Mathematical Constants.
                                                              (line   6)
* constants, physical:                   Physical Constants.  (line   6)
* constants, prefixes:                   Prefixes.            (line   6)
* contacting the GSL developers:         Further Information. (line   6)
* conventions, used in manual:           Conventions used in this manual.
                                                              (line   6)
* convergence, accelerating a series:    Series Acceleration. (line   6)
* conversion of units:                   Physical Constants.  (line   6)
* cooling schedule:                      Simulated Annealing algorithm.
                                                              (line  23)
* COPY, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line  85)
* cosine function, special functions:    Circular Trigonometric Functions.
                                                              (line  12)
* cosine of complex number:              Complex Trigonometric Functions.
                                                              (line  11)
* cost function:                         Simulated Annealing. (line   6)
* Coulomb wave functions:                Coulomb Functions.   (line   6)
* coupling coefficients:                 Coupling Coefficients.
                                                              (line   6)
* covariance matrix, from linear regression: Linear regression.
                                                              (line   9)
* covariance matrix, linear fits:        Fitting Overview.    (line  21)
* covariance matrix, nonlinear fits:     Computing the covariance matrix of best fit parameters.
                                                              (line   6)
* covariance, of two datasets:           Covariance.          (line   6)
* CRAY random number generator, RANF:    Other random number generators.
                                                              (line  23)
* cubic equation, solving:               Cubic Equations.     (line   6)
* cubic splines:                         Interpolation Types. (line  20)
* cumulative distribution functions (CDFs): Random Number Distributions.
                                                              (line   6)
* Cylindrical Bessel Functions:          Regular Cylindrical Bessel Functions.
                                                              (line   6)
* Daubechies wavelets:                   DWT Initialization.  (line  20)
* Dawson function:                       Dawson Function.     (line   6)
* debugging numerical programs:          Using gdb.           (line   6)
* Debye functions:                       Debye Functions.     (line   6)
* denormalized form, IEEE format:        Representation of floating point numbers.
                                                              (line  14)
* deprecated functions:                  Deprecated Functions.
                                                              (line   6)
* derivatives, calculating numerically:  Numerical Differentiation.
                                                              (line   6)
* determinant of a matrix, by LU decomposition: LU Decomposition.
                                                              (line  82)
* Deuflhard and Bader, Bulirsch-Stoer method.: Stepping Functions.
                                                              (line  89)
* DFTs, see FFT:                         Fast Fourier Transforms.
                                                              (line   6)
* diagonal, of a matrix:                 Creating row and column views.
                                                              (line  36)
* differential equations, initial value problems: Ordinary Differential Equations.
                                                              (line   6)
* differentiation of functions, numeric: Numerical Differentiation.
                                                              (line   6)
* digamma function:                      Psi (Digamma) Function.
                                                              (line   6)
* dilogarithm:                           Dilogarithm.         (line   6)
* direction vector, random 2D:           Spherical Vector Distributions.
                                                              (line  14)
* direction vector, random 3D:           Spherical Vector Distributions.
                                                              (line  33)
* direction vector, random N-dimensional: Spherical Vector Distributions.
                                                              (line  42)
* Dirichlet distribution:                The Dirichlet Distribution.
                                                              (line   8)
* Discrete Fourier Transforms, see FFT:  Fast Fourier Transforms.
                                                              (line   6)
* discrete Hankel transforms:            Discrete Hankel Transforms.
                                                              (line   6)
* Discrete Newton algorithm for multidimensional roots: Algorithms without Derivatives.
                                                              (line  25)
* Discrete random numbers:               General Discrete Distributions.
                                                              (line  52)
* Discrete random numbers, preprocessing: General Discrete Distributions.
                                                              (line  52)
* divided differences, polynomials:      Divided Difference Representation of Polynomials.
                                                              (line   6)
* division by zero, IEEE exceptions:     Setting up your IEEE environment.
                                                              (line   6)
* dollar sign $, shell prompt:           Conventions used in this manual.
                                                              (line  11)
* DOT, Level-1 BLAS:                     Level 1 GSL BLAS Interface.
                                                              (line   8)
* double factorial:                      Factorials.          (line  22)
* double precision, IEEE format:         Representation of floating point numbers.
                                                              (line  40)
* downloading GSL:                       Obtaining GSL.       (line   6)
* DWT initialization:                    DWT Initialization.  (line   6)
* DWT, mathematical definition:          DWT Definitions.     (line   6)
* DWT, one dimensional:                  DWT in one dimension.
                                                              (line   6)
* DWT, see wavelet transforms:           Wavelet Transforms.  (line   6)
* DWT, two dimensional:                  DWT in two dimension.
                                                              (line   6)
* e, defined as a macro:                 Mathematical Constants.
                                                              (line  10)
* E1(x), E2(x), Ei(x):                   Exponential Integral.
                                                              (line   6)
* eigenvalues and eigenvectors:          Eigensystems.        (line   6)
* elementary functions:                  Mathematical Functions.
                                                              (line   6)
* elementary operations:                 Elementary Operations.
                                                              (line   6)
* elliptic functions (Jacobi):           Elliptic Functions (Jacobi).
                                                              (line   6)
* elliptic integrals:                    Elliptic Integrals.  (line   6)
* energy function:                       Simulated Annealing. (line   6)
* energy, units of:                      Thermal Energy and Power.
                                                              (line   6)
* erf(x):                                Error Functions.     (line   6)
* erfc(x):                               Error Functions.     (line   6)
* Erlang distribution:                   The Gamma Distribution.
                                                              (line  16)
* error codes:                           Error Codes.         (line  12)
* error function:                        Error Functions.     (line   6)
* Error handlers:                        Error Handlers.      (line   6)
* error handling:                        Error Handling.      (line   6)
* error handling macros:                 Using GSL error reporting in your own functions.
                                                              (line   6)
* Errors:                                Error Handling.      (line   6)
* estimated standard deviation:          Statistics.          (line   6)
* estimated variance:                    Statistics.          (line   6)
* euclidean distance function, hypot:    Elementary Functions.
                                                              (line  23)
* Euler's constant, defined as a macro:  Mathematical Constants.
                                                              (line  58)
* evaluation of polynomials:             Polynomial Evaluation.
                                                              (line   6)
* evaluation of polynomials, in divided difference form: Divided Difference Representation of Polynomials.
                                                              (line   6)
* examples, conventions used in:         Conventions used in this manual.
                                                              (line   6)
* exceptions, IEEE arithmetic:           Setting up your IEEE environment.
                                                              (line   6)
* exchanging permutation elements:       Accessing permutation elements.
                                                              (line  17)
* exp:                                   Exponential Functions.
                                                              (line   6)
* expm1:                                 Elementary Functions.
                                                              (line  18)
* exponent, IEEE format:                 Representation of floating point numbers.
                                                              (line   6)
* Exponential distribution:              The Exponential Distribution.
                                                              (line   8)
* exponential function:                  Exponential Functions.
                                                              (line   6)
* exponential integrals:                 Exponential Integrals.
                                                              (line   6)
* Exponential power distribution:        The Exponential Power Distribution.
                                                              (line   8)
* exponential, difference from 1 computed accurately: Elementary Functions.
                                                              (line  18)
* exponentiation of complex number:      Elementary Complex Functions.
                                                              (line  16)
* F-distribution:                        The F-distribution.  (line  16)
* factorial:                             Factorials.          (line  14)
* factorization of matrices:             Linear Algebra.      (line   6)
* false position algorithm for finding roots: Root Bracketing Algorithms.
                                                              (line  33)
* Fast Fourier Transforms, see FFT:      Fast Fourier Transforms.
                                                              (line   6)
* FDL, GNU Free Documentation License:   GNU Free Documentation License.
                                                              (line   6)
* Fehlberg method, differential equations: Stepping Functions.
                                                              (line  73)
* Fermi-Dirac function:                  Fermi-Dirac Function.
                                                              (line   6)
* FFT:                                   Fast Fourier Transforms.
                                                              (line   6)
* FFT mathematical definition:           Mathematical Definitions.
                                                              (line   6)
* FFT of complex data, mixed-radix algorithm: Mixed-radix FFT routines for complex data.
                                                              (line   6)
* FFT of complex data, radix-2 algorithm: Radix-2 FFT routines for complex data.
                                                              (line   6)
* FFT of real data:                      Overview of real data FFTs.
                                                              (line   6)
* FFT of real data, mixed-radix algorithm: Mixed-radix FFT routines for real data.
                                                              (line   6)
* FFT of real data, radix-2 algorithm:   Radix-2 FFT routines for real data.
                                                              (line   6)
* FFT, complex data:                     Overview of complex data FFTs.
                                                              (line   6)
* finding minima:                        One dimensional Minimization.
                                                              (line   6)
* finding roots:                         One dimensional Root-Finding.
                                                              (line   6)
* finding zeros:                         One dimensional Root-Finding.
                                                              (line   6)
* fits, multi-parameter linear:          Multi-parameter fitting.
                                                              (line   6)
* fitting:                               Least-Squares Fitting.
                                                              (line   6)
* fitting, using Chebyshev polynomials:  Chebyshev Approximations.
                                                              (line   6)
* Fj(x), Fermi-Dirac integral:           Complete Fermi-Dirac Integrals.
                                                              (line   6)
* Fj(x,b), incomplete Fermi-Dirac integral: Incomplete Fermi-Dirac Integrals.
                                                              (line   6)
* flat distribution:                     The Flat (Uniform) Distribution.
                                                              (line   8)
* Fletcher-Reeves conjugate gradient algorithm, minimization: Multimin Algorithms.
                                                              (line  12)
* floating point numbers, approximate comparison: Approximate Comparison of Floating Point Numbers.
                                                              (line  13)
* force and energy, units of:            Force and Energy.    (line   6)
* Fortran range checking, equivalent in gcc: Accessing vector elements.
                                                              (line   6)
* Four-tap Generalized Feedback Shift Register: Random number generator algorithms.
                                                              (line 164)
* Fourier integrals, numerical:          QAWF adaptive integration for Fourier integrals.
                                                              (line   6)
* Fourier Transforms, see FFT:           Fast Fourier Transforms.
                                                              (line   6)
* Fractional Order Bessel Functions:     Regular Bessel Function - Fractional Order.
                                                              (line   6)
* free documentation:                    Free Software Needs Free Documentation.
                                                              (line   6)
* free software, explanation of:         GSL is Free Software.
                                                              (line   6)
* frexp:                                 Elementary Functions.
                                                              (line  44)
* functions, numerical differentiation:  Numerical Differentiation.
                                                              (line   6)
* fundamental constants:                 Fundamental Constants.
                                                              (line   6)
* Gamma distribution:                    The Gamma Distribution.
                                                              (line   9)
* gamma functions:                       Gamma Functions.     (line   6)
* Gauss-Kronrod quadrature:              Integrands without weight functions.
                                                              (line   6)
* Gaussian distribution:                 The Gaussian Distribution.
                                                              (line   7)
* Gaussian distribution, bivariate:      The Bivariate Gaussian Distribution.
                                                              (line   9)
* Gaussian Tail distribution:            The Gaussian Tail Distribution.
                                                              (line   8)
* gcc extensions, range-checking:        Accessing vector elements.
                                                              (line   6)
* gcc warning options:                   GCC warning options for numerical programs.
                                                              (line   6)
* gdb:                                   Using gdb.           (line   6)
* Gear method, differential equations:   Stepping Functions.  (line  93)
* Gegenbauer functions:                  Gegenbauer Functions.
                                                              (line   6)
* GEMM, Level-3 BLAS:                    Level 3 GSL BLAS Interface.
                                                              (line  22)
* GEMV, Level-2 BLAS:                    Level 2 GSL BLAS Interface.
                                                              (line  20)
* general polynomial equations, solving: General Polynomial Equations.
                                                              (line   6)
* Geometric random variates <1>:         The Geometric Distribution.
                                                              (line   8)
* Geometric random variates:             The Hypergeometric Distribution.
                                                              (line   8)
* GER, Level-2 BLAS:                     Level 2 GSL BLAS Interface.
                                                              (line 104)
* GERC, Level-2 BLAS:                    Level 2 GSL BLAS Interface.
                                                              (line 113)
* GERU, Level-2 BLAS:                    Level 2 GSL BLAS Interface.
                                                              (line 104)
* Givens Rotation, BLAS:                 Level 1 GSL BLAS Interface.
                                                              (line 116)
* Givens Rotation, Modified, BLAS:       Level 1 GSL BLAS Interface.
                                                              (line 135)
* GNU General Public License:            Introduction.        (line   6)
* golden section algorithm for finding minima: Minimization Algorithms.
                                                              (line  14)
* gsl_sf_result:                         The gsl_sf_result struct.
                                                              (line   6)
* gsl_sf_result_e10:                     The gsl_sf_result struct.
                                                              (line   6)
* Gumbel distribution (Type 1):          The Type-1 Gumbel Distribution.
                                                              (line   8)
* Gumbel distribution (Type 2):          The Type-2 Gumbel Distribution.
                                                              (line   8)
* Haar wavelets:                         DWT Initialization.  (line  26)
* Hankel transforms, discrete:           Discrete Hankel Transforms.
                                                              (line   6)
* HAVE_INLINE:                           Inline functions.    (line   6)
* hazard function, normal distribution:  Probability functions.
                                                              (line  19)
* HBOOK:                                 Ntuple References and Further Reading.
                                                              (line   6)
* header files, including:               Compiling and Linking.
                                                              (line   6)
* heapsort:                              Sorting.             (line   6)
* HEMM, Level-3 BLAS:                    Level 3 GSL BLAS Interface.
                                                              (line  56)
* HEMV, Level-2 BLAS:                    Level 2 GSL BLAS Interface.
                                                              (line  85)
* HER, Level-2 BLAS:                     Level 2 GSL BLAS Interface.
                                                              (line 131)
* HER2, Level-2 BLAS:                    Level 2 GSL BLAS Interface.
                                                              (line 158)
* HER2K, Level-3 BLAS:                   Level 3 GSL BLAS Interface.
                                                              (line 180)
* HERK, Level-3 BLAS:                    Level 3 GSL BLAS Interface.
                                                              (line 139)
* hermitian matrix, complex, eigensystem: Complex Hermitian Matrices.
                                                              (line   6)
* histogram statistics:                  Histogram Statistics.
                                                              (line   6)
* histogram, from ntuple:                Histogramming ntuple values.
                                                              (line  33)
* histograms:                            Histograms.          (line   6)
* histograms, random sampling from:      The histogram probability distribution struct.
                                                              (line   6)
* Householder linear solver:             Householder solver for linear systems.
                                                              (line   6)
* Householder matrix:                    Householder Transformations.
                                                              (line   6)
* Householder transformation:            Householder Transformations.
                                                              (line   6)
* HYBRID algorithm, unscaled without derivatives: Algorithms without Derivatives.
                                                              (line  21)
* HYBRID algorithms for nonlinear systems: Algorithms using Derivatives.
                                                              (line  13)
* HYBRIDJ algorithm:                     Algorithms using Derivatives.
                                                              (line  67)
* HYBRIDS algorithm, scaled without derivatives: Algorithms without Derivatives.
                                                              (line  14)
* HYBRIDSJ algorithm:                    Algorithms using Derivatives.
                                                              (line  14)
* hydrogen atom:                         Coulomb Functions.   (line   6)
* hyperbolic cosine, inverse:            Elementary Functions.
                                                              (line  28)
* hyperbolic functions, complex numbers: Complex Hyperbolic Functions.
                                                              (line   6)
* hyperbolic integrals:                  Hyperbolic Integrals.
                                                              (line   6)
* hyperbolic sine, inverse:              Elementary Functions.
                                                              (line  32)
* hyperbolic space:                      Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* hyperbolic tangent, inverse:           Elementary Functions.
                                                              (line  36)
* hypergeometric functions:              Hypergeometric Functions.
                                                              (line   6)
* hypergeometric random variates:        The Hypergeometric Distribution.
                                                              (line   6)
* hypot:                                 Elementary Functions.
                                                              (line  23)
* hypot function, special functions:     Circular Trigonometric Functions.
                                                              (line  17)
* identity matrix:                       Initializing matrix elements.
                                                              (line   6)
* identity permutation:                  Permutation allocation.
                                                              (line  20)
* IEEE exceptions:                       Setting up your IEEE environment.
                                                              (line   6)
* IEEE floating point:                   IEEE floating-point arithmetic.
                                                              (line   6)
* IEEE format for floating point numbers: Representation of floating point numbers.
                                                              (line   6)
* IEEE infinity, defined as a macro:     Infinities and Not-a-number.
                                                              (line   6)
* IEEE NaN, defined as a macro:          Infinities and Not-a-number.
                                                              (line   6)
* illumination, units of:                Light and Illumination.
                                                              (line   6)
* imperial units:                        Imperial Units.      (line   6)
* importance sampling, VEGAS:            VEGAS.               (line   6)
* including GSL header files:            Compiling and Linking.
                                                              (line   6)
* incomplete Beta function, normalized:  Incomplete Beta Function.
                                                              (line   9)
* incomplete Fermi-Dirac integral:       Incomplete Fermi-Dirac Integrals.
                                                              (line   6)
* incomplete Gamma function:             Incomplete Gamma Functions.
                                                              (line  16)
* indirect sorting:                      Sorting objects.     (line  57)
* indirect sorting, of vector elements:  Sorting vectors.     (line  31)
* infinity, defined as a macro:          Infinities and Not-a-number.
                                                              (line   6)
* infinity, IEEE format:                 Representation of floating point numbers.
                                                              (line  27)
* info-gsl mailing list:                 Obtaining GSL.       (line   6)
* initial value problems, differential equations: Ordinary Differential Equations.
                                                              (line   6)
* initializing matrices:                 Initializing matrix elements.
                                                              (line   6)
* initializing vectors:                  Initializing vector elements.
                                                              (line   6)
* inline functions:                      Inline functions.    (line   6)
* integer powers:                        Power Function.      (line   6)
* integrals, exponential:                Exponential Integrals.
                                                              (line   6)
* integration, numerical (quadrature):   Numerical Integration.
                                                              (line   6)
* interpolation:                         Interpolation.       (line   6)
* interpolation, using Chebyshev polynomials: Chebyshev Approximations.
                                                              (line   6)
* inverse complex trigonometric functions: Inverse Complex Trigonometric Functions.
                                                              (line   6)
* inverse cumulative distribution functions: Random Number Distributions.
                                                              (line   6)
* inverse hyperbolic cosine:             Elementary Functions.
                                                              (line  28)
* inverse hyperbolic functions, complex numbers: Inverse Complex Hyperbolic Functions.
                                                              (line   6)
* inverse hyperbolic sine:               Elementary Functions.
                                                              (line  32)
* inverse hyperbolic tangent:            Elementary Functions.
                                                              (line  36)
* inverse of a matrix, by LU decomposition: LU Decomposition. (line  68)
* inverting a permutation:               Permutation functions.
                                                              (line  11)
* Irregular Cylindrical Bessel Functions: Irregular Cylindrical Bessel Functions.
                                                              (line   6)
* Irregular Modified Bessel Functions, Fractional Order: Irregular Modified Bessel Functions - Fractional Order.
                                                              (line   6)
* Irregular Modified Cylindrical Bessel Functions: Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* Irregular Modified Spherical Bessel Functions: Irregular Modified Spherical Bessel Functions.
                                                              (line   6)
* Irregular Spherical Bessel Functions:  Irregular Spherical Bessel Functions.
                                                              (line   6)
* iterating through combinations:        Combination functions.
                                                              (line   7)
* iterating through permutations:        Permutation functions.
                                                              (line  15)
* iterative refinement of solutions in linear systems: LU Decomposition.
                                                              (line  56)
* Jacobi elliptic functions:             Elliptic Functions (Jacobi).
                                                              (line   6)
* Laguerre functions:                    Laguerre Functions.  (line   6)
* Lambert function:                      Lambert W Functions. (line   6)
* Landau distribution:                   The Landau Distribution.
                                                              (line   8)
* LAPACK, recommended for linear algebra <1>: Linear Algebra. (line  11)
* LAPACK, recommended for linear algebra: Eigensystems.       (line  12)
* Laplace distribution:                  The Laplace Distribution.
                                                              (line   7)
* LD_LIBRARY_PATH:                       Shared Libraries.    (line   6)
* ldexp:                                 Elementary Functions.
                                                              (line  40)
* leading dimension, matrices:           Matrices.            (line   6)
* least squares fit:                     Least-Squares Fitting.
                                                              (line   6)
* least squares fitting, nonlinear:      Nonlinear Least-Squares Fitting.
                                                              (line   7)
* least squares, covariance of best-fit parameters: Computing the covariance matrix of best fit parameters.
                                                              (line   6)
* Legendre forms of elliptic integrals:  Definition of Legendre Forms.
                                                              (line   6)
* Legendre functions:                    Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* length, computed accurately using hypot: Elementary Functions.
                                                              (line  23)
* Levenberg-Marquardt algorithms:        Minimization Algorithms using Derivatives.
                                                              (line  12)
* Levin u-transform:                     Series Acceleration. (line   6)
* Levy distribution:                     The Levy alpha-Stable Distributions.
                                                              (line   9)
* Levy distribution, skew:               The Levy skew alpha-Stable Distribution.
                                                              (line   9)
* libraries, linking with:               Linking programs with the library.
                                                              (line   6)
* libraries, shared:                     Shared Libraries.    (line   6)
* license of GSL:                        Introduction.        (line   6)
* light, units of:                       Light and Illumination.
                                                              (line   6)
* linear algebra:                        Linear Algebra.      (line   6)
* linear algebra, BLAS:                  BLAS Support.        (line   6)
* linear interpolation:                  Interpolation Types. (line   9)
* linear regression:                     Linear regression.   (line   6)
* linear systems, refinement of solutions: LU Decomposition.  (line  56)
* linear systems, solution of:           LU Decomposition.    (line  39)
* linking with GSL libraries:            Linking programs with the library.
                                                              (line   6)
* LMDER algorithm:                       Minimization Algorithms using Derivatives.
                                                              (line  13)
* log1p:                                 Elementary Functions.
                                                              (line  13)
* logarithm and related functions:       Logarithm and Related Functions.
                                                              (line   6)
* logarithm of Beta function:            Beta Functions.      (line  15)
* logarithm of combinatorial factor C(m,n): Factorials.       (line  48)
* logarithm of complex number:           Elementary Complex Functions.
                                                              (line  29)
* logarithm of cosh function, special functions: Hyperbolic Trigonometric Functions.
                                                              (line  12)
* logarithm of double factorial:         Factorials.          (line  36)
* logarithm of factorial:                Factorials.          (line  29)
* logarithm of Gamma function:           Gamma Functions.     (line  25)
* logarithm of Pochhammer symbol:        Pochhammer Symbol.   (line  17)
* logarithm of sinh function, special functions: Hyperbolic Trigonometric Functions.
                                                              (line   8)
* logarithm of the determinant of a matrix: LU Decomposition. (line  90)
* logarithm, computed accurately near 1: Elementary Functions.
                                                              (line  13)
* Logarithmic random variates:           The Logarithmic Distribution.
                                                              (line   8)
* Logistic distribution:                 The Logistic Distribution.
                                                              (line   7)
* Lognormal distribution:                The Lognormal Distribution.
                                                              (line   8)
* long double:                           Long double.         (line   6)
* low discrepancy sequences:             Quasi-Random Sequences.
                                                              (line   6)
* Low-level CBLAS:                       GSL CBLAS Library.   (line   6)
* LU decomposition:                      LU Decomposition.    (line   6)
* macros for mathematical constants:     Mathematical Constants.
                                                              (line   6)
* magnitude of complex number:           Properties of complex numbers.
                                                              (line  11)
* mailing list archives:                 Further Information. (line   6)
* mailing list for GSL announcements:    Obtaining GSL.       (line   6)
* mailing list, bug-gsl:                 Reporting Bugs.      (line   6)
* mantissa, IEEE format:                 Representation of floating point numbers.
                                                              (line   6)
* mass, units of:                        Mass and Weight.     (line   6)
* mathematical constants, defined as macros: Mathematical Constants.
                                                              (line   6)
* mathematical functions, elementary:    Mathematical Functions.
                                                              (line   6)
* matrices <1>:                          Vectors and Matrices.
                                                              (line   6)
* matrices:                              Matrices.            (line   6)
* matrices, initializing:                Initializing matrix elements.
                                                              (line   6)
* matrices, range-checking:              Accessing matrix elements.
                                                              (line   6)
* matrix determinant:                    LU Decomposition.    (line  82)
* matrix diagonal:                       Creating row and column views.
                                                              (line  36)
* matrix factorization:                  Linear Algebra.      (line   6)
* matrix inverse:                        LU Decomposition.    (line  68)
* matrix square root, Cholesky decomposition: Cholesky Decomposition.
                                                              (line   6)
* matrix subdiagonal:                    Creating row and column views.
                                                              (line  48)
* matrix superdiagonal:                  Creating row and column views.
                                                              (line  60)
* matrix, constant:                      Initializing matrix elements.
                                                              (line   6)
* matrix, identity:                      Initializing matrix elements.
                                                              (line   6)
* matrix, operations:                    BLAS Support.        (line   6)
* matrix, zero:                          Initializing matrix elements.
                                                              (line   6)
* max:                                   Statistics.          (line   6)
* maximal phase, Daubechies wavelets:    DWT Initialization.  (line  20)
* maximization, see minimization:        One dimensional Minimization.
                                                              (line   6)
* maximum of two numbers:                Maximum and Minimum functions.
                                                              (line   7)
* maximum value, from histogram:         Histogram Statistics.
                                                              (line   6)
* mean:                                  Statistics.          (line   6)
* mean value, from histogram:            Histogram Statistics.
                                                              (line  24)
* Mill's ratio, inverse:                 Probability functions.
                                                              (line  19)
* min:                                   Statistics.          (line   6)
* minimization, BFGS conjugate gradient algorithm: Multimin Algorithms.
                                                              (line  36)
* minimization, caveats:                 Minimization Caveats.
                                                              (line   6)
* minimization, conjugate gradient algorithm: Multimin Algorithms.
                                                              (line  12)
* minimization, multidimensional:        Multidimensional Minimization.
                                                              (line   6)
* minimization, one-dimensional:         One dimensional Minimization.
                                                              (line   6)
* minimization, overview:                Minimization Overview.
                                                              (line   6)
* minimization, Polak-Ribiere algorithm: Multimin Algorithms. (line  29)
* minimization, providing a function to minimize: Providing the function to minimize.
                                                              (line   6)
* minimization, simplex algorithm:       Multimin Algorithms. (line  55)
* minimization, steepest descent algorithm: Multimin Algorithms.
                                                              (line  45)
* minimization, stopping parameters:     Minimization Stopping Parameters.
                                                              (line   6)
* minimum finding, Brent's method:       Minimization Algorithms.
                                                              (line  32)
* minimum finding, golden section algorithm: Minimization Algorithms.
                                                              (line  14)
* minimum of two numbers:                Maximum and Minimum functions.
                                                              (line  11)
* minimum value, from histogram:         Histogram Statistics.
                                                              (line   6)
* MINPACK, minimization algorithms <1>:  Minimization Algorithms using Derivatives.
                                                              (line  13)
* MINPACK, minimization algorithms:      Algorithms using Derivatives.
                                                              (line  14)
* MISCFUN:                               Special Functions References and Further Reading.
                                                              (line  11)
* MISER monte carlo integration:         MISER.               (line   6)
* Mixed-radix FFT, complex data:         Mixed-radix FFT routines for complex data.
                                                              (line   6)
* Mixed-radix FFT, real data:            Mixed-radix FFT routines for real data.
                                                              (line   6)
* Modified Bessel Functions, Fractional Order: Regular Modified Bessel Functions - Fractional Order.
                                                              (line   6)
* Modified Clenshaw-Curtis quadrature:   Integrands with weight functions.
                                                              (line   6)
* Modified Cylindrical Bessel Functions: Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* Modified Givens Rotation, BLAS:        Level 1 GSL BLAS Interface.
                                                              (line 135)
* Modified Newton's method for nonlinear systems: Algorithms using Derivatives.
                                                              (line  91)
* Modified Spherical Bessel Functions:   Regular Modified Spherical Bessel Functions.
                                                              (line   6)
* Monte Carlo integration:               Monte Carlo Integration.
                                                              (line   6)
* MRG, multiple recursive random number generator: Random number generator algorithms.
                                                              (line 111)
* MT19937 random number generator:       Random number generator algorithms.
                                                              (line  18)
* multi-parameter regression:            Multi-parameter fitting.
                                                              (line   6)
* multidimensional root finding, Broyden algorithm: Algorithms without Derivatives.
                                                              (line  44)
* multidimensional root finding, overview: Overview of Multidimensional Root Finding.
                                                              (line   6)
* multidimensional root finding, providing a function to solve: Providing the multidimensional system of equations to solve.
                                                              (line   6)
* Multimin, caveats:                     Multimin Caveats.    (line   6)
* Multinomial distribution:              The Multinomial Distribution.
                                                              (line   8)
* multiplication:                        Elementary Operations.
                                                              (line   6)
* N-dimensional random direction vector: Spherical Vector Distributions.
                                                              (line  42)
* NaN, defined as a macro:               Infinities and Not-a-number.
                                                              (line   6)
* nautical units:                        Speed and Nautical Units.
                                                              (line   6)
* Negative Binomial distribution, random variates: The Negative Binomial Distribution.
                                                              (line   8)
* Nelder-Mead simplex algorithm for minimization: Multimin Algorithms.
                                                              (line  55)
* Newton algorithm, discrete:            Algorithms without Derivatives.
                                                              (line  25)
* Newton algorithm, globally convergent: Algorithms using Derivatives.
                                                              (line  91)
* Newton's method for finding roots:     Root Finding Algorithms using Derivatives.
                                                              (line  16)
* Newton's method for systems of nonlinear equations: Algorithms using Derivatives.
                                                              (line  73)
* Niederreiter sequence:                 Quasi-Random Sequences.
                                                              (line   6)
* NIST Statistical Reference Datasets:   Fitting References and Further Reading.
                                                              (line  16)
* non-normalized incomplete Gamma function: Incomplete Gamma Functions.
                                                              (line   9)
* nonlinear equation, solutions of:      One dimensional Root-Finding.
                                                              (line   6)
* nonlinear fitting, stopping parameters: Search Stopping Parameters for Minimization Algorithms.
                                                              (line   6)
* nonlinear functions, minimization:     One dimensional Minimization.
                                                              (line   6)
* nonlinear least squares fitting:       Nonlinear Least-Squares Fitting.
                                                              (line   7)
* nonlinear least squares fitting, overview: Overview of Nonlinear Least-Squares Fitting.
                                                              (line   6)
* nonlinear systems of equations, solution of: Multidimensional Root-Finding.
                                                              (line   6)
* normalized form, IEEE format:          Representation of floating point numbers.
                                                              (line  14)
* normalized incomplete Beta function:   Incomplete Beta Function.
                                                              (line   9)
* Not-a-number, defined as a macro:      Infinities and Not-a-number.
                                                              (line   6)
* NRM2, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line  36)
* ntuples:                               N-tuples.            (line   6)
* nuclear physics, constants:            Atomic and Nuclear Physics.
                                                              (line   6)
* numerical constants, defined as macros: Mathematical Constants.
                                                              (line   6)
* numerical derivatives:                 Numerical Differentiation.
                                                              (line   6)
* numerical integration (quadrature):    Numerical Integration.
                                                              (line   6)
* obtaining GSL:                         Obtaining GSL.       (line   6)
* ODEs, initial value problems:          Ordinary Differential Equations.
                                                              (line   6)
* optimization, combinatorial:           Simulated Annealing. (line   6)
* optimization, see minimization:        One dimensional Minimization.
                                                              (line   6)
* optimized functions, alternatives:     Alternative optimized functions.
                                                              (line   6)
* ordinary differential equations, initial value problem: Ordinary Differential Equations.
                                                              (line   6)
* oscillatory functions, numerical integration of: QAWO adaptive integration for oscillatory functions.
                                                              (line   6)
* overflow, IEEE exceptions:             Setting up your IEEE environment.
                                                              (line   6)
* Pareto distribution:                   The Pareto Distribution.
                                                              (line   8)
* PAW:                                   Ntuple References and Further Reading.
                                                              (line   6)
* permutations:                          Permutations.        (line   6)
* physical constants:                    Physical Constants.  (line   6)
* physical dimension, matrices:          Matrices.            (line   6)
* pi, defined as a macro:                Mathematical Constants.
                                                              (line  28)
* plain monte carlo:                     PLAIN Monte Carlo.   (line   6)
* Pochhammer symbol:                     Pochhammer Symbol.   (line   9)
* Poisson random numbers:                The Poisson Distribution.
                                                              (line   8)
* Polak-Ribiere algorithm, minimization: Multimin Algorithms. (line  29)
* polar form of complex numbers:         Complex numbers.     (line   6)
* polar to rectangular conversion:       Conversion Functions.
                                                              (line   6)
* polygamma functions:                   Psi (Digamma) Function.
                                                              (line   6)
* polynomial evaluation:                 Polynomial Evaluation.
                                                              (line   6)
* polynomial interpolation:              Interpolation Types. (line  13)
* polynomials, roots of:                 Polynomials.         (line   6)
* power function:                        Power Function.      (line   6)
* power of complex number:               Elementary Complex Functions.
                                                              (line  16)
* power, units of:                       Thermal Energy and Power.
                                                              (line   6)
* precision, IEEE arithmetic:            Setting up your IEEE environment.
                                                              (line   6)
* prefixes:                              Prefixes.            (line   6)
* pressure, units of:                    Pressure.            (line   6)
* Prince-Dormand, Runge-Kutta method:    Stepping Functions.  (line  80)
* printers units:                        Printers Units.      (line   6)
* probability distribution, from histogram: The histogram probability distribution struct.
                                                              (line   6)
* probability distributions, from histograms: Resampling from histograms.
                                                              (line   6)
* projection of ntuples:                 Histogramming ntuple values.
                                                              (line  33)
* psi function:                          Psi (Digamma) Function.
                                                              (line   6)
* QR decomposition:                      QR Decomposition.    (line   6)
* QR decomposition with column pivoting: QR Decomposition with Column Pivoting.
                                                              (line   6)
* QUADPACK:                              Numerical Integration.
                                                              (line   6)
* quadratic equation, solving:           Quadratic Equations. (line   6)
* quadrature:                            Numerical Integration.
                                                              (line   6)
* quantile functions:                    Random Number Distributions.
                                                              (line   6)
* quasi-random sequences:                Quasi-Random Sequences.
                                                              (line   6)
* R250 shift-register random number generator: Other random number generators.
                                                              (line  60)
* Racah coefficients:                    Coupling Coefficients.
                                                              (line   6)
* radioactivity, units of:               Radioactivity.       (line   6)
* Radix-2 FFT for real data:             Radix-2 FFT routines for real data.
                                                              (line   6)
* Radix-2 FFT, complex data:             Radix-2 FFT routines for complex data.
                                                              (line   6)
* rand, BSD random number generator:     Unix random number generators.
                                                              (line  17)
* rand48 random number generator:        Unix random number generators.
                                                              (line  59)
* random number distributions:           Random Number Distributions.
                                                              (line   6)
* random number generators:              Random Number Generation.
                                                              (line   6)
* random sampling from histograms:       The histogram probability distribution struct.
                                                              (line   6)
* RANDU random number generator:         Other random number generators.
                                                              (line 105)
* RANF random number generator:          Other random number generators.
                                                              (line  23)
* range:                                 Statistics.          (line   6)
* range-checking for matrices:           Accessing matrix elements.
                                                              (line   6)
* range-checking for vectors:            Accessing vector elements.
                                                              (line   6)
* RANLUX random number generator:        Random number generator algorithms.
                                                              (line  63)
* RANLXD random number generator:        Random number generator algorithms.
                                                              (line  57)
* RANLXS random number generator:        Random number generator algorithms.
                                                              (line  43)
* RANMAR random number generator:        Other random number generators.
                                                              (line 114)
* Rayleigh distribution:                 The Rayleigh Distribution.
                                                              (line   7)
* Rayleigh Tail distribution:            The Rayleigh Tail Distribution.
                                                              (line   8)
* real symmetric matrix, eigensystem:    Real Symmetric Matrices.
                                                              (line   6)
* Reciprocal Gamma function:             Gamma Functions.     (line  50)
* rectangular to polar conversion:       Conversion Functions.
                                                              (line   6)
* recursive stratified sampling, MISER:  MISER.               (line   6)
* reduction of angular variables:        Restriction Functions.
                                                              (line   6)
* refinement of solutions in linear systems: LU Decomposition.
                                                              (line  56)
* regression, least squares:             Least-Squares Fitting.
                                                              (line   6)
* Regular Bessel Functions, Fractional Order: Regular Bessel Function - Fractional Order.
                                                              (line   6)
* Regular Bessel Functions, Zeros of:    Zeros of Regular Bessel Functions.
                                                              (line   6)
* Regular Cylindrical Bessel Functions:  Regular Cylindrical Bessel Functions.
                                                              (line   6)
* Regular Modified Bessel Functions, Fractional Order: Regular Modified Bessel Functions - Fractional Order.
                                                              (line   6)
* Regular Modified Cylindrical Bessel Functions: Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* Regular Modified Spherical Bessel Functions: Regular Modified Spherical Bessel Functions.
                                                              (line   6)
* Regular Spherical Bessel Functions:    Regular Spherical Bessel Functions.
                                                              (line   6)
* Regulated Gamma function:              Gamma Functions.     (line  41)
* relative Pochhammer symbol:            Pochhammer Symbol.   (line  30)
* reporting bugs in GSL:                 Reporting Bugs.      (line   6)
* representations of complex numbers:    Complex numbers.     (line   6)
* resampling from histograms:            Resampling from histograms.
                                                              (line   6)
* residual, in nonlinear systems of equations <1>: Search Stopping Parameters for the multidimensional solver.
                                                              (line  31)
* residual, in nonlinear systems of equations: Search Stopping Parameters for Minimization Algorithms.
                                                              (line  30)
* reversing a permutation:               Permutation functions.
                                                              (line   7)
* RK2, Runge-Kutta method:               Stepping Functions.  (line  67)
* RK4, Runge-Kutta method:               Stepping Functions.  (line  70)
* RKF45, Runge-Kutta-Fehlberg method:    Stepping Functions.  (line  73)
* root finding:                          One dimensional Root-Finding.
                                                              (line   6)
* root finding, bisection algorithm:     Root Bracketing Algorithms.
                                                              (line  17)
* root finding, Brent's method:          Root Bracketing Algorithms.
                                                              (line  50)
* root finding, caveats:                 Root Finding Caveats.
                                                              (line   6)
* root finding, false position algorithm: Root Bracketing Algorithms.
                                                              (line  33)
* root finding, initial guess:           Search Bounds and Guesses.
                                                              (line   6)
* root finding, Newton's method:         Root Finding Algorithms using Derivatives.
                                                              (line  16)
* root finding, overview:                Root Finding Overview.
                                                              (line   6)
* root finding, providing a function to solve: Providing the function to solve.
                                                              (line   6)
* root finding, search bounds:           Search Bounds and Guesses.
                                                              (line   6)
* root finding, secant method:           Root Finding Algorithms using Derivatives.
                                                              (line  30)
* root finding, Steffenson's method:     Root Finding Algorithms using Derivatives.
                                                              (line  61)
* root finding, stopping parameters <1>: Search Stopping Parameters.
                                                              (line   6)
* root finding, stopping parameters:     Search Stopping Parameters for the multidimensional solver.
                                                              (line   6)
* roots:                                 One dimensional Root-Finding.
                                                              (line   6)
* ROTG, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line 116)
* rounding mode:                         Setting up your IEEE environment.
                                                              (line   6)
* Runge-Kutta Cash-Karp method:          Stepping Functions.  (line  77)
* Runge-Kutta methods, ordinary differential equations: Stepping Functions.
                                                              (line  67)
* Runge-Kutta Prince-Dormand method:     Stepping Functions.  (line  80)
* safe comparison of floating point numbers: Approximate Comparison of Floating Point Numbers.
                                                              (line  13)
* sampling from histograms <1>:          The histogram probability distribution struct.
                                                              (line   6)
* sampling from histograms:              Resampling from histograms.
                                                              (line   6)
* SCAL, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line 109)
* schedule, cooling:                     Simulated Annealing algorithm.
                                                              (line  23)
* secant method for finding roots:       Root Finding Algorithms using Derivatives.
                                                              (line  30)
* selection function, ntuples:           Histogramming ntuple values.
                                                              (line  13)
* series, acceleration:                  Series Acceleration. (line   6)
* shared libraries:                      Shared Libraries.    (line   6)
* shell prompt:                          Conventions used in this manual.
                                                              (line   6)
* Shi(x):                                Hyperbolic Integrals.
                                                              (line   6)
* shift-register random number generator: Other random number generators.
                                                              (line  60)
* Si(x):                                 Trigonometric Integrals.
                                                              (line   6)
* sign bit, IEEE format:                 Representation of floating point numbers.
                                                              (line   6)
* sign of the determinant of a matrix:   LU Decomposition.    (line  98)
* simplex algorithm, minimization:       Multimin Algorithms. (line  55)
* simulated annealing:                   Simulated Annealing. (line   6)
* sin, of complex number:                Complex Trigonometric Functions.
                                                              (line   7)
* sine function, special functions:      Circular Trigonometric Functions.
                                                              (line   8)
* single precision, IEEE format:         Representation of floating point numbers.
                                                              (line  31)
* singular functions, numerical integration of: QAWS adaptive integration for singular functions.
                                                              (line   6)
* singular points, specifying positions in quadrature: QAGP adaptive integration with known singular points.
                                                              (line   6)
* singular value decomposition:          Singular Value Decomposition.
                                                              (line   6)
* Skew Levy distribution:                The Levy skew alpha-Stable Distribution.
                                                              (line   9)
* slope, see numerical derivative:       Numerical Differentiation.
                                                              (line   6)
* Sobol sequence:                        Quasi-Random Sequences.
                                                              (line   6)
* solution of linear system by Householder transformations: Householder solver for linear systems.
                                                              (line   6)
* solution of linear systems, Ax=b:      Linear Algebra.      (line   6)
* solving a nonlinear equation:          One dimensional Root-Finding.
                                                              (line   6)
* solving nonlinear systems of equations: Multidimensional Root-Finding.
                                                              (line   6)
* sorting:                               Sorting.             (line   6)
* sorting eigenvalues and eigenvectors:  Sorting Eigenvalues and Eigenvectors.
                                                              (line   6)
* sorting vector elements:               Sorting vectors.     (line  23)
* source code, reuse in applications:    Code Reuse.          (line   6)
* special functions:                     Special Functions.   (line   6)
* Spherical Bessel Functions:            Regular Spherical Bessel Functions.
                                                              (line   6)
* spherical harmonics:                   Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* spherical random variates, 2D:         Spherical Vector Distributions.
                                                              (line  14)
* spherical random variates, 3D:         Spherical Vector Distributions.
                                                              (line  33)
* spherical random variates, N-dimensional: Spherical Vector Distributions.
                                                              (line  42)
* spline:                                Interpolation.       (line   6)
* square root of a matrix, Cholesky decomposition: Cholesky Decomposition.
                                                              (line   6)
* square root of complex number:         Elementary Complex Functions.
                                                              (line   7)
* standard deviation:                    Statistics.          (line   6)
* standard deviation, from histogram:    Histogram Statistics.
                                                              (line  30)
* standards conformance, ANSI C:         Using the library.   (line   6)
* Statistical Reference Datasets (StRD): Fitting References and Further Reading.
                                                              (line  16)
* statistics:                            Statistics.          (line   6)
* statistics, from histogram:            Histogram Statistics.
                                                              (line   6)
* steepest descent algorithm, minimization: Multimin Algorithms.
                                                              (line  45)
* Steffenson's method for finding roots: Root Finding Algorithms using Derivatives.
                                                              (line  61)
* stratified sampling in monte carlo integration: Monte Carlo Integration.
                                                              (line   6)
* stride, of vector index:               Vectors.             (line   6)
* Student t-distribution:                The t-distribution.  (line  15)
* subdiagonal, of a matrix:              Creating row and column views.
                                                              (line  48)
* summation, acceleration:               Series Acceleration. (line   6)
* superdiagonal, matrix:                 Creating row and column views.
                                                              (line  60)
* SVD:                                   Singular Value Decomposition.
                                                              (line   6)
* SWAP, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line  76)
* swapping permutation elements:         Accessing permutation elements.
                                                              (line  17)
* SYMM, Level-3 BLAS:                    Level 3 GSL BLAS Interface.
                                                              (line  41)
* symmetric matrix, real, eigensystem:   Real Symmetric Matrices.
                                                              (line   6)
* SYMV, Level-2 BLAS:                    Level 2 GSL BLAS Interface.
                                                              (line  71)
* synchrotron functions:                 Synchrotron Functions.
                                                              (line   6)
* SYR, Level-2 BLAS:                     Level 2 GSL BLAS Interface.
                                                              (line 120)
* SYR2, Level-2 BLAS:                    Level 2 GSL BLAS Interface.
                                                              (line 144)
* SYR2K, Level-3 BLAS:                   Level 3 GSL BLAS Interface.
                                                              (line 163)
* SYRK, Level-3 BLAS:                    Level 3 GSL BLAS Interface.
                                                              (line 125)
* systems of equations, nonlinear:       Multidimensional Root-Finding.
                                                              (line   6)
* t-distribution:                        The t-distribution.  (line  15)
* t-test:                                Statistics.          (line   6)
* tangent of complex number:             Complex Trigonometric Functions.
                                                              (line  15)
* Tausworthe random number generator:    Random number generator algorithms.
                                                              (line 129)
* Taylor coefficients, computation of:   Factorials.          (line  54)
* testing combination for validity:      Combination properties.
                                                              (line  18)
* testing permutation for validity:      Permutation properties.
                                                              (line  14)
* thermal energy, units of:              Thermal Energy and Power.
                                                              (line   6)
* time units:                            Measurement of Time. (line   6)
* trailing dimension, matrices:          Matrices.            (line   6)
* transformation, Householder:           Householder Transformations.
                                                              (line   6)
* transforms, Hankel:                    Discrete Hankel Transforms.
                                                              (line   6)
* transforms, wavelet:                   Wavelet Transforms.  (line   6)
* transport functions:                   Transport Functions. (line   6)
* traveling salesman problem:            Traveling Salesman Problem.
                                                              (line   6)
* tridiagonal decomposition <1>:         Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line   6)
* tridiagonal decomposition:             Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line   6)
* tridiagonal systems:                   Tridiagonal Systems. (line   6)
* trigonometric functions:               Trigonometric Functions.
                                                              (line   6)
* trigonometric functions of complex numbers: Complex Trigonometric Functions.
                                                              (line   6)
* trigonometric integrals:               Trigonometric Integrals.
                                                              (line   6)
* TRMM, Level-3 BLAS:                    Level 3 GSL BLAS Interface.
                                                              (line  78)
* TRMV, Level-2 BLAS:                    Level 2 GSL BLAS Interface.
                                                              (line  36)
* TRSM, Level-3 BLAS:                    Level 3 GSL BLAS Interface.
                                                              (line 102)
* TRSV, Level-2 BLAS:                    Level 2 GSL BLAS Interface.
                                                              (line  57)
* TSP:                                   Traveling Salesman Problem.
                                                              (line   6)
* TT800 random number generator:         Other random number generators.
                                                              (line  75)
* two dimensional Gaussian distribution: The Bivariate Gaussian Distribution.
                                                              (line   9)
* two dimensional histograms:            Two dimensional histograms.
                                                              (line   6)
* two-sided exponential distribution:    The Laplace Distribution.
                                                              (line   7)
* Type 1 Gumbel distribution, random variates: The Type-1 Gumbel Distribution.
                                                              (line   8)
* Type 2 Gumbel distribution:            The Type-2 Gumbel Distribution.
                                                              (line   8)
* u-transform for series:                Series Acceleration. (line   6)
* underflow, IEEE exceptions:            Setting up your IEEE environment.
                                                              (line   6)
* uniform distribution:                  The Flat (Uniform) Distribution.
                                                              (line   8)
* units, conversion of:                  Physical Constants.  (line   6)
* units, imperial:                       Imperial Units.      (line   6)
* Unix random number generators, rand:   Unix random number generators.
                                                              (line  17)
* Unix random number generators, rand48: Unix random number generators.
                                                              (line  17)
* unnormalized incomplete Gamma function: Incomplete Gamma Functions.
                                                              (line   9)
* unweighted linear fits:                Least-Squares Fitting.
                                                              (line   6)
* usage, compiling application programs: Using the library.   (line   6)
* value function, ntuples:               Histogramming ntuple values.
                                                              (line  23)
* Van der Pol oscillator, example:       ODE Example programs.
                                                              (line   6)
* variance:                              Statistics.          (line   6)
* variance, from histogram:              Histogram Statistics.
                                                              (line  30)
* variance-covariance matrix, linear fits: Fitting Overview.  (line  41)
* VAX random number generator:           Other random number generators.
                                                              (line  87)
* vector, operations:                    BLAS Support.        (line   6)
* vector, sorting elements of:           Sorting vectors.     (line  23)
* vectors <1>:                           Vectors.             (line   6)
* vectors:                               Vectors and Matrices.
                                                              (line   6)
* vectors, initializing:                 Initializing vector elements.
                                                              (line   6)
* vectors, range-checking:               Accessing vector elements.
                                                              (line   6)
* VEGAS monte carlo integration:         VEGAS.               (line   6)
* viscosity, units of:                   Viscosity.           (line   6)
* volume units:                          Volume Area and Length.
                                                              (line   6)
* W function:                            Lambert W Functions. (line   6)
* warning options:                       GCC warning options for numerical programs.
                                                              (line   6)
* warranty (none):                       No Warranty.         (line   6)
* wavelet transforms:                    Wavelet Transforms.  (line   6)
* website, developer information:        Further Information. (line   6)
* Weibull distribution:                  The Weibull Distribution.
                                                              (line   8)
* weight, units of:                      Mass and Weight.     (line   6)
* weighted linear fits:                  Least-Squares Fitting.
                                                              (line   6)
* Wigner coefficients:                   Coupling Coefficients.
                                                              (line   6)
* zero finding:                          One dimensional Root-Finding.
                                                              (line   6)
* zero matrix:                           Initializing matrix elements.
                                                              (line   6)
* zero, IEEE format:                     Representation of floating point numbers.
                                                              (line  27)
* Zeros of Regular Bessel Functions:     Zeros of Regular Bessel Functions.
                                                              (line   6)
* Zeta functions:                        Zeta Functions.      (line   6)



Tag Table:
Node: Top937
Node: Introduction3507
Node: Routines available in GSL4159
Node: GSL is Free Software6074
Node: Obtaining GSL8466
Node: No Warranty9607
Node: Reporting Bugs10113
Node: Further Information10997
Node: Conventions used in this manual12010
Node: Using the library12774
Node: An Example Program13360
Node: Compiling and Linking14064
Node: Linking programs with the library15134
Node: Linking with an alternative BLAS library16279
Node: Shared Libraries17209
Node: ANSI C Compliance18625
Node: Inline functions19718
Node: Long double20761
Node: Portability functions22159
Node: Alternative optimized functions23556
Node: Support for different numeric types25014
Node: Compatibility with C++28077
Node: Aliasing of arrays28649
Node: Thread-safety29400
Node: Deprecated Functions30487
Node: Code Reuse31112
Node: Error Handling31756
Node: Error Reporting32506
Node: Error Codes34398
Node: Error Handlers36167
Node: Using GSL error reporting in your own functions39804
Node: Error Reporting Examples41762
Node: Mathematical Functions42968
Node: Mathematical Constants43729
Node: Infinities and Not-a-number44859
Node: Elementary Functions45927
Node: Small integer powers48182
Node: Testing the Sign of Numbers49741
Node: Testing for Odd and Even Numbers50172
Node: Maximum and Minimum functions50728
Node: Approximate Comparison of Floating Point Numbers52810
Node: Complex Numbers54018
Ref: Complex Numbers-Footnote-155326
Node: Complex numbers55390
Node: Properties of complex numbers57193
Node: Complex arithmetic operators58180
Node: Elementary Complex Functions60926
Node: Complex Trigonometric Functions62746
Node: Inverse Complex Trigonometric Functions63967
Node: Complex Hyperbolic Functions66573
Node: Inverse Complex Hyperbolic Functions67869
Node: Complex Number References and Further Reading69709
Node: Polynomials71105
Node: Polynomial Evaluation71872
Node: Divided Difference Representation of Polynomials72332
Node: Quadratic Equations74001
Node: Cubic Equations75941
Node: General Polynomial Equations77312
Node: Roots of Polynomials Examples79603
Node: Roots of Polynomials References and Further Reading80992
Node: Special Functions81937
Node: Special Function Usage83909
Node: The gsl_sf_result struct85092
Node: Special Function Modes86356
Node: Airy Functions and Derivatives87292
Node: Airy Functions87988
Node: Derivatives of Airy Functions89348
Node: Zeros of Airy Functions90860
Node: Zeros of Derivatives of Airy Functions91579
Node: Bessel Functions92338
Node: Regular Cylindrical Bessel Functions93530
Node: Irregular Cylindrical Bessel Functions94854
Node: Regular Modified Cylindrical Bessel Functions96310
Node: Irregular Modified Cylindrical Bessel Functions99137
Node: Regular Spherical Bessel Functions102105
Node: Irregular Spherical Bessel Functions104317
Node: Regular Modified Spherical Bessel Functions106004
Node: Irregular Modified Spherical Bessel Functions108007
Node: Regular Bessel Function - Fractional Order110070
Node: Irregular Bessel Functions - Fractional Order111068
Node: Regular Modified Bessel Functions - Fractional Order111650
Node: Irregular Modified Bessel Functions - Fractional Order112575
Node: Zeros of Regular Bessel Functions113811
Node: Clausen Functions114914
Node: Coulomb Functions115518
Node: Normalized Hydrogenic Bound States115967
Node: Coulomb Wave Functions117080
Node: Coulomb Wave Function Normalization Constant120490
Node: Coupling Coefficients121243
Node: 3-j Symbols121966
Node: 6-j Symbols122570
Node: 9-j Symbols123194
Node: Dawson Function123900
Node: Debye Functions124467
Node: Dilogarithm126242
Node: Real Argument126537
Node: Complex Argument127106
Node: Elementary Operations127575
Node: Elliptic Integrals128399
Node: Definition of Legendre Forms128872
Node: Definition of Carlson Forms129711
Node: Legendre Form of Complete Elliptic Integrals130408
Node: Legendre Form of Incomplete Elliptic Integrals131292
Node: Carlson Forms133032
Node: Elliptic Functions (Jacobi)134579
Node: Error Functions135160
Node: Error Function135611
Node: Complementary Error Function135984
Node: Log Complementary Error Function136444
Node: Probability functions136896
Node: Exponential Functions138149
Node: Exponential Function138532
Node: Relative Exponential Functions139755
Node: Exponentiation With Error Estimate141426
Node: Exponential Integrals142635
Node: Exponential Integral143117
Node: Ei(x)143767
Node: Hyperbolic Integrals144215
Node: Ei_3(x)144871
Node: Trigonometric Integrals145263
Node: Arctangent Integral145862
Node: Fermi-Dirac Function146262
Node: Complete Fermi-Dirac Integrals146632
Node: Incomplete Fermi-Dirac Integrals149093
Node: Gamma and Beta Functions149736
Node: Gamma Functions150172
Node: Factorials153260
Node: Pochhammer Symbol155813
Node: Incomplete Gamma Functions157304
Node: Beta Functions158553
Node: Incomplete Beta Function159240
Node: Gegenbauer Functions159762
Node: Hypergeometric Functions161412
Node: Laguerre Functions165645
Node: Lambert W Functions167208
Node: Legendre Functions and Spherical Harmonics168246
Node: Legendre Polynomials168835
Node: Associated Legendre Polynomials and Spherical Harmonics170915
Node: Conical Functions173718
Node: Radial Functions for Hyperbolic Space175841
Node: Logarithm and Related Functions178018
Node: Power Function179639
Node: Psi (Digamma) Function180580
Node: Digamma Function181100
Node: Trigamma Function181930
Node: Polygamma Function182507
Node: Synchrotron Functions182891
Node: Transport Functions183654
Node: Trigonometric Functions184801
Node: Circular Trigonometric Functions185424
Node: Trigonometric Functions for Complex Arguments186433
Node: Hyperbolic Trigonometric Functions187467
Node: Conversion Functions188070
Node: Restriction Functions188853
Node: Trigonometric Functions With Error Estimates189482
Node: Zeta Functions190392
Node: Riemann Zeta Function190850
Node: Riemann Zeta Function Minus One191529
Node: Hurwitz Zeta Function192300
Node: Eta Function192805
Node: Special Functions Examples193364
Node: Special Functions References and Further Reading195058
Node: Vectors and Matrices196165
Node: Data types196924
Node: Blocks198127
Node: Block allocation199043
Node: Reading and writing blocks200447
Node: Example programs for blocks202511
Node: Vectors203138
Node: Vector allocation205024
Node: Accessing vector elements206560
Node: Initializing vector elements208802
Node: Reading and writing vectors209496
Node: Vector views211586
Node: Copying vectors219028
Node: Exchanging elements219888
Node: Vector operations220446
Node: Finding maximum and minimum elements of vectors221960
Node: Vector properties223485
Node: Example programs for vectors223848
Node: Matrices226027
Node: Matrix allocation228885
Node: Accessing matrix elements230470
Node: Initializing matrix elements232178
Node: Reading and writing matrices232974
Node: Matrix views235077
Node: Creating row and column views242217
Node: Copying matrices245388
Node: Copying rows and columns245984
Node: Exchanging rows and columns247661
Node: Matrix operations249139
Node: Finding maximum and minimum elements of matrices250779
Node: Matrix properties252683
Node: Example programs for matrices253049
Node: Vector and Matrix References and Further Reading257008
Node: Permutations257494
Node: The Permutation struct258782
Node: Permutation allocation259286
Node: Accessing permutation elements260720
Node: Permutation properties261483
Node: Permutation functions262182
Node: Applying Permutations263439
Node: Reading and writing permutations265364
Node: Permutations in cyclic form267546
Node: Permutation Examples270971
Node: Permutation References and Further Reading273388
Node: Combinations274062
Node: The Combination struct274907
Node: Combination allocation275459
Node: Accessing combination elements277244
Node: Combination properties277823
Node: Combination functions278709
Node: Reading and writing combinations279649
Node: Combination Examples281849
Node: Combination References and Further Reading283157
Node: Sorting283561
Node: Sorting objects284511
Node: Sorting vectors287422
Node: Selecting the k smallest or largest elements289997
Node: Computing the rank293339
Node: Sorting Examples294540
Node: Sorting References and Further Reading296120
Node: BLAS Support296666
Node: GSL BLAS Interface299641
Node: Level 1 GSL BLAS Interface300128
Node: Level 2 GSL BLAS Interface306871
Node: Level 3 GSL BLAS Interface316170
Node: BLAS Examples327299
Node: BLAS References and Further Reading328720
Node: Linear Algebra330079
Node: LU Decomposition331333
Node: QR Decomposition336541
Node: QR Decomposition with Column Pivoting342343
Node: Singular Value Decomposition346753
Node: Cholesky Decomposition350119
Node: Tridiagonal Decomposition of Real Symmetric Matrices352040
Node: Tridiagonal Decomposition of Hermitian Matrices353888
Node: Bidiagonalization355819
Node: Householder Transformations358235
Node: Householder solver for linear systems360081
Node: Tridiagonal Systems360986
Node: Linear Algebra Examples363497
Node: Linear Algebra References and Further Reading365467
Node: Eigensystems366957
Node: Real Symmetric Matrices368014
Node: Complex Hermitian Matrices370246
Node: Sorting Eigenvalues and Eigenvectors372764
Node: Eigenvalue and Eigenvector Examples374095
Node: Eigenvalue and Eigenvector References376390
Node: Fast Fourier Transforms377098
Node: Mathematical Definitions378307
Node: Overview of complex data FFTs380712
Node: Radix-2 FFT routines for complex data383655
Node: Mixed-radix FFT routines for complex data387867
Node: Overview of real data FFTs397196
Node: Radix-2 FFT routines for real data399630
Node: Mixed-radix FFT routines for real data402469
Node: FFT References and Further Reading413051
Node: Numerical Integration415920
Node: Numerical Integration Introduction417465
Node: Integrands without weight functions419399
Node: Integrands with weight functions420228
Node: Integrands with singular weight functions420939
Node: QNG non-adaptive Gauss-Kronrod integration421857
Node: QAG adaptive integration423128
Node: QAGS adaptive integration with singularities425759
Node: QAGP adaptive integration with known singular points427559
Node: QAGI adaptive integration on infinite intervals428873
Node: QAWC adaptive integration for Cauchy principal values431166
Node: QAWS adaptive integration for singular functions432341
Node: QAWO adaptive integration for oscillatory functions435582
Node: QAWF adaptive integration for Fourier integrals439361
Node: Numerical integration error codes441986
Node: Numerical integration examples442754
Node: Numerical integration References and Further Reading444755
Node: Random Number Generation445484
Node: General comments on random numbers446986
Node: The Random Number Generator Interface448936
Node: Random number generator initialization450294
Node: Sampling from a random number generator452310
Node: Auxiliary random number generator functions455477
Node: Random number environment variables457797
Node: Copying random number generator state460297
Node: Reading and writing random number generator state461272
Node: Random number generator algorithms462684
Node: Unix random number generators472146
Node: Other random number generators475877
Node: Random Number Generator Performance485218
Node: Random Number Generator Examples486536
Node: Random Number References and Further Reading488092
Node: Random Number Acknowledgements489389
Node: Quasi-Random Sequences489875
Node: Quasi-random number generator initialization490982
Node: Sampling from a quasi-random number generator491997
Node: Auxiliary quasi-random number generator functions492614
Node: Saving and resorting quasi-random number generator state493560
Node: Quasi-random number generator algorithms494368
Node: Quasi-random number generator examples495143
Node: Quasi-random number references496128
Node: Random Number Distributions496654
Node: Random Number Distribution Introduction499853
Node: The Gaussian Distribution501648
Node: The Gaussian Tail Distribution504310
Node: The Bivariate Gaussian Distribution505979
Node: The Exponential Distribution507302
Node: The Laplace Distribution508441
Node: The Exponential Power Distribution509538
Node: The Cauchy Distribution510806
Node: The Rayleigh Distribution512055
Node: The Rayleigh Tail Distribution513227
Node: The Landau Distribution514099
Node: The Levy alpha-Stable Distributions515054
Node: The Levy skew alpha-Stable Distribution516112
Node: The Gamma Distribution517724
Node: The Flat (Uniform) Distribution519253
Node: The Lognormal Distribution520405
Node: The Chi-squared Distribution521753
Node: The F-distribution523157
Node: The t-distribution524793
Node: The Beta Distribution526187
Node: The Logistic Distribution527346
Node: The Pareto Distribution528475
Node: Spherical Vector Distributions529645
Node: The Weibull Distribution532479
Node: The Type-1 Gumbel Distribution533673
Node: The Type-2 Gumbel Distribution534910
Node: The Dirichlet Distribution536141
Node: General Discrete Distributions537809
Node: The Poisson Distribution541674
Node: The Bernoulli Distribution542682
Node: The Binomial Distribution543433
Node: The Multinomial Distribution544638
Node: The Negative Binomial Distribution546290
Node: The Pascal Distribution547655
Node: The Geometric Distribution548813
Node: The Hypergeometric Distribution550061
Node: The Logarithmic Distribution551714
Node: Shuffling and Sampling552505
Node: Random Number Distribution Examples555324
Node: Random Number Distribution References and Further Reading558469
Node: Statistics560535
Node: Mean and standard deviation and variance561814
Node: Absolute deviation564859
Node: Higher moments (skewness and kurtosis)566151
Node: Autocorrelation568284
Node: Covariance569098
Node: Weighted Samples570047
Node: Maximum and Minimum values575291
Node: Median and Percentiles578030
Node: Example statistical programs580445
Node: Statistics References and Further Reading583104
Node: Histograms584312
Node: The histogram struct586072
Node: Histogram allocation587875
Node: Copying Histograms590836
Node: Updating and accessing histogram elements591512
Node: Searching histogram ranges594779
Node: Histogram Statistics595778
Node: Histogram Operations597638
Node: Reading and writing histograms599710
Node: Resampling from histograms602752
Node: The histogram probability distribution struct603550
Node: Example programs for histograms606581
Node: Two dimensional histograms608644
Node: The 2D histogram struct609365
Node: 2D Histogram allocation611173
Node: Copying 2D Histograms613244
Node: Updating and accessing 2D histogram elements613949
Node: Searching 2D histogram ranges617604
Node: 2D Histogram Statistics618623
Node: 2D Histogram Operations621482
Node: Reading and writing 2D histograms623656
Node: Resampling from 2D histograms627283
Node: Example programs for 2D histograms630302
Node: N-tuples632030
Node: The ntuple struct633286
Node: Creating ntuples633764
Node: Opening an existing ntuple file634431
Node: Writing ntuples635059
Node: Reading ntuples635520
Node: Closing an ntuple file635851
Node: Histogramming ntuple values636191
Node: Example ntuple programs638197
Node: Ntuple References and Further Reading641500
Node: Monte Carlo Integration641821
Node: Monte Carlo Interface643066
Node: PLAIN Monte Carlo645689
Node: MISER648127
Node: VEGAS653965
Node: Monte Carlo Examples662300
Node: Monte Carlo Integration References and Further Reading668216
Node: Simulated Annealing668998
Node: Simulated Annealing algorithm670205
Node: Simulated Annealing functions671359
Node: Examples with Simulated Annealing675885
Node: Trivial example676439
Node: Traveling Salesman Problem678999
Node: Simulated Annealing References and Further Reading682349
Node: Ordinary Differential Equations682760
Node: Defining the ODE System683581
Node: Stepping Functions685773
Node: Adaptive Step-size Control689741
Node: Evolution695200
Node: ODE Example programs697761
Node: ODE References and Further Reading702831
Node: Interpolation703470
Node: Introduction to Interpolation704483
Node: Interpolation Functions704924
Node: Interpolation Types706100
Node: Index Look-up and Acceleration708710
Node: Evaluation of Interpolating Functions710340
Node: Higher-level Interface712724
Node: Interpolation Example programs714763
Node: Interpolation References and Further Reading718010
Node: Numerical Differentiation718583
Node: Numerical Differentiation functions719172
Node: Numerical Differentiation Examples722028
Node: Numerical Differentiation References723441
Node: Chebyshev Approximations723992
Node: Chebyshev Definitions725008
Node: Creation and Calculation of Chebyshev Series725798
Node: Chebyshev Series Evaluation726770
Node: Derivatives and Integrals728154
Node: Chebyshev Approximation examples729396
Node: Chebyshev Approximation References and Further Reading730892
Node: Series Acceleration731341
Node: Acceleration functions732106
Node: Acceleration functions without error estimation734348
Node: Example of accelerating a series736957
Node: Series Acceleration References739303
Node: Wavelet Transforms740191
Node: DWT Definitions740732
Node: DWT Initialization741684
Node: DWT Transform Functions744263
Node: DWT in one dimension744798
Node: DWT in two dimension746817
Node: DWT Examples751418
Node: DWT References753073
Node: Discrete Hankel Transforms755236
Node: Discrete Hankel Transform Definition755704
Node: Discrete Hankel Transform Functions757843
Node: Discrete Hankel Transform References759374
Node: One dimensional Root-Finding759778
Node: Root Finding Overview761038
Node: Root Finding Caveats762896
Node: Initializing the Solver764663
Node: Providing the function to solve767296
Node: Search Bounds and Guesses771062
Node: Root Finding Iteration771923
Node: Search Stopping Parameters773772
Node: Root Bracketing Algorithms776286
Node: Root Finding Algorithms using Derivatives779584
Node: Root Finding Examples783087
Node: Root Finding References and Further Reading790302
Node: One dimensional Minimization790941
Node: Minimization Overview792243
Node: Minimization Caveats793949
Node: Initializing the Minimizer795286
Node: Providing the function to minimize797527
Node: Minimization Iteration798005
Node: Minimization Stopping Parameters800141
Node: Minimization Algorithms801750
Node: Minimization Examples804189
Node: Minimization References and Further Reading807096
Node: Multidimensional Root-Finding807552
Node: Overview of Multidimensional Root Finding809041
Node: Initializing the Multidimensional Solver811228
Node: Providing the multidimensional system of equations to solve814426
Node: Iteration of the multidimensional solver819331
Node: Search Stopping Parameters for the multidimensional solver821516
Node: Algorithms using Derivatives823253
Node: Algorithms without Derivatives827999
Node: Example programs for Multidimensional Root finding831017
Node: References and Further Reading for Multidimensional Root Finding839630
Node: Multidimensional Minimization840869
Node: Multimin Overview842161
Node: Multimin Caveats844133
Node: Initializing the Multidimensional Minimizer844882
Node: Providing a function to minimize847795
Node: Multimin Iteration851460
Node: Multimin Stopping Criteria853451
Node: Multimin Algorithms855003
Node: Multimin Examples859331
Node: Multimin References and Further Reading865948
Node: Least-Squares Fitting866627
Node: Fitting Overview867626
Node: Linear regression869679
Node: Linear fitting without a constant term872110
Node: Multi-parameter fitting874301
Node: Fitting Examples879122
Node: Fitting References and Further Reading885777
Node: Nonlinear Least-Squares Fitting886599
Node: Overview of Nonlinear Least-Squares Fitting888054
Node: Initializing the Nonlinear Least-Squares Solver889521
Node: Providing the Function to be Minimized892367
Node: Iteration of the Minimization Algorithm895304
Node: Search Stopping Parameters for Minimization Algorithms897154
Node: Minimization Algorithms using Derivatives899249
Node: Minimization Algorithms without Derivatives902497
Node: Computing the covariance matrix of best fit parameters902889
Node: Example programs for Nonlinear Least-Squares Fitting904923
Node: References and Further Reading for Nonlinear Least-Squares Fitting912590
Node: Physical Constants913326
Node: Fundamental Constants914754
Node: Astronomy and Astrophysics915893
Node: Atomic and Nuclear Physics916558
Node: Measurement of Time918195
Node: Imperial Units918625
Node: Speed and Nautical Units919067
Node: Printers Units919571
Node: Volume Area and Length919894
Node: Mass and Weight920580
Node: Thermal Energy and Power921399
Node: Pressure921822
Node: Viscosity922435
Node: Light and Illumination922711
Node: Radioactivity923303
Node: Force and Energy923638
Node: Prefixes924042
Node: Physical Constant Examples924785
Node: Physical Constant References and Further Reading926574
Node: IEEE floating-point arithmetic927285
Node: Representation of floating point numbers927871
Node: Setting up your IEEE environment932352
Node: IEEE References and Further Reading938961
Node: Debugging Numerical Programs940116
Node: Using gdb940600
Node: Examining floating point registers943943
Node: Handling floating point exceptions945228
Node: GCC warning options for numerical programs946642
Node: Debugging References950835
Node: Contributors to GSL951547
Node: Autoconf Macros953775
Node: GSL CBLAS Library957760
Node: Level 1 CBLAS Functions958287
Node: Level 2 CBLAS Functions963589
Node: Level 3 CBLAS Functions980259
Node: GSL CBLAS Examples989921
Node: Free Software Needs Free Documentation991483
Node: GNU General Public License996552
Node: GNU Free Documentation License1015822
Node: Function Index1038253
Node: Variable Index1266319
Node: Type Index1267491
Node: Concept Index1271064

End Tag Table
