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

INFO-DIR-SECTION Software libraries
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, 2007, 2008, 2009, 2010, 2011, 2012, 2013 The GSL Team.

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


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
1.16 of the library.  Please report any errors in this manual to
<bug-gsl@gnu.org>.

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

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

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

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

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


* Menu:

* Introduction::
* Using the library::
* Error Handling::
* Mathematical Functions::
* Complex Numbers::
* Polynomials::
* Special Functions::
* Vectors and Matrices::
* Permutations::
* Combinations::
* Multisets::
* 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::
* Basis Splines::
* Physical Constants::
* IEEE floating-point arithmetic::
* Debugging Numerical Programs::
* Contributors to GSL::
* Autoconf Macros::
* GSL CBLAS Library::
* 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               
     Basis Splines                    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.

   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 or online in the GSL bug tracker.(1)  Details of
compilation problems can be found in the `INSTALL' file.

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

   All bug reports should include:

   * 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.

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

   (1) `http://savannah.gnu.org/bugs/?group=gsl'


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 (3rd Ed.), ISBN
0954612078'.

   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,(1)

     J0(5) = -1.775967713143382920e-01

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

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

   (1) The last few digits may vary slightly depending on the compiler
and platform used--this is normal.


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.

   The option `-lm' links with the system math library. On some systems
it is not needed.(1)

   For a tutorial introduction to the GNU C Compiler and related
programs, see `An Introduction to GCC' (ISBN 0954161793).(2)

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

   (1) It is not needed on MacOS X.

   (2) `http://www.network-theory.co.uk/gcc/intro/'


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 `libcblas.a',

     $ gcc example.o -lgsl -lcblas -lm

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

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

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

   For more information about BLAS functions see *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, either modify the system dynamic linker
configuration(1) or define the shell variable `LD_LIBRARY_PATH' to
include the directory where the library is installed.

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

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

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

     % setenv LD_LIBRARY_PATH /usr/local/lib

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

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

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

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

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

   (1) `/etc/ld.so.conf' on GNU/Linux systems.


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)
so the library does not export any inline function definitions by
default.  Inline functions were introduced officially in the newer C99
standard but most C89 compilers have also included `inline' as an
extension for a long time.

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

     $ gcc -Wall -c -DHAVE_INLINE example.c

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

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

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


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

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

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

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

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

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

     checking whether printf works with long double... no

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

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


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. Error codes above 1024 are reserved
for applications, and are not used by the library.  Many of the error
codes use the same base name as the corresponding error code in the C
library.  Here are some of the most common error codes,

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

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

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

 -- Macro: int GSL_EINVAL
     Invalid argument.  This is used to indicate various kinds of
     problems with passing the wrong argument to a library function
     (like EINVAL in the C library).

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

 -- Function: const char * gsl_strerror (const int GSL_ERRNO)
     This function returns a pointer to a string describing the error
     code GSL_ERRNO. For example,

          printf ("error: %s\n", gsl_strerror (status));

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


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, stride, n);

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

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


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.(1)

 -- 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.

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

   (1) Note that the C99 standard only requires the system `isinf'
function to return a non-zero value, without the sign of the infinity.
The implementation in some earlier versions of GSL used the system
`isinf' function and may have this behavior on some platforms.
Therefore, it is advisable to test the sign of X separately, if needed,
rather than relying the sign of the return value from `gsl_isinf()'.


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_hypot3 (const double X, const double Y, const
          double Z)
     This function computes the value of \sqrt{x^2 + y^2 + z^2} in a
     way that avoids overflow.

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

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

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

 -- Function: double gsl_ldexp (double X, int E)
     This function computes the value of x * 2^e. It provides an
     alternative to the standard math function `ldexp(x,e)'.

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


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

 -- Function: double gsl_pow_int (double X, int N)
 -- Function: double gsl_pow_uint (double X, unsigned int N)
     These routines computes the power x^n for integer N.  The power is
     computed efficiently--for example, x^8 is computed as ((x^2)^2)^2,
     requiring only 3 multiplications.  A version of this function
     which also computes the numerical error in the result is available
     as `gsl_sf_pow_int_e'.

 -- Function: double gsl_pow_2 (const double X)
 -- Function: double gsl_pow_3 (const double X)
 -- Function: double gsl_pow_4 (const double X)
 -- Function: double gsl_pow_5 (const double X)
 -- Function: double gsl_pow_6 (const double X)
 -- Function: double gsl_pow_7 (const double X)
 -- Function: double gsl_pow_8 (const double X)
 -- Function: double gsl_pow_9 (const double X)
     These functions can be used to compute small integer powers x^2,
     x^3, etc. efficiently. The functions will be inlined when
     `HAVE_INLINE' is defined, so that use of these functions should be
     as efficient as explicitly writing the corresponding product
     expression.

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


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
=================================

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

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

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

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

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

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

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


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.

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

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


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:

* Representation of 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: Representation of complex numbers,  Next: Properties of complex numbers,  Up: Complex Numbers

5.1 Representation of complex numbers
=====================================

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

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

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

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

 -- Function: gsl_complex gsl_complex_rect (double X, double Y)
     This function uses the rectangular Cartesian components (X,Y) to
     return the complex number z = x + i y.  An inline version of this
     function is used when `HAVE_INLINE' is defined.

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

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

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

          GSL_SET_COMPLEX(&z, 3, 4)

     sets Z to be 3 + 4i.

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


File: gsl-ref.info,  Node: Properties of complex numbers,  Next: Complex arithmetic operators,  Prev: Representation of 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.  Note that in this case we use the negative
     square root in formula 4.6.21 of Abramowitz & Stegun giving
     \arccosh(z)=\log(z-\sqrt{z^2-1}).

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

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

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

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

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

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


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
=========================

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

 -- Function: double gsl_poly_eval (const double C[], const int LEN,
          const double X)
     This function evaluates a polynomial with real coefficients for
     the real variable X.

 -- Function: gsl_complex gsl_poly_complex_eval (const double C[],
          const int LEN, const gsl_complex Z)
     This function evaluates a polynomial with real coefficients for
     the complex variable Z.

 -- Function: gsl_complex gsl_complex_poly_complex_eval (const
          gsl_complex C[], const int LEN, const gsl_complex Z)
     This function evaluates a polynomial with complex coefficients for
     the complex variable Z.

 -- Function: int gsl_poly_eval_derivs (const double C[], const size_t
          LENC, const double X, double RES[], const size_t LENRES)
     This function evaluates a polynomial and its derivatives storing
     the results in the array RES of size LENRES.  The output array
     contains the values of d^k P/d x^k for the specified value of X
     starting with k = 0.


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, and
Burden and Faires, chapter 3, and discussed briefly below.

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

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

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

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

   where the elements of z = \x_0,x_0,x_1,x_1,...,x_n,x_n\ are defined
by z_2k = z_2k+1 = x_k. The divided-differences [z_0,z_1,...,z_k] are
discussed in Burden and Faires, section 3.4.

 -- Function: int gsl_poly_dd_init (double DD[], const double XA[],
          const double YA[], size_t SIZE)
     This function computes a divided-difference representation of the
     interpolating polynomial for the points (X, Y) stored in the
     arrays XA and YA of length SIZE.  On output the
     divided-differences of (XA,YA) are stored in the array DD, also of
     length SIZE. Using the notation above, dd[k] = [x_0,x_1,...,x_k].

 -- Function: double gsl_poly_dd_eval (const double DD[], const double
          XA[], const size_t SIZE, const double X)
     This function evaluates the polynomial stored in
     divided-difference form in the arrays DD and XA of length SIZE at
     the point X.  An inline version of this function is used when
     `HAVE_INLINE' is defined.

 -- Function: int gsl_poly_dd_taylor (double C[], double XP, const
          double DD[], const double XA[], size_t SIZE, double W[])
     This function converts the divided-difference representation of a
     polynomial to a Taylor expansion.  The divided-difference
     representation is supplied in the arrays DD and XA of length SIZE.
     On output the Taylor coefficients of the polynomial expanded about
     the point XP are stored in the array C also of length SIZE.  A
     workspace of length SIZE must be provided in the array W.

 -- Function: int gsl_poly_dd_hermite_init (double DD[], double ZA[],
          const double XA[], const double YA[], const double DYA[],
          const size_t SIZE)
     This function computes a divided-difference representation of the
     interpolating Hermite polynomial for the points (X, Y) stored in
     the arrays XA and YA of length SIZE. Hermite interpolation
     constructs polynomials which also match first derivatives dy/dx
     which are provided in the array DYA also of length SIZE. The first
     derivatives can be incorported into the usual divided-difference
     algorithm by forming a new dataset z = \x_0,x_0,x_1,x_1,...\,
     which is stored in the array ZA of length 2*SIZE on output. On
     output the divided-differences of the Hermite representation are
     stored in the array DD, also of length 2*SIZE. Using the notation
     above, dd[k] = [z_0,z_1,...,z_k]. The resulting Hermite polynomial
     can be evaluated by calling `gsl_poly_dd_eval' and using ZA for
     the input argument XA.


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

 -- Function: int gsl_poly_complex_solve_cubic (double A, double B,
          double C, gsl_complex * Z0, gsl_complex * Z1, gsl_complex *
          Z2)
     This function finds the complex roots of the cubic equation,

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

     The number of complex roots is returned (always three) and the
     locations of the roots are stored in Z0, Z1 and Z2.  The roots are
     returned in ascending order, sorted first by their real components
     and then by their imaginary components.



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


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 the following texts,

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

     R. L. Burden and J. D. Faires, `Numerical Analysis', 9th edition,
     ISBN 0-538-73351-9, 2011.


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::
* Mathieu 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 + xt) + \sin((1/3) t^3 + xt)) dt

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

* 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 * RESULT_ARRAY)
     This routine uses Steed's method to compute the values of the
     regular spherical Bessel functions j_l(x) for l from 0 to LMAX
     inclusive for lmax >= 0 and x >= 0, storing the results in the
     array RESULT_ARRAY.  The Steed/Barnett algorithm is described in
     `Comp. Phys. Comm.' 21, 297 (1981).  Steed's method is more stable
     than the recurrence used in the other functions but is also slower.


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 sixth-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.

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


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'.  Further information about the elliptic integrals
can be found in Abramowitz & Stegun, Chapter 17.

* 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
\Pi(\phi,k,n) are defined by,

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

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

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

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

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


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

 -- Function: double gsl_sf_ellint_Ecomp (double K, gsl_mode_t MODE)
 -- Function: int gsl_sf_ellint_Ecomp_e (double K, gsl_mode_t MODE,
          gsl_sf_result * RESULT)
     These routines compute the complete elliptic integral E(k) to the
     accuracy specified by the mode variable MODE.  Note that
     Abramowitz & Stegun define this function in terms of the parameter
     m = k^2.

 -- Function: double gsl_sf_ellint_Pcomp (double K, double N,
          gsl_mode_t MODE)
 -- Function: int gsl_sf_ellint_Pcomp_e (double K, double N, gsl_mode_t
          MODE, gsl_sf_result * RESULT)
     These routines compute the complete elliptic integral \Pi(k,n) to
     the accuracy specified by the mode variable MODE.  Note that
     Abramowitz & Stegun define this function in terms of the
     parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of
     sign n \to -n.


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

 -- Function: double gsl_sf_ellint_E (double PHI, double K, gsl_mode_t
          MODE)
 -- Function: int gsl_sf_ellint_E_e (double PHI, double K, gsl_mode_t
          MODE, gsl_sf_result * RESULT)
     These routines compute the incomplete elliptic integral E(\phi,k)
     to the accuracy specified by the mode variable MODE.  Note that
     Abramowitz & Stegun define this function in terms of the parameter
     m = k^2.

 -- Function: double gsl_sf_ellint_P (double PHI, double K, double N,
          gsl_mode_t MODE)
 -- Function: int gsl_sf_ellint_P_e (double PHI, double K, double N,
          gsl_mode_t MODE, gsl_sf_result * RESULT)
     These routines compute the incomplete elliptic integral
     \Pi(\phi,k,n) to the accuracy specified by the mode variable MODE.
     Note that Abramowitz & Stegun define this function in terms of the
     parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of
     sign n \to -n.

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

          D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
     The argument N is not used and will be removed in a future release.



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 Mills' ratio, is defined as,

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

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

 -- Function: double gsl_sf_hazard (double X)
 -- Function: int gsl_sf_hazard_e (double X, gsl_sf_result * RESULT)
     These routines compute the hazard function for the normal
     distribution.


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

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


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.



 -- Function: double gsl_sf_expint_En (int N, double X)
 -- Function: int gsl_sf_expint_En_e (int N, double X, gsl_sf_result *
          RESULT)
     These routines compute the exponential integral E_n(x) of order n,

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




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/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
   Note that the Fermi-Dirac integral is sometimes defined without the
normalisation factor in other texts.

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

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

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

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

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

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

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

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


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
=============================

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

* 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.

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

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

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

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

          \Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))
                      = (1 + (1/12x) + ...)  for x \to \infty
     and is a useful suggestion of Temme.

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

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


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

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

 -- Function: int gsl_sf_lnpoch_sgn_e (double A, double X,
          gsl_sf_result * RESULT, double * SGN)
     These routines compute the sign of the Pochhammer symbol and the
     logarithm of its magnitude.  The computed parameters are result =
     \log(|(a)_x|) with a corresponding error term, and sgn =
     \sgn((a)_x) where (a)_x = \Gamma(a + x)/\Gamma(a).

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


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) subject to a and b not being
     negative integers.

 -- Function: double gsl_sf_lnbeta (double A, double B)
 -- Function: int gsl_sf_lnbeta_e (double A, double B, gsl_sf_result *
          RESULT)
     These routines compute the logarithm of the Beta Function,
     \log(B(a,b)) subject to a and b not being negative integers.


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


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

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

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

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

 -- Function: double gsl_sf_hyperg_2F1_conj (double AR, double AI,
          double C, double X)
 -- Function: int gsl_sf_hyperg_2F1_conj_e (double AR, double AI,
          double C, double X, gsl_sf_result * RESULT)
     These routines compute the Gauss hypergeometric function 2F1(a_R +
     i a_I, a_R - i a_I, c, x) with complex parameters for |x| < 1.

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

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

 -- Function: double gsl_sf_hyperg_2F0 (double A, double B, double X)
 -- Function: int gsl_sf_hyperg_2F0_e (double A, double B, double X,
          gsl_sf_result * RESULT)
     These routines compute the hypergeometric function 2F0(a,b,x).
     The series representation is a divergent hypergeometric series.
     However, for x < 0 we have 2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)


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 arrays of Legendre polynomials P_l(x) and
     derivatives dP_l(x)/dx, for l = 0, \dots, lmax, |x| <= 1

 -- Function: double gsl_sf_legendre_Q0 (double X)
 -- Function: int gsl_sf_legendre_Q0_e (double X, gsl_sf_result *
          RESULT)
     These routines compute the Legendre function Q_0(x) for x > -1, x
     != 1.

 -- Function: double gsl_sf_legendre_Q1 (double X)
 -- Function: int gsl_sf_legendre_Q1_e (double X, gsl_sf_result *
          RESULT)
     These routines compute the Legendre function Q_1(x) for x > -1, x
     != 1.

 -- Function: double gsl_sf_legendre_Ql (int L, double X)
 -- Function: int gsl_sf_legendre_Ql_e (int L, double X, gsl_sf_result
          * RESULT)
     These routines compute the Legendre function Q_l(x) for x > -1, x
     != 1 and l >= 0.


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 arrays of Legendre polynomials P_l^m(x)
     and derivatives dP_l^m(x)/dx, for m >= 0, l = |m|, ..., lmax, |x|
     <= 1.

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

 -- Function: int gsl_sf_legendre_sphPlm_array (int LMAX, int M, double
          X, double RESULT_ARRAY[])
 -- Function: int gsl_sf_legendre_sphPlm_deriv_array (int LMAX, int M,
          double X, double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
     These functions compute arrays of normalized associated Legendre
     functions \sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x), and
     derivatives, for m >= 0, l = |m|, ..., lmax, |x| <= 1.0

 -- Function: int gsl_sf_legendre_array_size (const int LMAX, const int
          M)
     This function returns the size of RESULT_ARRAY[] needed for the
     array versions of P_l^m(x), LMAX - M + 1.  An inline version of
     this function is used when `HAVE_INLINE' is defined.


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: Mathieu Functions,  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: Mathieu Functions,  Next: Power Function,  Prev: Logarithm and Related Functions,  Up: Special Functions

7.26 Mathieu Functions
======================

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

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

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

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

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

* Menu:

* Mathieu Function Workspace::
* Mathieu Function Characteristic Values::
* Angular Mathieu Functions::
* Radial Mathieu Functions::


File: gsl-ref.info,  Node: Mathieu Function Workspace,  Next: Mathieu Function Characteristic Values,  Up: Mathieu Functions

7.26.1 Mathieu Function Workspace
---------------------------------

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

 -- Function: gsl_sf_mathieu_workspace * gsl_sf_mathieu_alloc (size_t
          N, double QMAX)
     This function returns a workspace for the array versions of the
     Mathieu routines.  The arguments N and QMAX specify the maximum
     order and q-value of Mathieu functions which can be computed with
     this workspace.


 -- Function: void gsl_sf_mathieu_free (gsl_sf_mathieu_workspace * WORK)
     This function frees the workspace WORK.


File: gsl-ref.info,  Node: Mathieu Function Characteristic Values,  Next: Angular Mathieu Functions,  Prev: Mathieu Function Workspace,  Up: Mathieu Functions

7.26.2 Mathieu Function Characteristic Values
---------------------------------------------

 -- Function: int gsl_sf_mathieu_a (int N, double Q, gsl_sf_result *
          RESULT)
 -- Function: int gsl_sf_mathieu_b (int N, double Q, gsl_sf_result *
          RESULT)
     These routines compute the characteristic values a_n(q), b_n(q) of
     the Mathieu functions ce_n(q,x) and se_n(q,x), respectively.

 -- Function: int gsl_sf_mathieu_a_array (int ORDER_MIN, int ORDER_MAX,
          double Q, gsl_sf_mathieu_workspace * WORK, double
          RESULT_ARRAY[])
 -- Function: int gsl_sf_mathieu_b_array (int ORDER_MIN, int ORDER_MAX,
          double Q, gsl_sf_mathieu_workspace * WORK, double
          RESULT_ARRAY[])
     These routines compute a series of Mathieu characteristic values
     a_n(q), b_n(q) for n from ORDER_MIN to ORDER_MAX inclusive,
     storing the results in the array RESULT_ARRAY.


File: gsl-ref.info,  Node: Angular Mathieu Functions,  Next: Radial Mathieu Functions,  Prev: Mathieu Function Characteristic Values,  Up: Mathieu Functions

7.26.3 Angular Mathieu Functions
--------------------------------

 -- Function: int gsl_sf_mathieu_ce (int N, double Q, double X,
          gsl_sf_result * RESULT)
 -- Function: int gsl_sf_mathieu_se (int N, double Q, double X,
          gsl_sf_result * RESULT)
     These routines compute the angular Mathieu functions ce_n(q,x) and
     se_n(q,x), respectively.

 -- Function: int gsl_sf_mathieu_ce_array (int NMIN, int NMAX, double
          Q, double X, gsl_sf_mathieu_workspace * WORK, double
          RESULT_ARRAY[])
 -- Function: int gsl_sf_mathieu_se_array (int NMIN, int NMAX, double
          Q, double X, gsl_sf_mathieu_workspace * WORK, double
          RESULT_ARRAY[])
     These routines compute a series of the angular Mathieu functions
     ce_n(q,x) and se_n(q,x) of order n from NMIN to NMAX inclusive,
     storing the results in the array RESULT_ARRAY.


File: gsl-ref.info,  Node: Radial Mathieu Functions,  Prev: Angular Mathieu Functions,  Up: Mathieu Functions

7.26.4 Radial Mathieu Functions
-------------------------------

 -- Function: int gsl_sf_mathieu_Mc (int J, int N, double Q, double X,
          gsl_sf_result * RESULT)
 -- Function: int gsl_sf_mathieu_Ms (int J, int N, double Q, double X,
          gsl_sf_result * RESULT)
     These routines compute the radial J-th kind Mathieu functions
     Mc_n^{(j)}(q,x) and Ms_n^{(j)}(q,x) of order N.

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

 -- Function: int gsl_sf_mathieu_Mc_array (int J, int NMIN, int NMAX,
          double Q, double X, gsl_sf_mathieu_workspace * WORK, double
          RESULT_ARRAY[])
 -- Function: int gsl_sf_mathieu_Ms_array (int J, int NMIN, int NMAX,
          double Q, double X, gsl_sf_mathieu_workspace * WORK, double
          RESULT_ARRAY[])
     These routines compute a series of the radial Mathieu functions of
     kind J, with order from NMIN to NMAX inclusive, storing the
     results in the array RESULT_ARRAY.


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

7.27 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

7.28 Psi (Digamma) Function
===========================

The polygamma functions of order n are defined by

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

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

* Menu:

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


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

7.28.1 Digamma Function
-----------------------

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

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

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


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

7.28.2 Trigamma Function
------------------------

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

 -- Function: double gsl_sf_psi_1 (double X)
 -- Function: int gsl_sf_psi_1_e (double X, gsl_sf_result * RESULT)
     These routines compute the Trigamma function \psi'(x) for general
     x.


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

7.28.3 Polygamma Function
-------------------------

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


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

7.29 Synchrotron Functions
==========================

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

 -- Function: double gsl_sf_synchrotron_1 (double X)
 -- Function: int gsl_sf_synchrotron_1_e (double X, gsl_sf_result *
          RESULT)
     These routines compute the first synchrotron function x
     \int_x^\infty dt K_{5/3}(t) for x >= 0.

 -- Function: double gsl_sf_synchrotron_2 (double X)
 -- Function: int gsl_sf_synchrotron_2_e (double X, gsl_sf_result *
          RESULT)
     These routines compute the second synchrotron function x
     K_{2/3}(x) for x >= 0.


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

7.30 Transport Functions
========================

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

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

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

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

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


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

7.31 Trigonometric Functions
============================

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

* 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

7.31.1 Circular Trigonometric Functions
---------------------------------------

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

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

 -- Function: double gsl_sf_hypot (double X, double Y)
 -- Function: int gsl_sf_hypot_e (double X, double Y, gsl_sf_result *
          RESULT)
     These routines compute the hypotenuse function \sqrt{x^2 + y^2}
     avoiding overflow and underflow.

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


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

7.31.2 Trigonometric Functions for Complex Arguments
----------------------------------------------------

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

 -- Function: int gsl_sf_complex_cos_e (double ZR, double ZI,
          gsl_sf_result * CZR, gsl_sf_result * CZI)
     This function computes the complex cosine, \cos(z_r + i z_i)
     storing the real and imaginary parts in CZR, CZI.

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


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

7.31.3 Hyperbolic Trigonometric Functions
-----------------------------------------

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

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


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

7.31.4 Conversion Functions
---------------------------

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

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


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

7.31.5 Restriction Functions
----------------------------

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

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

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

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



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

7.31.6 Trigonometric Functions With Error Estimates
---------------------------------------------------

 -- Function: int gsl_sf_sin_err_e (double X, double DX, gsl_sf_result
          * RESULT)
     This routine computes the sine of an angle X with an associated
     absolute error DX, \sin(x \pm dx).  Note that this function is
     provided in the error-handling form only since its purpose is to
     compute the propagated error.

 -- Function: int gsl_sf_cos_err_e (double X, double DX, gsl_sf_result
          * RESULT)
     This routine computes the cosine of an angle X with an associated
     absolute error DX, \cos(x \pm dx).  Note that this function is
     provided in the error-handling form only since its purpose is to
     compute the propagated error.


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

7.32 Zeta Functions
===================

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

* 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

7.32.1 Riemann Zeta Function
----------------------------

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

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

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


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

7.32.2 Riemann Zeta Function Minus One
--------------------------------------

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

 -- Function: double gsl_sf_zetam1_int (int N)
 -- Function: int gsl_sf_zetam1_int_e (int N, gsl_sf_result * RESULT)
     These routines compute \zeta(n) - 1 for integer N, n \ne 1.

 -- Function: double gsl_sf_zetam1 (double S)
 -- Function: int gsl_sf_zetam1_e (double S, gsl_sf_result * RESULT)
     These routines compute \zeta(s) - 1 for arbitrary S, s \ne 1.


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

7.32.3 Hurwitz Zeta Function
----------------------------

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

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


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

7.32.4 Eta Function
-------------------

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

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

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


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

7.33 Examples
=============

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

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

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

       double expected = -0.17759677131433830434739701;

       int status = gsl_sf_bessel_J0_e (x, &result);

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

Here are the results of running the program,

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

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

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

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

       double y = gsl_sf_bessel_J0 (x);

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

The results of the function are the same,

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


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

7.34 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, 
     Allan J. MacLeod, 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)

     N. M. Temme, Special Functions: An Introduction to the Classical
     Functions of Mathematical Physics (1996), ISBN 978-0471113133.

     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.(1)  The functions
`gsl_vector_get' and `gsl_vector_set' can perform portable range
checking for you and report an error if you attempt to access elements
outside the allowed range.

   The functions for accessing the elements of a vector or matrix are
defined in `gsl_vector.h' and declared `extern inline' to eliminate
function-call overhead.  You must compile your program with the
preprocessor macro `HAVE_INLINE' defined to use these functions.

   If necessary you can turn off range checking completely without
modifying any source files by recompiling your program with the
preprocessor definition `GSL_RANGE_CHECK_OFF'.  Provided your compiler
supports inline functions the effect of turning off range checking is
to replace calls to `gsl_vector_get(v,i)' by `v->data[i*v->stride]' and
calls to `gsl_vector_set(v,i,x)' by `v->data[i*v->stride]=x'.  Thus
there should be no performance penalty for using the range checking
functions when range checking is turned off.

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

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

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

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

 -- Function: double * gsl_vector_ptr (gsl_vector * V, size_t I)
 -- Function: const double * gsl_vector_const_ptr (const gsl_vector *
          V, size_t I)
     These functions return a pointer to the I-th element of a vector
     V.  If I lies outside the allowed range of 0 to N-1 then the error
     handler is invoked and a null pointer is returned.  Inline
     versions of these functions are used when `HAVE_INLINE' is defined.

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

   (1) Range checking is available in the GNU C Compiler
bounds-checking extension, but it is not part of the default
installation of GCC. Memory accesses can also be checked with Valgrind
or the `gcc -fmudflap' memory protection option.


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
-----------------------

 -- Function: int gsl_vector_add (gsl_vector * A, const gsl_vector * B)
     This function adds the elements of vector B to the elements of
     vector A.  The result a_i \leftarrow a_i + b_i is stored in A and
     B remains unchanged.  The two vectors must have the same length.

 -- Function: int gsl_vector_sub (gsl_vector * A, const gsl_vector * B)
     This function subtracts the elements of vector B from the elements
     of vector A.  The result a_i \leftarrow a_i - b_i is stored in A
     and B remains unchanged.  The two vectors must have the same
     length.

 -- Function: int gsl_vector_mul (gsl_vector * A, const gsl_vector * B)
     This function multiplies the elements of vector A by the elements
     of vector B.  The result a_i \leftarrow a_i * b_i is stored in A
     and B remains unchanged. The two vectors must have the same length.

 -- Function: int gsl_vector_div (gsl_vector * A, const gsl_vector * B)
     This function divides the elements of vector A by the elements of
     vector B.  The result a_i \leftarrow a_i / b_i is stored in A and
     B remains unchanged. The two vectors must have the same length.

 -- Function: int gsl_vector_scale (gsl_vector * A, const double X)
     This function multiplies the elements of vector A by the constant
     factor X.  The result a_i \leftarrow x a_i is stored in A.

 -- Function: int gsl_vector_add_constant (gsl_vector * A, const double
          X)
     This function adds the constant value X to the elements of the
     vector A.  The result a_i \leftarrow a_i + x is stored in A.


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
-----------------------------------------------------

The following operations are only defined for real vectors.

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

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

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

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

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

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


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
------------------------

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

 -- Function: int gsl_vector_isnull (const gsl_vector * V)
 -- Function: int gsl_vector_ispos (const gsl_vector * V)
 -- Function: int gsl_vector_isneg (const gsl_vector * V)
 -- Function: int gsl_vector_isnonneg (const gsl_vector * V)
     These functions return 1 if all the elements of the vector V are
     zero, strictly positive, strictly negative, or non-negative
     respectively, and 0 otherwise.

 -- Function: int gsl_vector_equal (const gsl_vector * U, const
          gsl_vector * V)
     This function returns 1 if the vectors U and V are equal (by
     comparison of element values) and 0 otherwise.


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

       gsl_vector_free (v);
       return 0;
     }

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

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

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

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

     int
     main (void)
     {
       int i;
       gsl_vector * v = gsl_vector_alloc (100);

       for (i = 0; i < 100; i++)
         {
           gsl_vector_set (v, i, 1.23 + i);
         }

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

       gsl_vector_free (v);
       return 0;
     }

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

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

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

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

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

       gsl_vector_free (v);
       return 0;
     }


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

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

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

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


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

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

 -- Function: double * gsl_matrix_ptr (gsl_matrix * M, size_t I, size_t
          J)
 -- Function: const double * gsl_matrix_const_ptr (const gsl_matrix *
          M, size_t I, size_t J)
     These functions return a pointer to the (i,j)-th element of a
     matrix M.  If I or J lie outside the allowed range of 0 to N1-1
     and 0 to N2-1 then the error handler is invoked and a null pointer
     is returned.  Inline versions of these functions are used when
     `HAVE_INLINE' is defined.


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_subrow (gsl_matrix * M, size_t
          I, size_t OFFSET, size_t N)
 -- Function: gsl_vector_const_view gsl_matrix_const_subrow (const
          gsl_matrix * M, size_t I, size_t OFFSET, size_t N)
     These functions return a vector view of the I-th row of the matrix
     M beginning at OFFSET elements past the first column and
     containing N elements. The `data' pointer of the new vector is set
     to null if I, OFFSET, or N are out of range.

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

 -- Function: gsl_vector_view gsl_matrix_subcolumn (gsl_matrix * M,
          size_t J, size_t OFFSET, size_t N)
 -- Function: gsl_vector_const_view gsl_matrix_const_subcolumn (const
          gsl_matrix * M, size_t J, size_t OFFSET, size_t N)
     These functions return a vector view of the J-th column of the
     matrix M beginning at OFFSET elements past the first row and
     containing N elements. The `data' pointer of the new vector is set
     to null if J, OFFSET, or N are out of range.

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

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

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

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

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

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

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


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.  The result a(i,j) \leftarrow a(i,j) + b(i,j) is stored
     in A and B remains unchanged. The two matrices must have the same
     dimensions.

 -- Function: int gsl_matrix_sub (gsl_matrix * A, const gsl_matrix * B)
     This function subtracts the elements of matrix B from the elements
     of matrix A.  The result a(i,j) \leftarrow a(i,j) - b(i,j) is
     stored in A and B remains unchanged. The two matrices must have
     the same dimensions.

 -- Function: int gsl_matrix_mul_elements (gsl_matrix * A, const
          gsl_matrix * B)
     This function multiplies the elements of matrix A by the elements
     of matrix B.  The result a(i,j) \leftarrow a(i,j) * b(i,j) is
     stored in A and B remains unchanged.  The two matrices must have
     the same dimensions.

 -- Function: int gsl_matrix_div_elements (gsl_matrix * A, const
          gsl_matrix * B)
     This function divides the elements of matrix A by the elements of
     matrix B.  The result a(i,j) \leftarrow a(i,j) / b(i,j) is stored
     in A and B remains unchanged. The two matrices must have the same
     dimensions.

 -- Function: int gsl_matrix_scale (gsl_matrix * A, const double X)
     This function multiplies the elements of matrix A by the constant
     factor X.  The result a(i,j) \leftarrow x a(i,j) is stored in A.

 -- Function: int gsl_matrix_add_constant (gsl_matrix * A, const double
          X)
     This function adds the constant value X to the elements of the
     matrix A.  The result a(i,j) \leftarrow a(i,j) + x is stored in A.


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
------------------------

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

 -- Function: int gsl_matrix_isnull (const gsl_matrix * M)
 -- Function: int gsl_matrix_ispos (const gsl_matrix * M)
 -- Function: int gsl_matrix_isneg (const gsl_matrix * M)
 -- Function: int gsl_matrix_isnonneg (const gsl_matrix * M)
     These functions return 1 if all the elements of the matrix M are
     zero, strictly positive, strictly negative, or non-negative
     respectively, and 0 otherwise. To test whether a matrix is
     positive-definite, use the Cholesky decomposition (*note Cholesky
     Decomposition::).

 -- Function: int gsl_matrix_equal (const gsl_matrix * A, const
          gsl_matrix * B)
     This function returns 1 if the matrices A and B are equal (by
     comparison of element values) and 0 otherwise.


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

       gsl_matrix_free (m);

       return 0;
     }

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

     $ ./a.out
     m(0,0) = 0.23
     m(0,1) = 1.23
     m(0,2) = 2.23
     m(1,0) = 100.23
     m(1,1) = 101.23
     m(1,2) = 102.23
     ...
     m(9,2) = 902.23
     gsl: matrix_source.c:13: ERROR: first index out of range
     Default GSL error handler invoked.
     Aborted (core dumped)

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

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

     int
     main (void)
     {
       int i, j, k = 0;
       gsl_matrix * m = gsl_matrix_alloc (100, 100);
       gsl_matrix * a = gsl_matrix_alloc (100, 100);

       for (i = 0; i < 100; i++)
         for (j = 0; j < 100; j++)
           gsl_matrix_set (m, i, j, 0.23 + i + j);

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

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

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

       gsl_matrix_free (m);
       gsl_matrix_free (a);

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

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

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

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

     int
     main (void)
     {
       size_t i,j;

       gsl_matrix *m = gsl_matrix_alloc (10, 10);

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

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

           d = gsl_blas_dnrm2 (&column.vector);

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

       gsl_matrix_free (m);

       return 0;
     }

Here is the output of the program,

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

The results can be confirmed using GNU OCTAVE,

     $ octave
     GNU Octave, version 2.0.16.92
     octave> m = sin(0:9)' * ones(1,10)
                    + ones(10,1) * cos(0:9);
     octave> sqrt(sum(m.^2))
     ans =
       4.3146  3.1205  2.1932  3.2611  2.5342  2.5728
       4.2047  3.6520  2.0852  3.0731


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

 -- Function: int gsl_permutation_swap (gsl_permutation * P, const
          size_t I, const size_t J)
     This function exchanges the I-th and J-th elements of the
     permutation P.


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 (const gsl_permutation * P)
     This function checks that the permutation P is valid.  The N
     elements should contain each of the numbers 0 to N-1 once and only
     once.


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.  In ISO C99
     the type modifier `z' represents `size_t', so `"%zu\n"' is a
     suitable format.(1) 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.

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

   (1) In versions of the GNU C library prior to the ISO C99 standard,
the type modifier `Z' was used instead.


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");

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

       return 0;
     }

Here is the output from the program,

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

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

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

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

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

       gsl_permutation_init (p);

       do
        {
           gsl_permutation_fprintf (stdout, p, " %u");
           printf ("\n");
        }
       while (gsl_permutation_next(p) == GSL_SUCCESS);

       gsl_permutation_free (p);

       return 0;
     }

Here is the output from the program,

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

The permutations are generated in lexicographic order.  To reverse the
sequence, begin with the final permutation (which is the reverse of the
identity) and replace `gsl_permutation_next' with
`gsl_permutation_prev'.


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: Multisets,  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.  An inline
     version of this function is used when `HAVE_INLINE' is defined.


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.  In ISO C99
     the type modifier `z' represents `size_t', so `"%zu\n"' is a
     suitable format.(1)  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.

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

   (1) In versions of the GNU C library prior to the ISO C99 standard,
the type modifier `Z' was used instead.


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: Multisets,  Next: Sorting,  Prev: Combinations,  Up: Top

11 Multisets
************

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

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

* Menu:

* The Multiset struct::
* Multiset allocation::
* Accessing multiset elements::
* Multiset properties::
* Multiset functions::
* Reading and writing multisets::
* Multiset Examples::


File: gsl-ref.info,  Node: The Multiset struct,  Next: Multiset allocation,  Up: Multisets

11.1 The Multiset struct
========================

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

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



File: gsl-ref.info,  Node: Multiset allocation,  Next: Accessing multiset elements,  Prev: The Multiset struct,  Up: Multisets

11.2 Multiset allocation
========================

 -- Function: gsl_multiset * gsl_multiset_alloc (size_t N, size_t K)
     This function allocates memory for a new multiset with parameters
     N, K.  The multiset is not initialized and its elements are
     undefined.  Use the function `gsl_multiset_calloc' if you want to
     create a multiset which is initialized to the lexicographically
     first multiset element. A null pointer is returned if insufficient
     memory is available to create the multiset.

 -- Function: gsl_multiset * gsl_multiset_calloc (size_t N, size_t K)
     This function allocates memory for a new multiset with parameters
     N, K and initializes it to the lexicographically first multiset
     element. A null pointer is returned if insufficient memory is
     available to create the multiset.

 -- Function: void gsl_multiset_init_first (gsl_multiset * C)
     This function initializes the multiset C to the lexicographically
     first multiset element, i.e. 0 repeated k times.

 -- Function: void gsl_multiset_init_last (gsl_multiset * C)
     This function initializes the multiset C to the lexicographically
     last multiset element, i.e. n-1 repeated k times.

 -- Function: void gsl_multiset_free (gsl_multiset * C)
     This function frees all the memory used by the multiset C.

 -- Function: int gsl_multiset_memcpy (gsl_multiset * DEST, const
          gsl_multiset * SRC)
     This function copies the elements of the multiset SRC into the
     multiset DEST.  The two multisets must have the same size.


File: gsl-ref.info,  Node: Accessing multiset elements,  Next: Multiset properties,  Prev: Multiset allocation,  Up: Multisets

11.3 Accessing multiset elements
================================

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

 -- Function: size_t gsl_multiset_get (const gsl_multiset * C, const
          size_t I)
     This function returns the value of the I-th element of the
     multiset C.  If I lies outside the allowed range of 0 to K-1 then
     the error handler is invoked and 0 is returned.  An inline version
     of this function is used when `HAVE_INLINE' is defined.


File: gsl-ref.info,  Node: Multiset properties,  Next: Multiset functions,  Prev: Accessing multiset elements,  Up: Multisets

11.4 Multiset properties
========================

 -- Function: size_t gsl_multiset_n (const gsl_multiset * C)
     This function returns the range (n) of the multiset C.

 -- Function: size_t gsl_multiset_k (const gsl_multiset * C)
     This function returns the number of elements (k) in the multiset C.

 -- Function: size_t * gsl_multiset_data (const gsl_multiset * C)
     This function returns a pointer to the array of elements in the
     multiset C.

 -- Function: int gsl_multiset_valid (gsl_multiset * C)
     This function checks that the multiset C is valid.  The K elements
     should lie in the range 0 to N-1, with each value occurring in
     nondecreasing order.


File: gsl-ref.info,  Node: Multiset functions,  Next: Reading and writing multisets,  Prev: Multiset properties,  Up: Multisets

11.5 Multiset functions
=======================

 -- Function: int gsl_multiset_next (gsl_multiset * C)
     This function advances the multiset C to the next multiset element
     in lexicographic order and returns `GSL_SUCCESS'.  If no further
     multisets elements are available it returns `GSL_FAILURE' and
     leaves C unmodified.  Starting with the first multiset and
     repeatedly applying this function will iterate through all
     possible multisets of a given order.

 -- Function: int gsl_multiset_prev (gsl_multiset * C)
     This function steps backwards from the multiset C to the previous
     multiset element in lexicographic order, returning `GSL_SUCCESS'.
     If no previous multiset is available it returns `GSL_FAILURE' and
     leaves C unmodified.


File: gsl-ref.info,  Node: Reading and writing multisets,  Next: Multiset Examples,  Prev: Multiset functions,  Up: Multisets

11.6 Reading and writing multisets
==================================

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

 -- Function: int gsl_multiset_fwrite (FILE * STREAM, const
          gsl_multiset * C)
     This function writes the elements of the multiset C to the stream
     STREAM in binary format.  The function returns `GSL_EFAILED' if
     there was a problem writing to the file.  Since the data is
     written in the native binary format it may not be portable between
     different architectures.

 -- Function: int gsl_multiset_fread (FILE * STREAM, gsl_multiset * C)
     This function reads elements from the open stream STREAM into the
     multiset C in binary format.  The multiset C must be preallocated
     with correct values of n and k since the function uses the size of
     C to determine how many bytes to read.  The function returns
     `GSL_EFAILED' if there was a problem reading from the file.  The
     data is assumed to have been written in the native binary format
     on the same architecture.

 -- Function: int gsl_multiset_fprintf (FILE * STREAM, const
          gsl_multiset * C, const char * FORMAT)
     This function writes the elements of the multiset C line-by-line
     to the stream STREAM using the format specifier FORMAT, which
     should be suitable for a type of SIZE_T.  In ISO C99 the type
     modifier `z' represents `size_t', so `"%zu\n"' is a suitable
     format.(1)  The function returns `GSL_EFAILED' if there was a
     problem writing to the file.

 -- Function: int gsl_multiset_fscanf (FILE * STREAM, gsl_multiset * C)
     This function reads formatted data from the stream STREAM into the
     multiset C.  The multiset C must be preallocated with correct
     values of n and k since the function uses the size of C to
     determine how many numbers to read.  The function returns
     `GSL_EFAILED' if there was a problem reading from the file.

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

   (1) In versions of the GNU C library prior to the ISO C99 standard,
the type modifier `Z' was used instead.


File: gsl-ref.info,  Node: Multiset Examples,  Prev: Reading and writing multisets,  Up: Multisets

11.7 Examples
=============

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

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

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

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

       return 0;
     }

Here is the output from the program,

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

All 70 multisets are generated and sorted lexicographically.


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

12 Sorting
**********

This chapter describes functions for sorting data, both directly and
indirectly (using an index).  All the functions use the "heapsort"
algorithm.  Heapsort is an O(N \log N) algorithm which operates
in-place and does not require any additional storage.  It also provides
consistent performance, the running time for its worst-case (ordered
data) being not significantly longer than the average and best cases.
Note that the heapsort algorithm does not preserve the relative ordering
of equal elements--it is an "unstable" sort.  However the resulting
order of equal elements will be consistent across different platforms
when using these functions.

* 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

12.1 Sorting objects
====================

The following function provides a simple alternative to the standard
library function `qsort'.  It is intended for systems lacking `qsort',
not as a replacement for it.  The function `qsort' should be used
whenever possible, as it will be faster and can provide stable ordering
of equal elements.  Documentation for `qsort' is available in the `GNU
C Library Reference Manual'.

   The functions described in this section are defined in the header
file `gsl_heapsort.h'.

 -- Function: void gsl_heapsort (void * ARRAY, size_t COUNT, size_t
          SIZE, gsl_comparison_fn_t COMPARE)
     This function sorts the COUNT elements of the array ARRAY, each of
     size SIZE, into ascending order using the comparison function
     COMPARE.  The type of the comparison function is defined by,

          int (*gsl_comparison_fn_t) (const void * a,
                                      const void * b)

     A comparison function should return a negative integer if the first
     argument is less than the second argument, `0' if the two arguments
     are equal and a positive integer if the first argument is greater
     than the second argument.

     For example, the following function can be used to sort doubles
     into ascending numerical order.

          int
          compare_doubles (const double * a,
                           const double * b)
          {
              if (*a > *b)
                 return 1;
              else if (*a < *b)
                 return -1;
              else
                 return 0;
          }

     The appropriate function call to perform the sort is,

          gsl_heapsort (array, count, sizeof(double),
                        compare_doubles);

     Note that unlike `qsort' the heapsort algorithm cannot be made into
     a stable sort by pointer arithmetic.  The trick of comparing
     pointers for equal elements in the comparison function does not
     work for the heapsort algorithm.  The heapsort algorithm performs
     an internal rearrangement of the data which destroys its initial
     ordering.

 -- Function: int gsl_heapsort_index (size_t * P, const void * ARRAY,
          size_t COUNT, size_t SIZE, gsl_comparison_fn_t COMPARE)
     This function indirectly sorts the COUNT elements of the array
     ARRAY, each of size SIZE, into ascending order using the
     comparison function COMPARE.  The resulting permutation is stored
     in P, an array of length N.  The elements of P give the index of
     the array element which would have been stored in that position if
     the array had been sorted in place.  The first element of P gives
     the index of the least element in ARRAY, and the last element of P
     gives the index of the greatest element in ARRAY.  The array
     itself is not changed.


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

12.2 Sorting vectors
====================

The following functions will sort the elements of an array or vector,
either directly or indirectly.  They are defined for all real and
integer types using the normal suffix rules.  For example, the `float'
versions of the array functions are `gsl_sort_float' and
`gsl_sort_float_index'.  The corresponding vector functions are
`gsl_sort_vector_float' and `gsl_sort_vector_float_index'.  The
prototypes are available in the header files `gsl_sort_float.h'
`gsl_sort_vector_float.h'.  The complete set of prototypes can be
included using the header files `gsl_sort.h' and `gsl_sort_vector.h'.

   There are no functions for sorting complex arrays or vectors, since
the ordering of complex numbers is not uniquely defined.  To sort a
complex vector by magnitude compute a real vector containing the
magnitudes of the complex elements, and sort this vector indirectly.
The resulting index gives the appropriate ordering of the original
complex vector.

 -- Function: void gsl_sort (double * DATA, const size_t STRIDE, size_t
          N)
     This function sorts the N elements of the array DATA with stride
     STRIDE into ascending numerical order.

 -- Function: void gsl_sort2 (double * DATA1, const size_t STRIDE1,
          double * DATA2, const size_t STRIDE2, size_t N)
     This function sorts the N elements of the array DATA1 with stride
     STRIDE1 into ascending numerical order, while making the same
     rearrangement of the array DATA2 with stride STRIDE2, also of size
     N.

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

 -- Function: void gsl_sort_vector2 (gsl_vector * V1, gsl_vector * V2)
     This function sorts the elements of the vector V1 into ascending
     numerical order, while making the same rearrangement of the vector
     V2.

 -- Function: void gsl_sort_index (size_t * P, const double * DATA,
          size_t STRIDE, size_t N)
     This function indirectly sorts the N elements of the array DATA
     with stride STRIDE into ascending order, storing the resulting
     permutation in P.  The array P must be allocated with a sufficient
     length to store the N elements of the permutation.  The elements
     of P give the index of the array element which would have been
     stored in that position if the array had been sorted in place.
     The array DATA is not changed.

 -- Function: int gsl_sort_vector_index (gsl_permutation * P, const
          gsl_vector * V)
     This function indirectly sorts the elements of the vector V into
     ascending order, storing the resulting permutation in P.  The
     elements of P give the index of the vector element which would
     have been stored in that position if the vector had been sorted in
     place.  The first element of P gives the index of the least element
     in V, and the last element of P gives the index of the greatest
     element in V.  The vector V is not changed.


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

12.3 Selecting the k smallest or largest elements
=================================================

The functions described in this section select the k smallest or
largest elements of a data set of size N.  The routines use an O(kN)
direct insertion algorithm which is suited to subsets that are small
compared with the total size of the dataset. For example, the routines
are useful for selecting the 10 largest values from one million data
points, but not for selecting the largest 100,000 values.  If the
subset is a significant part of the total dataset it may be faster to
sort all the elements of the dataset directly with an O(N \log N)
algorithm and obtain the smallest or largest values that way.

 -- Function: int gsl_sort_smallest (double * DEST, size_t K, const
          double * SRC, size_t STRIDE, size_t N)
     This function copies the K smallest elements of the array SRC, of
     size N and stride STRIDE, in ascending numerical order into the
     array DEST.  The size K of the subset must be less than or equal
     to N.  The data SRC is not modified by this operation.

 -- Function: int gsl_sort_largest (double * DEST, size_t K, const
          double * SRC, size_t STRIDE, size_t N)
     This function copies the K largest elements of the array SRC, of
     size N and stride STRIDE, in descending numerical order into the
     array DEST. K must be less than or equal to N. The data SRC is not
     modified by this operation.

 -- Function: int gsl_sort_vector_smallest (double * DEST, size_t K,
          const gsl_vector * V)
 -- Function: int gsl_sort_vector_largest (double * DEST, size_t K,
          const gsl_vector * V)
     These functions copy the K smallest or largest elements of the
     vector V into the array DEST. K must be less than or equal to the
     length of the vector V.

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

 -- Function: int gsl_sort_smallest_index (size_t * P, size_t K, const
          double * SRC, size_t STRIDE, size_t N)
     This function stores the indices of the K smallest elements of the
     array SRC, of size N and stride STRIDE, in the array P.  The
     indices are chosen so that the corresponding data is in ascending
     numerical order.  K must be less than or equal to N. The data SRC
     is not modified by this operation.

 -- Function: int gsl_sort_largest_index (size_t * P, size_t K, const
          double * SRC, size_t STRIDE, size_t N)
     This function stores the indices of the K largest elements of the
     array SRC, of size N and stride STRIDE, in the array P.  The
     indices are chosen so that the corresponding data is in descending
     numerical order.  K must be less than or equal to N. The data SRC
     is not modified by this operation.

 -- Function: int gsl_sort_vector_smallest_index (size_t * P, size_t K,
          const gsl_vector * V)
 -- Function: int gsl_sort_vector_largest_index (size_t * P, size_t K,
          const gsl_vector * V)
     These functions store the indices of the K smallest or largest
     elements of the vector V in the array P. K must be less than or
     equal to the length of the vector V.


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

12.4 Computing the rank
=======================

The "rank" of an element is its order in the sorted data.  The rank is
the inverse of the index permutation, P.  It can be computed using the
following algorithm,

     for (i = 0; i < p->size; i++)
     {
         size_t pi = p->data[i];
         rank->data[pi] = i;
     }

This can be computed directly from the function
`gsl_permutation_inverse(rank,p)'.

   The following function will print the rank of each element of the
vector V,

     void
     print_rank (gsl_vector * v)
     {
       size_t i;
       size_t n = v->size;
       gsl_permutation * perm = gsl_permutation_alloc(n);
       gsl_permutation * rank = gsl_permutation_alloc(n);

       gsl_sort_vector_index (perm, v);
       gsl_permutation_inverse (rank, perm);

       for (i = 0; i < n; i++)
        {
         double vi = gsl_vector_get(v, i);
         printf ("element = %d, value = %g, rank = %d\n",
                  i, vi, rank->data[i]);
        }

       gsl_permutation_free (perm);
       gsl_permutation_free (rank);
     }


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

12.5 Examples
=============

The following example shows how to use the permutation P to print the
elements of the vector V in ascending order,

     gsl_sort_vector_index (p, v);

     for (i = 0; i < v->size; i++)
     {
         double vpi = gsl_vector_get (v, p->data[i]);
         printf ("order = %d, value = %g\n", i, vpi);
     }

The next example uses the function `gsl_sort_smallest' to select the 5
smallest numbers from 100000 uniform random variates stored in an array,

     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_sort_double.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       size_t i, k = 5, N = 100000;

       double * x = malloc (N * sizeof(double));
       double * small = malloc (k * sizeof(double));

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (i = 0; i < N; i++)
         {
           x[i] = gsl_rng_uniform(r);
         }

       gsl_sort_smallest (small, k, x, 1, N);

       printf ("%d smallest values from %d\n", k, N);

       for (i = 0; i < k; i++)
         {
           printf ("%d: %.18f\n", i, small[i]);
         }

       free (x);
       free (small);
       gsl_rng_free (r);
       return 0;
     }
   The output lists the 5 smallest values, in ascending order,

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


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

12.6 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

13 BLAS Support
***************

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

   The library provides a low-level layer which corresponds directly to
the C-language BLAS standard, referred to here as "CBLAS", and a
higher-level interface for operations on GSL vectors and matrices.
Users who are interested in simple operations on GSL vector and matrix
objects should use the high-level layer described in this chapter.  The
functions are declared in the file `gsl_blas.h' and should satisfy the
needs of most users.

   Note that GSL matrices are implemented using dense-storage so the
interface only includes the corresponding dense-storage BLAS functions.
The full BLAS functionality for band-format and packed-format matrices
is available through the low-level CBLAS interface.  Similarly, GSL
vectors are restricted to positive strides, whereas the low-level CBLAS
interface supports negative strides as specified in the BLAS
standard.(1)

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

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

* Menu:

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

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

   (1) In the low-level CBLAS interface, a negative stride accesses the
vector elements in reverse order, i.e. the i-th element is given by
(N-i)*|incx| for incx < 0.


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

13.1 GSL BLAS Interface
=======================

GSL provides dense vector and matrix objects, based on the relevant
built-in types.  The library provides an interface to the BLAS
operations which apply to these objects.  The interface to this
functionality is given in the file `gsl_blas.h'.

* 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

13.1.1 Level 1
--------------

 -- Function: int gsl_blas_sdsdot (float ALPHA, const gsl_vector_float
          * X, const gsl_vector_float * Y, float * RESULT)
     This function computes the sum \alpha + x^T y for the vectors X
     and Y, returning the result in RESULT.

 -- Function: int gsl_blas_sdot (const gsl_vector_float * X, const
          gsl_vector_float * Y, float * RESULT)
 -- Function: int gsl_blas_dsdot (const gsl_vector_float * X, const
          gsl_vector_float * Y, double * RESULT)
 -- Function: int gsl_blas_ddot (const gsl_vector * X, const gsl_vector
          * Y, double * RESULT)
     These functions compute the scalar product x^T y for the vectors X
     and Y, returning the result in RESULT.

 -- Function: int gsl_blas_cdotu (const gsl_vector_complex_float * X,
          const gsl_vector_complex_float * Y, gsl_complex_float * DOTU)
 -- Function: int gsl_blas_zdotu (const gsl_vector_complex * X, const
          gsl_vector_complex * Y, gsl_complex * DOTU)
     These functions compute the complex scalar product x^T y for the
     vectors X and Y, returning the result in DOTU

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

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

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

          ||x||_2 = \sqrt {\sum (\Re(x_i)^2 + \Im(x_i)^2)}.

 -- Function: float gsl_blas_sasum (const gsl_vector_float * X)
 -- Function: double gsl_blas_dasum (const gsl_vector * X)
     These functions compute the absolute sum \sum |x_i| of the
     elements of the vector X.

 -- Function: float gsl_blas_scasum (const gsl_vector_complex_float * X)
 -- Function: double gsl_blas_dzasum (const gsl_vector_complex * X)
     These functions compute the sum of the magnitudes of the real and
     imaginary parts of the complex vector X, \sum |\Re(x_i)| +
     |\Im(x_i)|.

 -- Function: CBLAS_INDEX_t gsl_blas_isamax (const gsl_vector_float * X)
 -- Function: CBLAS_INDEX_t gsl_blas_idamax (const gsl_vector * X)
 -- Function: CBLAS_INDEX_t gsl_blas_icamax (const
          gsl_vector_complex_float * X)
 -- Function: CBLAS_INDEX_t gsl_blas_izamax (const gsl_vector_complex *
          X)
     These functions return the index of the largest element of the
     vector X. The largest element is determined by its absolute
     magnitude for real vectors and by the sum of the magnitudes of the
     real and imaginary parts |\Re(x_i)| + |\Im(x_i)| for complex
     vectors.  If the largest value occurs several times then the index
     of the first occurrence is returned.

 -- Function: int gsl_blas_sswap (gsl_vector_float * X,
          gsl_vector_float * Y)
 -- Function: int gsl_blas_dswap (gsl_vector * X, gsl_vector * Y)
 -- Function: int gsl_blas_cswap (gsl_vector_complex_float * X,
          gsl_vector_complex_float * Y)
 -- Function: int gsl_blas_zswap (gsl_vector_complex * X,
          gsl_vector_complex * Y)
     These functions exchange the elements of the vectors X and Y.

 -- Function: int gsl_blas_scopy (const gsl_vector_float * X,
          gsl_vector_float * Y)
 -- Function: int gsl_blas_dcopy (const gsl_vector * X, gsl_vector * Y)
 -- Function: int gsl_blas_ccopy (const gsl_vector_complex_float * X,
          gsl_vector_complex_float * Y)
 -- Function: int gsl_blas_zcopy (const gsl_vector_complex * X,
          gsl_vector_complex * Y)
     These functions copy the elements of the vector X into the vector
     Y.

 -- Function: int gsl_blas_saxpy (float ALPHA, const gsl_vector_float *
          X, gsl_vector_float * Y)
 -- Function: int gsl_blas_daxpy (double ALPHA, const gsl_vector * X,
          gsl_vector * Y)
 -- Function: int gsl_blas_caxpy (const gsl_complex_float ALPHA, const
          gsl_vector_complex_float * X, gsl_vector_complex_float * Y)
 -- Function: int gsl_blas_zaxpy (const gsl_complex ALPHA, const
          gsl_vector_complex * X, gsl_vector_complex * Y)
     These functions compute the sum y = \alpha x + y for the vectors X
     and Y.

 -- Function: void gsl_blas_sscal (float ALPHA, gsl_vector_float * X)
 -- Function: void gsl_blas_dscal (double ALPHA, gsl_vector * X)
 -- Function: void gsl_blas_cscal (const gsl_complex_float ALPHA,
          gsl_vector_complex_float * X)
 -- Function: void gsl_blas_zscal (const gsl_complex ALPHA,
          gsl_vector_complex * X)
 -- Function: void gsl_blas_csscal (float ALPHA,
          gsl_vector_complex_float * X)
 -- Function: void gsl_blas_zdscal (double ALPHA, gsl_vector_complex *
          X)
     These functions rescale the vector X by the multiplicative factor
     ALPHA.

 -- Function: int gsl_blas_srotg (float A[], float B[], float C[],
          float S[])
 -- Function: int gsl_blas_drotg (double A[], double B[], double C[],
          double S[])
     These functions compute a Givens rotation (c,s) which zeroes the
     vector (a,b),

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

     The variables A and B are overwritten by the routine.

 -- Function: int gsl_blas_srot (gsl_vector_float * X, gsl_vector_float
          * Y, float C, float S)
 -- Function: int gsl_blas_drot (gsl_vector * X, gsl_vector * Y, const
          double C, const double S)
     These functions apply a Givens rotation (x', y') = (c x + s y, -s
     x + c y) to the vectors X, Y.

 -- Function: int gsl_blas_srotmg (float D1[], float D2[], float B1[],
          float B2, float P[])
 -- Function: int gsl_blas_drotmg (double D1[], double D2[], double
          B1[], double B2, double P[])
     These functions compute a modified Givens transformation.  The
     modified Givens transformation is defined in the original Level-1
     BLAS specification, given in the references.

 -- Function: int gsl_blas_srotm (gsl_vector_float * X,
          gsl_vector_float * Y, const float P[])
 -- Function: int gsl_blas_drotm (gsl_vector * X, gsl_vector * Y, const
          double P[])
     These functions apply a modified Givens transformation.


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

13.1.2 Level 2
--------------

 -- Function: int gsl_blas_sgemv (CBLAS_TRANSPOSE_t TRANSA, float
          ALPHA, const gsl_matrix_float * A, const gsl_vector_float *
          X, float BETA, gsl_vector_float * Y)
 -- Function: int gsl_blas_dgemv (CBLAS_TRANSPOSE_t TRANSA, double
          ALPHA, const gsl_matrix * A, const gsl_vector * X, double
          BETA, gsl_vector * Y)
 -- Function: int gsl_blas_cgemv (CBLAS_TRANSPOSE_t TRANSA, const
          gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
          const gsl_vector_complex_float * X, const gsl_complex_float
          BETA, gsl_vector_complex_float * Y)
 -- Function: int gsl_blas_zgemv (CBLAS_TRANSPOSE_t TRANSA, const
          gsl_complex ALPHA, const gsl_matrix_complex * A, const
          gsl_vector_complex * X, const gsl_complex BETA,
          gsl_vector_complex * Y)
     These functions compute the matrix-vector product and sum y =
     \alpha op(A) x + \beta y, where op(A) = A, A^T, A^H for TRANSA =
     `CblasNoTrans', `CblasTrans', `CblasConjTrans'.

 -- Function: int gsl_blas_strmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_float * A,
          gsl_vector_float * X)
 -- Function: int gsl_blas_dtrmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix * A, gsl_vector *
          X)
 -- Function: int gsl_blas_ctrmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex_float *
          A, gsl_vector_complex_float * X)
 -- Function: int gsl_blas_ztrmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex * A,
          gsl_vector_complex * X)
     These functions compute the matrix-vector product x = op(A) x for
     the triangular matrix A, where op(A) = A, A^T, A^H for TRANSA =
     `CblasNoTrans', `CblasTrans', `CblasConjTrans'.  When UPLO is
     `CblasUpper' then the upper triangle of A is used, and when UPLO
     is `CblasLower' then the lower triangle of A is used.  If DIAG is
     `CblasNonUnit' then the diagonal of the matrix is used, but if
     DIAG is `CblasUnit' then the diagonal elements of the matrix A are
     taken as unity and are not referenced.

 -- Function: int gsl_blas_strsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_float * A,
          gsl_vector_float * X)
 -- Function: int gsl_blas_dtrsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix * A, gsl_vector *
          X)
 -- Function: int gsl_blas_ctrsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex_float *
          A, gsl_vector_complex_float * X)
 -- Function: int gsl_blas_ztrsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex * A,
          gsl_vector_complex * X)
     These functions compute inv(op(A)) x for X, where op(A) = A, A^T,
     A^H for TRANSA = `CblasNoTrans', `CblasTrans', `CblasConjTrans'.
     When UPLO is `CblasUpper' then the upper triangle of A is used,
     and when UPLO is `CblasLower' then the lower triangle of A is
     used.  If DIAG is `CblasNonUnit' then the diagonal of the matrix
     is used, but if DIAG is `CblasUnit' then the diagonal elements of
     the matrix A are taken as unity and are not referenced.

 -- Function: int gsl_blas_ssymv (CBLAS_UPLO_t UPLO, float ALPHA, const
          gsl_matrix_float * A, const gsl_vector_float * X, float BETA,
          gsl_vector_float * Y)
 -- Function: int gsl_blas_dsymv (CBLAS_UPLO_t UPLO, double ALPHA,
          const gsl_matrix * A, const gsl_vector * X, double BETA,
          gsl_vector * Y)
     These functions compute the matrix-vector product and sum y =
     \alpha A x + \beta y for the symmetric matrix A.  Since the matrix
     A is symmetric only its upper half or lower half need to be
     stored.  When UPLO is `CblasUpper' then the upper triangle and
     diagonal of A are used, and when UPLO is `CblasLower' then the
     lower triangle and diagonal of A are used.

 -- Function: int gsl_blas_chemv (CBLAS_UPLO_t UPLO, const
          gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
          const gsl_vector_complex_float * X, const gsl_complex_float
          BETA, gsl_vector_complex_float * Y)
 -- Function: int gsl_blas_zhemv (CBLAS_UPLO_t UPLO, const gsl_complex
          ALPHA, const gsl_matrix_complex * A, const gsl_vector_complex
          * X, const gsl_complex BETA, gsl_vector_complex * Y)
     These functions compute the matrix-vector product and sum y =
     \alpha A x + \beta y for the hermitian matrix A.  Since the matrix
     A is hermitian only its upper half or lower half need to be
     stored.  When UPLO is `CblasUpper' then the upper triangle and
     diagonal of A are used, and when UPLO is `CblasLower' then the
     lower triangle and diagonal of A are used.  The imaginary elements
     of the diagonal are automatically assumed to be zero and are not
     referenced.

 -- Function: int gsl_blas_sger (float ALPHA, const gsl_vector_float *
          X, const gsl_vector_float * Y, gsl_matrix_float * A)
 -- Function: int gsl_blas_dger (double ALPHA, const gsl_vector * X,
          const gsl_vector * Y, gsl_matrix * A)
 -- Function: int gsl_blas_cgeru (const gsl_complex_float ALPHA, const
          gsl_vector_complex_float * X, const gsl_vector_complex_float
          * Y, gsl_matrix_complex_float * A)
 -- Function: int gsl_blas_zgeru (const gsl_complex ALPHA, const
          gsl_vector_complex * X, const gsl_vector_complex * Y,
          gsl_matrix_complex * A)
     These functions compute the rank-1 update A = \alpha x y^T + A of
     the matrix A.

 -- Function: int gsl_blas_cgerc (const gsl_complex_float ALPHA, const
          gsl_vector_complex_float * X, const gsl_vector_complex_float
          * Y, gsl_matrix_complex_float * A)
 -- Function: int gsl_blas_zgerc (const gsl_complex ALPHA, const
          gsl_vector_complex * X, const gsl_vector_complex * Y,
          gsl_matrix_complex * A)
     These functions compute the conjugate rank-1 update A = \alpha x
     y^H + A of the matrix A.

 -- Function: int gsl_blas_ssyr (CBLAS_UPLO_t UPLO, float ALPHA, const
          gsl_vector_float * X, gsl_matrix_float * A)
 -- Function: int gsl_blas_dsyr (CBLAS_UPLO_t UPLO, double ALPHA, const
          gsl_vector * X, gsl_matrix * A)
     These functions compute the symmetric rank-1 update A = \alpha x
     x^T + A of the symmetric matrix A.  Since the matrix A is
     symmetric only its upper half or lower half need to be stored.
     When UPLO is `CblasUpper' then the upper triangle and diagonal of
     A are used, and when UPLO is `CblasLower' then the lower triangle
     and diagonal of A are used.

 -- Function: int gsl_blas_cher (CBLAS_UPLO_t UPLO, float ALPHA, const
          gsl_vector_complex_float * X, gsl_matrix_complex_float * A)
 -- Function: int gsl_blas_zher (CBLAS_UPLO_t UPLO, double ALPHA, const
          gsl_vector_complex * X, gsl_matrix_complex * A)
     These functions compute the hermitian rank-1 update A = \alpha x
     x^H + A of the hermitian matrix A.  Since the matrix A is
     hermitian only its upper half or lower half need to be stored.
     When UPLO is `CblasUpper' then the upper triangle and diagonal of
     A are used, and when UPLO is `CblasLower' then the lower triangle
     and diagonal of A are used.  The imaginary elements of the
     diagonal are automatically set to zero.

 -- Function: int gsl_blas_ssyr2 (CBLAS_UPLO_t UPLO, float ALPHA, const
          gsl_vector_float * X, const gsl_vector_float * Y,
          gsl_matrix_float * A)
 -- Function: int gsl_blas_dsyr2 (CBLAS_UPLO_t UPLO, double ALPHA,
          const gsl_vector * X, const gsl_vector * Y, gsl_matrix * A)
     These functions compute the symmetric rank-2 update A = \alpha x
     y^T + \alpha y x^T + A of the symmetric matrix A.  Since the
     matrix A is symmetric only its upper half or lower half need to be
     stored.  When UPLO is `CblasUpper' then the upper triangle and
     diagonal of A are used, and when UPLO is `CblasLower' then the
     lower triangle and diagonal of A are used.

 -- Function: int gsl_blas_cher2 (CBLAS_UPLO_t UPLO, const
          gsl_complex_float ALPHA, const gsl_vector_complex_float * X,
          const gsl_vector_complex_float * Y, gsl_matrix_complex_float
          * A)
 -- Function: int gsl_blas_zher2 (CBLAS_UPLO_t UPLO, const gsl_complex
          ALPHA, const gsl_vector_complex * X, const gsl_vector_complex
          * Y, gsl_matrix_complex * A)
     These functions compute the hermitian rank-2 update A = \alpha x
     y^H + \alpha^* y x^H + A of the hermitian matrix A.  Since the
     matrix A is hermitian only its upper half or lower half need to be
     stored.  When UPLO is `CblasUpper' then the upper triangle and
     diagonal of A are used, and when UPLO is `CblasLower' then the
     lower triangle and diagonal of A are used.  The imaginary elements
     of the diagonal are automatically set to zero.


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

13.1.3 Level 3
--------------

 -- Function: int gsl_blas_sgemm (CBLAS_TRANSPOSE_t TRANSA,
          CBLAS_TRANSPOSE_t TRANSB, float ALPHA, const gsl_matrix_float
          * A, const gsl_matrix_float * B, float BETA, gsl_matrix_float
          * C)
 -- Function: int gsl_blas_dgemm (CBLAS_TRANSPOSE_t TRANSA,
          CBLAS_TRANSPOSE_t TRANSB, double ALPHA, const gsl_matrix * A,
          const gsl_matrix * B, double BETA, gsl_matrix * C)
 -- Function: int gsl_blas_cgemm (CBLAS_TRANSPOSE_t TRANSA,
          CBLAS_TRANSPOSE_t TRANSB, const gsl_complex_float ALPHA,
          const gsl_matrix_complex_float * A, const
          gsl_matrix_complex_float * B, const gsl_complex_float BETA,
          gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zgemm (CBLAS_TRANSPOSE_t TRANSA,
          CBLAS_TRANSPOSE_t TRANSB, const gsl_complex ALPHA, const
          gsl_matrix_complex * A, const gsl_matrix_complex * B, const
          gsl_complex BETA, gsl_matrix_complex * C)
     These functions compute the matrix-matrix product and sum C =
     \alpha op(A) op(B) + \beta C where op(A) = A, A^T, A^H for TRANSA
     = `CblasNoTrans', `CblasTrans', `CblasConjTrans' and similarly for
     the parameter TRANSB.

 -- Function: int gsl_blas_ssymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          float ALPHA, const gsl_matrix_float * A, const
          gsl_matrix_float * B, float BETA, gsl_matrix_float * C)
 -- Function: int gsl_blas_dsymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          double ALPHA, const gsl_matrix * A, const gsl_matrix * B,
          double BETA, gsl_matrix * C)
 -- Function: int gsl_blas_csymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          const gsl_complex_float ALPHA, const gsl_matrix_complex_float
          * A, const gsl_matrix_complex_float * B, const
          gsl_complex_float BETA, gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zsymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          const gsl_complex ALPHA, const gsl_matrix_complex * A, const
          gsl_matrix_complex * B, const gsl_complex BETA,
          gsl_matrix_complex * C)
     These functions compute the matrix-matrix product and sum C =
     \alpha A B + \beta C for SIDE is `CblasLeft' and C = \alpha B A +
     \beta C for SIDE is `CblasRight', where the matrix A is symmetric.
     When UPLO is `CblasUpper' then the upper triangle and diagonal of
     A are used, and when UPLO is `CblasLower' then the lower triangle
     and diagonal of A are used.

 -- Function: int gsl_blas_chemm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          const gsl_complex_float ALPHA, const gsl_matrix_complex_float
          * A, const gsl_matrix_complex_float * B, const
          gsl_complex_float BETA, gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zhemm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          const gsl_complex ALPHA, const gsl_matrix_complex * A, const
          gsl_matrix_complex * B, const gsl_complex BETA,
          gsl_matrix_complex * C)
     These functions compute the matrix-matrix product and sum C =
     \alpha A B + \beta C for SIDE is `CblasLeft' and C = \alpha B A +
     \beta C for SIDE is `CblasRight', where the matrix A is hermitian.
     When UPLO is `CblasUpper' then the upper triangle and diagonal of
     A are used, and when UPLO is `CblasLower' then the lower triangle
     and diagonal of A are used.  The imaginary elements of the
     diagonal are automatically set to zero.

 -- Function: int gsl_blas_strmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, float ALPHA,
          const gsl_matrix_float * A, gsl_matrix_float * B)
 -- Function: int gsl_blas_dtrmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, double ALPHA,
          const gsl_matrix * A, gsl_matrix * B)
 -- Function: int gsl_blas_ctrmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
          gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
          gsl_matrix_complex_float * B)
 -- Function: int gsl_blas_ztrmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
          gsl_complex ALPHA, const gsl_matrix_complex * A,
          gsl_matrix_complex * B)
     These functions compute the matrix-matrix product B = \alpha op(A)
     B for SIDE is `CblasLeft' and B = \alpha B op(A) for SIDE is
     `CblasRight'.  The matrix A is triangular and op(A) = A, A^T, A^H
     for TRANSA = `CblasNoTrans', `CblasTrans', `CblasConjTrans'. When
     UPLO is `CblasUpper' then the upper triangle of A is used, and
     when UPLO is `CblasLower' then the lower triangle of A is used.
     If DIAG is `CblasNonUnit' then the diagonal of A is used, but if
     DIAG is `CblasUnit' then the diagonal elements of the matrix A are
     taken as unity and are not referenced.

 -- Function: int gsl_blas_strsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, float ALPHA,
          const gsl_matrix_float * A, gsl_matrix_float * B)
 -- Function: int gsl_blas_dtrsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, double ALPHA,
          const gsl_matrix * A, gsl_matrix * B)
 -- Function: int gsl_blas_ctrsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
          gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
          gsl_matrix_complex_float * B)
 -- Function: int gsl_blas_ztrsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
          gsl_complex ALPHA, const gsl_matrix_complex * A,
          gsl_matrix_complex * B)
     These functions compute the inverse-matrix matrix product B =
     \alpha op(inv(A))B for SIDE is `CblasLeft' and B = \alpha B
     op(inv(A)) for SIDE is `CblasRight'.  The matrix A is triangular
     and op(A) = A, A^T, A^H for TRANSA = `CblasNoTrans', `CblasTrans',
     `CblasConjTrans'. When UPLO is `CblasUpper' then the upper
     triangle of A is used, and when UPLO is `CblasLower' then the
     lower triangle of A is used.  If DIAG is `CblasNonUnit' then the
     diagonal of A is used, but if DIAG is `CblasUnit' then the
     diagonal elements of the matrix A are taken as unity and are not
     referenced.

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

 -- Function: int gsl_blas_cherk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, float ALPHA, const gsl_matrix_complex_float * A, float
          BETA, gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zherk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, double ALPHA, const gsl_matrix_complex * A, double
          BETA, gsl_matrix_complex * C)
     These functions compute a rank-k update of the hermitian matrix C,
     C = \alpha A A^H + \beta C when TRANS is `CblasNoTrans' and C =
     \alpha A^H A + \beta C when TRANS is `CblasConjTrans'.  Since the
     matrix C is hermitian only its upper half or lower half need to be
     stored.  When UPLO is `CblasUpper' then the upper triangle and
     diagonal of C are used, and when UPLO is `CblasLower' then the
     lower triangle and diagonal of C are used.  The imaginary elements
     of the diagonal are automatically set to zero.

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

 -- Function: int gsl_blas_cher2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, const gsl_complex_float ALPHA, const
          gsl_matrix_complex_float * A, const gsl_matrix_complex_float
          * B, float BETA, gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zher2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
          TRANS, const gsl_complex ALPHA, const gsl_matrix_complex * A,
          const gsl_matrix_complex * B, double BETA, gsl_matrix_complex
          * C)
     These functions compute a rank-2k update of the hermitian matrix C,
     C = \alpha A B^H + \alpha^* B A^H + \beta C when TRANS is
     `CblasNoTrans' and C = \alpha A^H B + \alpha^* B^H A + \beta C when
     TRANS is `CblasConjTrans'.  Since the matrix C is hermitian only
     its upper half or lower half need to be stored.  When UPLO is
     `CblasUpper' then the upper triangle and diagonal of C are used,
     and when UPLO is `CblasLower' then the lower triangle and diagonal
     of C are used.  The imaginary elements of the diagonal are
     automatically set to zero.


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

13.2 Examples
=============

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

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

The matrices are stored in row major order, according to the C
convention for arrays.

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

     int
     main (void)
     {
       double a[] = { 0.11, 0.12, 0.13,
                      0.21, 0.22, 0.23 };

       double b[] = { 1011, 1012,
                      1021, 1022,
                      1031, 1032 };

       double c[] = { 0.00, 0.00,
                      0.00, 0.00 };

       gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
       gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
       gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);

       /* Compute C = A B */

       gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
                       1.0, &A.matrix, &B.matrix,
                       0.0, &C.matrix);

       printf ("[ %g, %g\n", c[0], c[1]);
       printf ("  %g, %g ]\n", c[2], c[3]);

       return 0;
     }

Here is the output from the program,

     $ ./a.out
     [ 367.76, 368.12
       674.06, 674.72 ]


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

13.3 References and Further Reading
===================================

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

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

     `BLAS Technical Forum'
     `http://www.netlib.org/blas/blast-forum/'

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

14 Linear Algebra
*****************

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

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

* 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::
* Hessenberg Decomposition of Real Matrices::
* Hessenberg-Triangular Decomposition of Real Matrices::
* Bidiagonalization::
* Householder Transformations::
* Householder solver for linear systems::
* Tridiagonal Systems::
* Balancing::
* Linear Algebra Examples::
* Linear Algebra References and Further Reading::


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

14.1 LU Decomposition
=====================

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

     P A = L U

where P is a permutation matrix, L is unit lower triangular matrix and
U is upper triangular matrix. For square matrices this decomposition
can be used to convert the linear system A x = b into a pair of
triangular systems (L y = P b, U x = y), which can be solved by forward
and back-substitution.  Note that the LU decomposition is valid for
singular matrices.

 -- Function: int gsl_linalg_LU_decomp (gsl_matrix * A, gsl_permutation
          * P, int * SIGNUM)
 -- Function: int gsl_linalg_complex_LU_decomp (gsl_matrix_complex * A,
          gsl_permutation * P, int * SIGNUM)
     These functions factorize the square matrix A into the LU
     decomposition PA = LU.  On output the diagonal and upper
     triangular part of the input matrix A contain the matrix U. The
     lower triangular part of the input matrix (excluding the diagonal)
     contains L.  The diagonal elements of L are unity, and are not
     stored.

     The permutation matrix P is encoded in the permutation P. The j-th
     column of the matrix P is given by the k-th column of the identity
     matrix, where k = p_j the j-th element of the permutation vector.
     The sign of the permutation is given by SIGNUM. It has the value
     (-1)^n, where n is the number of interchanges in the permutation.

     The algorithm used in the decomposition is Gaussian Elimination
     with partial pivoting (Golub & Van Loan, `Matrix Computations',
     Algorithm 3.4.1).

 -- Function: int gsl_linalg_LU_solve (const gsl_matrix * LU, const
          gsl_permutation * P, const gsl_vector * B, gsl_vector * X)
 -- Function: int gsl_linalg_complex_LU_solve (const gsl_matrix_complex
          * LU, const gsl_permutation * P, const gsl_vector_complex *
          B, gsl_vector_complex * X)
     These functions solve the square system A x = b using the LU
     decomposition of A into (LU, P) given by `gsl_linalg_LU_decomp' or
     `gsl_linalg_complex_LU_decomp' as input.

 -- Function: int gsl_linalg_LU_svx (const gsl_matrix * LU, const
          gsl_permutation * P, gsl_vector * X)
 -- Function: int gsl_linalg_complex_LU_svx (const gsl_matrix_complex *
          LU, const gsl_permutation * P, gsl_vector_complex * X)
     These functions solve the square system A x = b in-place using the
     precomputed LU decomposition of A into (LU,P). On input X should
     contain the right-hand side b, which is replaced by the solution
     on output.

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

 -- Function: int gsl_linalg_LU_invert (const gsl_matrix * LU, const
          gsl_permutation * P, gsl_matrix * INVERSE)
 -- Function: int gsl_linalg_complex_LU_invert (const
          gsl_matrix_complex * LU, const gsl_permutation * P,
          gsl_matrix_complex * INVERSE)
     These functions compute the inverse of a matrix A from its LU
     decomposition (LU,P), storing the result in the matrix INVERSE.
     The inverse is computed by solving the system A x = b for each
     column of the identity matrix.  It is preferable to avoid direct
     use of the inverse whenever possible, as the linear solver
     functions can obtain the same result more efficiently and reliably
     (consult any introductory textbook on numerical linear algebra for
     details).

 -- Function: double gsl_linalg_LU_det (gsl_matrix * LU, int SIGNUM)
 -- Function: gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex
          * LU, int SIGNUM)
     These functions compute the determinant of a matrix A from its LU
     decomposition, LU. The determinant is computed as the product of
     the diagonal elements of U and the sign of the row permutation
     SIGNUM.

 -- Function: double gsl_linalg_LU_lndet (gsl_matrix * LU)
 -- Function: double gsl_linalg_complex_LU_lndet (gsl_matrix_complex *
          LU)
     These functions compute the logarithm of the absolute value of the
     determinant of a matrix A, \ln|\det(A)|, from its LU
     decomposition, LU.  This function may be useful if the direct
     computation of the determinant would overflow or underflow.

 -- Function: int gsl_linalg_LU_sgndet (gsl_matrix * LU, int SIGNUM)
 -- Function: gsl_complex gsl_linalg_complex_LU_sgndet
          (gsl_matrix_complex * LU, int SIGNUM)
     These functions compute the sign or phase factor of the
     determinant of a matrix A, \det(A)/|\det(A)|, from its LU
     decomposition, LU.


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

14.2 QR Decomposition
=====================

A general rectangular M-by-N matrix A has a QR decomposition into the
product of an orthogonal M-by-M square matrix Q (where Q^T Q = I) and
an M-by-N right-triangular matrix R,

     A = Q R

This decomposition can be used to convert the linear system A x = b
into the triangular system R x = Q^T b, which can be solved by
back-substitution. Another use of the QR decomposition is to compute an
orthonormal basis for a set of vectors. The first N columns of Q form
an orthonormal basis for the range of A, ran(A), when A has full column
rank.

 -- Function: int gsl_linalg_QR_decomp (gsl_matrix * A, gsl_vector *
          TAU)
     This function factorizes the M-by-N matrix A into the QR
     decomposition A = Q R.  On output the diagonal and upper
     triangular part of the input matrix contain the matrix R. The
     vector TAU and the columns of the lower triangular part of the
     matrix A contain the Householder coefficients and Householder
     vectors which encode the orthogonal matrix Q.  The vector TAU must
     be of length k=\min(M,N). The matrix Q is related to these
     components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i
     v_i^T and v_i is the Householder vector v_i =
     (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the same storage
     scheme as used by LAPACK.

     The algorithm used to perform the decomposition is Householder QR
     (Golub & Van Loan, `Matrix Computations', Algorithm 5.2.1).

 -- Function: int gsl_linalg_QR_solve (const gsl_matrix * QR, const
          gsl_vector * TAU, const gsl_vector * B, gsl_vector * X)
     This function solves the square system A x = b using the QR
     decomposition of A held in (QR, TAU) which must have been computed
     previously with `gsl_linalg_QR_decomp'.  The least-squares
     solution for rectangular systems can be found using
     `gsl_linalg_QR_lssolve'.

 -- Function: int gsl_linalg_QR_svx (const gsl_matrix * QR, const
          gsl_vector * TAU, gsl_vector * X)
     This function solves the square system A x = b in-place using the
     QR decomposition of A held in (QR,TAU) which must have been
     computed previously by `gsl_linalg_QR_decomp'.  On input X should
     contain the right-hand side b, which is replaced by the solution
     on output.

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

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

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

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

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

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

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

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

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

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

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


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

14.3 QR Decomposition with Column Pivoting
==========================================

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

     A P = Q R

The first r columns of Q form an orthonormal basis for the range of A
for a matrix with column rank r.  This decomposition can also be used
to convert the linear system A x = b into the triangular system R y =
Q^T b, x = P y, which can be solved by back-substitution and
permutation.  We denote the QR decomposition with column pivoting by
QRP^T since A = Q R P^T.

 -- Function: int gsl_linalg_QRPT_decomp (gsl_matrix * A, gsl_vector *
          TAU, gsl_permutation * P, int * SIGNUM, gsl_vector * NORM)
     This function factorizes the M-by-N matrix A into the QRP^T
     decomposition A = Q R P^T.  On output the diagonal and upper
     triangular part of the input matrix contain the matrix R. The
     permutation matrix P is stored in the permutation P.  The sign of
     the permutation is given by SIGNUM. It has the value (-1)^n, where
     n is the number of interchanges in the permutation. The vector TAU
     and the columns of the lower triangular part of the matrix A
     contain the Householder coefficients and vectors which encode the
     orthogonal matrix Q.  The vector TAU must be of length
     k=\min(M,N). The matrix Q is related to these components by, Q =
     Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the
     Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)).
     This is the same storage scheme as used by LAPACK.  The vector
     NORM is a workspace of length N used for column pivoting.

     The algorithm used to perform the decomposition is Householder QR
     with column pivoting (Golub & Van Loan, `Matrix Computations',
     Algorithm 5.4.1).

 -- Function: int gsl_linalg_QRPT_decomp2 (const gsl_matrix * A,
          gsl_matrix * Q, gsl_matrix * R, gsl_vector * TAU,
          gsl_permutation * P, int * SIGNUM, gsl_vector * NORM)
     This function factorizes the matrix A into the decomposition A = Q
     R P^T without modifying A itself and storing the output in the
     separate matrices Q and R.

 -- Function: int gsl_linalg_QRPT_solve (const gsl_matrix * QR, const
          gsl_vector * TAU, const gsl_permutation * P, const gsl_vector
          * B, gsl_vector * X)
     This function solves the square system A x = b using the QRP^T
     decomposition of A held in (QR, TAU, P) which must have been
     computed previously by `gsl_linalg_QRPT_decomp'.

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

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

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

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

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


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

14.4 Singular Value Decomposition
=================================

A general rectangular M-by-N matrix A has a singular value
decomposition (SVD) into the product of an M-by-N orthogonal matrix U,
an N-by-N diagonal matrix of singular values S and the transpose of an
N-by-N orthogonal square matrix V,

     A = U S V^T

The singular values \sigma_i = S_{ii} are all non-negative and are
generally chosen to form a non-increasing sequence \sigma_1 >= \sigma_2
>= ... >= \sigma_N >= 0.

   The singular value decomposition of a matrix has many practical uses.
The condition number of the matrix is given by the ratio of the largest
singular value to the smallest singular value. The presence of a zero
singular value indicates that the matrix is singular. The number of
non-zero singular values indicates the rank of the matrix.  In practice
singular value decomposition of a rank-deficient matrix will not produce
exact zeroes for singular values, due to finite numerical precision.
Small singular values should be edited by choosing a suitable tolerance.

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

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

 -- Function: int gsl_linalg_SV_decomp (gsl_matrix * A, gsl_matrix * V,
          gsl_vector * S, gsl_vector * WORK)
     This function factorizes the M-by-N matrix A into the singular
     value decomposition A = U S V^T for M >= N.  On output the matrix
     A is replaced by U. The diagonal elements of the singular value
     matrix S are stored in the vector S. The singular values are
     non-negative and form a non-increasing sequence from S_1 to S_N.
     The matrix V contains the elements of V in untransposed form. To
     form the product U S V^T it is necessary to take the transpose of
     V.  A workspace of length N is required in WORK.

     This routine uses the Golub-Reinsch SVD algorithm.

 -- Function: int gsl_linalg_SV_decomp_mod (gsl_matrix * A, gsl_matrix
          * X, gsl_matrix * V, gsl_vector * S, gsl_vector * WORK)
     This function computes the SVD using the modified Golub-Reinsch
     algorithm, which is faster for M>>N.  It requires the vector WORK
     of length N and the N-by-N matrix X as additional working space.

 -- Function: int gsl_linalg_SV_decomp_jacobi (gsl_matrix * A,
          gsl_matrix * V, gsl_vector * S)
     This function computes the SVD of the M-by-N matrix A using
     one-sided Jacobi orthogonalization for M >= N.  The Jacobi method
     can compute singular values to higher relative accuracy than
     Golub-Reinsch algorithms (see references for details).

 -- Function: int gsl_linalg_SV_solve (const gsl_matrix * U, const
          gsl_matrix * V, const gsl_vector * S, const gsl_vector * B,
          gsl_vector * X)
     This function solves the system A x = b using the singular value
     decomposition (U, S, V) of A which must have been computed
     previously with `gsl_linalg_SV_decomp'.

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

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

 -- Function: int gsl_linalg_SV_leverage (const gsl_matrix * U,
          gsl_vector * H)
     This function computes the statistical leverage values h_i of a
     matrix A using its singular value decomposition (U, S, V)
     previously computed with `gsl_linalg_SV_decomp'. h_i are the
     diagonal values of the matrix A (A^T A)^-1 A^T and depend only on
     the matrix U which is the input to this function.


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

14.5 Cholesky Decomposition
===========================

A symmetric, positive definite square matrix A has a Cholesky
decomposition into a product of a lower triangular matrix L and its
transpose L^T,

     A = L L^T

This is sometimes referred to as taking the square-root of a matrix. The
Cholesky decomposition can only be carried out when all the eigenvalues
of the matrix are positive.  This decomposition can be used to convert
the linear system A x = b into a pair of triangular systems (L y = b,
L^T x = y), which can be solved by forward and back-substitution.

 -- Function: int gsl_linalg_cholesky_decomp (gsl_matrix * A)
 -- Function: int gsl_linalg_complex_cholesky_decomp
          (gsl_matrix_complex * A)
     These functions factorize the symmetric, positive-definite square
     matrix A into the Cholesky decomposition A = L L^T (or A = L L^H
     for the complex case). On input, the values from the diagonal and
     lower-triangular part of the matrix A are used (the upper
     triangular part is ignored).  On output the diagonal and lower
     triangular part of the input matrix A contain the matrix L, while
     the upper triangular part of the input matrix is overwritten with
     L^T (the diagonal terms being identical for both L and L^T).  If
     the matrix is not positive-definite then the decomposition will
     fail, returning the error code `GSL_EDOM'.

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

 -- Function: int gsl_linalg_cholesky_solve (const gsl_matrix *
          CHOLESKY, const gsl_vector * B, gsl_vector * X)
 -- Function: int gsl_linalg_complex_cholesky_solve (const
          gsl_matrix_complex * CHOLESKY, const gsl_vector_complex * B,
          gsl_vector_complex * X)
     These functions solve the system A x = b using the Cholesky
     decomposition of A held in the matrix CHOLESKY which must have
     been previously computed by `gsl_linalg_cholesky_decomp' or
     `gsl_linalg_complex_cholesky_decomp'.

 -- Function: int gsl_linalg_cholesky_svx (const gsl_matrix * CHOLESKY,
          gsl_vector * X)
 -- Function: int gsl_linalg_complex_cholesky_svx (const
          gsl_matrix_complex * CHOLESKY, gsl_vector_complex * X)
     These functions solve the system A x = b in-place using the
     Cholesky decomposition of A held in the matrix CHOLESKY which must
     have been previously computed by `gsl_linalg_cholesky_decomp' or
     `gsl_linalg_complex_cholesky_decomp'. On input X should contain
     the right-hand side b, which is replaced by the solution on output.

 -- Function: int gsl_linalg_cholesky_invert (gsl_matrix * CHOLESKY)
 -- Function: int gsl_linalg_complex_cholesky_invert
          (gsl_matrix_complex * CHOLESKY)
     These functions compute the inverse of a matrix from its Cholesky
     decomposition CHOLESKY, which must have been previously computed
     by `gsl_linalg_cholesky_decomp' or
     `gsl_linalg_complex_cholesky_decomp'.  On output, the inverse is
     stored in-place in CHOLESKY.


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

14.6 Tridiagonal Decomposition of Real Symmetric Matrices
=========================================================

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

     A = Q T Q^T

where Q is an orthogonal matrix and T is a symmetric tridiagonal matrix.

 -- Function: int gsl_linalg_symmtd_decomp (gsl_matrix * A, gsl_vector
          * TAU)
     This function factorizes the symmetric square matrix A into the
     symmetric tridiagonal decomposition Q T Q^T.  On output the
     diagonal and subdiagonal part of the input matrix A contain the
     tridiagonal matrix T.  The remaining lower triangular part of the
     input matrix contains the Householder vectors which, together with
     the Householder coefficients TAU, encode the orthogonal matrix Q.
     This storage scheme is the same as used by LAPACK.  The upper
     triangular part of A is not referenced.

 -- Function: int gsl_linalg_symmtd_unpack (const gsl_matrix * A, const
          gsl_vector * TAU, gsl_matrix * Q, gsl_vector * DIAG,
          gsl_vector * SUBDIAG)
     This function unpacks the encoded symmetric tridiagonal
     decomposition (A, TAU) obtained from `gsl_linalg_symmtd_decomp'
     into the orthogonal matrix Q, the vector of diagonal elements DIAG
     and the vector of subdiagonal elements SUBDIAG.

 -- Function: int gsl_linalg_symmtd_unpack_T (const gsl_matrix * A,
          gsl_vector * DIAG, gsl_vector * SUBDIAG)
     This function unpacks the diagonal and subdiagonal of the encoded
     symmetric tridiagonal decomposition (A, TAU) obtained from
     `gsl_linalg_symmtd_decomp' into the vectors DIAG and SUBDIAG.


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

14.7 Tridiagonal Decomposition of Hermitian Matrices
====================================================

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

     A = U T U^T

where U is a unitary matrix and T is a real symmetric tridiagonal
matrix.

 -- Function: int gsl_linalg_hermtd_decomp (gsl_matrix_complex * A,
          gsl_vector_complex * TAU)
     This function factorizes the hermitian matrix A into the symmetric
     tridiagonal decomposition U T U^T.  On output the real parts of
     the diagonal and subdiagonal part of the input matrix A contain
     the tridiagonal matrix T.  The remaining lower triangular part of
     the input matrix contains the Householder vectors which, together
     with the Householder coefficients TAU, encode the unitary matrix
     U. This storage scheme is the same as used by LAPACK.  The upper
     triangular part of A and imaginary parts of the diagonal are not
     referenced.

 -- Function: int gsl_linalg_hermtd_unpack (const gsl_matrix_complex *
          A, const gsl_vector_complex * TAU, gsl_matrix_complex * U,
          gsl_vector * DIAG, gsl_vector * SUBDIAG)
     This function unpacks the encoded tridiagonal decomposition (A,
     TAU) obtained from `gsl_linalg_hermtd_decomp' into the unitary
     matrix U, the real vector of diagonal elements DIAG and the real
     vector of subdiagonal elements SUBDIAG.

 -- Function: int gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex
          * A, gsl_vector * DIAG, gsl_vector * SUBDIAG)
     This function unpacks the diagonal and subdiagonal of the encoded
     tridiagonal decomposition (A, TAU) obtained from the
     `gsl_linalg_hermtd_decomp' into the real vectors DIAG and SUBDIAG.


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

14.8 Hessenberg Decomposition of Real Matrices
==============================================

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

     A = U H U^T

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

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

 -- Function: int gsl_linalg_hessenberg_unpack (gsl_matrix * H,
          gsl_vector * TAU, gsl_matrix * U)
     This function constructs the orthogonal matrix U from the
     information stored in the Hessenberg matrix H along with the
     vector TAU. H and TAU are outputs from
     `gsl_linalg_hessenberg_decomp'.

 -- Function: int gsl_linalg_hessenberg_unpack_accum (gsl_matrix * H,
          gsl_vector * TAU, gsl_matrix * V)
     This function is similar to `gsl_linalg_hessenberg_unpack', except
     it accumulates the matrix U into V, so that V' = VU.  The matrix V
     must be initialized prior to calling this function.  Setting V to
     the identity matrix provides the same result as
     `gsl_linalg_hessenberg_unpack'. If H is order N, then V must have
     N columns but may have any number of rows.

 -- Function: int gsl_linalg_hessenberg_set_zero (gsl_matrix * H)
     This function sets the lower triangular portion of H, below the
     subdiagonal, to zero. It is useful for clearing out the
     Householder vectors after calling `gsl_linalg_hessenberg_decomp'.


File: gsl-ref.info,  Node: Hessenberg-Triangular Decomposition of Real Matrices,  Next: Bidiagonalization,  Prev: Hessenberg Decomposition of Real Matrices,  Up: Linear Algebra

14.9 Hessenberg-Triangular Decomposition of Real Matrices
=========================================================

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

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

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

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


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

14.10 Bidiagonalization
=======================

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

     A = U B V^T

where U and V are orthogonal matrices and B is a N-by-N bidiagonal
matrix with non-zero entries only on the diagonal and superdiagonal.
The size of U is M-by-N and the size of V is N-by-N.

 -- Function: int gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector
          * TAU_U, gsl_vector * TAU_V)
     This function factorizes the M-by-N matrix A into bidiagonal form
     U B V^T.  The diagonal and superdiagonal of the matrix B are
     stored in the diagonal and superdiagonal of A.  The orthogonal
     matrices U and V are stored as compressed Householder vectors in
     the remaining elements of A.  The Householder coefficients are
     stored in the vectors TAU_U and TAU_V.  The length of TAU_U must
     equal the number of elements in the diagonal of A and the length
     of TAU_V should be one element shorter.

 -- Function: int gsl_linalg_bidiag_unpack (const gsl_matrix * A, const
          gsl_vector * TAU_U, gsl_matrix * U, const gsl_vector * TAU_V,
          gsl_matrix * V, gsl_vector * DIAG, gsl_vector * SUPERDIAG)
     This function unpacks the bidiagonal decomposition of A produced by
     `gsl_linalg_bidiag_decomp', (A, TAU_U, TAU_V) into the separate
     orthogonal matrices U, V and the diagonal vector DIAG and
     superdiagonal SUPERDIAG.  Note that U is stored as a compact
     M-by-N orthogonal matrix satisfying U^T U = I for efficiency.

 -- Function: int gsl_linalg_bidiag_unpack2 (gsl_matrix * A, gsl_vector
          * TAU_U, gsl_vector * TAU_V, gsl_matrix * V)
     This function unpacks the bidiagonal decomposition of A produced by
     `gsl_linalg_bidiag_decomp', (A, TAU_U, TAU_V) into the separate
     orthogonal matrices U, V and the diagonal vector DIAG and
     superdiagonal SUPERDIAG.  The matrix U is stored in-place in A.

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


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

14.11 Householder Transformations
=================================

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

     P = I - \tau v v^T

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

 -- Function: double gsl_linalg_householder_transform (gsl_vector * V)
 -- Function: gsl_complex gsl_linalg_complex_householder_transform
          (gsl_vector_complex * V)
     This function prepares a Householder transformation P = I - \tau v
     v^T which can be used to zero all the elements of the input vector
     except the first.  On output the transformation is stored in the
     vector V and the scalar \tau is returned.

 -- Function: int gsl_linalg_householder_hm (double TAU, const
          gsl_vector * V, gsl_matrix * A)
 -- Function: int gsl_linalg_complex_householder_hm (gsl_complex TAU,
          const gsl_vector_complex * V, gsl_matrix_complex * A)
     This function applies the Householder matrix P defined by the
     scalar TAU and the vector V to the left-hand side of the matrix A.
     On output the result P A is stored in A.

 -- Function: int gsl_linalg_householder_mh (double TAU, const
          gsl_vector * V, gsl_matrix * A)
 -- Function: int gsl_linalg_complex_householder_mh (gsl_complex TAU,
          const gsl_vector_complex * V, gsl_matrix_complex * A)
     This function applies the Householder matrix P defined by the
     scalar TAU and the vector V to the right-hand side of the matrix
     A. On output the result A P is stored in A.

 -- Function: int gsl_linalg_householder_hv (double TAU, const
          gsl_vector * V, gsl_vector * W)
 -- Function: int gsl_linalg_complex_householder_hv (gsl_complex TAU,
          const gsl_vector_complex * V, gsl_vector_complex * W)
     This function applies the Householder transformation P defined by
     the scalar TAU and the vector V to the vector W.  On output the
     result P w is stored in W.


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

14.12 Householder solver for linear systems
===========================================

 -- Function: int gsl_linalg_HH_solve (gsl_matrix * A, const gsl_vector
          * B, gsl_vector * X)
     This function solves the system A x = b directly using Householder
     transformations. On output the solution is stored in X and B is
     not modified. The matrix A is destroyed by the Householder
     transformations.

 -- Function: int gsl_linalg_HH_svx (gsl_matrix * A, gsl_vector * X)
     This function solves the system A x = b in-place using Householder
     transformations.  On input X should contain the right-hand side b,
     which is replaced by the solution on output.  The matrix A is
     destroyed by the Householder transformations.


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

14.13 Tridiagonal Systems
=========================

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

 -- Function: int gsl_linalg_solve_tridiag (const gsl_vector * DIAG,
          const gsl_vector * E, const gsl_vector * F, const gsl_vector
          * B, gsl_vector * X)
     This function solves the general N-by-N system A x = b where A is
     tridiagonal (N >= 2). The super-diagonal and sub-diagonal vectors
     E and F must be one element shorter than the diagonal vector DIAG.
     The form of A for the 4-by-4 case is shown below,

          A = ( d_0 e_0  0   0  )
              ( f_0 d_1 e_1  0  )
              (  0  f_1 d_2 e_2 )
              (  0   0  f_2 d_3 )


 -- Function: int gsl_linalg_solve_symm_tridiag (const gsl_vector *
          DIAG, const gsl_vector * E, const gsl_vector * B, gsl_vector
          * X)
     This function solves the general N-by-N system A x = b where A is
     symmetric tridiagonal (N >= 2).  The off-diagonal vector E must be
     one element shorter than the diagonal vector DIAG.  The form of A
     for the 4-by-4 case is shown below,

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

 -- Function: int gsl_linalg_solve_cyc_tridiag (const gsl_vector *
          DIAG, const gsl_vector * E, const gsl_vector * F, const
          gsl_vector * B, gsl_vector * X)
     This function solves the general N-by-N system A x = b where A is
     cyclic tridiagonal (N >= 3).  The cyclic super-diagonal and
     sub-diagonal vectors E and F must have the same number of elements
     as the diagonal vector DIAG.  The form of A for the 4-by-4 case is
     shown below,

          A = ( d_0 e_0  0  f_3 )
              ( f_0 d_1 e_1  0  )
              (  0  f_1 d_2 e_2 )
              ( e_3  0  f_2 d_3 )

 -- Function: int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector *
          DIAG, const gsl_vector * E, const gsl_vector * B, gsl_vector
          * X)
     This function solves the general N-by-N system A x = b where A is
     symmetric cyclic tridiagonal (N >= 3).  The cyclic off-diagonal
     vector E must have the same number of elements as the diagonal
     vector DIAG.  The form of A for the 4-by-4 case is shown below,

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


File: gsl-ref.info,  Node: Balancing,  Next: Linear Algebra Examples,  Prev: Tridiagonal Systems,  Up: Linear Algebra

14.14 Balancing
===============

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

     A' = D^(-1) A D

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

 -- Function: int gsl_linalg_balance_matrix (gsl_matrix * A, gsl_vector
          * D)
     This function replaces the matrix A with its balanced counterpart
     and stores the diagonal elements of the similarity transformation
     into the vector D.


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

14.15 Examples
==============

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

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

and the solution is found using LU decomposition of the matrix A.

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

     int
     main (void)
     {
       double a_data[] = { 0.18, 0.60, 0.57, 0.96,
                           0.41, 0.24, 0.99, 0.58,
                           0.14, 0.30, 0.97, 0.66,
                           0.51, 0.13, 0.19, 0.85 };

       double b_data[] = { 1.0, 2.0, 3.0, 4.0 };

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

       gsl_vector_view b
         = gsl_vector_view_array (b_data, 4);

       gsl_vector *x = gsl_vector_alloc (4);

       int s;

       gsl_permutation * p = gsl_permutation_alloc (4);

       gsl_linalg_LU_decomp (&m.matrix, p, &s);

       gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);

       printf ("x = \n");
       gsl_vector_fprintf (stdout, x, "%g");

       gsl_permutation_free (p);
       gsl_vector_free (x);
       return 0;
     }

Here is the output from the program,

     x = -4.05205
     -12.6056
     1.66091
     8.69377

This can be verified by multiplying the solution x by the original
matrix A using GNU OCTAVE,

     octave> A = [ 0.18, 0.60, 0.57, 0.96;
                   0.41, 0.24, 0.99, 0.58;
                   0.14, 0.30, 0.97, 0.66;
                   0.51, 0.13, 0.19, 0.85 ];

     octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];

     octave> A * x
     ans =
       1.0000
       2.0000
       3.0000
       4.0000

This reproduces the original right-hand side vector, b, in accordance
with the equation A x = b.


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

14.16 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 (1975), p 74-76

     J.C. Nash and S. Shlien "Simple algorithms for the partial singular
     value decomposition", `Computer Journal', Volume 30 (1987), p
     268-275.

     James Demmel, Kres<imir 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

15 Eigensystems
***************

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

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

* Menu:

* Real Symmetric Matrices::
* Complex Hermitian Matrices::
* Real Nonsymmetric Matrices::
* Real Generalized Symmetric-Definite Eigensystems::
* Complex Generalized Hermitian-Definite Eigensystems::
* Real Generalized Nonsymmetric Eigensystems::
* 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

15.1 Real Symmetric Matrices
============================

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

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

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

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

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

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

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


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

15.2 Complex Hermitian Matrices
===============================

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

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

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

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

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

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

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


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

15.3 Real Nonsymmetric Matrices
===============================

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

     A = Z T Z^T

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

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

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

 -- Function: void gsl_eigen_nonsymm_params (const int COMPUTE_T, const
          int BALANCE, gsl_eigen_nonsymm_workspace * W)
     This function sets some parameters which determine how the
     eigenvalue problem is solved in subsequent calls to
     `gsl_eigen_nonsymm'.

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

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

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

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

          T = Z^(-1) A Z

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

 -- Function: int gsl_eigen_nonsymm (gsl_matrix * A, gsl_vector_complex
          * EVAL, gsl_eigen_nonsymm_workspace * W)
     This function computes the eigenvalues of the real nonsymmetric
     matrix A and stores them in the vector EVAL. If T is desired, it
     is stored in the upper portion of A on output.  Otherwise, on
     output, the diagonal of A will contain the 1-by-1 real eigenvalues
     and 2-by-2 complex conjugate eigenvalue systems, and the rest of A
     is destroyed. In rare cases, this function may fail to find all
     eigenvalues. If this happens, an error code is returned and the
     number of converged eigenvalues is stored in `w->n_evals'.  The
     converged eigenvalues are stored in the beginning of EVAL.

 -- Function: int gsl_eigen_nonsymm_Z (gsl_matrix * A,
          gsl_vector_complex * EVAL, gsl_matrix * Z,
          gsl_eigen_nonsymm_workspace * W)
     This function is identical to `gsl_eigen_nonsymm' except that it
     also computes the Schur vectors and stores them into Z.

 -- Function: gsl_eigen_nonsymmv_workspace * gsl_eigen_nonsymmv_alloc
          (const size_t N)
     This function allocates a workspace for computing eigenvalues and
     eigenvectors of N-by-N real nonsymmetric matrices. The size of the
     workspace is O(5n).

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

 -- Function: void gsl_eigen_nonsymmv_params (const int BALANCE,
          gsl_eigen_nonsymm_workspace * W)
     This function sets parameters which determine how the eigenvalue
     problem is solved in subsequent calls to `gsl_eigen_nonsymmv'.  If
     BALANCE is set to 1, a balancing transformation is applied to the
     matrix. See `gsl_eigen_nonsymm_params' for more information.
     Balancing is turned off by default since it does not preserve the
     orthogonality of the Schur vectors.

 -- Function: int gsl_eigen_nonsymmv (gsl_matrix * A,
          gsl_vector_complex * EVAL, gsl_matrix_complex * EVEC,
          gsl_eigen_nonsymmv_workspace * W)
     This function computes eigenvalues and right eigenvectors of the
     N-by-N real nonsymmetric matrix A. It first calls
     `gsl_eigen_nonsymm' to compute the eigenvalues, Schur form T, and
     Schur vectors. Then it finds eigenvectors of T and backtransforms
     them using the Schur vectors. The Schur vectors are destroyed in
     the process, but can be saved by using `gsl_eigen_nonsymmv_Z'. The
     computed eigenvectors are normalized to have unit magnitude. On
     output, the upper portion of A contains the Schur form T. If
     `gsl_eigen_nonsymm' fails, no eigenvectors are computed, and an
     error code is returned.

 -- Function: int gsl_eigen_nonsymmv_Z (gsl_matrix * A,
          gsl_vector_complex * EVAL, gsl_matrix_complex * EVEC,
          gsl_matrix * Z, gsl_eigen_nonsymmv_workspace * W)
     This function is identical to `gsl_eigen_nonsymmv' except that it
     also saves the Schur vectors into Z.


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

15.4 Real Generalized Symmetric-Definite Eigensystems
=====================================================

The real generalized symmetric-definite eigenvalue problem is to find
eigenvalues \lambda and eigenvectors x such that
     A x = \lambda B x
   where A and B are symmetric matrices, and B is positive-definite.
This problem reduces to the standard symmetric eigenvalue problem by
applying the Cholesky decomposition to B:
                           A x = \lambda B x
                           A x = \lambda L L^t x
     ( L^{-1} A L^{-t} ) L^t x = \lambda L^t x
   Therefore, the problem becomes C y = \lambda y where C = L^{-1} A
L^{-t} is symmetric, and y = L^t x. The standard symmetric eigensolver
can be applied to the matrix C.  The resulting eigenvectors are
backtransformed to find the vectors of the original problem. The
eigenvalues and eigenvectors of the generalized symmetric-definite
eigenproblem are always real.

 -- Function: gsl_eigen_gensymm_workspace * gsl_eigen_gensymm_alloc
          (const size_t N)
     This function allocates a workspace for computing eigenvalues of
     N-by-N real generalized symmetric-definite eigensystems. The size
     of the workspace is O(2n).

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

 -- Function: int gsl_eigen_gensymm (gsl_matrix * A, gsl_matrix * B,
          gsl_vector * EVAL, gsl_eigen_gensymm_workspace * W)
     This function computes the eigenvalues of the real generalized
     symmetric-definite matrix pair (A, B), and stores them in EVAL,
     using the method outlined above. On output, B contains its
     Cholesky decomposition and A is destroyed.

 -- Function: gsl_eigen_gensymmv_workspace * gsl_eigen_gensymmv_alloc
          (const size_t N)
     This function allocates a workspace for computing eigenvalues and
     eigenvectors of N-by-N real generalized symmetric-definite
     eigensystems. The size of the workspace is O(4n).

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

 -- Function: int gsl_eigen_gensymmv (gsl_matrix * A, gsl_matrix * B,
          gsl_vector * EVAL, gsl_matrix * EVEC,
          gsl_eigen_gensymmv_workspace * W)
     This function computes the eigenvalues and eigenvectors of the real
     generalized symmetric-definite matrix pair (A, B), and stores them
     in EVAL and EVEC respectively. The computed eigenvectors are
     normalized to have unit magnitude. On output, B contains its
     Cholesky decomposition and A is destroyed.


File: gsl-ref.info,  Node: Complex Generalized Hermitian-Definite Eigensystems,  Next: Real Generalized Nonsymmetric Eigensystems,  Prev: Real Generalized Symmetric-Definite Eigensystems,  Up: Eigensystems

15.5 Complex Generalized Hermitian-Definite Eigensystems
========================================================

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

 -- Function: gsl_eigen_genherm_workspace * gsl_eigen_genherm_alloc
          (const size_t N)
     This function allocates a workspace for computing eigenvalues of
     N-by-N complex generalized hermitian-definite eigensystems. The
     size of the workspace is O(3n).

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

 -- Function: int gsl_eigen_genherm (gsl_matrix_complex * A,
          gsl_matrix_complex * B, gsl_vector * EVAL,
          gsl_eigen_genherm_workspace * W)
     This function computes the eigenvalues of the complex generalized
     hermitian-definite matrix pair (A, B), and stores them in EVAL,
     using the method outlined above. On output, B contains its
     Cholesky decomposition and A is destroyed.

 -- Function: gsl_eigen_genhermv_workspace * gsl_eigen_genhermv_alloc
          (const size_t N)
     This function allocates a workspace for computing eigenvalues and
     eigenvectors of N-by-N complex generalized hermitian-definite
     eigensystems. The size of the workspace is O(5n).

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

 -- Function: int gsl_eigen_genhermv (gsl_matrix_complex * A,
          gsl_matrix_complex * B, gsl_vector * EVAL, gsl_matrix_complex
          * EVEC, gsl_eigen_genhermv_workspace * W)
     This function computes the eigenvalues and eigenvectors of the
     complex generalized hermitian-definite matrix pair (A, B), and
     stores them in EVAL and EVEC respectively. The computed
     eigenvectors are normalized to have unit magnitude. On output, B
     contains its Cholesky decomposition and A is destroyed.


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

15.6 Real Generalized Nonsymmetric Eigensystems
===============================================

Given two square matrices (A, B), the generalized nonsymmetric
eigenvalue problem is to find eigenvalues \lambda and eigenvectors x
such that
     A x = \lambda B x
   We may also define the problem as finding eigenvalues \mu and
eigenvectors y such that
     \mu A y = B y
   Note that these two problems are equivalent (with \lambda = 1/\mu)
if neither \lambda nor \mu is zero. If say, \lambda is zero, then it is
still a well defined eigenproblem, but its alternate problem involving
\mu is not. Therefore, to allow for zero (and infinite) eigenvalues,
the problem which is actually solved is
     \beta A x = \alpha B x
   The eigensolver routines below will return two values \alpha and
\beta and leave it to the user to perform the divisions \lambda =
\alpha / \beta and \mu = \beta / \alpha.

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

   The solution of the real generalized nonsymmetric eigensystem
problem for a matrix pair (A, B) involves computing the generalized
Schur decomposition
     A = Q S Z^T
     B = Q T Z^T
   where Q and Z are orthogonal matrices of left and right Schur
vectors respectively, and (S, T) is the generalized Schur form whose
diagonal elements give the \alpha and \beta values. The algorithm used
is the QZ method due to Moler and Stewart (see references).

 -- Function: gsl_eigen_gen_workspace * gsl_eigen_gen_alloc (const
          size_t N)
     This function allocates a workspace for computing eigenvalues of
     N-by-N real generalized nonsymmetric eigensystems. The size of the
     workspace is O(n).

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

 -- Function: void gsl_eigen_gen_params (const int COMPUTE_S, const int
          COMPUTE_T, const int BALANCE, gsl_eigen_gen_workspace * W)
     This function sets some parameters which determine how the
     eigenvalue problem is solved in subsequent calls to
     `gsl_eigen_gen'.

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

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

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

 -- Function: int gsl_eigen_gen (gsl_matrix * A, gsl_matrix * B,
          gsl_vector_complex * ALPHA, gsl_vector * BETA,
          gsl_eigen_gen_workspace * W)
     This function computes the eigenvalues of the real generalized
     nonsymmetric matrix pair (A, B), and stores them as pairs in
     (ALPHA, BETA), where ALPHA is complex and BETA is real. If \beta_i
     is non-zero, then \lambda = \alpha_i / \beta_i is an eigenvalue.
     Likewise, if \alpha_i is non-zero, then \mu = \beta_i / \alpha_i
     is an eigenvalue of the alternate problem \mu A y = B y. The
     elements of BETA are normalized to be non-negative.

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

 -- Function: int gsl_eigen_gen_QZ (gsl_matrix * A, gsl_matrix * B,
          gsl_vector_complex * ALPHA, gsl_vector * BETA, gsl_matrix *
          Q, gsl_matrix * Z, gsl_eigen_gen_workspace * W)
     This function is identical to `gsl_eigen_gen' except that it also
     computes the left and right Schur vectors and stores them into Q
     and Z respectively.

 -- Function: gsl_eigen_genv_workspace * gsl_eigen_genv_alloc (const
          size_t N)
     This function allocates a workspace for computing eigenvalues and
     eigenvectors of N-by-N real generalized nonsymmetric eigensystems.
     The size of the workspace is O(7n).

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

 -- Function: int gsl_eigen_genv (gsl_matrix * A, gsl_matrix * B,
          gsl_vector_complex * ALPHA, gsl_vector * BETA,
          gsl_matrix_complex * EVEC, gsl_eigen_genv_workspace * W)
     This function computes eigenvalues and right eigenvectors of the
     N-by-N real generalized nonsymmetric matrix pair (A, B). The
     eigenvalues are stored in (ALPHA, BETA) and the eigenvectors are
     stored in EVEC. It first calls `gsl_eigen_gen' to compute the
     eigenvalues, Schur forms, and Schur vectors. Then it finds
     eigenvectors of the Schur forms and backtransforms them using the
     Schur vectors. The Schur vectors are destroyed in the process, but
     can be saved by using `gsl_eigen_genv_QZ'. The computed
     eigenvectors are normalized to have unit magnitude. On output, (A,
     B) contains the generalized Schur form (S, T). If `gsl_eigen_gen'
     fails, no eigenvectors are computed, and an error code is returned.

 -- Function: int gsl_eigen_genv_QZ (gsl_matrix * A, gsl_matrix * B,
          gsl_vector_complex * ALPHA, gsl_vector * BETA,
          gsl_matrix_complex * EVEC, gsl_matrix * Q, gsl_matrix * Z,
          gsl_eigen_genv_workspace * W)
     This function is identical to `gsl_eigen_genv' except that it also
     computes the left and right Schur vectors and stores them into Q
     and Z respectively.


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

15.7 Sorting Eigenvalues and Eigenvectors
=========================================

 -- Function: int gsl_eigen_symmv_sort (gsl_vector * EVAL, gsl_matrix *
          EVEC, gsl_eigen_sort_t SORT_TYPE)
     This function simultaneously sorts the eigenvalues stored in the
     vector EVAL and the corresponding real eigenvectors stored in the
     columns of the matrix EVEC into ascending or descending order
     according to the value of the parameter SORT_TYPE,

    `GSL_EIGEN_SORT_VAL_ASC'
          ascending order in numerical value

    `GSL_EIGEN_SORT_VAL_DESC'
          descending order in numerical value

    `GSL_EIGEN_SORT_ABS_ASC'
          ascending order in magnitude

    `GSL_EIGEN_SORT_ABS_DESC'
          descending order in magnitude


 -- Function: int gsl_eigen_hermv_sort (gsl_vector * EVAL,
          gsl_matrix_complex * EVEC, gsl_eigen_sort_t SORT_TYPE)
     This function simultaneously sorts the eigenvalues stored in the
     vector EVAL and the corresponding complex eigenvectors stored in
     the columns of the matrix EVEC into ascending or descending order
     according to the value of the parameter SORT_TYPE as shown above.

 -- Function: int gsl_eigen_nonsymmv_sort (gsl_vector_complex * EVAL,
          gsl_matrix_complex * EVEC, gsl_eigen_sort_t SORT_TYPE)
     This function simultaneously sorts the eigenvalues stored in the
     vector EVAL and the corresponding complex eigenvectors stored in
     the columns of the matrix EVEC into ascending or descending order
     according to the value of the parameter SORT_TYPE as shown above.
     Only `GSL_EIGEN_SORT_ABS_ASC' and `GSL_EIGEN_SORT_ABS_DESC' are
     supported due to the eigenvalues being complex.

 -- Function: int gsl_eigen_gensymmv_sort (gsl_vector * EVAL,
          gsl_matrix * EVEC, gsl_eigen_sort_t SORT_TYPE)
     This function simultaneously sorts the eigenvalues stored in the
     vector EVAL and the corresponding real eigenvectors stored in the
     columns of the matrix EVEC into ascending or descending order
     according to the value of the parameter SORT_TYPE as shown above.

 -- Function: int gsl_eigen_genhermv_sort (gsl_vector * EVAL,
          gsl_matrix_complex * EVEC, gsl_eigen_sort_t SORT_TYPE)
     This function simultaneously sorts the eigenvalues stored in the
     vector EVAL and the corresponding complex eigenvectors stored in
     the columns of the matrix EVEC into ascending or descending order
     according to the value of the parameter SORT_TYPE as shown above.

 -- Function: int gsl_eigen_genv_sort (gsl_vector_complex * ALPHA,
          gsl_vector * BETA, gsl_matrix_complex * EVEC,
          gsl_eigen_sort_t SORT_TYPE)
     This function simultaneously sorts the eigenvalues stored in the
     vectors (ALPHA, BETA) and the corresponding complex eigenvectors
     stored in the columns of the matrix EVEC into ascending or
     descending order according to the value of the parameter SORT_TYPE
     as shown above. Only `GSL_EIGEN_SORT_ABS_ASC' and
     `GSL_EIGEN_SORT_ABS_DESC' are supported due to the eigenvalues
     being complex.


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

15.8 Examples
=============

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

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

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

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

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

       gsl_eigen_symmv_workspace * w =
         gsl_eigen_symmv_alloc (4);

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

       gsl_eigen_symmv_free (w);

       gsl_eigen_symmv_sort (eval, evec,
                             GSL_EIGEN_SORT_ABS_ASC);

       {
         int i;

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

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

       gsl_vector_free (eval);
       gsl_matrix_free (evec);

       return 0;
     }

Here is the beginning of the output from the program,

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

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

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

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

     octave> v
     v =

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

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

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

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

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

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

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

       gsl_eigen_nonsymmv_workspace * w =
         gsl_eigen_nonsymmv_alloc (4);

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

       gsl_eigen_nonsymmv_free (w);

       gsl_eigen_nonsymmv_sort (eval, evec,
                                GSL_EIGEN_SORT_ABS_DESC);

       {
         int i, j;

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

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

       gsl_vector_complex_free(eval);
       gsl_matrix_complex_free(evec);

       return 0;
     }

Here is the beginning of the output from the program,

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

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

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

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

     octave> v
     v =

      Columns 1 through 3:

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

      Column 4:

       -0.14493 + 0.00000i
        0.35660 + 0.00000i
        0.91937 + 0.00000i
        0.08118 + 0.00000i
   Note that the eigenvectors corresponding to the eigenvalue 5.54555 +
3.08545i differ by the multiplicative constant 0.9999984 + 0.0017674i
which is an arbitrary phase factor of magnitude 1.


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

15.9 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.

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

     C. Moler, G. Stewart, "An Algorithm for Generalized Matrix
     Eigenvalue Problems", SIAM J. Numer. Anal., Vol 10, No 2, 1973.

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

16 Fast Fourier Transforms (FFTs)
*********************************

This chapter describes functions for performing Fast Fourier Transforms
(FFTs).  The library includes radix-2 routines (for lengths which are a
power of two) and mixed-radix routines (which work for any length).  For
efficiency there are separate versions of the routines for real data and
for complex data.  The mixed-radix routines are a reimplementation of
the FFTPACK library of Paul Swarztrauber.  Fortran code for FFTPACK is
available on Netlib (FFTPACK also includes some routines for sine and
cosine transforms but these are currently not available in GSL).  For
details and derivations of the underlying algorithms consult the
document `GSL FFT Algorithms' (*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

16.1 Mathematical Definitions
=============================

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

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

   The DFT usually arises as an approximation to the continuous Fourier
transform when functions are sampled at discrete intervals in space or
time.  The naive evaluation of the discrete Fourier transform is a
matrix-vector multiplication W\vec{z}. A general matrix-vector
multiplication takes O(n^2) operations for n data-points.  Fast Fourier
transform algorithms use a divide-and-conquer strategy to factorize the
matrix W into smaller sub-matrices, corresponding to the integer
factors of the length n.  If n can be factorized into a product of
integers f_1 f_2 ... f_m then the DFT can be computed in O(n \sum f_i)
operations.  For a radix-2 FFT this gives an operation count of O(n
\log_2 n).

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

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

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

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

The factor of 1/n makes this a true inverse.  For example, a call to
`gsl_fft_complex_forward' followed by a call to
`gsl_fft_complex_inverse' should return the original data (within
numerical errors).

   In general there are two possible choices for the sign of the
exponential in the transform/ inverse-transform pair. GSL follows the
same convention as FFTPACK, using a negative exponential for the forward
transform.  The advantage of this convention is that the inverse
transform recreates the original function with simple Fourier
synthesis.  Numerical Recipes uses the opposite convention, a positive
exponential in the forward transform.

   The "backwards FFT" is simply our terminology for an unscaled
version of the inverse FFT,

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

When the overall scale of the result is unimportant it is often
convenient to use the backwards FFT instead of the inverse to save
unnecessary divisions.


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

16.2 Overview of complex data FFTs
==================================

The inputs and outputs for the complex FFT routines are "packed arrays"
of floating point numbers.  In a packed array the real and imaginary
parts of each complex number are placed in alternate neighboring
elements.  For example, the following definition of a packed array of
length 6,

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

can be used to hold an array of three complex numbers, `z[3]', in the
following way,

     data[0] = Re(z[0])
     data[1] = Im(z[0])
     data[2] = Re(z[1])
     data[3] = Im(z[1])
     data[4] = Re(z[2])
     data[5] = Im(z[2])

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

   A "stride" parameter allows the user to perform transforms on the
elements `z[stride*i]' instead of `z[i]'.  A stride greater than 1 can
be used to take an in-place FFT of the column of a matrix. A stride of
1 accesses the array without any additional spacing between elements.

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

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

   For physical applications it is important to remember that the index
appearing in the DFT does not correspond directly to a physical
frequency.  If the time-step of the DFT is \Delta then the
frequency-domain includes both positive and negative frequencies,
ranging from -1/(2\Delta) through 0 to +1/(2\Delta).  The positive
frequencies are stored from the beginning of the array up to the
middle, and the negative frequencies are stored backwards from the end
of the array.

   Here is a table which shows the layout of the array DATA, and the
correspondence between the time-domain data z, and the frequency-domain
data x.

     index    z               x = FFT(z)

     0        z(t = 0)        x(f = 0)
     1        z(t = 1)        x(f = 1/(n Delta))
     2        z(t = 2)        x(f = 2/(n Delta))
     .        ........        ..................
     n/2      z(t = n/2)      x(f = +1/(2 Delta),
                                    -1/(2 Delta))
     .        ........        ..................
     n-3      z(t = n-3)      x(f = -3/(n Delta))
     n-2      z(t = n-2)      x(f = -2/(n Delta))
     n-1      z(t = n-1)      x(f = -1/(n Delta))

When n is even the location n/2 contains the most positive and negative
frequencies (+1/(2 \Delta), -1/(2 \Delta)) which are equivalent.  If n
is odd then general structure of the table above still applies, but n/2
does not appear.


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

16.3 Radix-2 FFT routines for complex data
==========================================

The radix-2 algorithms described in this section are simple and compact,
although not necessarily the most efficient.  They use the Cooley-Tukey
algorithm to compute in-place complex FFTs for lengths which are a power
of 2--no additional storage is required.  The corresponding
self-sorting mixed-radix routines offer better performance at the
expense of requiring additional working space.

   All the functions described in this section are declared in the
header file `gsl_fft_complex.h'.

 -- Function: int gsl_fft_complex_radix2_forward
          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
 -- Function: int gsl_fft_complex_radix2_transform
          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N,
          gsl_fft_direction SIGN)
 -- Function: int gsl_fft_complex_radix2_backward
          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
 -- Function: int gsl_fft_complex_radix2_inverse
          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
     These functions compute forward, backward and inverse FFTs of
     length N with stride STRIDE, on the packed complex array DATA
     using an in-place radix-2 decimation-in-time algorithm.  The
     length of the transform N is restricted to powers of two.  For the
     `transform' version of the function the SIGN argument can be
     either `forward' (-1) or `backward' (+1).

     The functions return a value of `GSL_SUCCESS' if no errors were
     detected, or `GSL_EDOM' if the length of the data N is not a power
     of two.

 -- Function: int gsl_fft_complex_radix2_dif_forward
          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
 -- Function: int gsl_fft_complex_radix2_dif_transform
          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N,
          gsl_fft_direction SIGN)
 -- Function: int gsl_fft_complex_radix2_dif_backward
          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
 -- Function: int gsl_fft_complex_radix2_dif_inverse
          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
     These are decimation-in-frequency versions of the radix-2 FFT
     functions.


   Here is an example program which computes the FFT of a short pulse
in a sample of length 128.  To make the resulting Fourier transform
real the pulse is defined for equal positive and negative times (-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

16.4 Mixed-radix FFT routines for complex data
==============================================

This section describes mixed-radix FFT algorithms for complex data.  The
mixed-radix functions work for FFTs of any length.  They are a
reimplementation of Paul Swarztrauber's Fortran FFTPACK library.  The
theory is explained in the review article `Self-sorting Mixed-radix
FFTs' by Clive Temperton.  The routines here use the same indexing
scheme and basic algorithms as FFTPACK.

   The mixed-radix algorithm is based on sub-transform modules--highly
optimized small length FFTs which are combined to create larger FFTs.
There are efficient modules for factors of 2, 3, 4, 5, 6 and 7.  The
modules for the composite factors of 4 and 6 are faster than combining
the modules for 2*2 and 2*3.

   For factors which are not implemented as modules there is a
fall-back to a general length-n module which uses Singleton's method for
efficiently computing a DFT. This module is O(n^2), and slower than a
dedicated module would be but works for any length n.  Of course,
lengths which use the general length-n module will still be factorized
as much as possible.  For example, a length of 143 will be factorized
into 11*13.  Large prime factors are the worst case scenario, e.g. as
found in n=2*3*99991, and should be avoided because their O(n^2)
scaling will dominate the run-time (consult the document `GSL FFT
Algorithms' included in the GSL distribution if you encounter this
problem).

   The mixed-radix initialization function
`gsl_fft_complex_wavetable_alloc' returns the list of factors chosen by
the library for a given length n.  It can be used to check how well the
length has been factorized, and estimate the run-time.  To a first
approximation the run-time scales as n \sum f_i, where the f_i are the
factors of n.  For programs under user control you may wish to issue a
warning that the transform will be slow when the length is poorly
factorized.  If you frequently encounter data lengths which cannot be
factorized using the existing small-prime modules consult `GSL FFT
Algorithms' for details on adding support for other factors.

   All the functions described in this section are declared in the
header file `gsl_fft_complex.h'.

 -- Function: gsl_fft_complex_wavetable *
gsl_fft_complex_wavetable_alloc (size_t N)
     This function prepares a trigonometric lookup table for a complex
     FFT of length N. The function returns a pointer to the newly
     allocated `gsl_fft_complex_wavetable' if no errors were detected,
     and a null pointer in the case of error.  The length N is
     factorized into a product of subtransforms, and the factors and
     their trigonometric coefficients are stored in the wavetable. The
     trigonometric coefficients are computed using direct calls to
     `sin' and `cos', for accuracy.  Recursion relations could be used
     to compute the lookup table faster, but if an application performs
     many FFTs of the same length then this computation is a one-off
     overhead which does not affect the final throughput.

     The wavetable structure can be used repeatedly for any transform
     of the same length.  The table is not modified by calls to any of
     the other FFT functions.  The same wavetable can be used for both
     forward and backward (or inverse) transforms of a given length.

 -- Function: void gsl_fft_complex_wavetable_free
          (gsl_fft_complex_wavetable * WAVETABLE)
     This function frees the memory associated with the wavetable
     WAVETABLE.  The wavetable can be freed if no further FFTs of the
     same length will be needed.

These functions operate on a `gsl_fft_complex_wavetable' structure
which contains internal parameters for the FFT.  It is not necessary to
set any of the components directly but it can sometimes be useful to
examine them.  For example, the chosen factorization of the FFT length
is given and can be used to provide an estimate of the run-time or
numerical error. The wavetable structure is declared in the header file
`gsl_fft_complex.h'.

 -- Data Type: gsl_fft_complex_wavetable
     This is a structure that holds the factorization and trigonometric
     lookup tables for the mixed radix fft algorithm.  It has the
     following components:

    `size_t n'
          This is the number of complex data points

    `size_t nf'
          This is the number of factors that the length `n' was
          decomposed into.

    `size_t factor[64]'
          This is the array of factors.  Only the first `nf' elements
          are used.

    `gsl_complex * trig'
          This is a pointer to a preallocated trigonometric lookup
          table of `n' complex elements.

    `gsl_complex * twiddle[64]'
          This is an array of pointers into `trig', giving the twiddle
          factors for each pass.

The mixed radix algorithms require additional working space to hold the
intermediate steps of the transform.

 -- Function: gsl_fft_complex_workspace *
gsl_fft_complex_workspace_alloc (size_t N)
     This function allocates a workspace for a complex transform of
     length N.

 -- Function: void gsl_fft_complex_workspace_free
          (gsl_fft_complex_workspace * WORKSPACE)
     This function frees the memory associated with the workspace
     WORKSPACE. The workspace can be freed if no further FFTs of the
     same length will be needed.

The following functions compute the transform,

 -- Function: int gsl_fft_complex_forward (gsl_complex_packed_array
          DATA, size_t STRIDE, size_t N, const
          gsl_fft_complex_wavetable * WAVETABLE,
          gsl_fft_complex_workspace * WORK)
 -- Function: int gsl_fft_complex_transform (gsl_complex_packed_array
          DATA, size_t STRIDE, size_t N, const
          gsl_fft_complex_wavetable * WAVETABLE,
          gsl_fft_complex_workspace * WORK, gsl_fft_direction SIGN)
 -- Function: int gsl_fft_complex_backward (gsl_complex_packed_array
          DATA, size_t STRIDE, size_t N, const
          gsl_fft_complex_wavetable * WAVETABLE,
          gsl_fft_complex_workspace * WORK)
 -- Function: int gsl_fft_complex_inverse (gsl_complex_packed_array
          DATA, size_t STRIDE, size_t N, const
          gsl_fft_complex_wavetable * WAVETABLE,
          gsl_fft_complex_workspace * WORK)
     These functions compute forward, backward and inverse FFTs of
     length N with stride STRIDE, on the packed complex array DATA,
     using a mixed radix decimation-in-frequency algorithm.  There is
     no restriction on the length N.  Efficient modules are provided
     for subtransforms of length 2, 3, 4, 5, 6 and 7.  Any remaining
     factors are computed with a slow, O(n^2), general-n module. The
     caller must supply a WAVETABLE containing the trigonometric lookup
     tables and a workspace WORK.  For the `transform' version of the
     function the SIGN argument can be either `forward' (-1) or
     `backward' (+1).

     The functions return a value of `0' if no errors were detected. The
     following `gsl_errno' conditions are defined for these functions:

    `GSL_EDOM'
          The length of the data N is not a positive integer (i.e. N is
          zero).

    `GSL_EINVAL'
          The length of the data N and the length used to compute the
          given WAVETABLE do not match.

   Here is an example program which computes the FFT of a short pulse
in a sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.

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

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

     int
     main (void)
     {
       int i;
       const int n = 630;
       double data[2*n];

       gsl_fft_complex_wavetable * wavetable;
       gsl_fft_complex_workspace * workspace;

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

       data[0] = 1.0;

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

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

       wavetable = gsl_fft_complex_wavetable_alloc (n);
       workspace = gsl_fft_complex_workspace_alloc (n);

       for (i = 0; i < wavetable->nf; i++)
         {
            printf ("# factor %d: %d\n", i,
                    wavetable->factor[i]);
         }

       gsl_fft_complex_forward (data, 1, n,
                                wavetable, workspace);

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

       gsl_fft_complex_wavetable_free (wavetable);
       gsl_fft_complex_workspace_free (workspace);
       return 0;
     }

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


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

16.5 Overview of real data FFTs
===============================

The functions for real data are similar to those for complex data.
However, there is an important difference between forward and inverse
transforms.  The Fourier transform of a real sequence is not real.  It
is a complex sequence with a special symmetry:

     z_k = z_{n-k}^*

A sequence with this symmetry is called "conjugate-complex" or
"half-complex".  This different structure requires different storage
layouts for the forward transform (from real to half-complex) and
inverse transform (from half-complex back to real).  As a consequence
the routines are divided into two sets: functions in `gsl_fft_real'
which operate on real sequences and functions in `gsl_fft_halfcomplex'
which operate on half-complex sequences.

   Functions in `gsl_fft_real' compute the frequency coefficients of a
real sequence.  The half-complex coefficients c of a real sequence x
are given by Fourier analysis,

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

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

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

The symmetry of the half-complex sequence implies that only half of the
complex numbers in the output need to be stored.  The remaining half can
be reconstructed using the half-complex symmetry condition. This works
for all lengths, even and odd--when the length is even the middle value
where k=n/2 is also real.  Thus only N real numbers are required to
store the half-complex sequence, and the transform of a real sequence
can be stored in the same size array as the original data.

   The precise storage arrangements depend on the algorithm, and are
different for radix-2 and mixed-radix routines.  The radix-2 function
operates in-place, which constrains the locations where each element can
be stored.  The restriction forces real and imaginary parts to be stored
far apart.  The mixed-radix algorithm does not have this restriction,
and it stores the real and imaginary parts of a given term in
neighboring locations (which is desirable for better locality of memory
accesses).


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

16.6 Radix-2 FFT routines for real data
=======================================

This section describes radix-2 FFT algorithms for real data.  They use
the Cooley-Tukey algorithm to compute in-place FFTs for lengths which
are a power of 2.

   The radix-2 FFT functions for real data are declared in the header
files `gsl_fft_real.h'

 -- Function: int gsl_fft_real_radix2_transform (double DATA[], size_t
          STRIDE, size_t N)
     This function computes an in-place radix-2 FFT of length N and
     stride STRIDE on the real array DATA.  The output is a
     half-complex sequence, which is stored in-place.  The arrangement
     of the half-complex terms uses the following scheme: for k < n/2
     the real part of the k-th term is stored in location k, and the
     corresponding imaginary part is stored in location n-k.  Terms
     with k > n/2 can be reconstructed using the symmetry z_k =
     z^*_{n-k}.  The terms for k=0 and k=n/2 are both purely real, and
     count as a special case.  Their real parts are stored in locations
     0 and n/2 respectively, while their imaginary parts which are zero
     are not stored.

     The following table shows the correspondence between the output
     DATA and the equivalent results obtained by considering the input
     data as a complex sequence with zero imaginary part (assuming
     STRIDE=1),

          complex[0].real    =    data[0]
          complex[0].imag    =    0
          complex[1].real    =    data[1]
          complex[1].imag    =    data[n-1]
          ...............         ................
          complex[k].real    =    data[k]
          complex[k].imag    =    data[n-k]
          ...............         ................
          complex[n/2].real  =    data[n/2]
          complex[n/2].imag  =    0
          ...............         ................
          complex[k'].real   =    data[k]        k' = n - k
          complex[k'].imag   =   -data[n-k]
          ...............         ................
          complex[n-1].real  =    data[1]
          complex[n-1].imag  =   -data[n-1]
     Note that the output data can be converted into the full complex
     sequence using the function `gsl_fft_halfcomplex_radix2_unpack'
     described below.

   The radix-2 FFT functions for halfcomplex data are declared in the
header file `gsl_fft_halfcomplex.h'.

 -- Function: int gsl_fft_halfcomplex_radix2_inverse (double DATA[],
          size_t STRIDE, size_t N)
 -- Function: int gsl_fft_halfcomplex_radix2_backward (double DATA[],
          size_t STRIDE, size_t N)
     These functions compute the inverse or backwards in-place radix-2
     FFT of length N and stride STRIDE on the half-complex sequence
     DATA stored according the output scheme used by
     `gsl_fft_real_radix2'.  The result is a real array stored in
     natural order.

 -- Function: int gsl_fft_halfcomplex_radix2_unpack (const double
          HALFCOMPLEX_COEFFICIENT[], gsl_complex_packed_array
          COMPLEX_COEFFICIENT, size_t STRIDE, size_t N)
     This function converts HALFCOMPLEX_COEFFICIENT, an array of
     half-complex coefficients as returned by
     `gsl_fft_real_radix2_transform', into an ordinary complex array,
     COMPLEX_COEFFICIENT.  It fills in the complex array using the
     symmetry z_k = z_{n-k}^* to reconstruct the redundant elements.
     The algorithm for the conversion is,

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

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

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


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

16.7 Mixed-radix FFT routines for real data
===========================================

This section describes mixed-radix FFT algorithms for real data.  The
mixed-radix functions work for FFTs of any length.  They are a
reimplementation of the real-FFT routines in the Fortran FFTPACK library
by Paul Swarztrauber.  The theory behind the algorithm is explained in
the article `Fast Mixed-Radix Real Fourier Transforms' by Clive
Temperton.  The routines here use the same indexing scheme and basic
algorithms as FFTPACK.

   The functions use the FFTPACK storage convention for half-complex
sequences.  In this convention the half-complex transform of a real
sequence is stored with frequencies in increasing order, starting at
zero, with the real and imaginary parts of each frequency in neighboring
locations.  When a value is known to be real the imaginary part is not
stored.  The imaginary part of the zero-frequency component is never
stored.  It is known to be zero (since the zero frequency component is
simply the sum of the input data (all real)).  For a sequence of even
length the imaginary part of the frequency n/2 is not stored either,
since the symmetry z_k = z_{n-k}^* implies that this is purely real too.

   The storage scheme is best shown by some examples.  The table below
shows the output for an odd-length sequence, n=5.  The two columns give
the correspondence between the 5 values in the half-complex sequence
returned by `gsl_fft_real_transform', HALFCOMPLEX[] and the values
COMPLEX[] that would be returned if the same real input sequence were
passed to `gsl_fft_complex_backward' as a complex sequence (with
imaginary parts set to `0'),

     complex[0].real  =  halfcomplex[0]
     complex[0].imag  =  0
     complex[1].real  =  halfcomplex[1]
     complex[1].imag  =  halfcomplex[2]
     complex[2].real  =  halfcomplex[3]
     complex[2].imag  =  halfcomplex[4]
     complex[3].real  =  halfcomplex[3]
     complex[3].imag  = -halfcomplex[4]
     complex[4].real  =  halfcomplex[1]
     complex[4].imag  = -halfcomplex[2]

The upper elements of the COMPLEX array, `complex[3]' and `complex[4]'
are filled in using the symmetry condition.  The imaginary part of the
zero-frequency term `complex[0].imag' is known to be zero by the
symmetry.

   The next table shows the output for an even-length sequence, n=6.
In the even case there are two values which are purely real,

     complex[0].real  =  halfcomplex[0]
     complex[0].imag  =  0
     complex[1].real  =  halfcomplex[1]
     complex[1].imag  =  halfcomplex[2]
     complex[2].real  =  halfcomplex[3]
     complex[2].imag  =  halfcomplex[4]
     complex[3].real  =  halfcomplex[5]
     complex[3].imag  =  0
     complex[4].real  =  halfcomplex[3]
     complex[4].imag  = -halfcomplex[4]
     complex[5].real  =  halfcomplex[1]
     complex[5].imag  = -halfcomplex[2]

The upper elements of the COMPLEX array, `complex[4]' and `complex[5]'
are filled in using the symmetry condition.  Both `complex[0].imag' and
`complex[3].imag' are known to be zero.

   All these functions are declared in the header files
`gsl_fft_real.h' and `gsl_fft_halfcomplex.h'.

 -- Function: gsl_fft_real_wavetable * gsl_fft_real_wavetable_alloc
          (size_t N)
 -- Function: gsl_fft_halfcomplex_wavetable *
gsl_fft_halfcomplex_wavetable_alloc (size_t N)
     These functions prepare trigonometric lookup tables for an FFT of
     size n real elements.  The functions return a pointer to the newly
     allocated struct if no errors were detected, and a null pointer in
     the case of error.  The length N is factorized into a product of
     subtransforms, and the factors and their trigonometric
     coefficients are stored in the wavetable. The trigonometric
     coefficients are computed using direct calls to `sin' and `cos',
     for accuracy.  Recursion relations could be used to compute the
     lookup table faster, but if an application performs many FFTs of
     the same length then computing the wavetable is a one-off overhead
     which does not affect the final throughput.

     The wavetable structure can be used repeatedly for any transform
     of the same length.  The table is not modified by calls to any of
     the other FFT functions.  The appropriate type of wavetable must
     be used for forward real or inverse half-complex transforms.

 -- Function: void gsl_fft_real_wavetable_free (gsl_fft_real_wavetable
          * WAVETABLE)
 -- Function: void gsl_fft_halfcomplex_wavetable_free
          (gsl_fft_halfcomplex_wavetable * WAVETABLE)
     These functions free the memory associated with the wavetable
     WAVETABLE. The wavetable can be freed if no further FFTs of the
     same length will be needed.

The mixed radix algorithms require additional working space to hold the
intermediate steps of the transform,

 -- Function: gsl_fft_real_workspace * gsl_fft_real_workspace_alloc
          (size_t N)
     This function allocates a workspace for a real transform of length
     N.  The same workspace can be used for both forward real and
     inverse halfcomplex transforms.

 -- Function: void gsl_fft_real_workspace_free (gsl_fft_real_workspace
          * WORKSPACE)
     This function frees the memory associated with the workspace
     WORKSPACE. The workspace can be freed if no further FFTs of the
     same length will be needed.

The following functions compute the transforms of real and half-complex
data,

 -- Function: int gsl_fft_real_transform (double DATA[], size_t STRIDE,
          size_t N, const gsl_fft_real_wavetable * WAVETABLE,
          gsl_fft_real_workspace * WORK)
 -- Function: int gsl_fft_halfcomplex_transform (double DATA[], size_t
          STRIDE, size_t N, const gsl_fft_halfcomplex_wavetable *
          WAVETABLE, gsl_fft_real_workspace * WORK)
     These functions compute the FFT of DATA, a real or half-complex
     array of length N, using a mixed radix decimation-in-frequency
     algorithm.  For `gsl_fft_real_transform' DATA is an array of
     time-ordered real data.  For `gsl_fft_halfcomplex_transform' DATA
     contains Fourier coefficients in the half-complex ordering
     described above.  There is no restriction on the length N.
     Efficient modules are provided for subtransforms of length 2, 3, 4
     and 5.  Any remaining factors are computed with a slow, O(n^2),
     general-n module.  The caller must supply a WAVETABLE containing
     trigonometric lookup tables and a workspace WORK.

 -- Function: int gsl_fft_real_unpack (const double REAL_COEFFICIENT[],
          gsl_complex_packed_array COMPLEX_COEFFICIENT, size_t STRIDE,
          size_t N)
     This function converts a single real array, REAL_COEFFICIENT into
     an equivalent complex array, COMPLEX_COEFFICIENT, (with imaginary
     part set to zero), suitable for `gsl_fft_complex' routines.  The
     algorithm for the conversion is simply,

          for (i = 0; i < n; i++)
            {
              complex_coefficient[i*stride].real
                = real_coefficient[i*stride];
              complex_coefficient[i*stride].imag
                = 0.0;
            }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       gsl_fft_real_wavetable_free (real);

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

       hc = gsl_fft_halfcomplex_wavetable_alloc (n);

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

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

       gsl_fft_real_workspace_free (work);
       return 0;
     }


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

16.8 References and Further Reading
===================================

A good starting point for learning more about the FFT is the review
article `Fast Fourier Transforms: A Tutorial Review and A State of the
Art' by Duhamel and Vetterli,

     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

17 Numerical Integration
************************

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

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

* 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::
* CQUAD doubly-adaptive integration::
* Fixed order Gauss-Legendre integration::
* 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

17.1 Introduction
=================

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

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

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

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

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

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

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

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

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

     `Q' - quadrature routine

     `N' - non-adaptive integrator
     `A' - adaptive integrator

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

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

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

* 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

17.1.1 Integrands without weight functions
------------------------------------------

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

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

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


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

17.1.2 Integrands with weight functions
---------------------------------------

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

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


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

17.1.3 Integrands with singular weight functions
------------------------------------------------

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

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


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

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

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

 -- Function: int gsl_integration_qng (const gsl_function * F, double
          A, double B, double EPSABS, double EPSREL, double * RESULT,
          double * ABSERR, size_t * NEVAL)
     This function applies the Gauss-Kronrod 10-point, 21-point,
     43-point and 87-point integration rules in succession until an
     estimate of the integral of f over (a,b) is achieved within the
     desired absolute and relative error limits, EPSABS and EPSREL.  The
     function returns the final approximation, RESULT, an estimate of
     the absolute error, ABSERR and the number of function evaluations
     used, NEVAL.  The Gauss-Kronrod rules are designed in such a way
     that each rule uses all the results of its predecessors, in order
     to minimize the total number of function evaluations.


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

17.3 QAG adaptive integration
=============================

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

 -- Function: gsl_integration_workspace *
gsl_integration_workspace_alloc (size_t N)
     This function allocates a workspace sufficient to hold N double
     precision intervals, their integration results and error estimates.
     One workspace may be used multiple times as all necessary
     reinitialization is performed automatically by the integration
     routines.

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

 -- Function: int gsl_integration_qag (const gsl_function * F, double
          A, double B, double EPSABS, double EPSREL, size_t LIMIT, int
          KEY, gsl_integration_workspace * WORKSPACE, double * RESULT,
          double * ABSERR)
     This function applies an integration rule adaptively until an
     estimate of the integral of f over (a,b) is achieved within the
     desired absolute and relative error limits, EPSABS and EPSREL.
     The function returns the final approximation, RESULT, and an
     estimate of the absolute error, ABSERR.  The integration rule is
     determined by the value of KEY, which should be chosen from the
     following symbolic names,

          GSL_INTEG_GAUSS15  (key = 1)
          GSL_INTEG_GAUSS21  (key = 2)
          GSL_INTEG_GAUSS31  (key = 3)
          GSL_INTEG_GAUSS41  (key = 4)
          GSL_INTEG_GAUSS51  (key = 5)
          GSL_INTEG_GAUSS61  (key = 6)

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

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


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

17.4 QAGS adaptive integration with singularities
=================================================

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

 -- Function: int gsl_integration_qags (const gsl_function * F, double
          A, double B, double EPSABS, double EPSREL, size_t LIMIT,
          gsl_integration_workspace * WORKSPACE, double * RESULT,
          double * ABSERR)
     This function applies the Gauss-Kronrod 21-point integration rule
     adaptively until an estimate of the integral of f over (a,b) is
     achieved within the desired absolute and relative error limits,
     EPSABS and EPSREL.  The results are extrapolated using the
     epsilon-algorithm, which accelerates the convergence of the
     integral in the presence of discontinuities and integrable
     singularities.  The function returns the final approximation from
     the extrapolation, RESULT, and an estimate of the absolute error,
     ABSERR.  The subintervals and their results are stored in the
     memory provided by WORKSPACE.  The maximum number of subintervals
     is given by LIMIT, which may not exceed the allocated size of the
     workspace.



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

17.5 QAGP adaptive integration with known singular points
=========================================================

 -- Function: int gsl_integration_qagp (const gsl_function * F, double
          * PTS, size_t NPTS, double EPSABS, double EPSREL, size_t
          LIMIT, gsl_integration_workspace * WORKSPACE, double *
          RESULT, double * ABSERR)
     This function applies the adaptive integration algorithm QAGS
     taking account of the user-supplied locations of singular points.
     The array PTS of length NPTS should contain the endpoints of the
     integration ranges defined by the integration region and locations
     of the singularities.  For example, to integrate over the region
     (a,b) with break-points at x_1, x_2, x_3 (where a < x_1 < x_2 <
     x_3 < b) the following PTS array should be used

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

     with NPTS = 5.

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



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

17.6 QAGI adaptive integration on infinite intervals
====================================================

 -- Function: int gsl_integration_qagi (gsl_function * F, double
          EPSABS, double EPSREL, size_t LIMIT,
          gsl_integration_workspace * WORKSPACE, double * RESULT,
          double * ABSERR)
     This function computes the integral of the function F over the
     infinite interval (-\infty,+\infty).  The integral is mapped onto
     the semi-open interval (0,1] using the transformation x = (1-t)/t,

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

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

 -- Function: int gsl_integration_qagiu (gsl_function * F, double A,
          double EPSABS, double EPSREL, size_t LIMIT,
          gsl_integration_workspace * WORKSPACE, double * RESULT,
          double * ABSERR)
     This function computes the integral of the function F over the
     semi-infinite interval (a,+\infty).  The integral is mapped onto
     the semi-open interval (0,1] using the transformation x = a +
     (1-t)/t,

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

     and then integrated using the QAGS algorithm.

 -- Function: int gsl_integration_qagil (gsl_function * F, double B,
          double EPSABS, double EPSREL, size_t LIMIT,
          gsl_integration_workspace * WORKSPACE, double * RESULT,
          double * ABSERR)
     This function computes the integral of the function F over the
     semi-infinite interval (-\infty,b).  The integral is mapped onto
     the semi-open interval (0,1] using the transformation x = b -
     (1-t)/t,

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

     and then integrated using the QAGS algorithm.


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

17.7 QAWC adaptive integration for Cauchy principal values
==========================================================

 -- Function: int gsl_integration_qawc (gsl_function * F, double A,
          double B, double C, double EPSABS, double EPSREL, size_t
          LIMIT, gsl_integration_workspace * WORKSPACE, double *
          RESULT, double * ABSERR)
     This function computes the Cauchy principal value of the integral
     of f over (a,b), with a singularity at C,

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

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



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

17.8 QAWS adaptive integration for singular functions
=====================================================

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

 -- Function: gsl_integration_qaws_table *
gsl_integration_qaws_table_alloc (double ALPHA, double BETA, int MU,
          int NU)
     This function allocates space for a `gsl_integration_qaws_table'
     struct describing a singular weight function W(x) with the
     parameters (\alpha, \beta, \mu, \nu),

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

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

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

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

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

 -- Function: int gsl_integration_qaws_table_set
          (gsl_integration_qaws_table * T, double ALPHA, double BETA,
          int MU, int NU)
     This function modifies the parameters (\alpha, \beta, \mu, \nu) of
     an existing `gsl_integration_qaws_table' struct T.

 -- Function: void gsl_integration_qaws_table_free
          (gsl_integration_qaws_table * T)
     This function frees all the memory associated with the
     `gsl_integration_qaws_table' struct T.

 -- Function: int gsl_integration_qaws (gsl_function * F, const double
          A, const double B, gsl_integration_qaws_table * T, const
          double EPSABS, const double EPSREL, const size_t LIMIT,
          gsl_integration_workspace * WORKSPACE, double * RESULT,
          double * ABSERR)
     This function computes the integral of the function f(x) over the
     interval (a,b) with the singular weight function (x-a)^\alpha
     (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x).  The parameters of the
     weight function (\alpha, \beta, \mu, \nu) are taken from the table
     T.  The integral is,

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

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



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

17.9 QAWO adaptive integration for oscillatory functions
========================================================

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

 -- Function: gsl_integration_qawo_table *
gsl_integration_qawo_table_alloc (double OMEGA, double L, enum
          gsl_integration_qawo_enum SINE, size_t N)
     This function allocates space for a `gsl_integration_qawo_table'
     struct and its associated workspace describing a sine or cosine
     weight function W(x) with the parameters (\omega, L),

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

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

          GSL_INTEG_COSINE
          GSL_INTEG_SINE

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


 -- Function: int gsl_integration_qawo_table_set
          (gsl_integration_qawo_table * T, double OMEGA, double L, enum
          gsl_integration_qawo_enum SINE)
     This function changes the parameters OMEGA, L and SINE of the
     existing workspace T.

 -- Function: int gsl_integration_qawo_table_set_length
          (gsl_integration_qawo_table * T, double L)
     This function allows the length parameter L of the workspace T to
     be changed.

 -- Function: void gsl_integration_qawo_table_free
          (gsl_integration_qawo_table * T)
     This function frees all the memory associated with the workspace T.

 -- Function: int gsl_integration_qawo (gsl_function * F, const double
          A, const double EPSABS, const double EPSREL, const size_t
          LIMIT, gsl_integration_workspace * WORKSPACE,
          gsl_integration_qawo_table * WF, double * RESULT, double *
          ABSERR)
     This function uses an adaptive algorithm to compute the integral of
     f over (a,b) with the weight function \sin(\omega x) or
     \cos(\omega x) defined by the table WF,

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

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

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



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

17.10 QAWF adaptive integration for Fourier integrals
=====================================================

 -- Function: int gsl_integration_qawf (gsl_function * F, const double
          A, const double EPSABS, const size_t LIMIT,
          gsl_integration_workspace * WORKSPACE,
          gsl_integration_workspace * CYCLE_WORKSPACE,
          gsl_integration_qawo_table * WF, double * RESULT, double *
          ABSERR)
     This function attempts to compute a Fourier integral of the
     function F over the semi-infinite interval [a,+\infty).

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

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

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

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

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

          TOL_k = u_k abserr

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

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

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

     where E_k is the estimated error on the interval C_k.

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



File: gsl-ref.info,  Node: CQUAD doubly-adaptive integration,  Next: Fixed order Gauss-Legendre integration,  Prev: QAWF adaptive integration for Fourier integrals,  Up: Numerical Integration

17.11 CQUAD doubly-adaptive integration
=======================================

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

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

 -- Function: gsl_integration_cquad_workspace *
gsl_integration_cquad_workspace_alloc (size_t N)
     This function allocates a workspace sufficient to hold the data for
     N intervals. The number N is not the maximum number of intervals
     that will be evaluated. If the workspace is full, intervals with
     smaller error estimates will be discarded. A minimum of 3
     intervals is required and for most functions, a workspace of size
     100 is sufficient.

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

 -- Function: int gsl_integration_cquad (const gsl_function * F, double
          A, double B, double EPSABS, double EPSREL,
          gsl_integration_cquad_workspace * WORKSPACE, double * RESULT,
          double * ABSERR, size_t * NEVALS)
     This function computes the integral of f over (a,b) within the
     desired absolute and relative error limits, EPSABS and EPSREL
     using the CQUAD algorithm.  The function returns the final
     approximation, RESULT, an estimate of the absolute error, ABSERR,
     and the number of function evaluations required, NEVALS.

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

     The subintervals and their results are stored in the memory
     provided by WORKSPACE. If the error estimate or the number of
     function evaluations is not needed, the pointers ABSERR and NEVALS
     can be set to `NULL'.


File: gsl-ref.info,  Node: Fixed order Gauss-Legendre integration,  Next: Numerical integration error codes,  Prev: CQUAD doubly-adaptive integration,  Up: Numerical Integration

17.12 Gauss-Legendre integration
================================

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

 -- Function: gsl_integration_glfixed_table *
gsl_integration_glfixed_table_alloc (size_t N)
     This function determines the Gauss-Legendre abscissae and weights
     necessary for an n-point fixed order integration scheme.  If
     possible, high precision precomputed coefficients are used.  If
     precomputed weights are not available, lower precision
     coefficients are computed on the fly.

 -- Function: double gsl_integration_glfixed (const gsl_function * F,
          double A, double B, const gsl_integration_glfixed_table * T)
     This function applies the Gauss-Legendre integration rule
     contained in table T and returns the result.

 -- Function: int gsl_integration_glfixed_point (double A, double B,
          size_t I, double * XI, double * WI, const
          gsl_integration_glfixed_table * T)
     For I in [0, ..., t->n - 1], this function obtains the I-th
     Gauss-Legendre point XI and weight WI on the interval [A,B].  The
     points and weights are ordered by increasing point value.  A
     function f may be integrated on [A,B] by summing wi * f(xi) over I.

 -- Function: void gsl_integration_glfixed_table_free
          (gsl_integration_glfixed_table * T)
     This function frees the memory associated with the table T.


File: gsl-ref.info,  Node: Numerical integration error codes,  Next: Numerical integration examples,  Prev: Fixed order Gauss-Legendre integration,  Up: Numerical Integration

17.13 Error codes
=================

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

`GSL_EMAXITER'
     the maximum number of subdivisions was exceeded.

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

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

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


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

17.14 Examples
==============

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

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

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

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_integration.h>

     double f (double x, void * params) {
       double alpha = *(double *) params;
       double f = log(alpha*x) / sqrt(x);
       return f;
     }

     int
     main (void)
     {
       gsl_integration_workspace * w
         = gsl_integration_workspace_alloc (1000);

       double result, error;
       double expected = -4.0;
       double alpha = 1.0;

       gsl_function F;
       F.function = &f;
       F.params = &alpha;

       gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
                             w, &result, &error);

       printf ("result          = % .18f\n", result);
       printf ("exact result    = % .18f\n", expected);
       printf ("estimated error = % .18f\n", error);
       printf ("actual error    = % .18f\n", result - expected);
       printf ("intervals =  %d\n", w->size);

       gsl_integration_workspace_free (w);

       return 0;
     }

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

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

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


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

17.15 References and Further Reading
====================================

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

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

The CQUAD integration algorithm is described in the following paper:

     P. Gonnet, "Increasing the Reliability of Adaptive Quadrature Using
     Explicit Interpolants", `ACM Transactions on Mathematical
     Software', Volume 37 (2010), Issue 3, Article 26.


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

18 Random Number Generation
***************************

The library provides a large collection of random number generators
which can be accessed through a uniform interface.  Environment
variables allow you to select different generators and seeds at runtime,
so that you can easily switch between generators without needing to
recompile your program.  Each instance of a generator keeps track of its
own state, allowing the generators to be used in multi-threaded
programs.  Additional functions are available for transforming uniform
random numbers into samples from continuous or discrete probability
distributions such as the Gaussian, log-normal or Poisson distributions.

   These functions are declared in the header file `gsl_rng.h'.

* 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

18.1 General comments on random numbers
=======================================

In 1988, Park and Miller wrote a paper entitled "Random number
generators: good ones are hard to find." [Commun. ACM, 31, 1192-1201].
Fortunately, some excellent random number generators are available,
though poor ones are still in common use.  You may be happy with the
system-supplied random number generator on your computer, but you should
be aware that as computers get faster, requirements on random number
generators increase.  Nowadays, a simulation that calls a random number
generator millions of times can often finish before you can make it down
the hall to the coffee machine and back.

   A very nice review of random number generators was written by Pierre
L'Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks,
ed. (Wiley, 1997).  The chapter is available in postscript from
L'Ecuyer's ftp site (see references).  Knuth's volume on Seminumerical
Algorithms (originally published in 1968) devotes 170 pages to random
number generators, and has recently been updated in its 3rd edition
(1997).  It is brilliant, a classic.  If you don't own it, you should
stop reading right now, run to the nearest bookstore, and buy it.

   A good random number generator will satisfy both theoretical and
statistical properties.  Theoretical properties are often hard to obtain
(they require real math!), but one prefers a random number generator
with a long period, low serial correlation, and a tendency _not_ to
"fall mainly on the planes."  Statistical tests are performed with
numerical simulations.  Generally, a random number generator is used to
estimate some quantity for which the theory of probability provides an
exact answer.  Comparison to this exact answer provides a measure of
"randomness".


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

18.2 The Random Number Generator Interface
==========================================

It is important to remember that a random number generator is not a
"real" function like sine or cosine.  Unlike real functions, successive
calls to a random number generator yield different return values.  Of
course that is just what you want for a random number generator, but to
achieve this effect, the generator must keep track of some kind of
"state" variable.  Sometimes this state is just an integer (sometimes
just the value of the previously generated random number), but often it
is more complicated than that and may involve a whole array of numbers,
possibly with some indices thrown in.  To use the random number
generators, you do not need to know the details of what comprises the
state, and besides that varies from algorithm to algorithm.

   The random number generator library uses two special structs,
`gsl_rng_type' which holds static information about each type of
generator and `gsl_rng' which describes an instance of a generator
created from a given `gsl_rng_type'.

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


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

18.3 Random number generator initialization
===========================================

 -- Function: gsl_rng * gsl_rng_alloc (const gsl_rng_type * T)
     This function returns a pointer to a newly-created instance of a
     random number generator of type T.  For example, the following
     code creates an instance of the Tausworthe generator,

          gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);

     If there is insufficient memory to create the generator then the
     function returns a null pointer and the error handler is invoked
     with an error code of `GSL_ENOMEM'.

     The generator is automatically initialized with the default seed,
     `gsl_rng_default_seed'.  This is zero by default but can be changed
     either directly or by using the environment variable `GSL_RNG_SEED'
     (*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 >= 1 are supplied, then the generated streams of random
     numbers should be completely different.  If the seed S is zero
     then the standard seed from the original implementation is used
     instead.  For example, the original Fortran source code for the
     `ranlux' generator used a seed of 314159265, and so choosing S
     equal to zero reproduces this when using `gsl_rng_ranlux'.

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

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

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


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

18.4 Sampling from a random number generator
============================================

The following functions return uniformly distributed random numbers,
either as integers or double precision floating point numbers.  Inline
versions of these functions are used when `HAVE_INLINE' is defined.  To
obtain non-uniform distributions *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 be determined using the auxiliary functions
     `gsl_rng_max (r)' and `gsl_rng_min (r)'.

 -- Function: double gsl_rng_uniform (const gsl_rng * R)
     This function returns a double precision floating point number
     uniformly distributed in the range [0,1).  The range includes 0.0
     but excludes 1.0.  The value is typically obtained by dividing the
     result of `gsl_rng_get(r)' by `gsl_rng_max(r) + 1.0' in double
     precision.  Some generators compute this ratio internally so that
     they can provide floating point numbers with more than 32 bits of
     randomness (the maximum number of bits that can be portably
     represented in a single `unsigned long int').

 -- Function: double gsl_rng_uniform_pos (const gsl_rng * R)
     This function returns a positive double precision floating point
     number uniformly distributed in the range (0,1), excluding both
     0.0 and 1.0.  The number is obtained by sampling the generator
     with the algorithm of `gsl_rng_uniform' until a non-zero value is
     obtained.  You can use this function if you need to avoid a
     singularity at 0.0.

 -- Function: unsigned long int gsl_rng_uniform_int (const gsl_rng * R,
          unsigned long int N)
     This function returns a random integer from 0 to n-1 inclusive by
     scaling down and/or discarding samples from the generator R.  All
     integers in the range [0,n-1] are produced with equal probability.
     For generators with a non-zero minimum value an offset is applied
     so that zero is returned with the correct probability.

     Note that this function is designed for sampling from ranges
     smaller than the range of the underlying generator.  The parameter
     N must be less than or equal to the range of the generator R.  If
     N is larger than the range of the generator then the function
     calls the error handler with an error code of `GSL_EINVAL' and
     returns zero.

     In particular, this function is not intended for generating the
     full range of unsigned integer values [0,2^32-1]. Instead choose a
     generator with the maximal integer range and zero minimum value,
     such as `gsl_rng_ranlxd1', `gsl_rng_mt19937' or `gsl_rng_taus',
     and sample it directly using `gsl_rng_get'.  The range of each
     generator can be found using the auxiliary functions described in
     the next section.


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

18.5 Auxiliary random number generator functions
================================================

The following functions provide information about an existing
generator.  You should use them in preference to hard-coding the
generator parameters into your own code.

 -- Function: const char * gsl_rng_name (const gsl_rng * R)
     This function returns a pointer to the name of the generator.  For
     example,

          printf ("r is a '%s' generator\n",
                  gsl_rng_name (r));

     would print something like `r is a 'taus' generator'.

 -- Function: unsigned long int gsl_rng_max (const gsl_rng * R)
     `gsl_rng_max' returns the largest value that `gsl_rng_get' can
     return.

 -- Function: unsigned long int gsl_rng_min (const gsl_rng * R)
     `gsl_rng_min' returns the smallest value that `gsl_rng_get' can
     return.  Usually this value is zero.  There are some generators
     with algorithms that cannot return zero, and for these generators
     the minimum value is 1.

 -- Function: void * gsl_rng_state (const gsl_rng * R)
 -- Function: size_t gsl_rng_size (const gsl_rng * R)
     These functions return a pointer to the state of generator R and
     its size.  You can use this information to access the state
     directly.  For example, the following code will write the state of
     a generator to a stream,

          void * state = gsl_rng_state (r);
          size_t n = gsl_rng_size (r);
          fwrite (state, n, 1, stream);

 -- Function: const gsl_rng_type ** gsl_rng_types_setup (void)
     This function returns a pointer to an array of all the available
     generator types, terminated by a null pointer. The function should
     be called once at the start of the program, if needed.  The
     following code fragment shows how to iterate over the array of
     generator types to print the names of the available algorithms,

          const gsl_rng_type **t, **t0;

          t0 = gsl_rng_types_setup ();

          printf ("Available generators:\n");

          for (t = t0; *t != 0; t++)
            {
              printf ("%s\n", (*t)->name);
            }


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

18.6 Random number environment variables
========================================

The library allows you to choose a default generator and seed from the
environment variables `GSL_RNG_TYPE' and `GSL_RNG_SEED' and the
function `gsl_rng_env_setup'.  This makes it easy try out different
generators and seeds without having to recompile your program.

 -- Function: const gsl_rng_type * gsl_rng_env_setup (void)
     This function reads the environment variables `GSL_RNG_TYPE' and
     `GSL_RNG_SEED' and uses their values to set the corresponding
     library variables `gsl_rng_default' and `gsl_rng_default_seed'.
     These global variables are defined as follows,

          extern const gsl_rng_type *gsl_rng_default
          extern unsigned long int gsl_rng_default_seed

     The environment variable `GSL_RNG_TYPE' should be the name of a
     generator, such as `taus' or `mt19937'.  The environment variable
     `GSL_RNG_SEED' should contain the desired seed value.  It is
     converted to an `unsigned long int' using the C library function
     `strtoul'.

     If you don't specify a generator for `GSL_RNG_TYPE' then
     `gsl_rng_mt19937' is used as the default.  The initial value of
     `gsl_rng_default_seed' is zero.


Here is a short program which shows how to create a global generator
using the environment variables `GSL_RNG_TYPE' and `GSL_RNG_SEED',

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

     gsl_rng * r;  /* global generator */

     int
     main (void)
     {
       const gsl_rng_type * T;

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       printf ("generator type: %s\n", gsl_rng_name (r));
       printf ("seed = %lu\n", gsl_rng_default_seed);
       printf ("first value = %lu\n", gsl_rng_get (r));

       gsl_rng_free (r);
       return 0;
     }

Running the program without any environment variables uses the initial
defaults, an `mt19937' generator with a seed of 0,

     $ ./a.out
     generator type: mt19937
     seed = 0
     first value = 4293858116

By setting the two variables on the command line we can change the
default generator and the seed,

     $ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out
     GSL_RNG_TYPE=taus
     GSL_RNG_SEED=123
     generator type: taus
     seed = 123
     first value = 2720986350


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

18.7 Copying random number generator state
==========================================

The above methods do not expose the random number `state' which changes
from call to call.  It is often useful to be able to save and restore
the state.  To permit these practices, a few somewhat more advanced
functions are supplied.  These include:

 -- Function: int gsl_rng_memcpy (gsl_rng * DEST, const gsl_rng * SRC)
     This function copies the random number generator SRC into the
     pre-existing generator DEST, making DEST into an exact copy of
     SRC.  The two generators must be of the same type.

 -- Function: gsl_rng * gsl_rng_clone (const gsl_rng * R)
     This function returns a pointer to a newly created generator which
     is an exact copy of the generator R.


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

18.8 Reading and writing random number generator state
======================================================

The library provides functions for reading and writing the random
number state to a file as binary data.

 -- Function: int gsl_rng_fwrite (FILE * STREAM, const gsl_rng * R)
     This function writes the random number state of the random number
     generator R to the stream STREAM in binary format.  The return
     value is 0 for success and `GSL_EFAILED' if there was a problem
     writing to the file.  Since the data is written in the native
     binary format it may not be portable between different
     architectures.

 -- Function: int gsl_rng_fread (FILE * STREAM, gsl_rng * R)
     This function reads the random number state into the random number
     generator R from the open stream STREAM in binary format.  The
     random number generator R must be preinitialized with the correct
     random number generator type since type information is not saved.
     The return value is 0 for success and `GSL_EFAILED' if there was a
     problem reading from the file.  The data is assumed to have been
     written in the native binary format on the same architecture.


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

18.9 Random number generator algorithms
=======================================

The functions described above make no reference to the actual algorithm
used.  This is deliberate so that you can switch algorithms without
having to change any of your application source code.  The library
provides a large number of generators of different types, including
simulation quality generators, generators provided for compatibility
with other libraries and historical generators from the past.

   The following generators are recommended for use in simulation.  They
have extremely long periods, low correlation and pass most statistical
tests.  For the most reliable source of uncorrelated numbers, the
second-generation RANLUX generators have the strongest proof of
randomness.

 -- Generator: gsl_rng_mt19937
     The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
     variant of the twisted generalized feedback shift-register
     algorithm, and is known as the "Mersenne Twister" generator.  It
     has a Mersenne prime period of 2^19937 - 1 (about 10^6000) and is
     equi-distributed in 623 dimensions.  It has passed the DIEHARD
     statistical tests.  It uses 624 words of state per generator and is
     comparable in speed to the other generators.  The original
     generator used a default seed of 4357 and choosing S equal to zero
     in `gsl_rng_set' reproduces this.  Later versions switched to 5489
     as the default seed, you can choose this explicitly via
     `gsl_rng_set' instead if you require it.

     For more information see,
          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',
     in increasing order of strength.  It uses double-precision
     floating point arithmetic internally and can be significantly
     faster than the integer version of `ranlux', particularly on
     64-bit architectures.  The period of the generator is about
     10^171.  The algorithm has mathematically proven properties and
     can provide truly decorrelated numbers at a known level of
     randomness.  The higher luxury levels provide increased
     decorrelation between samples as an additional safety margin.

     Note that the range of allowed seeds for this generator is
     [0,2^31-1]. Higher seed values are wrapped modulo 2^31.

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

 -- Generator: gsl_rng_ranlux
 -- Generator: gsl_rng_ranlux389
     The `ranlux' generator is an implementation of the original
     algorithm developed by 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

18.10 Unix random number generators
===================================

The standard Unix random number generators `rand', `random' and
`rand48' are provided as part of GSL. Although these generators are
widely available individually often they aren't all available on the
same platform.  This makes it difficult to write portable code using
them and so we have included the complete set of Unix generators in GSL
for convenience.  Note that these generators don't produce high-quality
randomness and aren't suitable for work requiring accurate statistics.
However, if you won't be measuring statistical quantities and just want
to introduce some variation into your program then these generators are
quite acceptable.

 -- Generator: gsl_rng_rand
     This is the BSD `rand' generator.  Its sequence is

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

     with a = 1103515245, c = 12345 and m = 2^31.  The seed specifies
     the initial value, x_1.  The period of this generator is 2^31, and
     it uses 1 word of storage per generator.

 -- Generator: gsl_rng_random_bsd
 -- Generator: gsl_rng_random_libc5
 -- Generator: gsl_rng_random_glibc2
     These generators implement the `random' family of functions, a set
     of linear feedback shift register generators originally used in BSD
     Unix.  There are several versions of `random' in use today: the
     original BSD version (e.g. on SunOS4), a libc5 version (found on
     older GNU/Linux systems) and a glibc2 version.  Each version uses a
     different seeding procedure, and thus produces different sequences.

     The original BSD routines accepted a variable length buffer for the
     generator state, with longer buffers providing higher-quality
     randomness.  The `random' function implemented algorithms for
     buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm
     with the largest length that would fit into the user-supplied
     buffer was used.  To support these algorithms additional
     generators are available with the following names,

          gsl_rng_random8_bsd
          gsl_rng_random32_bsd
          gsl_rng_random64_bsd
          gsl_rng_random128_bsd
          gsl_rng_random256_bsd

     where the numeric suffix indicates the buffer length.  The
     original BSD `random' function used a 128-byte default buffer and
     so `gsl_rng_random_bsd' has been made equivalent to
     `gsl_rng_random128_bsd'.  Corresponding versions of the `libc5'
     and `glibc2' generators are also available, with the names
     `gsl_rng_random8_libc5', `gsl_rng_random8_glibc2', etc.

 -- Generator: gsl_rng_rand48
     This is the Unix `rand48' generator.  Its sequence is

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

     defined on 48-bit unsigned integers with a = 25214903917, c = 11
     and m = 2^48.  The seed specifies the upper 32 bits of the initial
     value, x_1, with the lower 16 bits set to `0x330E'.  The function
     `gsl_rng_get' returns the upper 32 bits from each term of the
     sequence.  This does not have a direct parallel in the original
     `rand48' functions, but forcing the result to type `long int'
     reproduces the output of `mrand48'.  The function
     `gsl_rng_uniform' uses the full 48 bits of internal state to return
     the double precision number x_n/m, which is equivalent to the
     function `drand48'.  Note that some versions of the GNU C Library
     contained a bug in `mrand48' function which caused it to produce
     different results (only the lower 16-bits of the return value were
     set).


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

18.11 Other random number generators
====================================

The generators in this section are provided for compatibility with
existing libraries.  If you are converting an existing program to use
GSL then you can select these generators to check your new
implementation against the original one, using the same random number
generator.  After verifying that your new program reproduces the
original results you can then switch to a higher-quality generator.

   Note that most of the generators in this section are based on single
linear congruence relations, which are the least sophisticated type of
generator.  In particular, linear congruences have poor properties when
used with a non-prime modulus, as several of these routines do (e.g.
with a power of two modulus, 2^31 or 2^32).  This leads to periodicity
in the least significant bits of each number, with only the higher bits
having any randomness.  Thus if you want to produce a random bitstream
it is best to avoid using the least significant bits.

 -- Generator: gsl_rng_ranf
     This is the CRAY random number generator `RANF'.  Its sequence is

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

     defined on 48-bit unsigned integers with a = 44485709377909 and m
     = 2^48.  The seed specifies the lower 32 bits of the initial value,
     x_1, with the lowest bit set to prevent the seed taking an even
     value.  The upper 16 bits of x_1 are set to 0. A consequence of
     this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc.
     produce the same sequences.

     The generator compatible with the CRAY MATHLIB routine RANF. It
     produces double precision floating point numbers which should be
     identical to those from the original RANF.

     There is a subtlety in the implementation of the seeding.  The
     initial state is reversed through one step, by multiplying by the
     modular inverse of a mod m.  This is done for compatibility with
     the original CRAY implementation.

     Note that you can only seed the generator with integers up to
     2^32, while the original CRAY implementation uses non-portable
     wide integers which can cover all 2^48 states of the generator.

     The function `gsl_rng_get' returns the upper 32 bits from each term
     of the sequence.  The function `gsl_rng_uniform' uses the full 48
     bits to return the double precision number x_n/m.

     The period of this generator is 2^46.

 -- Generator: gsl_rng_ranmar
     This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman
     and Tsang.  It is a 24-bit generator, originally designed for
     single-precision IEEE floating point numbers.  It was included in
     the CERNLIB high-energy physics library.

 -- Generator: gsl_rng_r250
     This is the shift-register generator of Kirkpatrick and Stoll.  The
     sequence is based on the recurrence

          x_n = x_{n-103} ^^ x_{n-250}

     where ^^ denotes "exclusive-or", defined on 32-bit words.  The
     period of this generator is about 2^250 and it uses 250 words of
     state per generator.

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

     For more information see,
          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_knuthran2
     This is a second-order multiple recursive generator described by
     Knuth in `Seminumerical Algorithms', 3rd Ed., page 108.  Its
     sequence is,

          x_n = (a_1 x_{n-1} + a_2 x_{n-2}) mod m

     with a_1 = 271828183, a_2 = 314159269, and m = 2^31 - 1.

 -- Generator: gsl_rng_knuthran2002
 -- Generator: gsl_rng_knuthran
     This is a second-order multiple recursive generator described by
     Knuth in `Seminumerical Algorithms', 3rd Ed., Section 3.6.  Knuth
     provides its C code.  The updated routine `gsl_rng_knuthran2002'
     is from the revised 9th printing and corrects some weaknesses in
     the earlier version, which is implemented as `gsl_rng_knuthran'.

 -- Generator: gsl_rng_borosh13
 -- Generator: gsl_rng_fishman18
 -- Generator: gsl_rng_fishman20
 -- Generator: gsl_rng_lecuyer21
 -- Generator: gsl_rng_waterman14
     These multiplicative generators are taken from Knuth's
     `Seminumerical Algorithms', 3rd Ed., pages 106-108. Their sequence
     is,

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

     where the seed specifies the initial value, x_1.  The parameters a
     and m are as follows, Borosh-Niederreiter: a = 1812433253, m =
     2^32, Fishman18: a = 62089911, m = 2^31 - 1, Fishman20: a = 48271,
     m = 2^31 - 1, L'Ecuyer: a = 40692, m = 2^31 - 249, Waterman: a =
     1566083941, m = 2^32.

 -- Generator: gsl_rng_fishman2x
     This is the L'Ecuyer-Fishman random number generator. It is taken
     from Knuth's `Seminumerical Algorithms', 3rd Ed., page 108. Its
     sequence is,

          z_{n+1} = (x_n - y_n) mod m

     with m = 2^31 - 1.  x_n and y_n are given by the `fishman20' and
     `lecuyer21' algorithms.  The seed specifies the initial value, x_1.


 -- Generator: gsl_rng_coveyou
     This is the Coveyou random number generator. It is taken from
     Knuth's `Seminumerical Algorithms', 3rd Ed., Section 3.2.2. Its
     sequence is,

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

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


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

18.12 Performance
=================

The following table shows the relative performance of a selection the
available random number generators.  The fastest simulation quality
generators are `taus', `gfsr4' and `mt19937'.  The generators which
offer the best mathematically-proven quality are those based on the
RANLUX algorithm.

     1754 k ints/sec,    870 k doubles/sec, taus
     1613 k ints/sec,    855 k doubles/sec, gfsr4
     1370 k ints/sec,    769 k doubles/sec, mt19937
      565 k ints/sec,    571 k doubles/sec, ranlxs0
      400 k ints/sec,    405 k doubles/sec, ranlxs1
      490 k ints/sec,    389 k doubles/sec, mrg
      407 k ints/sec,    297 k doubles/sec, ranlux
      243 k ints/sec,    254 k doubles/sec, ranlxd1
      251 k ints/sec,    253 k doubles/sec, ranlxs2
      238 k ints/sec,    215 k doubles/sec, cmrg
      247 k ints/sec,    198 k doubles/sec, ranlux389
      141 k ints/sec,    140 k doubles/sec, ranlxd2


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

18.13 Examples
==============

The following program demonstrates the use of a random number generator
to produce uniform random numbers in the range [0.0, 1.0),

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

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       int i, n = 10;

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (i = 0; i < n; i++)
         {
           double u = gsl_rng_uniform (r);
           printf ("%.5f\n", u);
         }

       gsl_rng_free (r);

       return 0;
     }

Here is the output of the program,

     $ ./a.out
     0.99974
     0.16291
     0.28262
     0.94720
     0.23166
     0.48497
     0.95748
     0.74431
     0.54004
     0.73995

The numbers depend on the seed used by the generator.  The default seed
can be changed with the `GSL_RNG_SEED' environment variable to produce
a different stream of numbers.  The generator itself can be changed
using the environment variable `GSL_RNG_TYPE'.  Here is the output of
the program using a seed value of 123 and the multiple-recursive
generator `mrg',

     $ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out
     GSL_RNG_TYPE=mrg
     GSL_RNG_SEED=123
     0.33050
     0.86631
     0.32982
     0.67620
     0.53391
     0.06457
     0.16847
     0.70229
     0.04371
     0.86374


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

18.14 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

18.15 Acknowledgements
======================

Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for
making the source code to their generators (MT19937, MM&TN; TT800,
MM&YK) available under the GNU General Public License.  Thanks to Martin
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

19 Quasi-Random Sequences
*************************

This chapter describes functions for generating quasi-random sequences
in arbitrary dimensions.  A quasi-random sequence progressively covers a
d-dimensional space with a set of points that are uniformly
distributed.  Quasi-random sequences are also known as low-discrepancy
sequences.  The quasi-random sequence generators use an interface that
is similar to the interface for random number generators, except that
seeding is not required--each generator produces a single sequence.

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

* 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

19.1 Quasi-random number generator initialization
=================================================

 -- Function: gsl_qrng * gsl_qrng_alloc (const gsl_qrng_type * T,
          unsigned int D)
     This function returns a pointer to a newly-created instance of a
     quasi-random sequence generator of type T and dimension D.  If
     there is insufficient memory to create the generator then the
     function returns a null pointer and the error handler is invoked
     with an error code of `GSL_ENOMEM'.

 -- Function: void gsl_qrng_free (gsl_qrng * Q)
     This function frees all the memory associated with the generator Q.

 -- Function: void gsl_qrng_init (gsl_qrng * Q)
     This function reinitializes the generator Q to its starting point.
     Note that quasi-random sequences do not use a seed and always
     produce the same set of values.


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

19.2 Sampling from a quasi-random number generator
==================================================

 -- Function: int gsl_qrng_get (const gsl_qrng * Q, double X[])
     This function stores the next point from the sequence generator Q
     in the array X.  The space available for X must match the
     dimension of the generator.  The point X will lie in the range 0 <
     x_i < 1 for each x_i.  An inline version of this function is used
     when `HAVE_INLINE' is defined.


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

19.3 Auxiliary quasi-random number generator functions
======================================================

 -- Function: const char * gsl_qrng_name (const gsl_qrng * Q)
     This function returns a pointer to the name of the generator.

 -- Function: size_t gsl_qrng_size (const gsl_qrng * Q)
 -- Function: void * gsl_qrng_state (const gsl_qrng * Q)
     These functions return a pointer to the state of generator R and
     its size.  You can use this information to access the state
     directly.  For example, the following code will write the state of
     a generator to a stream,

          void * state = gsl_qrng_state (q);
          size_t n = gsl_qrng_size (q);
          fwrite (state, n, 1, stream);


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

19.4 Saving and resorting quasi-random number generator state
=============================================================

 -- Function: int gsl_qrng_memcpy (gsl_qrng * DEST, const gsl_qrng *
          SRC)
     This function copies the quasi-random sequence generator SRC into
     the pre-existing generator DEST, making DEST into an exact copy of
     SRC.  The two generators must be of the same type.

 -- Function: gsl_qrng * gsl_qrng_clone (const gsl_qrng * Q)
     This function returns a pointer to a newly created generator which
     is an exact copy of the generator Q.


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

19.5 Quasi-random number generator algorithms
=============================================

The following quasi-random sequence algorithms are available,

 -- Generator: gsl_qrng_niederreiter_2
     This generator uses the algorithm described in Bratley, Fox,
     Niederreiter, `ACM Trans. Model. Comp. Sim.' 2, 195 (1992). It is
     valid up to 12 dimensions.

 -- Generator: gsl_qrng_sobol
     This generator uses the Sobol sequence described in Antonov,
     Saleev, `USSR Comput. Maths. Math. Phys.' 19, 252 (1980). It is
     valid up to 40 dimensions.

 -- Generator: gsl_qrng_halton
 -- Generator: gsl_qrng_reversehalton
     These generators use the Halton and reverse Halton sequences
     described in J.H. Halton, `Numerische Mathematik' 2, 84-90 (1960)
     and B. Vandewoestyne and R. Cools `Computational and Applied
     Mathematics' 189, 1&2, 341-361 (2006).  They are valid up to 1229
     dimensions.


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

19.6 Examples
=============

The following program prints the first 1024 points of the 2-dimensional
Sobol sequence.

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

     int
     main (void)
     {
       int i;
       gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);

       for (i = 0; i < 1024; i++)
         {
           double v[2];
           gsl_qrng_get (q, v);
           printf ("%.5f %.5f\n", v[0], v[1]);
         }

       gsl_qrng_free (q);
       return 0;
     }

Here is the output from the program,

     $ ./a.out
     0.50000 0.50000
     0.75000 0.25000
     0.25000 0.75000
     0.37500 0.37500
     0.87500 0.87500
     0.62500 0.12500
     0.12500 0.62500
     ....

It can be seen that successive points progressively fill-in the spaces
between previous points.


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

19.7 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

20 Random Number Distributions
******************************

This chapter describes functions for generating random variates and
computing their probability distributions.  Samples from the
distributions described in this chapter can be obtained using any of the
random number generators in the library as an underlying source of
randomness.

   In the simplest cases a non-uniform distribution can be obtained
analytically from the uniform distribution of a random number generator
by applying an appropriate transformation.  This method uses one call to
the random number generator.  More complicated distributions are created
by the "acceptance-rejection" method, which compares the desired
distribution against a distribution which is similar and known
analytically.  This usually requires several samples from the generator.

   The library also provides cumulative distribution functions and
inverse cumulative distribution functions, sometimes referred to as
quantile functions.  The cumulative distribution functions and their
inverses are computed separately for the upper and lower tails of the
distribution, allowing full accuracy to be retained for small results.

   The functions for random variates and probability density functions
described in this section are declared in `gsl_randist.h'.  The
corresponding cumulative distribution functions are declared in
`gsl_cdf.h'.

   Note that the discrete random variate functions always return a
value of type `unsigned int', and on most platforms this has a maximum
value of 2^32-1 ~=~ 4.29e9. They should only be called with a safe
range of parameters (where there is a negligible probability of a
variate exceeding this limit) to prevent incorrect results due to
overflow.

* 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

20.1 Introduction
=================

Continuous random number distributions are defined by a probability
density function, p(x), such that the probability of x occurring in the
infinitesimal range x to x+dx is p dx.

   The cumulative distribution function for the lower tail P(x) is
defined by the integral,

     P(x) = \int_{-\infty}^{x} dx' p(x')

and gives the probability of a variate taking a value less than x.

   The cumulative distribution function for the upper tail Q(x) is
defined by the integral,

     Q(x) = \int_{x}^{+\infty} dx' p(x')

and gives the probability of a variate taking a value greater than x.

   The upper and lower cumulative distribution functions are related by
P(x) + Q(x) = 1 and satisfy 0 <= P(x) <= 1, 0 <= Q(x) <= 1.

   The inverse cumulative distributions, x=P^{-1}(P) and x=Q^{-1}(Q)
give the values of x which correspond to a specific value of P or Q.
They can be used to find confidence limits from probability values.

   For discrete distributions the probability of sampling the integer
value k is given by p(k), where \sum_k p(k) = 1.  The cumulative
distribution for the lower tail P(k) of a discrete distribution is
defined as,

     P(k) = \sum_{i <= k} p(i)

where the sum is over the allowed range of the distribution less than
or equal to k.

   The cumulative distribution for the upper tail of a discrete
distribution Q(k) is defined as

     Q(k) = \sum_{i > k} p(i)

giving the sum of probabilities for all values greater than k.  These
two definitions satisfy the identity P(k)+Q(k)=1.

   If the range of the distribution is 1 to n inclusive then P(n)=1,
Q(n)=0 while P(1) = p(1), Q(1)=1-p(1).


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

20.2 The Gaussian Distribution
==============================

 -- Function: double gsl_ran_gaussian (const gsl_rng * R, double SIGMA)
     This function returns a Gaussian random variate, with mean zero and
     standard deviation SIGMA.  The probability distribution for
     Gaussian random variates is,

          p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx

     for x in the range -\infty to +\infty.  Use the transformation z =
     \mu + x on the numbers returned by `gsl_ran_gaussian' to obtain a
     Gaussian distribution with mean \mu.  This function uses the
     Box-Muller algorithm which requires two calls to the random number
     generator R.

 -- Function: double gsl_ran_gaussian_pdf (double X, double SIGMA)
     This function computes the probability density p(x) at X for a
     Gaussian distribution with standard deviation SIGMA, using the
     formula given above.


 -- Function: double gsl_ran_gaussian_ziggurat (const gsl_rng * R,
          double SIGMA)
 -- Function: double gsl_ran_gaussian_ratio_method (const gsl_rng * R,
          double SIGMA)
     This function computes a Gaussian random variate using the
     alternative Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva
     ratio methods.  The Ziggurat algorithm is the fastest available
     algorithm in most cases.

 -- Function: double gsl_ran_ugaussian (const gsl_rng * R)
 -- Function: double gsl_ran_ugaussian_pdf (double X)
 -- Function: double gsl_ran_ugaussian_ratio_method (const gsl_rng * R)
     These functions compute results for the unit Gaussian
     distribution.  They are equivalent to the functions above with a
     standard deviation of one, SIGMA = 1.

 -- Function: double gsl_cdf_gaussian_P (double X, double SIGMA)
 -- Function: double gsl_cdf_gaussian_Q (double X, double SIGMA)
 -- Function: double gsl_cdf_gaussian_Pinv (double P, double SIGMA)
 -- Function: double gsl_cdf_gaussian_Qinv (double Q, double SIGMA)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the Gaussian distribution with
     standard deviation SIGMA.

 -- Function: double gsl_cdf_ugaussian_P (double X)
 -- Function: double gsl_cdf_ugaussian_Q (double X)
 -- Function: double gsl_cdf_ugaussian_Pinv (double P)
 -- Function: double gsl_cdf_ugaussian_Qinv (double Q)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the unit Gaussian distribution.


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

20.3 The Gaussian Tail Distribution
===================================

 -- Function: double gsl_ran_gaussian_tail (const gsl_rng * R, double
          A, double SIGMA)
     This function provides random variates from the upper tail of a
     Gaussian distribution with standard deviation SIGMA.  The values
     returned are larger than the lower limit A, which must be
     positive.  The method is based on Marsaglia's famous
     rectangle-wedge-tail algorithm (Ann.  Math. Stat. 32, 894-899
     (1961)), with this aspect explained in Knuth, v2, 3rd ed, p139,586
     (exercise 11).

     The probability distribution for Gaussian tail random variates is,

          p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2/(2 \sigma^2)) dx

     for x > a where N(a;\sigma) is the normalization constant,

          N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).


 -- Function: double gsl_ran_gaussian_tail_pdf (double X, double A,
          double SIGMA)
     This function computes the probability density p(x) at X for a
     Gaussian tail distribution with standard deviation SIGMA and lower
     limit A, using the formula given above.


 -- Function: double gsl_ran_ugaussian_tail (const gsl_rng * R, double
          A)
 -- Function: double gsl_ran_ugaussian_tail_pdf (double X, double A)
     These functions compute results for the tail of a unit Gaussian
     distribution.  They are equivalent to the functions above with a
     standard deviation of one, SIGMA = 1.


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

20.4 The Bivariate Gaussian Distribution
========================================

 -- Function: void gsl_ran_bivariate_gaussian (const gsl_rng * R,
          double SIGMA_X, double SIGMA_Y, double RHO, double * X,
          double * Y)
     This function generates a pair of correlated Gaussian variates,
     with mean zero, correlation coefficient RHO and standard deviations
     SIGMA_X and SIGMA_Y in the x and y directions.  The probability
     distribution for bivariate Gaussian random variates is,

          p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy

     for x,y in the range -\infty to +\infty.  The correlation
     coefficient RHO should lie between 1 and -1.

 -- Function: double gsl_ran_bivariate_gaussian_pdf (double X, double
          Y, double SIGMA_X, double SIGMA_Y, double RHO)
     This function computes the probability density p(x,y) at (X,Y) for
     a bivariate Gaussian distribution with standard deviations
     SIGMA_X, SIGMA_Y and correlation coefficient RHO, using the
     formula given above.



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

20.5 The Exponential Distribution
=================================

 -- Function: double gsl_ran_exponential (const gsl_rng * R, double MU)
     This function returns a random variate from the exponential
     distribution with mean MU. The distribution is,

          p(x) dx = {1 \over \mu} \exp(-x/\mu) dx

     for x >= 0.

 -- Function: double gsl_ran_exponential_pdf (double X, double MU)
     This function computes the probability density p(x) at X for an
     exponential distribution with mean MU, using the formula given
     above.


 -- Function: double gsl_cdf_exponential_P (double X, double MU)
 -- Function: double gsl_cdf_exponential_Q (double X, double MU)
 -- Function: double gsl_cdf_exponential_Pinv (double P, double MU)
 -- Function: double gsl_cdf_exponential_Qinv (double Q, double MU)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the exponential distribution
     with mean MU.


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

20.6 The Laplace Distribution
=============================

 -- Function: double gsl_ran_laplace (const gsl_rng * R, double A)
     This function returns a random variate from the Laplace
     distribution with width A.  The distribution is,

          p(x) dx = {1 \over 2 a}  \exp(-|x/a|) dx

     for -\infty < x < \infty.

 -- Function: double gsl_ran_laplace_pdf (double X, double A)
     This function computes the probability density p(x) at X for a
     Laplace distribution with width A, using the formula given above.


 -- Function: double gsl_cdf_laplace_P (double X, double A)
 -- Function: double gsl_cdf_laplace_Q (double X, double A)
 -- Function: double gsl_cdf_laplace_Pinv (double P, double A)
 -- Function: double gsl_cdf_laplace_Qinv (double Q, double A)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the Laplace distribution with
     width A.


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

20.7 The Exponential Power Distribution
=======================================

 -- Function: double gsl_ran_exppow (const gsl_rng * R, double A,
          double B)
     This function returns a random variate from the exponential power
     distribution with scale parameter A and exponent B.  The
     distribution is,

          p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx

     for x >= 0.  For b = 1 this reduces to the Laplace distribution.
     For b = 2 it has the same form as a Gaussian distribution, but
     with a = \sqrt{2} \sigma.

 -- Function: double gsl_ran_exppow_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for an
     exponential power distribution with scale parameter A and exponent
     B, using the formula given above.


 -- Function: double gsl_cdf_exppow_P (double X, double A, double B)
 -- Function: double gsl_cdf_exppow_Q (double X, double A, double B)
     These functions compute the cumulative distribution functions
     P(x), Q(x) for the exponential power distribution with parameters
     A and B.


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

20.8 The Cauchy Distribution
============================

 -- Function: double gsl_ran_cauchy (const gsl_rng * R, double A)
     This function returns a random variate from the Cauchy
     distribution with scale parameter A.  The probability distribution
     for Cauchy random variates is,

          p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx

     for x in the range -\infty to +\infty.  The Cauchy distribution is
     also known as the Lorentz distribution.

 -- Function: double gsl_ran_cauchy_pdf (double X, double A)
     This function computes the probability density p(x) at X for a
     Cauchy distribution with scale parameter A, using the formula
     given above.


 -- Function: double gsl_cdf_cauchy_P (double X, double A)
 -- Function: double gsl_cdf_cauchy_Q (double X, double A)
 -- Function: double gsl_cdf_cauchy_Pinv (double P, double A)
 -- Function: double gsl_cdf_cauchy_Qinv (double Q, double A)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the Cauchy distribution with
     scale parameter A.


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

20.9 The Rayleigh Distribution
==============================

 -- Function: double gsl_ran_rayleigh (const gsl_rng * R, double SIGMA)
     This function returns a random variate from the Rayleigh
     distribution with scale parameter SIGMA.  The distribution is,

          p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx

     for x > 0.

 -- Function: double gsl_ran_rayleigh_pdf (double X, double SIGMA)
     This function computes the probability density p(x) at X for a
     Rayleigh distribution with scale parameter SIGMA, using the
     formula given above.


 -- Function: double gsl_cdf_rayleigh_P (double X, double SIGMA)
 -- Function: double gsl_cdf_rayleigh_Q (double X, double SIGMA)
 -- Function: double gsl_cdf_rayleigh_Pinv (double P, double SIGMA)
 -- Function: double gsl_cdf_rayleigh_Qinv (double Q, double SIGMA)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the Rayleigh distribution with
     scale parameter SIGMA.


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

20.10 The Rayleigh Tail Distribution
====================================

 -- Function: double gsl_ran_rayleigh_tail (const gsl_rng * R, double
          A, double SIGMA)
     This function returns a random variate from the tail of the
     Rayleigh distribution with scale parameter SIGMA and a lower limit
     of A.  The distribution is,

          p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx

     for x > a.

 -- Function: double gsl_ran_rayleigh_tail_pdf (double X, double A,
          double SIGMA)
     This function computes the probability density p(x) at X for a
     Rayleigh tail distribution with scale parameter SIGMA and lower
     limit A, using the formula given above.



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

20.11 The Landau Distribution
=============================

 -- Function: double gsl_ran_landau (const gsl_rng * R)
     This function returns a random variate from the Landau
     distribution.  The probability distribution for Landau random
     variates is defined analytically by the complex integral,

          p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s)
     For numerical purposes it is more convenient to use the following
     equivalent form of the integral,

          p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).

 -- Function: double gsl_ran_landau_pdf (double X)
     This function computes the probability density p(x) at X for the
     Landau distribution using an approximation to the formula given
     above.



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

20.12 The Levy alpha-Stable Distributions
=========================================

 -- Function: double gsl_ran_levy (const gsl_rng * R, double C, double
          ALPHA)
     This function returns a random variate from the Levy symmetric
     stable distribution with scale C and exponent ALPHA.  The symmetric
     stable probability distribution is defined by a Fourier transform,

          p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)

     There is no explicit solution for the form of p(x) and the library
     does not define a corresponding `pdf' function.  For \alpha = 1
     the distribution reduces to the Cauchy distribution.  For \alpha =
     2 it is a Gaussian distribution with \sigma = \sqrt{2} c.  For
     \alpha < 1 the tails of the distribution become extremely wide.

     The algorithm only works for 0 < alpha <= 2.



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

20.13 The Levy skew alpha-Stable Distribution
=============================================

 -- Function: double gsl_ran_levy_skew (const gsl_rng * R, double C,
          double ALPHA, double BETA)
     This function returns a random variate from the Levy skew stable
     distribution with scale C, exponent ALPHA and skewness parameter
     BETA.  The skewness parameter must lie in the range [-1,1].  The
     Levy skew stable probability distribution is defined by a Fourier
     transform,

          p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))

     When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by
     -(2/\pi)\log|t|.  There is no explicit solution for the form of
     p(x) and the library does not define a corresponding `pdf'
     function.  For \alpha = 2 the distribution reduces to a Gaussian
     distribution with \sigma = \sqrt{2} c and the skewness parameter
     has no effect.  For \alpha < 1 the tails of the distribution
     become extremely wide.  The symmetric distribution corresponds to
     \beta = 0.

     The algorithm only works for 0 < alpha <= 2.

   The Levy alpha-stable distributions have the property that if N
alpha-stable variates are drawn from the distribution p(c, \alpha,
\beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be
distributed as an alpha-stable variate, p(N^(1/\alpha) c, \alpha,
\beta).



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

20.14 The Gamma Distribution
============================

 -- Function: double gsl_ran_gamma (const gsl_rng * R, double A, double
          B)
     This function returns a random variate from the gamma
     distribution.  The distribution function is,

          p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx

     for x > 0.

     The gamma distribution with an integer parameter A is known as the
     Erlang distribution.

     The variates are computed using the Marsaglia-Tsang fast gamma
     method.  This function for this method was previously called
     `gsl_ran_gamma_mt' and can still be accessed using this name.

 -- Function: double gsl_ran_gamma_knuth (const gsl_rng * R, double A,
          double B)
     This function returns a gamma variate using the algorithms from
     Knuth (vol 2).

 -- Function: double gsl_ran_gamma_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a
     gamma distribution with parameters A and B, using the formula
     given above.


 -- Function: double gsl_cdf_gamma_P (double X, double A, double B)
 -- Function: double gsl_cdf_gamma_Q (double X, double A, double B)
 -- Function: double gsl_cdf_gamma_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_gamma_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the gamma distribution with
     parameters A and B.


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

20.15 The Flat (Uniform) Distribution
=====================================

 -- Function: double gsl_ran_flat (const gsl_rng * R, double A, double
          B)
     This function returns a random variate from the flat (uniform)
     distribution from A to B. The distribution is,

          p(x) dx = {1 \over (b-a)} dx

     if a <= x < b and 0 otherwise.

 -- Function: double gsl_ran_flat_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a
     uniform distribution from A to B, using the formula given above.


 -- Function: double gsl_cdf_flat_P (double X, double A, double B)
 -- Function: double gsl_cdf_flat_Q (double X, double A, double B)
 -- Function: double gsl_cdf_flat_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_flat_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for a uniform distribution from A to
     B.


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

20.16 The Lognormal Distribution
================================

 -- Function: double gsl_ran_lognormal (const gsl_rng * R, double ZETA,
          double SIGMA)
     This function returns a random variate from the lognormal
     distribution.  The distribution function is,

          p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2} } \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx

     for x > 0.

 -- Function: double gsl_ran_lognormal_pdf (double X, double ZETA,
          double SIGMA)
     This function computes the probability density p(x) at X for a
     lognormal distribution with parameters ZETA and SIGMA, using the
     formula given above.


 -- Function: double gsl_cdf_lognormal_P (double X, double ZETA, double
          SIGMA)
 -- Function: double gsl_cdf_lognormal_Q (double X, double ZETA, double
          SIGMA)
 -- Function: double gsl_cdf_lognormal_Pinv (double P, double ZETA,
          double SIGMA)
 -- Function: double gsl_cdf_lognormal_Qinv (double Q, double ZETA,
          double SIGMA)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the lognormal distribution with
     parameters ZETA and SIGMA.


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

20.17 The Chi-squared Distribution
==================================

The chi-squared distribution arises in statistics.  If Y_i are n
independent Gaussian random variates with unit variance then the
sum-of-squares,

     X_i = \sum_i Y_i^2

has a chi-squared distribution with n degrees of freedom.

 -- Function: double gsl_ran_chisq (const gsl_rng * R, double NU)
     This function returns a random variate from the chi-squared
     distribution with NU degrees of freedom. The distribution function
     is,

          p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx

     for x >= 0.

 -- Function: double gsl_ran_chisq_pdf (double X, double NU)
     This function computes the probability density p(x) at X for a
     chi-squared distribution with NU degrees of freedom, using the
     formula given above.


 -- Function: double gsl_cdf_chisq_P (double X, double NU)
 -- Function: double gsl_cdf_chisq_Q (double X, double NU)
 -- Function: double gsl_cdf_chisq_Pinv (double P, double NU)
 -- Function: double gsl_cdf_chisq_Qinv (double Q, double NU)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the chi-squared distribution
     with NU degrees of freedom.


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

20.18 The F-distribution
========================

The F-distribution arises in statistics.  If Y_1 and Y_2 are
chi-squared deviates with \nu_1 and \nu_2 degrees of freedom then the
ratio,

     X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }

has an F-distribution F(x;\nu_1,\nu_2).

 -- Function: double gsl_ran_fdist (const gsl_rng * R, double NU1,
          double NU2)
     This function returns a random variate from the F-distribution
     with degrees of freedom NU1 and NU2. The distribution function is,

          p(x) dx =
             { \Gamma((\nu_1 + \nu_2)/2)
                  \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) }
             \nu_1^{\nu_1/2} \nu_2^{\nu_2/2}
             x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}

     for x >= 0.

 -- Function: double gsl_ran_fdist_pdf (double X, double NU1, double
          NU2)
     This function computes the probability density p(x) at X for an
     F-distribution with NU1 and NU2 degrees of freedom, using the
     formula given above.


 -- Function: double gsl_cdf_fdist_P (double X, double NU1, double NU2)
 -- Function: double gsl_cdf_fdist_Q (double X, double NU1, double NU2)
 -- Function: double gsl_cdf_fdist_Pinv (double P, double NU1, double
          NU2)
 -- Function: double gsl_cdf_fdist_Qinv (double Q, double NU1, double
          NU2)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the F-distribution with NU1 and
     NU2 degrees of freedom.


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

20.19 The t-distribution
========================

The t-distribution arises in statistics.  If Y_1 has a normal
distribution and Y_2 has a chi-squared distribution with \nu degrees of
freedom then the ratio,

     X = { Y_1 \over \sqrt{Y_2 / \nu} }

has a t-distribution t(x;\nu) with \nu degrees of freedom.

 -- Function: double gsl_ran_tdist (const gsl_rng * R, double NU)
     This function returns a random variate from the t-distribution.
     The distribution function is,

          p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
             (1 + x^2/\nu)^{-(\nu + 1)/2} dx

     for -\infty < x < +\infty.

 -- Function: double gsl_ran_tdist_pdf (double X, double NU)
     This function computes the probability density p(x) at X for a
     t-distribution with NU degrees of freedom, using the formula given
     above.


 -- Function: double gsl_cdf_tdist_P (double X, double NU)
 -- Function: double gsl_cdf_tdist_Q (double X, double NU)
 -- Function: double gsl_cdf_tdist_Pinv (double P, double NU)
 -- Function: double gsl_cdf_tdist_Qinv (double Q, double NU)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the t-distribution with NU
     degrees of freedom.


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

20.20 The Beta Distribution
===========================

 -- Function: double gsl_ran_beta (const gsl_rng * R, double A, double
          B)
     This function returns a random variate from the beta distribution.
     The distribution function is,

          p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx

     for 0 <= x <= 1.

 -- Function: double gsl_ran_beta_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a
     beta distribution with parameters A and B, using the formula given
     above.


 -- Function: double gsl_cdf_beta_P (double X, double A, double B)
 -- Function: double gsl_cdf_beta_Q (double X, double A, double B)
 -- Function: double gsl_cdf_beta_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_beta_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the beta distribution with
     parameters A and B.


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

20.21 The Logistic Distribution
===============================

 -- Function: double gsl_ran_logistic (const gsl_rng * R, double A)
     This function returns a random variate from the logistic
     distribution.  The distribution function is,

          p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx

     for -\infty < x < +\infty.

 -- Function: double gsl_ran_logistic_pdf (double X, double A)
     This function computes the probability density p(x) at X for a
     logistic distribution with scale parameter A, using the formula
     given above.


 -- Function: double gsl_cdf_logistic_P (double X, double A)
 -- Function: double gsl_cdf_logistic_Q (double X, double A)
 -- Function: double gsl_cdf_logistic_Pinv (double P, double A)
 -- Function: double gsl_cdf_logistic_Qinv (double Q, double A)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the logistic distribution with
     scale parameter A.


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

20.22 The Pareto Distribution
=============================

 -- Function: double gsl_ran_pareto (const gsl_rng * R, double A,
          double B)
     This function returns a random variate from the Pareto
     distribution of order A.  The distribution function is,

          p(x) dx = (a/b) / (x/b)^{a+1} dx

     for x >= b.

 -- Function: double gsl_ran_pareto_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a
     Pareto distribution with exponent A and scale B, using the formula
     given above.


 -- Function: double gsl_cdf_pareto_P (double X, double A, double B)
 -- Function: double gsl_cdf_pareto_Q (double X, double A, double B)
 -- Function: double gsl_cdf_pareto_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_pareto_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the Pareto distribution with
     exponent A and scale B.


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

20.23 Spherical Vector Distributions
====================================

The spherical distributions generate random vectors, located on a
spherical surface.  They can be used as random directions, for example
in the steps of a random walk.

 -- Function: void gsl_ran_dir_2d (const gsl_rng * R, double * X,
          double * Y)
 -- Function: void gsl_ran_dir_2d_trig_method (const gsl_rng * R,
          double * X, double * Y)
     This function returns a random direction vector v = (X,Y) in two
     dimensions.  The vector is normalized such that |v|^2 = x^2 + y^2
     = 1.  The obvious way to do this is to take a uniform random
     number between 0 and 2\pi and let X and Y be the sine and cosine
     respectively.  Two trig functions would have been expensive in the
     old days, but with modern hardware implementations, this is
     sometimes the fastest way to go.  This is the case for the Pentium
     (but not the case for the Sun Sparcstation).  One can avoid the
     trig evaluations by choosing X and Y in the interior of a unit
     circle (choose them at random from the interior of the enclosing
     square, and then reject those that are outside the unit circle),
     and then dividing by \sqrt{x^2 + y^2}.  A much cleverer approach,
     attributed to von Neumann (See Knuth, v2, 3rd ed, p140, exercise
     23), requires neither trig nor a square root.  In this approach, U
     and V are chosen at random from the interior of a unit circle, and
     then x=(u^2-v^2)/(u^2+v^2) and y=2uv/(u^2+v^2).

 -- Function: void gsl_ran_dir_3d (const gsl_rng * R, double * X,
          double * Y, double * Z)
     This function returns a random direction vector v = (X,Y,Z) in
     three dimensions.  The vector is normalized such that |v|^2 = x^2
     + y^2 + z^2 = 1.  The method employed is due to Robert E. Knop
     (CACM 13, 326 (1970)), and explained in Knuth, v2, 3rd ed, p136.
     It uses the surprising fact that the distribution projected along
     any axis is actually uniform (this is only true for 3 dimensions).

 -- Function: void gsl_ran_dir_nd (const gsl_rng * R, size_t N, double
          * X)
     This function returns a random direction vector v =
     (x_1,x_2,...,x_n) in N dimensions.  The vector is normalized such
     that |v|^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1.  The method uses the
     fact that a multivariate Gaussian distribution is spherically
     symmetric.  Each component is generated to have a Gaussian
     distribution, and then the components are normalized.  The method
     is described by Knuth, v2, 3rd ed, p135-136, and attributed to G.
     W. Brown, Modern Mathematics for the Engineer (1956).


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

20.24 The Weibull Distribution
==============================

 -- Function: double gsl_ran_weibull (const gsl_rng * R, double A,
          double B)
     This function returns a random variate from the Weibull
     distribution.  The distribution function is,

          p(x) dx = {b \over a^b} x^{b-1}  \exp(-(x/a)^b) dx

     for x >= 0.

 -- Function: double gsl_ran_weibull_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a
     Weibull distribution with scale A and exponent B, using the
     formula given above.


 -- Function: double gsl_cdf_weibull_P (double X, double A, double B)
 -- Function: double gsl_cdf_weibull_Q (double X, double A, double B)
 -- Function: double gsl_cdf_weibull_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_weibull_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the Weibull distribution with
     scale A and exponent B.


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

20.25 The Type-1 Gumbel Distribution
====================================

 -- Function: double gsl_ran_gumbel1 (const gsl_rng * R, double A,
          double B)
     This function returns  a random variate from the Type-1 Gumbel
     distribution.  The Type-1 Gumbel distribution function is,

          p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx

     for -\infty < x < \infty.

 -- Function: double gsl_ran_gumbel1_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a
     Type-1 Gumbel distribution with parameters A and B, using the
     formula given above.


 -- Function: double gsl_cdf_gumbel1_P (double X, double A, double B)
 -- Function: double gsl_cdf_gumbel1_Q (double X, double A, double B)
 -- Function: double gsl_cdf_gumbel1_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_gumbel1_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the Type-1 Gumbel distribution
     with parameters A and B.


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

20.26 The Type-2 Gumbel Distribution
====================================

 -- Function: double gsl_ran_gumbel2 (const gsl_rng * R, double A,
          double B)
     This function returns a random variate from the Type-2 Gumbel
     distribution.  The Type-2 Gumbel distribution function is,

          p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx

     for 0 < x < \infty.

 -- Function: double gsl_ran_gumbel2_pdf (double X, double A, double B)
     This function computes the probability density p(x) at X for a
     Type-2 Gumbel distribution with parameters A and B, using the
     formula given above.


 -- Function: double gsl_cdf_gumbel2_P (double X, double A, double B)
 -- Function: double gsl_cdf_gumbel2_Q (double X, double A, double B)
 -- Function: double gsl_cdf_gumbel2_Pinv (double P, double A, double B)
 -- Function: double gsl_cdf_gumbel2_Qinv (double Q, double A, double B)
     These functions compute the cumulative distribution functions
     P(x), Q(x) and their inverses for the Type-2 Gumbel distribution
     with parameters A and B.


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

20.27 The Dirichlet Distribution
================================

 -- Function: void gsl_ran_dirichlet (const gsl_rng * R, size_t K,
          const double ALPHA[], double THETA[])
     This function returns an array of K random variates from a
     Dirichlet distribution of order K-1. The distribution function is

          p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K =
            (1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K

     for theta_i >= 0 and alpha_i > 0.  The delta function ensures that
     \sum \theta_i = 1.  The normalization factor Z is

          Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}

     The random variates are generated by sampling K values from gamma
     distributions with parameters a=alpha_i, b=1, and renormalizing.
     See A.M. Law, W.D. Kelton, `Simulation Modeling and Analysis'
     (1991).

 -- Function: double gsl_ran_dirichlet_pdf (size_t K, const double
          ALPHA[], const double THETA[])
     This function computes the probability density p(\theta_1, ... ,
     \theta_K) at THETA[K] for a Dirichlet distribution with parameters
     ALPHA[K], using the formula given above.

 -- Function: double gsl_ran_dirichlet_lnpdf (size_t K, const double
          ALPHA[], const double THETA[])
     This function computes the logarithm of the probability density
     p(\theta_1, ... , \theta_K) for a Dirichlet distribution with
     parameters ALPHA[K].


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

20.28 General Discrete Distributions
====================================

Given K discrete events with different probabilities P[k], produce a
random value k consistent with its probability.

   The obvious way to do this is to preprocess the probability list by
generating a cumulative probability array with K+1 elements:

       C[0] = 0
     C[k+1] = C[k]+P[k].

Note that this construction produces C[K]=1.  Now choose a uniform
deviate u between 0 and 1, and find the value of k such that C[k] <= u
< C[k+1].  Although this in principle requires of order \log K steps per
random number generation, they are fast steps, and if you use something
like \lfloor uK \rfloor as a starting point, you can often do pretty
well.

   But faster methods have been devised.  Again, the idea is to
preprocess the probability list, and save the result in some form of
lookup table; then the individual calls for a random discrete event can
go rapidly.  An approach invented by G. Marsaglia (Generating discrete
random variables in a computer, Comm ACM 6, 37-38 (1963)) is very
clever, and readers interested in examples of good algorithm design are
directed to this short and well-written paper.  Unfortunately, for
large K, Marsaglia's lookup table can be quite large.

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

   This method can be used to speed up some of the discrete random
number generators below, such as the binomial distribution.  To use it
for something like the Poisson Distribution, a modification would have
to be made, since it only takes a finite set of K outcomes.

 -- Function: gsl_ran_discrete_t * gsl_ran_discrete_preproc (size_t K,
          const double * P)
     This function returns a pointer to a structure that contains the
     lookup table for the discrete random number generator.  The array
     P[] contains the probabilities of the discrete events; these array
     elements must all be positive, but they needn't add up to one (so
     you can think of them more generally as "weights")--the
     preprocessor will normalize appropriately.  This return value is
     used as an argument for the `gsl_ran_discrete' function below.

 -- Function: size_t gsl_ran_discrete (const gsl_rng * R, const
          gsl_ran_discrete_t * G)
     After the preprocessor, above, has been called, you use this
     function to get the discrete random numbers.

 -- Function: double gsl_ran_discrete_pdf (size_t K, const
          gsl_ran_discrete_t * G)
     Returns the probability P[k] of observing the variable K.  Since
     P[k] is not stored as part of the lookup table, it must be
     recomputed; this computation takes O(K), so if K is large and you
     care about the original array P[k] used to create the lookup
     table, then you should just keep this original array P[k] around.

 -- Function: void gsl_ran_discrete_free (gsl_ran_discrete_t * G)
     De-allocates the lookup table pointed to by G.


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

20.29 The Poisson Distribution
==============================

 -- Function: unsigned int gsl_ran_poisson (const gsl_rng * R, double
          MU)
     This function returns a random integer from the Poisson
     distribution with mean MU.  The probability distribution for
     Poisson variates is,

          p(k) = {\mu^k \over k!} \exp(-\mu)

     for k >= 0.

 -- Function: double gsl_ran_poisson_pdf (unsigned int K, double MU)
     This function computes the probability p(k) of obtaining  K from a
     Poisson distribution with mean MU, using the formula given above.


 -- Function: double gsl_cdf_poisson_P (unsigned int K, double MU)
 -- Function: double gsl_cdf_poisson_Q (unsigned int K, double MU)
     These functions compute the cumulative distribution functions
     P(k), Q(k) for the Poisson distribution with parameter MU.


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

20.30 The Bernoulli Distribution
================================

 -- Function: unsigned int gsl_ran_bernoulli (const gsl_rng * R, double
          P)
     This function returns either 0 or 1, the result of a Bernoulli
     trial with probability P.  The probability distribution for a
     Bernoulli trial is,

          p(0) = 1 - p
          p(1) = p


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



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

20.31 The Binomial Distribution
===============================

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

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

     for 0 <= k <= n.

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


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


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

20.32 The Multinomial Distribution
==================================

 -- Function: void gsl_ran_multinomial (const gsl_rng * R, size_t K,
          unsigned int N, const double P[], unsigned int N[])
     This function computes a random sample N[] from the multinomial
     distribution formed by N trials from an underlying distribution
     P[K]. The distribution function for N[] is,

          P(n_1, n_2, ..., n_K) =
            (N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K

     where (n_1, n_2, ..., n_K) are nonnegative integers with
     sum_{k=1}^K n_k = N, and (p_1, p_2, ..., p_K) is a probability
     distribution with \sum p_i = 1.  If the array P[K] is not
     normalized then its entries will be treated as weights and
     normalized appropriately.  The arrays N[] and P[] must both be of
     length K.

     Random variates are generated using the conditional binomial
     method (see C.S. Davis, `The computer generation of multinomial
     random variates', Comp. Stat. Data Anal. 16 (1993) 205-217 for
     details).

 -- Function: double gsl_ran_multinomial_pdf (size_t K, const double
          P[], const unsigned int N[])
     This function computes the probability P(n_1, n_2, ..., n_K) of
     sampling N[K] from a multinomial distribution with parameters
     P[K], using the formula given above.

 -- Function: double gsl_ran_multinomial_lnpdf (size_t K, const double
          P[], const unsigned int N[])
     This function returns the logarithm of the probability for the
     multinomial distribution P(n_1, n_2, ..., n_K) with parameters
     P[K].


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

20.33 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

20.34 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

20.35 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

20.36 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

20.37 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

20.38 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

20.39 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");
       gsl_rng_free (r);
       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);
         }

       gsl_rng_free (r);
       return 0;
     }

Here is some output from the program, four 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

20.40 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.  Available online at
     `http://cg.scs.carleton.ca/~luc/rnbookindex.html'.

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

21 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'.   All the functions operate on C arrays with a
STRIDE parameter specifying the spacing between elements.

* Menu:

* Mean and standard deviation and variance::
* Absolute deviation::
* Higher moments (skewness and kurtosis)::
* Autocorrelation::
* Covariance::
* Correlation::
* 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

21.1 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_tss (const double DATA[], size_t STRIDE,
          size_t N)
 -- Function: double gsl_stats_tss_m (const double DATA[], size_t
          STRIDE, size_t N, double MEAN)
     These functions return the total sum of squares (TSS) of DATA about
     the mean.  For `gsl_stats_tss_m' the user-supplied value of MEAN
     is used, and for `gsl_stats_tss' it is computed using
     `gsl_stats_mean'.

          TSS =  \sum (x_i - mean)^2

 -- 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

21.2 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

21.3 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

21.4 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: Correlation,  Prev: Autocorrelation,  Up: Statistics

21.5 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
          STRIDE2, 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: Correlation,  Next: Weighted Samples,  Prev: Covariance,  Up: Statistics

21.6 Correlation
================

 -- Function: double gsl_stats_correlation (const double DATA1[], const
          size_t STRIDE1, const double DATA2[], const size_t STRIDE2,
          const size_t N)
     This function efficiently computes the Pearson correlation
     coefficient between the datasets DATA1 and DATA2 which must both
     be of the same length N.
          r = cov(x, y) / (\Hat\sigma_x \Hat\sigma_y)
            = {1/(n-1) \sum (x_i - \Hat x) (y_i - \Hat y)
               \over
               \sqrt{1/(n-1) \sum (x_i - \Hat x)^2} \sqrt{1/(n-1) \sum (y_i - \Hat y)^2}
              }

 -- Function: double gsl_stats_spearman (const double DATA1[], const
          size_t STRIDE1, const double DATA2[], const size_t STRIDE2,
          const size_t N, double WORK[])
     This function computes the Spearman rank correlation coefficient
     between the datasets DATA1 and DATA2 which must both be of the same
     length N. Additional workspace of size 2*N is required in WORK.
     The Spearman rank correlation between vectors x and y is
     equivalent to the Pearson correlation between the ranked vectors
     x_R and y_R, where ranks are defined to be the average of the
     positions of an element in the ascending order of the values.


File: gsl-ref.info,  Node: Weighted Samples,  Next: Maximum and Minimum values,  Prev: Correlation,  Up: Statistics

21.7 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 calculated 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 the
     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_wtss (const double W[], const size_t
          WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
 -- Function: double gsl_stats_wtss_m (const double W[], const size_t
          WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
          WMEAN)
     These functions return the weighted total sum of squares (TSS) of
     DATA about the weighted mean.  For `gsl_stats_wtss_m' the
     user-supplied value of WMEAN is used, and for `gsl_stats_wtss' it
     is computed using `gsl_stats_wmean'.

          TSS =  \sum w_i (x_i - wmean)^2

 -- 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 - \Hat x)/\Hat \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 - \Hat x)/\Hat \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

21.8 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

21.9 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

21.10 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 5.373
     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

21.11 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

22 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

22.1 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

22.2 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

22.3 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

22.4 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

22.5 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

22.6 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

22.7 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

22.8 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

22.9 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

22.10 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

22.11 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

22.12 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

22.13 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

22.14 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

22.15 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

22.16 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

22.17 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 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

22.18 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

22.19 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

22.20 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_histogram2d_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

22.21 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

22.22 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);
         }

         gsl_histogram2d_pdf_free (p);
       }

       gsl_histogram2d_free (h);
       gsl_rng_free (r);

       return 0;
     }



File: gsl-ref.info,  Node: N-tuples,  Next: Monte Carlo Integration,  Prev: Histograms,  Up: Top

23 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

23.1 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

23.2 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

23.3 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

23.4 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

23.5 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

23.6 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

23.7 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

23.8 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);
       gsl_rng_free (r);

       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

23.9 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

24 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

24.1 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

24.2 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^2 / N^2) \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,
          const double XL[], const 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

24.3 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,
          const double XL[], const 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 which can be
changed using the following two functions.(1)

 -- Function: void gsl_monte_miser_params_get (const
          gsl_monte_miser_state * S, gsl_monte_miser_params * PARAMS)
     This function copies the parameters of the integrator state into
     the user-supplied PARAMS structure.

 -- Function: void gsl_monte_miser_params_set (gsl_monte_miser_state *
          S, const gsl_monte_miser_params * PARAMS)
     This function sets the integrator parameters based on values
     provided in the PARAMS structure.

   Typically the values of the parameters are first read using
`gsl_monte_miser_params_get', the necessary changes are made to the
fields of the PARAMS structure, and the values are copied back into the
integrator state using `gsl_monte_miser_params_set'.  The functions use
the `gsl_monte_miser_params' structure which contains the following
fields:

 -- 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.

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

   (1) The previous method of accessing these fields directly through
the `gsl_monte_miser_state' struct is now deprecated.


File: gsl-ref.info,  Node: VEGAS,  Next: Monte Carlo Examples,  Prev: MISER,  Up: Monte Carlo Integration

24.4 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 convergence of the algorithm can be tested using the overall
chi-squared value of the results, which is available from the following
function:

 -- Function: double gsl_monte_vegas_chisq (const gsl_monte_vegas_state
          * S)
     This function returns the chi-squared per degree of freedom for the
     weighted estimate of the integral.  The returned value should be
     close to 1.  A value 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.

 -- Function: void gsl_monte_vegas_runval (const gsl_monte_vegas_state
          * S, double * RESULT, double * SIGMA)
     This function returns the raw (unaveraged) values of the integral
     RESULT and its error SIGMA from the most recent iteration of the
     algorithm.

   The VEGAS algorithm is highly configurable. Several parameters can
be changed using the following two functions.

 -- Function: void gsl_monte_vegas_params_get (const
          gsl_monte_vegas_state * S, gsl_monte_vegas_params * PARAMS)
     This function copies the parameters of the integrator state into
     the user-supplied PARAMS structure.

 -- Function: void gsl_monte_vegas_params_set (gsl_monte_vegas_state *
          S, const gsl_monte_vegas_params * PARAMS)
     This function sets the integrator parameters based on values
     provided in the PARAMS structure.

   Typically the values of the parameters are first read using
`gsl_monte_vegas_params_get', the necessary changes are made to the
fields of the PARAMS structure, and the values are copied back into the
integrator state using `gsl_monte_vegas_params_set'.  The functions use
the `gsl_monte_vegas_params' structure which contains the following
fields:

 -- 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.

   The above fields and the CHISQ value can also be accessed directly
in the `gsl_monte_vegas_state' but such use is deprecated.


File: gsl-ref.info,  Node: Monte Carlo Examples,  Next: Monte Carlo Integration References and Further Reading,  Prev: VEGAS,  Up: Monte Carlo Integration

24.5 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 = %.2g 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, gsl_monte_vegas_chisq (s));
           }
         while (fabs (gsl_monte_vegas_chisq (s) - 1.0) > 0.5);

         display_results ("vegas final", res, err);

         gsl_monte_vegas_free (s);
       }

       gsl_rng_free (r);

       return 0;
     }

With 500,000 function calls the plain Monte Carlo algorithm achieves a
fractional error of 1%.  The estimated error `sigma' is roughly
consistent with the actual error-the computed result differs from the
true result by about 1.4 standard deviations,

     plain ==================
     result =  1.412209
     sigma  =  0.013436
     exact  =  1.393204
     error  =  0.019005 = 1.4 sigma

The MISER algorithm reduces the error by a factor of four, and also
correctly estimates the error,

     miser ==================
     result =  1.391322
     sigma  =  0.003461
     exact  =  1.393204
     error  = -0.001882 = 0.54 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 =  1.392673
     sigma  =  0.003410
     exact  =  1.393204
     error  = -0.000531 = 0.16 sigma
     converging...
     result =  1.393281 sigma =  0.000362 chisq/dof = 1.5
     vegas final ==================
     result =  1.393281
     sigma  =  0.000362
     exact  =  1.393204
     error  =  0.000077 = 0.21 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

24.6 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

25 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

25.1 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

25.2 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:

          #-iter  #-evals  temperature  position  energy  best_energy

     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

25.3 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

25.3.1 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 <string.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 1000

     /* max step size in random walk */
     #define STEP_SIZE 1.0

     /* Boltzmann constant */
     #define K 1.0

     /* initial temperature */
     #define T_INITIAL 0.008

     /* damping factor for temperature */
     #define MU_T 1.003
     #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);

       gsl_rng_free (r);
       return 0;
     }

   Here are a couple of plots that are generated by running
`siman_test' in the following way:

     $ ./siman_test | awk '!/^#/ {print $1, $4}'
      | graph -y 1.34 1.4 -W0 -X generation -Y position
      | plot -Tps > siman-test.eps
     $ ./siman_test | awk '!/^#/ {print $1, $5}'
      | graph -y -0.88 -0.83 -W0 -X generation -Y energy
      | plot -Tps > siman-energy.eps


File: gsl-ref.info,  Node: Traveling Salesman Problem,  Prev: Trivial example,  Up: Examples with Simulated Annealing

25.3.2 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
      | awk '{print $1, $NF}'
      | graph -y 3300 6500 -W0 -X generation -Y distance
         -L "TSP - 12 southwest cities"
      | plot -Tps > 12-cities.eps
     $ grep initial_city_coord tsp.output
       | awk '{print $2, $3}'
       | graph -X "longitude (- means west)" -Y "latitude"
          -L "TSP - initial-order" -f 0.03 -S 1 0.1
       | plot -Tps > initial-route.eps
     $ grep final_city_coord tsp.output
       | awk '{print $2, $3}'
       | graph -X "longitude (- means west)" -Y "latitude"
          -L "TSP - final-order" -f 0.03 -S 1 0.1
       | plot -Tps > 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: -103.2 34.41 Clovis
     ###final_city_coord: -96.77 32.79 Dallas
     ###final_city_coord: -106.76 32.34 Las Cruces
     ###final_city_coord: -112.07 33.54 Phoenix
     ###final_city_coord: -108.74 35.52 Gallup
     ###final_city_coord: -108.58 37.35 Cortez
     ###final_city_coord: -107.87 37.29 Durango
     ###final_city_coord: -107.84 35.15 Grants
     ###final_city_coord: -106.62 35.12 Albuquerque
     ###final_city_coord: -106.28 35.89 Los Alamos
     ###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

25.4 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

26 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. A driver object can be used
as a high level wrapper for easy use of low level functions.

   These functions are declared in the header file `gsl_odeiv2.h'.
This is a new interface in version 1.15 and uses the prefix
`gsl_odeiv2' for all functions.  It is recommended over the previous
`gsl_odeiv' implementation defined in `gsl_odeiv.h' The old interface
has been retained under the original name for backwards compatibility.

* Menu:

* Defining the ODE System::
* Stepping Functions::
* Adaptive Step-size Control::
* Evolution::
* Driver::
* ODE Example programs::
* ODE References and Further Reading::


File: gsl-ref.info,  Node: Defining the ODE System,  Next: Stepping Functions,  Up: Ordinary Differential Equations

26.1 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_odeiv2_system' datatype.

 -- Data Type: gsl_odeiv2_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. A special return value `GSL_EBADFUNC' causes
          `gsl_odeiv2' routines to immediately stop and return. The
          user must call an appropriate reset function (e.g.
          `gsl_odeiv2_driver_reset' or `gsl_odeiv2_step_reset') before
          continuing. Use return values distinct from standard GSL
          error codes to distinguish your function as the source of the
          error.

    `int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);'
          This function should store the vector of derivative elements
          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.

          Not all of the stepper algorithms of `gsl_odeiv2' make use of
          the Jacobian matrix, so it may not be necessary to provide
          this function (the `jacobian' element of the struct can be
          replaced by a null pointer for those algorithms).

          The function should return `GSL_SUCCESS' if the calculation
          was completed successfully. Any other return value indicates
          an error. A special return value `GSL_EBADFUNC' causes
          `gsl_odeiv2' routines to immediately stop and return. The
          user must call an appropriate reset function (e.g.
          `gsl_odeiv2_driver_reset' or `gsl_odeiv2_step_reset') before
          continuing. Use return values distinct from standard GSL
          error codes to distinguish your function as the source of the
          error.

    `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

26.2 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_odeiv2_step * gsl_odeiv2_step_alloc (const
          gsl_odeiv2_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. Please
     note that if you use a stepper method that requires access to a
     driver object, it is advisable to use a driver allocation method,
     which automatically allocates a stepper, too.

 -- Function: int gsl_odeiv2_step_reset (gsl_odeiv2_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_odeiv2_step_free (gsl_odeiv2_step * S)
     This function frees all the memory associated with the stepping
     function S.

 -- Function: const char * gsl_odeiv2_step_name (const gsl_odeiv2_step
          * S)
     This function returns a pointer to the name of the stepping
     function.  For example,

          printf ("step method is '%s'\n",
                   gsl_odeiv2_step_name (s));

     would print something like `step method is 'rkf45''.

 -- Function: unsigned int gsl_odeiv2_step_order (const gsl_odeiv2_step
          * S)
     This function returns the order of the stepping function on the
     previous step. The order can vary if the stepping function itself
     is adaptive.

 -- Function: int gsl_odeiv2_step_set_driver (gsl_odeiv2_step * S,
          const gsl_odeiv2_driver * D)
     This function sets a pointer of the driver object D for stepper S,
     to allow the stepper to access control (and evolve) object through
     the driver object. This is a requirement for some steppers, to get
     the desired error level for internal iteration of stepper.
     Allocation of a driver object calls this function automatically.

 -- Function: int gsl_odeiv2_step_apply (gsl_odeiv2_step * S, double T,
          double H, double Y[], double YERR[], const double DYDT_IN[],
          double DYDT_OUT[], const gsl_odeiv2_system * SYS)
     This function applies the stepping function S to the system of
     equations defined by SYS, using the step-size H to advance the
     system from time T and state Y to time T+H.  The new state of the
     system is stored in Y on output, with an estimate of the absolute
     error in each component stored in YERR.  If the argument DYDT_IN
     is not null it should point an array containing the derivatives
     for the system at time T on input. This is optional as the
     derivatives will be computed internally if they are not provided,
     but allows the reuse of existing derivative information.  On
     output the new derivatives of the system at time T+H will be
     stored in DYDT_OUT if it is not null.

     The stepping function returns `GSL_FAILURE' if it is unable to
     compute the requested step. Also, if the user-supplied functions
     defined in the system SYS return a status other than `GSL_SUCCESS'
     the step will be aborted. In that 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. Failure may be due to a
     singularity in the system or too large step-size H. In that case
     the step should be attempted again with a smaller step-size, e.g.
     H/2.

     If the driver object is not appropriately set via
     `gsl_odeiv2_step_set_driver' for those steppers that need it, the
     stepping function returns `GSL_EFAULT'. If the user-supplied
     functions defined in the system SYS returns `GSL_EBADFUNC', the
     function returns immediately with the same return code. In this
     case the user must call `gsl_odeiv2_step_reset' before calling
     this function again.


   The following algorithms are available,

 -- Step Type: gsl_odeiv2_step_rk2
     Explicit embedded Runge-Kutta (2, 3) method.

 -- Step Type: gsl_odeiv2_step_rk4
     Explicit 4th order (classical) Runge-Kutta. Error estimation is
     carried out by the step doubling method. For more efficient
     estimate of the error, use the embedded methods described below.

 -- Step Type: gsl_odeiv2_step_rkf45
     Explicit embedded Runge-Kutta-Fehlberg (4, 5) method.  This method
     is a good general-purpose integrator.

 -- Step Type: gsl_odeiv2_step_rkck
     Explicit embedded Runge-Kutta Cash-Karp (4, 5) method.

 -- Step Type: gsl_odeiv2_step_rk8pd
     Explicit embedded Runge-Kutta Prince-Dormand (8, 9) method.

 -- Step Type: gsl_odeiv2_step_rk1imp
     Implicit Gaussian first order Runge-Kutta. Also known as implicit
     Euler or backward Euler method. Error estimation is carried out by
     the step doubling method. This algorithm requires the Jacobian and
     access to the driver object via `gsl_odeiv2_step_set_driver'.

 -- Step Type: gsl_odeiv2_step_rk2imp
     Implicit Gaussian second order Runge-Kutta. Also known as implicit
     mid-point rule. Error estimation is carried out by the step
     doubling method. This stepper requires the Jacobian and access to
     the driver object via `gsl_odeiv2_step_set_driver'.

 -- Step Type: gsl_odeiv2_step_rk4imp
     Implicit Gaussian 4th order Runge-Kutta. Error estimation is
     carried out by the step doubling method. This algorithm requires
     the Jacobian and access to the driver object via
     `gsl_odeiv2_step_set_driver'.

 -- Step Type: gsl_odeiv2_step_bsimp
     Implicit Bulirsch-Stoer method of Bader and Deuflhard. The method
     is generally suitable for stiff problems. This stepper requires the
     Jacobian.

 -- Step Type: gsl_odeiv2_step_msadams
     A variable-coefficient linear multistep Adams method in Nordsieck
     form. This stepper uses explicit Adams-Bashforth (predictor) and
     implicit Adams-Moulton (corrector) methods in P(EC)^m functional
     iteration mode. Method order varies dynamically between 1 and 12.
     This stepper requires the access to the driver object via
     `gsl_odeiv2_step_set_driver'.

 -- Step Type: gsl_odeiv2_step_msbdf
     A variable-coefficient linear multistep backward differentiation
     formula (BDF) method in Nordsieck form. This stepper uses the
     explicit BDF formula as predictor and implicit BDF formula as
     corrector. A modified Newton iteration method is used to solve the
     system of non-linear equations. Method order varies dynamically
     between 1 and 5. The method is generally suitable for stiff
     problems. This stepper requires the Jacobian and the access to the
     driver object via `gsl_odeiv2_step_set_driver'.


File: gsl-ref.info,  Node: Adaptive Step-size Control,  Next: Evolution,  Prev: Stepping Functions,  Up: Ordinary Differential Equations

26.3 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_odeiv2_control * gsl_odeiv2_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\prime_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_odeiv2_control * gsl_odeiv2_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_odeiv2_control * gsl_odeiv2_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_odeiv2_control * gsl_odeiv2_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_odeiv2_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\prime_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_odeiv2_control * gsl_odeiv2_control_alloc (const
          gsl_odeiv2_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_odeiv2_control_init (gsl_odeiv2_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_odeiv2_control_free (gsl_odeiv2_control * C)
     This function frees all the memory associated with the control
     function C.

 -- Function: int gsl_odeiv2_control_hadjust (gsl_odeiv2_control * C,
          gsl_odeiv2_step * S, const double Y[], 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_odeiv2_control_name (const
          gsl_odeiv2_control * C)
     This function returns a pointer to the name of the control
     function.  For example,

          printf ("control method is '%s'\n",
                  gsl_odeiv2_control_name (c));

     would print something like `control method is 'standard''

 -- Function: int gsl_odeiv2_control_errlevel (gsl_odeiv2_control * C,
          const double Y, const double DYDT, const double H, const
          size_t IND, double * ERRLEV)
     This function calculates the desired error level of the IND-th
     component to ERRLEV. It requires the value (Y) and value of the
     derivative (DYDT) of the component, and the current step size H.

 -- Function: int gsl_odeiv2_control_set_driver (gsl_odeiv2_control *
          C, const gsl_odeiv2_driver * D)
     This function sets a pointer of the driver object D for control
     object C.


File: gsl-ref.info,  Node: Evolution,  Next: Driver,  Prev: Adaptive Step-size Control,  Up: Ordinary Differential Equations

26.4 Evolution
==============

The evolution function combines the results of a stepping function and
control function to reliably advance the solution forward one step
using an acceptable step-size.

 -- Function: gsl_odeiv2_evolve * gsl_odeiv2_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_odeiv2_evolve_apply (gsl_odeiv2_evolve * E,
          gsl_odeiv2_control * CON, gsl_odeiv2_step * STEP, const
          gsl_odeiv2_system * SYS, double * T, double T1, double * H,
          double Y[])
     This function advances the system (E, SYS) 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. The control function CON is
     applied to check whether the local error estimated by the stepping
     function STEP using step-size H exceeds the required error
     tolerance. If the error is too high, the step is retried by
     calling STEP with a decreased step-size. This process is continued
     until an acceptable step-size is found. An estimate of the local
     error for the step can be obtained from the components of the
     array `E->yerr[]'.

     If the user-supplied functions defined in the system SYS returns
     `GSL_EBADFUNC', the function returns immediately with the same
     return code. In this case the user must call
     `gsl_odeiv2_step_reset' and `gsl_odeiv2_evolve_reset' before
     calling this function again.

     Otherwise, if the user-supplied functions defined in the system
     SYS or the stepping function STEP return a status other than
     `GSL_SUCCESS', the step is retried with a decreased step-size. If
     the step-size decreases below machine precision, a status of
     `GSL_FAILURE' is returned if the user functions returned
     `GSL_SUCCESS'. Otherwise the value returned by user function is
     returned. If no acceptable step can be made, T and Y will be
     restored to their pre-step values and H contains the final
     attempted step-size.

     If the step is successful the function returns a suggested
     step-size for the next step in H. The maximum time T1 is guaranteed
     not to be exceeded by the time-step. On the final time-step the
     value of T will be set to T1 exactly.

 -- Function: int gsl_odeiv2_evolve_apply_fixed_step (gsl_odeiv2_evolve
          * E, gsl_odeiv2_control * CON, gsl_odeiv2_step * STEP, const
          gsl_odeiv2_system * SYS, double * T, const double H, double
          Y[])
     This function advances the ODE-system (E, SYS, CON) from time T
     and position Y using the stepping function STEP by a specified
     step size H. If the local error estimated by the stepping function
     exceeds the desired error level, the step is not taken and the
     function returns `GSL_FAILURE'. Otherwise the value returned by
     user function is returned.

 -- Function: int gsl_odeiv2_evolve_reset (gsl_odeiv2_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_odeiv2_evolve_free (gsl_odeiv2_evolve * E)
     This function frees all the memory associated with the evolution
     function E.

 -- Function: int gsl_odeiv2_evolve_set_driver (gsl_odeiv2_evolve * E,
          const gsl_odeiv2_driver * D)
     This function sets a pointer of the driver object D for evolve
     object E.

   If a system has discontinuous changes in the derivatives at known
points, it is advisable to evolve the system between each discontinuity
in sequence.  For example, if a step-change in an external driving
force occurs at times t_a, t_b and t_c then evolution should be carried
out over the ranges (t_0,t_a), (t_a,t_b), (t_b,t_c), and (t_c,t_1)
separately and not directly over the range (t_0,t_1).


File: gsl-ref.info,  Node: Driver,  Next: ODE Example programs,  Prev: Evolution,  Up: Ordinary Differential Equations

26.5 Driver
===========

The driver object is a high level wrapper that combines the evolution,
control and stepper objects for easy use.

 -- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_y_new (const
          gsl_odeiv2_system * SYS, const gsl_odeiv2_step_type * T,
          const double HSTART, const double EPSABS, const double EPSREL)
 -- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_yp_new (const
          gsl_odeiv2_system * SYS, const gsl_odeiv2_step_type * T,
          const double HSTART, const double EPSABS, const double EPSREL)
 -- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_standard_new
          (const gsl_odeiv2_system * SYS, const gsl_odeiv2_step_type *
          T, const double HSTART, const double EPSABS, const double
          EPSREL, const double A_Y, const double A_DYDT)
 -- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_scaled_new
          (const gsl_odeiv2_system * SYS, const gsl_odeiv2_step_type *
          T, const double HSTART, const double EPSABS, const double
          EPSREL, const double A_Y, const double A_DYDT, const double
          SCALE_ABS[])
     These functions return a pointer to a newly allocated instance of a
     driver object. The functions automatically allocate and initialise
     the evolve, control and stepper objects for ODE system SYS using
     stepper type T. The initial step size is given in HSTART. The rest
     of the arguments follow the syntax and semantics of the control
     functions with same name (`gsl_odeiv2_control_*_new').

 -- Function: int gsl_odeiv2_driver_set_hmin (gsl_odeiv2_driver * D,
          const double HMIN)
     The function sets a minimum for allowed step size HMIN for driver
     D. Default value is 0.

 -- Function: int gsl_odeiv2_driver_set_hmax (gsl_odeiv2_driver * D,
          const double HMAX)
     The function sets a maximum for allowed step size HMAX for driver
     D. Default value is `GSL_DBL_MAX'.

 -- Function: int gsl_odeiv2_driver_set_nmax (gsl_odeiv2_driver * D,
          const unsigned long int NMAX)
     The function sets a maximum for allowed number of steps NMAX for
     driver D. Default value of 0 sets no limit for steps.

 -- Function: int gsl_odeiv2_driver_apply (gsl_odeiv2_driver * D,
          double * T, const double T1, double Y[])
     This function evolves the driver system D from T to T1. Initially
     vector Y should contain the values of dependent variables at point
     T. If the function is unable to complete the calculation, an error
     code from `gsl_odeiv2_evolve_apply' is returned, and T and Y
     contain the values from last successful step.

     If maximum number of steps is reached, a value of `GSL_EMAXITER'
     is returned. If the step size drops below minimum value, the
     function returns with `GSL_ENOPROG'. If the user-supplied functions
     defined in the system SYS returns `GSL_EBADFUNC', the function
     returns immediately with the same return code. In this case the
     user must call `gsl_odeiv2_driver_reset' before calling this
     function again.

 -- Function: int gsl_odeiv2_driver_apply_fixed_step (gsl_odeiv2_driver
          * D, double * T, const double H, const unsigned long int N,
          double Y[])
     This function evolves the driver system D from T with N steps of
     size H. If the function is unable to complete the calculation, an
     error code from `gsl_odeiv2_evolve_apply_fixed_step' is returned,
     and T and Y contain the values from last successful step.

 -- Function: int gsl_odeiv2_driver_reset (gsl_odeiv2_driver * D)
     This function resets the evolution and stepper objects.

 -- Function: int gsl_odeiv2_driver_reset_hstart (gsl_odeiv2_driver *
          D, const double HSTART)
     The routine resets the evolution and stepper objects and sets new
     initial step size to HSTART. This function can be used e.g. to
     change the direction of integration.

 -- Function: int gsl_odeiv2_driver_free (gsl_odeiv2_driver * D)
     This function frees the driver object, and the related evolution,
     stepper and control objects.


File: gsl-ref.info,  Node: ODE Example programs,  Next: ODE References and Further Reading,  Prev: Driver,  Up: Ordinary Differential Equations

26.6 Examples
=============

The following program solves the second-order nonlinear Van der Pol
oscillator equation,

     u''(t) + \mu u'(t) (u(t)^2 - 1) + u(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, v = u'(t),

     u' = v
     v' = -u + \mu v (1-u^2)

The program begins by defining functions for these derivatives and
their Jacobian. The main function uses driver level functions to solve
the problem. The program evolves the solution from (u, v) = (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 (u, v).  The loop in the example prints the solution at the
points t_i = 1, 2, \dots, 100.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_odeiv2.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)
     {
       double mu = 10;
       gsl_odeiv2_system sys = {func, jac, 2, &mu};

       gsl_odeiv2_driver * d =
         gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk8pd,
     				  1e-6, 1e-6, 0.0);
       int i;
       double t = 0.0, t1 = 100.0;
       double y[2] = { 1.0, 0.0 };

       for (i = 1; i <= 100; i++)
         {
           double ti = i * t1 / 100.0;
           int status = gsl_odeiv2_driver_apply (d, &t, ti, y);

           if (status != GSL_SUCCESS)
     	{
     	  printf ("error, return value=%d\n", status);
     	  break;
     	}

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv2_driver_free (d);
       return 0;
     }

The user can work with the lower level functions directly, as in the
following example. In this case an intermediate result is printed after
each successful step instead of equidistant time points.

     int
     main (void)
     {
       const gsl_odeiv2_step_type * T
         = gsl_odeiv2_step_rk8pd;

       gsl_odeiv2_step * s
         = gsl_odeiv2_step_alloc (T, 2);
       gsl_odeiv2_control * c
         = gsl_odeiv2_control_y_new (1e-6, 0.0);
       gsl_odeiv2_evolve * e
         = gsl_odeiv2_evolve_alloc (2);

       double mu = 10;
       gsl_odeiv2_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_odeiv2_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_odeiv2_evolve_free (e);
       gsl_odeiv2_control_free (c);
       gsl_odeiv2_step_free (s);
       return 0;
     }

For functions with multiple parameters, the appropriate information can
be passed in through the PARAMS argument in `gsl_odeiv2_system'
definition (MU in this example) by using a pointer to a struct.

It is also possible to work with a non-adaptive integrator, using only
the stepping function itself, `gsl_odeiv2_driver_apply_fixed_step' or
`gsl_odeiv2_evolve_apply_fixed_step'. The following program uses the
driver level function, with fourth-order Runge-Kutta stepping function
with a fixed stepsize of 0.001.

     int
     main (void)
     {
       double mu = 10;
       gsl_odeiv2_system sys = { func, jac, 2, &mu };

       gsl_odeiv2_driver *d =
         gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk4,
                                        1e-3, 1e-8, 1e-8);

       double t = 0.0;
       double y[2] = { 1.0, 0.0 };
       int i, s;

       for (i = 0; i < 100; i++)
         {
           s = gsl_odeiv2_driver_apply_fixed_step (d, &t, 1e-3, 1000, y);

           if (s != GSL_SUCCESS)
             {
               printf ("error: driver returned %d\n", s);
               break;
             }

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv2_driver_free (d);
       return s;
     }


File: gsl-ref.info,  Node: ODE References and Further Reading,  Prev: ODE Example programs,  Up: Ordinary Differential Equations

26.7 References and Further Reading
===================================

     Ascher, U.M., Petzold, L.R., `Computer Methods for Ordinary
     Differential and Differential-Algebraic Equations', SIAM,
     Philadelphia, 1998.

     Hairer, E., Norsett, S. P., Wanner, G., `Solving Ordinary
     Differential Equations I: Nonstiff Problems', Springer, Berlin,
     1993.

     Hairer, E., Wanner, G., `Solving Ordinary Differential Equations
     II: Stiff and Differential-Algebraic Problems', Springer, Berlin,
     1996.

   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.

The Adams and BDF multistep methods `msadams' and `msbdf' are based on
the following articles,

     G. D. Byrne and A. C. Hindmarsh, "A Polyalgorithm for the
     Numerical Solution of Ordinary Differential Equations.", ACM
     Trans. Math. Software, 1, 71-96, 1975.

     P. N. Brown, G. D. Byrne and A. C. Hindmarsh, "VODE: A
     Variable-coefficient ODE Solver.", SIAM J. Sci. Stat. Comput. 10,
     1038-1051, 1989.

     A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban,
     D. E. Shumaker and C. S. Woodward, "SUNDIALS: Suite of Nonlinear
     and Differential/Algebraic Equation Solvers.", ACM Trans. Math.
     Software 31, 363-396, 2005.


File: gsl-ref.info,  Node: Interpolation,  Next: Numerical Differentiation,  Prev: Ordinary Differential Equations,  Up: Top

27 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.

   These interpolation methods produce curves that pass through each
datapoint.  To interpolate noisy data with a smoothing curve see *Note
Basis Splines::.

   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

27.1 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

27.2 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, with
     increasing x values; 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

27.3 Interpolation Types
========================

The interpolation library provides six 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)
 -- Function: unsigned int gsl_interp_type_min_size (const
          gsl_interp_type * T)
     These functions return the minimum number of points required by the
     interpolation object INTERP or interpolation type T.  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

27.4 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].  An inline version of this function
     is used when `HAVE_INLINE' is defined.

 -- 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]'.  An
     inline version of this function is used when `HAVE_INLINE' is
     defined.

 -- Function: int gsl_interp_accel_reset (gsl_interp_accel * ACC);
     This function reinitializes the accelerator object ACC.  It should
     be used when the cached information is no longer applicable--for
     example, when switching to a new dataset.

 -- 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

27.5 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.  When X is outside the range of XA,
     the error code `GSL_EDOM' is returned with a value of `GSL_NAN' for
     Y.

 -- 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

27.6 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

27.7 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 be 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: y[0] == y[3] 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

27.8 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

28 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

28.1 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+h, 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

28.2 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

28.3 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

29 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::
* Auxiliary Functions for 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

29.1 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: Auxiliary Functions for Chebyshev Series,  Prev: Chebyshev Definitions,  Up: Chebyshev Approximations

29.2 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: Auxiliary Functions for Chebyshev Series,  Next: Chebyshev Series Evaluation,  Prev: Creation and Calculation of Chebyshev Series,  Up: Chebyshev Approximations

29.3 Auxiliary Functions
========================

The following functions provide information about an existing Chebyshev
series.

 -- Function: size_t gsl_cheb_order (const gsl_cheb_series * CS)
     This function returns the order of Chebyshev series CS.

 -- Function: size_t gsl_cheb_size (const gsl_cheb_series * CS)
 -- Function: double * gsl_cheb_coeffs (const gsl_cheb_series * CS)
     These functions return the size of the Chebyshev coefficient array
     `c[]' and a pointer to its location in memory for the Chebyshev
     series CS.


File: gsl-ref.info,  Node: Chebyshev Series Evaluation,  Next: Derivatives and Integrals,  Prev: Auxiliary Functions for Chebyshev Series,  Up: Chebyshev Approximations

29.4 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
     X, 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

29.5 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

29.6 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

29.7 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

30 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

30.1 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: void 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.
     All the terms of the series passed in through ARRAY should be
     non-zero.


File: gsl-ref.info,  Node: Acceleration functions without error estimation,  Next: Example of accelerating a series,  Prev: Acceleration functions,  Up: Series Acceleration

30.2 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: void 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

30.3 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

30.4 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

31 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

31.1 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

31.2 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
     This 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. A null pointer is returned if insufficient
     memory is available.

 -- 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

31.3 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

31.3.1 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

31.3.2 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 form 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

31.4 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));

       FILE * f;
       gsl_wavelet *w;
       gsl_wavelet_workspace *work;

       w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
       work = gsl_wavelet_workspace_alloc (n);

       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]);
         }

       gsl_wavelet_free (w);
       gsl_wavelet_workspace_free (work);

       free (data);
       free (abscoeff);
       free (p);
       return 0;
     }

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

31.5 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

32 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

32.1 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 and
j_(\nu,m) the m-th zero of the Bessel function J_\nu(x).  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)t) / 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

32.2 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 XMAX.

 -- 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 XMAX.

 -- 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.

     Applying this function to its output gives the original data
     multiplied by (1/j_(\nu,M))^2, up to numerical errors.

 -- 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

32.3 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

33 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

33.1 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

33.2 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

33.3 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

33.4 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 as a function pointer:

     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 macro `GSL_FN_EVAL(&F,x)'
defined in `gsl_math.h'.

 -- 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 macro
`GSL_FN_FDF_EVAL_F(&FDF,x)' and the derivative f'(x) can be evaluated
using the macro `GSL_FN_FDF_EVAL_DF(&FDF,x)'.  Both the function y =
f(x) and its derivative dy = f'(x) can be evaluated at the same time
using the macro `GSL_FN_FDF_EVAL_F_DF(&FDF,x,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

33.5 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 an 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

33.6 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

33.7 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

33.8 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

33.9 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"(1) 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.

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

   (1) J.F. Steffensen (1873-1961). The spelling used in the name of
the function is slightly incorrect, but has been preserved to avoid
incompatibility.


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

33.10 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);

       gsl_root_fsolver_free (s);

       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);

       gsl_root_fdfsolver_free (s);
       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

33.11 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

34 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

34.1 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

34.2 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

34.3 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

34.4 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

34.5 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

34.6 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

34.7 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.

 -- Minimizer: gsl_min_fminimizer_quad_golden
     This is a variant of Brent's algorithm which uses the safeguarded
     step-length algorithm of Gill and Murray.


File: gsl-ref.info,  Node: Minimization Examples,  Next: Minimization References and Further Reading,  Prev: Minimization Algorithms,  Up: One dimensional Minimization

34.8 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\n",
                   iter, a, b,
                   m, m - m_expected, b - a);
         }
       while (status == GSL_CONTINUE && iter < max_iter);

       gsl_min_fminimizer_free (s);

       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

34.9 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

35 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

35.1 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

35.2 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, const gsl_vector * X)
 -- Function: int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver
          * S, gsl_multiroot_function_fdf * FDF, const gsl_vector * X)
     These functions set, or reset, an existing solver S to use the
     function F or function and derivative FDF, and the initial guess
     X.  Note that the initial position is copied from X, this argument
     is not modified by subsequent iterations.

 -- 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

35.3 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

35.4 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 `s->x' and
its function value `s->f' 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, given by `s->x'.

 -- 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, given by `s->f'.

 -- 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,
     given by `s->dx'.


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

35.5 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

35.6 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.  If the Jacobian matrix is singular, an
     error code of `GSL_EDOM' is returned.

 -- 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

35.7 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'.  Note that this step size will not be
     suitable for all problems.

 -- 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

35.8 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

35.9 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

36 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 with Derivatives::
* Multimin Algorithms without Derivatives::
* Multimin Examples::
* Multimin References and Further Reading::


File: gsl-ref.info,  Node: Multimin Overview,  Next: Multimin Caveats,  Up: Multidimensional Minimization

36.1 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.

   Algorithms which do not require the gradient of the function use
different strategies.  For example, the Nelder-Mead Simplex algorithm
maintains n+1 trial parameter vectors as the vertices of a
n-dimensional simplex.  On each iteration it tries to improve the worst
vertex of the simplex by geometrical transformations.  The iterations
are continued until the overall size of the simplex has decreased
sufficiently.

   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

36.2 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

36.3 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)
 -- Function: int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer
          * S, gsl_multimin_function * F, const gsl_vector * X, const
          gsl_vector * STEP_SIZE)
     The function `gsl_multimin_fdfminimizer_set' 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|.  A TOL value of 0.1 is suitable for most purposes, since
     line minimization only needs to be carried out approximately.
     Note that setting TOL to zero will force the use of "exact"
     line-searches, which are extremely expensive.

     The function `gsl_multimin_fminimizer_set' 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

36.4 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.  If the function cannot be
          computed, an error value of `GSL_NAN' should be returned.

    `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.  If the function cannot be
          computed, an error value of `GSL_NAN' should be returned.

    `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 two-dimensional
paraboloid with five parameters,

     /* Paraboloid centered on (p[0],p[1]), with
        scale factors (p[2],p[3]) and minimum p[4] */

     double
     my_f (const gsl_vector *v, void *params)
     {
       double x, y;
       double *p = (double *)params;

       x = gsl_vector_get(v, 0);
       y = gsl_vector_get(v, 1);

       return p[2] * (x - p[0]) * (x - p[0]) +
                p[3] * (y - p[1]) * (y - p[1]) + p[4];
     }

     /* 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 *p = (double *)params;

       x = gsl_vector_get(v, 0);
       y = gsl_vector_get(v, 1);

       gsl_vector_set(df, 0, 2.0 * p[2] * (x - p[0]));
       gsl_vector_set(df, 1, 2.0 * p[3] * (y - p[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;

     /* Paraboloid center at (1,2), scale factors (10, 20),
        minimum value 30 */
     double p[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };

     my_func.n = 2;  /* number of function components */
     my_func.f = &my_f;
     my_func.df = &my_df;
     my_func.fdf = &my_fdf;
     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

36.5 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 error code `GSL_ENOPROG' signifies that the
     minimizer is unable to improve on its current estimate, either due
     to numerical difficulty or because a genuine local minimum has been
     reached.

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 with Derivatives,  Prev: Multimin Iteration,  Up: Multidimensional Minimization

36.6 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 with Derivatives,  Next: Multimin Algorithms without Derivatives,  Prev: Multimin Stopping Criteria,  Up: Multidimensional Minimization

36.7 Algorithms with Derivatives
================================

There are several minimization methods available. The best choice of
algorithm depends on the problem.  The algorithms described in this
section use the value of the function and its gradient at each
evaluation point.

 -- Minimizer: gsl_multimin_fdfminimizer_conjugate_fr
     This is the Fletcher-Reeves conjugate gradient algorithm. The
     conjugate gradient algorithm proceeds as a succession of line
     minimizations. The sequence of search directions 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_bfgs2
 -- Minimizer: gsl_multimin_fdfminimizer_vector_bfgs
     These methods use the vector Broyden-Fletcher-Goldfarb-Shanno
     (BFGS) algorithm.  This 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.

     The `bfgs2' version of this minimizer is the most efficient
     version available, and is a faithful implementation of the line
     minimization scheme described in Fletcher's `Practical Methods of
     Optimization', Algorithms 2.6.2 and 2.6.4.  It supersedes the
     original `bfgs' routine and requires substantially fewer function
     and gradient evaluations.  The user-supplied tolerance TOL
     corresponds to the parameter \sigma used by Fletcher.  A value of
     0.1 is recommended for typical use (larger values correspond to
     less accurate line searches).


 -- 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.


File: gsl-ref.info,  Node: Multimin Algorithms without Derivatives,  Next: Multimin Examples,  Prev: Multimin Algorithms with Derivatives,  Up: Multidimensional Minimization

36.8 Algorithms without Derivatives
===================================

The algorithms described in this section use only the value of the
function at each evaluation point.

 -- Minimizer: gsl_multimin_fminimizer_nmsimplex2
 -- Minimizer: gsl_multimin_fminimizer_nmsimplex
     These methods use the Simplex algorithm of Nelder and Mead.
     Starting from the initial vector X = p_0, the algorithm constructs
     an additional n vectors p_i using the step size vector s =
     STEP_SIZE as follows:

          p_0 = (x_0, x_1, ... , x_n)
          p_1 = (x_0 + s_0, x_1, ... , x_n)
          p_2 = (x_0, x_1 + s_1, ... , x_n)
          ... = ...
          p_n = (x_0, x_1, ... , x_n + s_n)

     These vectors form the n+1 vertices of a simplex in n dimensions.
     On each iteration the algorithm uses simple geometrical
     transformations to update the vector corresponding to the highest
     function value.  The geometric transformations are reflection,
     reflection followed by expansion, contraction and multiple
     contraction.  Using these transformations the simplex moves through
     the 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 minimizer-specific characteristic size is calculated 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'.

     The `nmsimplex2' version of this minimiser is a new O(N) operations
     implementation of the earlier O(N^2) operations `nmsimplex'
     minimiser.  It uses the same underlying algorithm, but the simplex
     updates are computed more efficiently for high-dimensional
     problems.  In addition, the size of simplex is calculated as the
     RMS distance of each vertex from the center rather than the mean
     distance, allowing a linear update of this quantity on each step.
     The memory usage is O(N^2) for both algorithms.

 -- Minimizer: gsl_multimin_fminimizer_nmsimplex2rand
     This method is a variant of `nmsimplex2' which initialises the
     simplex around the starting point X using a randomly-oriented set
     of basis vectors instead of the fixed coordinate axes. The final
     dimensions of the simplex are scaled along the coordinate axes by
     the vector STEP_SIZE.  The randomisation uses a simple
     deterministic generator so that repeated calls to
     `gsl_multimin_fminimizer_set' for a given solver object will vary
     the orientation in a well-defined way.


File: gsl-ref.info,  Node: Multimin Examples,  Next: Multimin References and Further Reading,  Prev: Multimin Algorithms without Derivatives,  Up: Multidimensional Minimization

36.9 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), scale factors
          10,20, height 30. */
       double par[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };

       gsl_vector *x;
       gsl_multimin_function_fdf my_func;

       my_func.n = 2;
       my_func.f = my_f;
       my_func.df = my_df;
       my_func.fdf = my_fdf;
       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)
     {
       double par[5] = {1.0, 2.0, 10.0, 20.0, 30.0};

       const gsl_multimin_fminimizer_type *T =
         gsl_multimin_fminimizer_nmsimplex2;
       gsl_multimin_fminimizer *s = NULL;
       gsl_vector *ss, *x;
       gsl_multimin_function minex_func;

       size_t iter = 0;
       int status;
       double size;

       /* Starting point */
       x = gsl_vector_alloc (2);
       gsl_vector_set (x, 0, 5.0);
       gsl_vector_set (x, 1, 7.0);

       /* Set initial step sizes to 1 */
       ss = gsl_vector_alloc (2);
       gsl_vector_set_all (ss, 1.0);

       /* Initialize method and iterate */
       minex_func.n = 2;
       minex_func.f = my_f;
       minex_func.params = par;

       s = gsl_multimin_fminimizer_alloc (T, 2);
       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 %10.3e %10.3e f() = %7.3f size = %.3f\n",
                   iter,
                   gsl_vector_get (s->x, 0),
                   gsl_vector_get (s->x, 1),
                   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 = 1.130
         2  5.250e+00  4.000e+00 f() = 290.625 size = 1.409
         3  5.250e+00  4.000e+00 f() = 290.625 size = 1.409
         4  5.500e+00  1.000e+00 f() = 252.500 size = 1.409
         5  2.625e+00  3.500e+00 f() = 101.406 size = 1.847
         6  2.625e+00  3.500e+00 f() = 101.406 size = 1.847
         7  0.000e+00  3.000e+00 f() =  60.000 size = 1.847
         8  2.094e+00  1.875e+00 f() =  42.275 size = 1.321
         9  2.578e-01  1.906e+00 f() =  35.684 size = 1.069
        10  5.879e-01  2.445e+00 f() =  35.664 size = 0.841
        11  1.258e+00  2.025e+00 f() =  30.680 size = 0.476
        12  1.258e+00  2.025e+00 f() =  30.680 size = 0.367
        13  1.093e+00  1.849e+00 f() =  30.539 size = 0.300
        14  8.830e-01  2.004e+00 f() =  30.137 size = 0.172
        15  8.830e-01  2.004e+00 f() =  30.137 size = 0.126
        16  9.582e-01  2.060e+00 f() =  30.090 size = 0.106
        17  1.022e+00  2.004e+00 f() =  30.005 size = 0.063
        18  1.022e+00  2.004e+00 f() =  30.005 size = 0.043
        19  1.022e+00  2.004e+00 f() =  30.005 size = 0.043
        20  1.022e+00  2.004e+00 f() =  30.005 size = 0.027
        21  1.022e+00  2.004e+00 f() =  30.005 size = 0.022
        22  9.920e-01  1.997e+00 f() =  30.001 size = 0.016
        23  9.920e-01  1.997e+00 f() =  30.001 size = 0.013
     converged to minimum at
        24  9.920e-01  1.997e+00 f() =  30.001 size = 0.008

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

36.10 References and Further Reading
====================================

The conjugate gradient and BFGS methods are described in detail in the
following book,

     R. Fletcher, `Practical Methods of Optimization (Second Edition)'
     Wiley (1987), ISBN 0471915475.

   A brief description of multidimensional minimization algorithms and
more recent references can be found in,

     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), 308-313.



File: gsl-ref.info,  Node: Least-Squares Fitting,  Next: Nonlinear Least-Squares Fitting,  Prev: Multidimensional Minimization,  Up: Top

37 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::
* Robust linear regression::
* Troubleshooting::
* Fitting Examples::
* Fitting References and Further Reading::


File: gsl-ref.info,  Node: Fitting Overview,  Next: Linear regression,  Up: Least-Squares Fitting

37.1 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 < > 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

     \delta c_a = \sum_i (dc_a/dy_i) \delta y_i

allowing the covariance matrix to be written in terms of the errors on
the data,

     C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>

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

     C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i)

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}}.  The correlation coefficient of
the fit parameters c_a and c_b is given by \rho_{ab} = C_{ab} /
\sqrt{C_{aa} C_{bb}}.


File: gsl-ref.info,  Node: Linear regression,  Next: Linear fitting without a constant term,  Prev: Fitting Overview,  Up: Least-Squares Fitting

37.2 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.  Note: the correlation
     coefficient of the data can be computed using
     `gsl_stats_correlation' (*note Correlation::), it does not depend
     on the fit.

 -- 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 COV00, double COV01, double COV11, 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

37.3 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 COV11,
          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: Robust linear regression,  Prev: Linear fitting without a constant term,  Up: Least-Squares Fitting

37.4 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.  The chi-squared value is given by \chi^2 = \sum_i w_i (y_i
- \sum_j X_{ij} c_j)^2.

   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)
     This function computes the best-fit parameters C of the model y =
     X c for the observations Y and the matrix of predictor variables
     X, using the preallocated workspace provided in WORK.  The p-by-p
     variance-covariance matrix of the model parameters COV is set to
     \sigma^2 (X^T X)^-1, where \sigma is the standard deviation of the
     fit residuals.  The sum of squares of the residuals from the
     best-fit, \chi^2, is returned in CHISQ. If the coefficient of
     determination is desired, it can be computed from the expression
     R^2 = 1 - \chi^2 / TSS, where the total sum of squares (TSS) of
     the observations Y may be computed from `gsl_stats_tss'.

     The best-fit is found by singular value decomposition of the matrix
     X using the modified Golub-Reinsch SVD algorithm, with column
     scaling to improve the accuracy of the singular values. Any
     components which have zero singular value (to machine precision)
     are discarded from the fit.

 -- Function: int gsl_multifit_wlinear (const gsl_matrix * X, const
          gsl_vector * W, const gsl_vector * Y, gsl_vector * C,
          gsl_matrix * COV, double * CHISQ,
          gsl_multifit_linear_workspace * WORK)
     This function computes the best-fit parameters C of the weighted
     model y = X c for the observations Y with weights W and the matrix
     of predictor variables X, using the preallocated workspace
     provided in WORK.  The p-by-p covariance matrix of the model
     parameters COV is computed as (X^T W X)^-1. The weighted sum of
     squares of the residuals from the best-fit, \chi^2, is returned in
     CHISQ. If the coefficient of determination is desired, it can be
     computed from the expression R^2 = 1 - \chi^2 / WTSS, where the
     weighted total sum of squares (WTSS) of the observations Y may be
     computed from `gsl_stats_wtss'.

 -- 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)
 -- 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)
     In these functions components of the fit 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_usvd (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)
 -- Function: int gsl_multifit_wlinear_usvd (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)
     These functions compute the fit using an SVD without column
     scaling.

 -- 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.

 -- Function: int gsl_multifit_linear_residuals (const gsl_matrix * X,
          const gsl_vector * Y, const gsl_vector * C, gsl_vector * R)
     This function computes the vector of residuals r = y - X c for the
     observations Y, coefficients C and matrix of predictor variables X.


File: gsl-ref.info,  Node: Robust linear regression,  Next: Troubleshooting,  Prev: Multi-parameter fitting,  Up: Least-Squares Fitting

37.5 Robust linear regression
=============================

Ordinary least squares (OLS) models are often heavily influenced by the
presence of outliers.  Outliers are data points which do not follow the
general trend of the other observations, although there is strictly no
precise definition of an outlier. Robust linear regression refers to
regression algorithms which are robust to outliers. The most common
type of robust regression is M-estimation. The general M-estimator
minimizes the objective function

     \sum_i \rho(e_i) = \sum_i \rho (y_i - Y(c, x_i))

   where e_i = y_i - Y(c, x_i) is the residual of the ith data point,
and \rho(e_i) is a function which should have the following properties:
     \rho(e) \ge 0

     \rho(0) = 0

     \rho(-e) = \rho(e)

     \rho(e_1) > \rho(e_2) for |e_1| > |e_2|
   The special case of ordinary least squares is given by \rho(e_i) =
e_i^2.  Letting \psi = \rho' be the derivative of \rho, differentiating
the objective function with respect to the coefficients c and setting
the partial derivatives to zero produces the system of equations

     \sum_i \psi(e_i) X_i = 0

   where X_i is a vector containing row i of the design matrix X.
Next, we define a weight function w(e) = \psi(e)/e, and let w_i =
w(e_i):

     \sum_i w_i e_i X_i = 0

   This system of equations is equivalent to solving a weighted
ordinary least squares problem, minimizing \chi^2 = \sum_i w_i e_i^2.
The weights however, depend on the residuals e_i, which depend on the
coefficients c, which depend on the weights. Therefore, an iterative
solution is used, called Iteratively Reweighted Least Squares (IRLS).
  1. Compute initial estimates of the coefficients c^(0) using ordinary
     least squares

  2. For iteration k, form the residuals e_i^(k) = (y_i - X_i
     c^(k-1))/(t \sigma^(k) \sqrt1 - h_i), where t is a tuning constant
     depending on the choice of \psi, and h_i are the statistical
     leverages (diagonal elements of the matrix X (X^T X)^-1 X^T).
     Including t and h_i in the residual calculation has been shown to
     improve the convergence of the method.  The residual standard
     deviation is approximated as \sigma^(k) = MAD / 0.6745, where MAD
     is the Median-Absolute-Deviation of the n-p largest residuals from
     the previous iteration.

  3. Compute new weights w_i^(k) = \psi(e_i^(k))/e_i^(k).

  4. Compute new coefficients c^(k) by solving the weighted least
     squares problem with weights w_i^(k).

  5. Steps 2 through 4 are iterated until the coefficients converge or
     until some maximum iteration limit is reached. Coefficients are
     tested for convergence using the critera:

          |c_i^(k) - c_i^(k-1)| \le \epsilon \times max(|c_i^(k)|, |c_i^(k-1)|)

     for all 0 \le i < p where \epsilon is a small tolerance factor.
        The key to this method lies in selecting the function \psi(e_i)
to assign smaller weights to large residuals, and larger weights to
smaller residuals. As the iteration proceeds, outliers are assigned
smaller and smaller weights, eventually having very little or no effect
on the fitted model.

 -- Function: gsl_multifit_robust_workspace * gsl_multifit_robust_alloc
          (const gsl_multifit_robust_type * T, const size_t N, const
          size_t P)
     This function allocates a workspace for fitting a model to N
     observations using P parameters. The type T specifies the function
     \psi and can be selected from the following choices.

      -- Robust type: gsl_multifit_robust_default
          This specifies the `gsl_multifit_robust_bisquare' type (see
          below) and is a good general purpose choice for robust
          regression.

      -- Robust type: gsl_multifit_robust_bisquare
          This is Tukey's biweight (bisquare) function and is a good
          general purpose choice for robust regression. The weight
          function is given by

               w(e) = (1 - e^2)^2

          and the default tuning constant is t = 4.685.

      -- Robust type: gsl_multifit_robust_cauchy
          This is Cauchy's function, also known as the Lorentzian
          function.  This function does not guarantee a unique solution,
          meaning different choices of the coefficient vector C could
          minimize the objective function. Therefore this option should
          be used with care. The weight function is given by

               w(e) = 1 / (1 + e^2)

          and the default tuning constant is t = 2.385.

      -- Robust type: gsl_multifit_robust_fair
          This is the fair \rho function, which guarantees a unique
          solution and has continuous derivatives to three orders. The
          weight function is given by

               w(e) = 1 / (1 + |e|)

          and the default tuning constant is t = 1.400.

      -- Robust type: gsl_multifit_robust_huber
          This specifies Huber's \rho function, which is a parabola in
          the vicinity of zero and increases linearly for a given
          threshold |e| > t. This function is also considered an
          excellent general purpose robust estimator, however,
          occasional difficulties can be encountered due to the
          discontinuous first derivative of the \psi function.  The
          weight function is given by

               w(e) = 1/max(1,|e|)

          and the default tuning constant is t = 1.345.

      -- Robust type: gsl_multifit_robust_ols
          This specifies the ordinary least squares solution, which can
          be useful for quickly checking the difference between the
          various robust and OLS solutions. The weight function is
          given by

               w(e) = 1

          and the default tuning constant is t = 1.

      -- Robust type: gsl_multifit_robust_welsch
          This specifies the Welsch function which can perform well in
          cases where the residuals have an exponential distribution.
          The weight function is given by

               w(e) = \exp(-e^2)

          and the default tuning constant is t = 2.985.

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

 -- Function: const char * gsl_multifit_robust_name (const
          gsl_multifit_robust_workspace * W)
     This function returns the name of the robust type T specified to
     `gsl_multifit_robust_alloc'.

 -- Function: int gsl_multifit_robust_tune (const double TUNE,
          gsl_multifit_robust_workspace * W)
     This function sets the tuning constant t used to adjust the
     residuals at each iteration to TUNE.  Decreasing the tuning
     constant increases the downweight assigned to large residuals,
     while increasing the tuning constant decreases the downweight
     assigned to large residuals.

 -- Function: int gsl_multifit_robust (const gsl_matrix * X, const
          gsl_vector * Y, gsl_vector * C, gsl_matrix * COV,
          gsl_multifit_robust_workspace * W)
     This function computes the best-fit parameters C of the model y =
     X c for the observations Y and the matrix of predictor variables
     X, attemping to reduce the influence of outliers using the
     algorithm outlined above.  The p-by-p variance-covariance matrix
     of the model parameters COV is estimated as \sigma^2 (X^T X)^-1,
     where \sigma is an approximation of the residual standard
     deviation using the theory of robust regression. Special care must
     be taken when estimating \sigma and other statistics such as R^2,
     and so these are computed internally and are available by calling
     the function `gsl_multifit_robust_statistics'.

 -- Function: int gsl_multifit_robust_est (const gsl_vector * X, const
          gsl_vector * C, const gsl_matrix * COV, double * Y, double *
          Y_ERR)
     This function uses the best-fit robust 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.

 -- Function: gsl_multifit_robust_stats gsl_multifit_robust_statistics
          (const gsl_multifit_robust_workspace * W)
     This function returns a structure containing relevant statistics
     from a robust regression. The function `gsl_multifit_robust' must
     be called first to perform the regression and calculate these
     statistics.  The returned `gsl_multifit_robust_stats' structure
     contains the following fields.
          double `sigma_ols' This contains the standard deviation of
          the residuals as computed from ordinary least squares (OLS).

          double `sigma_mad' This contains an estimate of the standard
          deviation of the final residuals using the
          Median-Absolute-Deviation statistic

          double `sigma_rob' This contains an estimate of the standard
          deviation of the final residuals from the theory of robust
          regression (see Street et al, 1988).

          double `sigma' This contains an estimate of the standard
          deviation of the final residuals by attemping to reconcile
          `sigma_rob' and `sigma_ols' in a reasonable way.

          double `Rsq' This contains the R^2 coefficient of
          determination statistic using the estimate `sigma'.

          double `adj_Rsq' This contains the adjusted R^2 coefficient
          of determination statistic using the estimate `sigma'.

          double `rmse' This contains the root mean squared error of
          the final residuals

          double `sse' This contains the residual sum of squares taking
          into account the robust covariance matrix.

          size_t `dof' This contains the number of degrees of freedom n
          - p

          size_t `numit' Upon successful convergence, this contains the
          number of iterations performed

          gsl_vector * `weights' This contains the final weight vector
          of length N

          gsl_vector * `r' This contains the final residual vector of
          length N, r = y - X c


File: gsl-ref.info,  Node: Troubleshooting,  Next: Fitting Examples,  Prev: Robust linear regression,  Up: Least-Squares Fitting

37.6 Troubleshooting
====================

When using models based on polynomials, care should be taken when
constructing the design matrix X. If the x values are large, then the
matrix X could be ill-conditioned since its columns are powers of x,
leading to unstable least-squares solutions.  In this case it can often
help to center and scale the x values using the mean and standard
deviation:

     x' = (x - mu)/sigma

and then construct the X matrix using the transformed values x'.


File: gsl-ref.info,  Node: Fitting Examples,  Next: Fitting References and Further Reading,  Prev: Troubleshooting,  Up: Least-Squares Fitting

37.7 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);
       }

       gsl_matrix_free (X);
       gsl_vector_free (y);
       gsl_vector_free (w);
       gsl_vector_free (c);
       gsl_matrix_free (cov);

       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);
         }

       gsl_rng_free(r);

       return 0;
     }

The data can be prepared by running the resulting executable program,

     $ GSL_RNG_TYPE=mt19937_1999 ./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.

   The next program demonstrates the advantage of robust least squares
on a dataset with outliers. The program generates linear (x,y) data
pairs on the line y = 1.45 x + 3.88, adds some random noise, and
inserts 3 outliers into the dataset. Both the robust and ordinary least
squares (OLS) coefficients are computed for comparison.

     #include <stdio.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_randist.h>

     int
     dofit(const gsl_multifit_robust_type *T,
           const gsl_matrix *X, const gsl_vector *y,
           gsl_vector *c, gsl_matrix *cov)
     {
       int s;
       gsl_multifit_robust_workspace * work
         = gsl_multifit_robust_alloc (T, X->size1, X->size2);

       s = gsl_multifit_robust (X, y, c, cov, work);
       gsl_multifit_robust_free (work);

       return s;
     }

     int
     main (int argc, char **argv)
     {
       int i;
       size_t n;
       const size_t p = 2; /* linear fit */
       gsl_matrix *X, *cov;
       gsl_vector *x, *y, *c, *c_ols;
       const double a = 1.45; /* slope */
       const double b = 3.88; /* intercept */
       gsl_rng *r;

       if (argc != 2)
         {
           fprintf (stderr,"usage: robfit n\n");
           exit (-1);
         }

       n = atoi (argv[1]);

       X = gsl_matrix_alloc (n, p);
       x = gsl_vector_alloc (n);
       y = gsl_vector_alloc (n);

       c = gsl_vector_alloc (p);
       c_ols = gsl_vector_alloc (p);
       cov = gsl_matrix_alloc (p, p);

       r = gsl_rng_alloc(gsl_rng_default);

       /* generate linear dataset */
       for (i = 0; i < n - 3; i++)
         {
           double dx = 10.0 / (n - 1.0);
           double ei = gsl_rng_uniform(r);
           double xi = -5.0 + i * dx;
           double yi = a * xi + b;

           gsl_vector_set (x, i, xi);
           gsl_vector_set (y, i, yi + ei);
         }

       /* add a few outliers */
       gsl_vector_set(x, n - 3, 4.7);
       gsl_vector_set(y, n - 3, -8.3);

       gsl_vector_set(x, n - 2, 3.5);
       gsl_vector_set(y, n - 2, -6.7);

       gsl_vector_set(x, n - 1, 4.1);
       gsl_vector_set(y, n - 1, -6.0);

       /* construct design matrix X for linear fit */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);

           gsl_matrix_set (X, i, 0, 1.0);
           gsl_matrix_set (X, i, 1, xi);
         }

       /* perform robust and OLS fit */
       dofit(gsl_multifit_robust_ols, X, y, c_ols, cov);
       dofit(gsl_multifit_robust_bisquare, X, y, c, cov);

       /* output data and model */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);
           gsl_vector_view v = gsl_matrix_row(X, i);
           double y_ols, y_rob, y_err;

           gsl_multifit_robust_est(&v.vector, c, cov, &y_rob, &y_err);
           gsl_multifit_robust_est(&v.vector, c_ols, cov, &y_ols, &y_err);

           printf("%g %g %g %g\n", xi, yi, y_rob, y_ols);
         }

     #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\n",
                 C(0), C(1));

         printf ("# covariance matrix:\n");
         printf ("# [ %+.5e, %+.5e\n",
                    COV(0,0), COV(0,1));
         printf ("#   %+.5e, %+.5e\n",
                    COV(1,0), COV(1,1));
       }

       gsl_matrix_free (X);
       gsl_vector_free (x);
       gsl_vector_free (y);
       gsl_vector_free (c);
       gsl_vector_free (c_ols);
       gsl_matrix_free (cov);
       gsl_rng_free(r);

       return 0;
     }

   The output from the program is shown in the following plot.


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

37.8 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'.

The GSL implementation of robust linear regression closely follows the
publications

     DuMouchel, W. and F. O'Brien (1989), "Integrating a robust option
     into a multiple regression computing environment," Computer
     Science and Statistics:  Proceedings of the 21st Symposium on the
     Interface, American Statistical Association

     Street, J.O., R.J. Carroll, and D. Ruppert (1988), "A note on
     computing robust regression estimates via iteratively reweighted
     least squares," The American Statistician, v. 42, pp. 152-154.


File: gsl-ref.info,  Node: Nonlinear Least-Squares Fitting,  Next: Basis Splines,  Prev: Least-Squares Fitting,  Up: Top

38 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::
* Finite Difference Jacobian::
* Iteration of the Minimization Algorithm::
* Search Stopping Parameters for Minimization Algorithms::
* High Level Driver::
* 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

38.1 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

38.2 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, const 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_multifit_function_fdf * FDF, const 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_fsolver * 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: Finite Difference Jacobian,  Prev: Initializing the Nonlinear Least-Squares Solver,  Up: Nonlinear Least-Squares Fitting

38.3 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. If an analytic Jacobian is
          unavailable, or too expensive to compute, this function
          pointer may be set to NULL, in which case the Jacobian will
          be internally computed using finite difference approximations
          of the function F.

    `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. If an analytic
          Jacobian is unavailable, or too expensive to compute, this
          function pointer may be set to NULL, in which case the
          Jacobian will be internally computed using finite difference
          approximations of the function F.

    `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: Finite Difference Jacobian,  Next: Iteration of the Minimization Algorithm,  Prev: Providing the Function to be Minimized,  Up: Nonlinear Least-Squares Fitting

38.4 Finite Difference Jacobian
===============================

For the algorithms which require a Jacobian matrix of derivatives of
the fit functions, there are times when an analytic Jacobian may be
unavailable or too expensive to compute. Therefore GSL supports
approximating the Jacobian numerically using finite differences of the
fit functions. This is typically done by setting the relevant function
pointers of the `gsl_multifit_function_fdf' data type to NULL, however
the following functions allow the user to access the approximate
Jacobian directly if needed.

 -- Function: int gsl_multifit_fdfsolver_dif_df (const gsl_vector * X,
          gsl_multifit_function_fdf * FDF, const gsl_vector * F,
          gsl_matrix * J)
     This function takes as input the current position X with the
     function values computed at the current position F, along with FDF
     which specifies the fit function and parameters and approximates
     the N-by-P Jacobian J using forward finite differences: J_ij = d
     f_i(x,params) / d x_j = (f_i(x^*,params) - f_i(x,params)) / d x_j.
     where x^* has the jth element perturbed by \Delta x_j and \Delta
     x_j = \epsilon |x_j|, where \epsilon is the square root of the
     machine precision `GSL_DBL_EPSILON'.

 -- Function: int gsl_multifit_fdfsolver_dif_fdf (const gsl_vector * X,
          gsl_multifit_function_fdf * FDF, gsl_vector * F, gsl_matrix *
          J)
     This function computes the vector of function values F and the
     approximate Jacobian J at the position vector X using the system
     described in FDF. See `gsl_multifit_fdfsolver_dif_df' for a
     description of how the Jacobian is computed.


File: gsl-ref.info,  Node: Iteration of the Minimization Algorithm,  Next: Search Stopping Parameters for Minimization Algorithms,  Prev: Finite Difference Jacobian,  Up: Nonlinear Least-Squares Fitting

38.5 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: High Level Driver,  Prev: Iteration of the Minimization Algorithm,  Up: Nonlinear Least-Squares Fitting

38.6 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: High Level Driver,  Next: Minimization Algorithms using Derivatives,  Prev: Search Stopping Parameters for Minimization Algorithms,  Up: Nonlinear Least-Squares Fitting

38.7 High Level Driver
======================

These routines provide a high level wrapper that combine the iteration
and convergence testing for easy use.

 -- Function: int gsl_multifit_fsolver_driver (gsl_multifit_fsolver *
          S, const size_t MAXITER, const double EPSABS, const double
          EPSREL)
 -- Function: int gsl_multifit_fdfsolver_driver (gsl_multifit_fdfsolver
          * S, const size_t MAXITER, const double EPSABS, const double
          EPSREL)
     These functions iterate the solver S for a maximum of MAXITER
     iterations. After each iteration, the system is tested for
     convergence using `gsl_multifit_test_delta' with the error
     tolerances EPSABS and EPSREL.


File: gsl-ref.info,  Node: Minimization Algorithms using Derivatives,  Next: Minimization Algorithms without Derivatives,  Prev: High Level Driver,  Up: Nonlinear Least-Squares Fitting

38.8 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

    `GSL_ENOPROG'
          the routine has made 10 or more attempts to find a suitable
          trial step without success (but subsequent calls can be made
          to continue the search).(1)

     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.

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

   (1) The return code `GSL_CONTINUE' was used for this case in
versions prior to 1.14.


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

38.9 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

38.10 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

38.11 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;
       unsigned int 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: %u %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);
       gsl_matrix_free (covar);
       gsl_rng_free (r);
       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

38.12 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: Basis Splines,  Next: Physical Constants,  Prev: Nonlinear Least-Squares Fitting,  Up: Top

39 Basis Splines
****************

This chapter describes functions for the computation of smoothing basis
splines (B-splines). A smoothing spline differs from an interpolating
spline in that the resulting curve is not required to pass through each
datapoint.  *Note Interpolation::, for information about interpolating
splines.

   The header file `gsl_bspline.h' contains the prototypes for the
bspline functions and related declarations.

* Menu:

* Overview of B-splines::
* Initializing the B-splines solver::
* Constructing the knots vector::
* Evaluation of B-spline basis functions::
* Evaluation of B-spline basis function derivatives::
* Working with the Greville abscissae::
* Example programs for B-splines::
* References and Further Reading::


File: gsl-ref.info,  Node: Overview of B-splines,  Next: Initializing the B-splines solver,  Up: Basis Splines

39.1 Overview
=============

B-splines are commonly used as basis functions to fit smoothing curves
to large data sets. To do this, the abscissa axis is broken up into
some number of intervals, where the endpoints of each interval are
called "breakpoints". These breakpoints are then converted to "knots"
by imposing various continuity and smoothness conditions at each
interface. Given a nondecreasing knot vector t = {t_0, t_1, ...,
t_{n+k-1}}, the n basis splines of order k are defined by

     B_(i,1)(x) = (1, t_i <= x < t_(i+1)
                  (0, else
     B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x)
                   + [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x)

for i = 0, ..., n-1. The common case of cubic B-splines is given by k =
4. The above recurrence relation can be evaluated in a numerically
stable way by the de Boor algorithm.

   If we define appropriate knots on an interval [a,b] then the
B-spline basis functions form a complete set on that interval.
Therefore we can expand a smoothing function as

     f(x) = \sum_i c_i B_(i,k)(x)

given enough (x_j, f(x_j)) data pairs. The coefficients c_i can be
readily obtained from a least-squares fit.


File: gsl-ref.info,  Node: Initializing the B-splines solver,  Next: Constructing the knots vector,  Prev: Overview of B-splines,  Up: Basis Splines

39.2 Initializing the B-splines solver
======================================

The computation of B-spline functions requires a preallocated workspace
of type `gsl_bspline_workspace'. If B-spline derivatives are also
required, an additional `gsl_bspline_deriv_workspace' is needed.

 -- Function: gsl_bspline_workspace * gsl_bspline_alloc (const size_t
          K, const size_t NBREAK)
     This function allocates a workspace for computing B-splines of
     order K. The number of breakpoints is given by NBREAK. This leads
     to n = nbreak + k - 2 basis functions. Cubic B-splines are
     specified by k = 4. The size of the workspace is O(5k + nbreak).

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

 -- Function: gsl_bspline_deriv_workspace * gsl_bspline_deriv_alloc
          (const size_t K)
     This function allocates a workspace for computing the derivatives
     of a B-spline basis function of order K.  The size of the workspace
     is O(2k^2).

 -- Function: void gsl_bspline_deriv_free (gsl_bspline_deriv_workspace
          * W)
     This function frees the memory associated with the derivative
     workspace W.


File: gsl-ref.info,  Node: Constructing the knots vector,  Next: Evaluation of B-spline basis functions,  Prev: Initializing the B-splines solver,  Up: Basis Splines

39.3 Constructing the knots vector
==================================

 -- Function: int gsl_bspline_knots (const gsl_vector * BREAKPTS,
          gsl_bspline_workspace * W)
     This function computes the knots associated with the given
     breakpoints and stores them internally in `w->knots'.

 -- Function: int gsl_bspline_knots_uniform (const double A, const
          double B, gsl_bspline_workspace * W)
     This function assumes uniformly spaced breakpoints on [a,b] and
     constructs the corresponding knot vector using the previously
     specified NBREAK parameter. The knots are stored in `w->knots'.


File: gsl-ref.info,  Node: Evaluation of B-spline basis functions,  Next: Evaluation of B-spline basis function derivatives,  Prev: Constructing the knots vector,  Up: Basis Splines

39.4 Evaluation of B-splines
============================

 -- Function: int gsl_bspline_eval (const double X, gsl_vector * B,
          gsl_bspline_workspace * W)
     This function evaluates all B-spline basis functions at the
     position X and stores them in the vector B, so that the i-th
     element is B_i(x). The vector B must be of length n = nbreak + k -
     2.  This value may also be obtained by calling
     `gsl_bspline_ncoeffs'.  Computing all the basis functions at once
     is more efficient than computing them individually, due to the
     nature of the defining recurrence relation.

 -- Function: int gsl_bspline_eval_nonzero (const double X, gsl_vector
          * BK, size_t * ISTART, size_t * IEND, gsl_bspline_workspace *
          W)
     This function evaluates all potentially nonzero B-spline basis
     functions at the position X and stores them in the vector BK, so
     that the i-th element is B_(istart+i)(x).  The last element of BK
     is B_(iend)(x).  The vector BK must be of length k.  By returning
     only the nonzero basis functions, this function allows quantities
     involving linear combinations of the B_i(x) to be computed without
     unnecessary terms (such linear combinations occur, for example,
     when evaluating an interpolated function).

 -- Function: size_t gsl_bspline_ncoeffs (gsl_bspline_workspace * W)
     This function returns the number of B-spline coefficients given by
     n = nbreak + k - 2.


File: gsl-ref.info,  Node: Evaluation of B-spline basis function derivatives,  Next: Working with the Greville abscissae,  Prev: Evaluation of B-spline basis functions,  Up: Basis Splines

39.5 Evaluation of B-spline derivatives
=======================================

 -- Function: int gsl_bspline_deriv_eval (const double X, const size_t
          NDERIV, gsl_matrix * DB, gsl_bspline_workspace * W,
          gsl_bspline_deriv_workspace * DW)
     This function evaluates all B-spline basis function derivatives of
     orders 0 through nderiv (inclusive) at the position X and stores
     them in the matrix DB.  The (i,j)-th element of DB is
     d^jB_i(x)/dx^j.  The matrix DB must be of size n = nbreak + k - 2
     by nderiv + 1.  The value n may also be obtained by calling
     `gsl_bspline_ncoeffs'.  Note that function evaluations are
     included as the zeroth order derivatives in DB.  Computing all the
     basis function derivatives at once is more efficient than
     computing them individually, due to the nature of the defining
     recurrence relation.

 -- Function: int gsl_bspline_deriv_eval_nonzero (const double X, const
          size_t NDERIV, gsl_matrix * DB, size_t * ISTART, size_t *
          IEND, gsl_bspline_workspace * W, gsl_bspline_deriv_workspace
          * DW)
     This function evaluates all potentially nonzero B-spline basis
     function derivatives of orders 0 through nderiv (inclusive) at the
     position X and stores them in the matrix DB.  The (i,j)-th element
     of DB is d^j/dx^j B_(istart+i)(x).  The last row of DB contains
     d^j/dx^j B_(iend)(x).  The matrix DB must be of size k by at least
     nderiv + 1.  Note that function evaluations are included as the
     zeroth order derivatives in DB.  By returning only the nonzero
     basis functions, this function allows quantities involving linear
     combinations of the B_i(x) and their derivatives to be computed
     without unnecessary terms.


File: gsl-ref.info,  Node: Working with the Greville abscissae,  Next: Example programs for B-splines,  Prev: Evaluation of B-spline basis function derivatives,  Up: Basis Splines

39.6 Working with the Greville abscissae
========================================

The Greville abscissae are defined to be the mean location of k-1
consecutive knots in the knot vector for each basis spline function of
order k.  With the first and last knots in the `gsl_bspline_workspace'
knot vector excluded, there are `gsl_bspline_ncoeffs' Greville abscissae
for any given B-spline basis.  These values are often used in B-spline
collocation applications and may also be called Marsden-Schoenberg
points.

 -- Function: double gsl_bspline_greville_abscissa (size_t I,
          gsl_bspline_workspace *W);
     Returns the location of the i-th Greville abscissa for the given
     B-spline basis.  For the ill-defined case when k=1, the
     implementation chooses to return breakpoint interval midpoints.


File: gsl-ref.info,  Node: Example programs for B-splines,  Next: References and Further Reading,  Prev: Working with the Greville abscissae,  Up: Basis Splines

39.7 Examples
=============

The following program computes a linear least squares fit to data using
cubic B-spline basis functions with uniform breakpoints. The data is
generated from the curve y(x) = \cos(x) \exp(-x/10) on the interval [0,
15] with Gaussian noise added.

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>
     #include <gsl/gsl_bspline.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_statistics.h>

     /* number of data points to fit */
     #define N        200

     /* number of fit coefficients */
     #define NCOEFFS  12

     /* nbreak = ncoeffs + 2 - k = ncoeffs - 2 since k = 4 */
     #define NBREAK   (NCOEFFS - 2)

     int
     main (void)
     {
       const size_t n = N;
       const size_t ncoeffs = NCOEFFS;
       const size_t nbreak = NBREAK;
       size_t i, j;
       gsl_bspline_workspace *bw;
       gsl_vector *B;
       double dy;
       gsl_rng *r;
       gsl_vector *c, *w;
       gsl_vector *x, *y;
       gsl_matrix *X, *cov;
       gsl_multifit_linear_workspace *mw;
       double chisq, Rsq, dof, tss;

       gsl_rng_env_setup();
       r = gsl_rng_alloc(gsl_rng_default);

       /* allocate a cubic bspline workspace (k = 4) */
       bw = gsl_bspline_alloc(4, nbreak);
       B = gsl_vector_alloc(ncoeffs);

       x = gsl_vector_alloc(n);
       y = gsl_vector_alloc(n);
       X = gsl_matrix_alloc(n, ncoeffs);
       c = gsl_vector_alloc(ncoeffs);
       w = gsl_vector_alloc(n);
       cov = gsl_matrix_alloc(ncoeffs, ncoeffs);
       mw = gsl_multifit_linear_alloc(n, ncoeffs);

       printf("#m=0,S=0\n");
       /* this is the data to be fitted */
       for (i = 0; i < n; ++i)
         {
           double sigma;
           double xi = (15.0 / (N - 1)) * i;
           double yi = cos(xi) * exp(-0.1 * xi);

           sigma = 0.1 * yi;
           dy = gsl_ran_gaussian(r, sigma);
           yi += dy;

           gsl_vector_set(x, i, xi);
           gsl_vector_set(y, i, yi);
           gsl_vector_set(w, i, 1.0 / (sigma * sigma));

           printf("%f %f\n", xi, yi);
         }

       /* use uniform breakpoints on [0, 15] */
       gsl_bspline_knots_uniform(0.0, 15.0, bw);

       /* construct the fit matrix X */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);

           /* compute B_j(xi) for all j */
           gsl_bspline_eval(xi, B, bw);

           /* fill in row i of X */
           for (j = 0; j < ncoeffs; ++j)
             {
               double Bj = gsl_vector_get(B, j);
               gsl_matrix_set(X, i, j, Bj);
             }
         }

       /* do the fit */
       gsl_multifit_wlinear(X, w, y, c, cov, &chisq, mw);

       dof = n - ncoeffs;
       tss = gsl_stats_wtss(w->data, 1, y->data, 1, y->size);
       Rsq = 1.0 - chisq / tss;

       fprintf(stderr, "chisq/dof = %e, Rsq = %f\n",
                        chisq / dof, Rsq);

       /* output the smoothed curve */
       {
         double xi, yi, yerr;

         printf("#m=1,S=0\n");
         for (xi = 0.0; xi < 15.0; xi += 0.1)
           {
             gsl_bspline_eval(xi, B, bw);
             gsl_multifit_linear_est(B, c, cov, &yi, &yerr);
             printf("%f %f\n", xi, yi);
           }
       }

       gsl_rng_free(r);
       gsl_bspline_free(bw);
       gsl_vector_free(B);
       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_matrix_free(X);
       gsl_vector_free(c);
       gsl_vector_free(w);
       gsl_matrix_free(cov);
       gsl_multifit_linear_free(mw);

       return 0;
     } /* main() */

   The output can be plotted with GNU `graph'.

     $ ./a.out > bspline.dat
     chisq/dof = 1.118217e+00, Rsq = 0.989771
     $ graph -T ps -X x -Y y -x 0 15 -y -1 1.3 < bspline.dat > bspline.ps


File: gsl-ref.info,  Node: References and Further Reading,  Prev: Example programs for B-splines,  Up: Basis Splines

39.8 References and Further Reading
===================================

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

     C. de Boor, `A Practical Guide to Splines' (1978), Springer-Verlag,
     ISBN 0-387-90356-9.

   Further information of Greville abscissae and B-spline collocation
can be found in the following paper,

     Richard W. Johnson, Higher order B-spline collocation at the
     Greville abscissae.  `Applied Numerical Mathematics'. vol. 52,
     2005, 63-75.

A large collection of B-spline routines is available in the PPPACK
library available at `http://www.netlib.org/pppack', which is also part
of SLATEC.


File: gsl-ref.info,  Node: Physical Constants,  Next: IEEE floating-point arithmetic,  Prev: Basis Splines,  Up: Top

40 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

40.1 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

40.2 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

40.3 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_\infty by Ry = h c R_\infty.

`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

40.4 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

40.5 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

40.6 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

40.7 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

40.8 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

40.9 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

40.10 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

40.11 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

40.12 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

40.13 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

40.14 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

40.15 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

40.16 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

40.17 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

40.18 References and Further Reading
====================================

The authoritative sources for physical constants are the 2006 CODATA
recommended values, published in the article below. Further information
on the values of physical constants is also available from the NIST
website.

     P.J. Mohr, B.N. Taylor, D.B. Newell, "CODATA Recommended Values of
     the Fundamental Physical Constants: 2006", Reviews of Modern
     Physics, 80(2), pp. 633-730 (2008).

     `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

41 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

41.1 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

41.2 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.

     Note that on the x86 series of processors this function sets both
     the original x87 mode and the newer MXCSR mode, which controls SSE
     floating-point operations.  The SSE floating-point units do not
     have a precision-control bit, and always work in double-precision.
     The single-precision and extended-precision keywords have no
     effect in this case.

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 final result is
higher than the correct value (when we add each term to the sum the
final result is always rounded up, which increases the sum by at least
one tick until the added term underflows to zero).  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

41.3 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
       -Wconversion -Wshadow -Wpointer-arith
       -Wcast-qual -Wcast-align
       -Wwrite-strings -Wnested-externs
       -fshort-enums -fno-common -Dinline= -g -O2

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.

`-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 cyclic functions and the initial functions 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 and wrote the
     ode-initval2 routines.

*Lowell Johnson*
     Implementation of the Mathieu functions.

*Patrick Alken*
     Implementation of nonsymmetric and generalized eigensystems,
     B-splines, and robust linear regression.

*Rhys Ulerich*
     Wrote the multiset routines.

*Pavel Holoborodko*
     Wrote the fixed order Gauss-Legendre quadrature routines.

*Pedro Gonnet*
     Wrote the CQUAD integration routines.


   Thanks to Nigel Lowry for help in proofreading the manual.

   The non-symmetric eigensystems routines contain code based on the
LAPACK linear algebra library.  LAPACK is distributed under the
following license:


     Copyright (c) 1992-2006 The University of Tennessee.  All rights
     reserved.

     Redistribution and use in source and binary forms, with or without
     modification, are permitted provided that the following conditions
     are met:

     * Redistributions of source code must retain the above copyright
     notice, this list of conditions and the following disclaimer.

     * Redistributions in binary form must reproduce the above copyright
     notice, this list of conditions and the following disclaimer listed
     in this license in the documentation and/or other materials
     provided with the distribution.

     * Neither the name of the copyright holders nor the names of its
     contributors may be used to endorse or promote products derived
     from this software without specific prior written permission.

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


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],[cos])
     AC_CHECK_LIB([gslcblas],[cblas_dgemm])
     AC_CHECK_LIB([gsl],[gsl_blas_dgemm])

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 cos in -lm... yes
     checking for cblas_dgemm in -lgslcblas... yes
     checking for gsl_blas_dgemm 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:

     AX_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 `AX_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: GNU General Public License,  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: GNU General Public License,  Next: GNU Free Documentation License,  Prev: GSL CBLAS Library,  Up: Top

GNU General Public License
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     An "entity transaction" is a transaction transferring control of an
     organization, or substantially all assets of one, or subdividing an
     organization, or merging organizations.  If propagation of a
     covered work results from an entity transaction, each party to that
     transaction who receives a copy of the work also receives whatever
     licenses to the work the party's predecessor in interest had or
     could give under the previous paragraph, plus a right to
     possession of the Corresponding Source of the work from the
     predecessor in interest, if the predecessor has it or can get it
     with reasonable efforts.

     You may not impose any further restrictions on the exercise of the
     rights granted or affirmed under this License.  For example, you
     may not impose a license fee, royalty, or other charge for
     exercise of rights granted under this License, and you may not
     initiate litigation (including a cross-claim or counterclaim in a
     lawsuit) alleging that any patent claim is infringed by making,
     using, selling, offering for sale, or importing the Program or any
     portion of it.

 11. Patents.

     A "contributor" is a copyright holder who authorizes use under this
     License of the Program or a work on which the Program is based.
     The work thus licensed is called the contributor's "contributor
     version".

     A contributor's "essential patent claims" are all patent claims
     owned or controlled by the contributor, whether already acquired or
     hereafter acquired, that would be infringed by some manner,
     permitted by this License, of making, using, or selling its
     contributor version, but do not include claims that would be
     infringed only as a consequence of further modification of the
     contributor version.  For purposes of this definition, "control"
     includes the right to grant patent sublicenses in a manner
     consistent with the requirements of this License.

     Each contributor grants you a non-exclusive, worldwide,
     royalty-free patent license under the contributor's essential
     patent claims, to make, use, sell, offer for sale, import and
     otherwise run, modify and propagate the contents of its
     contributor version.

     In the following three paragraphs, a "patent license" is any
     express agreement or commitment, however denominated, not to
     enforce a patent (such as an express permission to practice a
     patent or covenant not to sue for patent infringement).  To
     "grant" such a patent license to a party means to make such an
     agreement or commitment not to enforce a patent against the party.

     If you convey a covered work, knowingly relying on a patent
     license, and the Corresponding Source of the work is not available
     for anyone to copy, free of charge and under the terms of this
     License, through a publicly available network server or other
     readily accessible means, then you must either (1) cause the
     Corresponding Source to be so available, or (2) arrange to deprive
     yourself of the benefit of the patent license for this particular
     work, or (3) arrange, in a manner consistent with the requirements
     of this License, to extend the patent license to downstream
     recipients.  "Knowingly relying" means you have actual knowledge
     that, but for the patent license, your conveying the covered work
     in a country, or your recipient's use of the covered work in a
     country, would infringe one or more identifiable patents in that
     country that you have reason to believe are valid.

     If, pursuant to or in connection with a single transaction or
     arrangement, you convey, or propagate by procuring conveyance of, a
     covered work, and grant a patent license to some of the parties
     receiving the covered work authorizing them to use, propagate,
     modify or convey a specific copy of the covered work, then the
     patent license you grant is automatically extended to all
     recipients of the covered work and works based on it.

     A patent license is "discriminatory" if it does not include within
     the scope of its coverage, prohibits the exercise of, or is
     conditioned on the non-exercise of one or more of the rights that
     are specifically granted under this License.  You may not convey a
     covered work if you are a party to an arrangement with a third
     party that is in the business of distributing software, under
     which you make payment to the third party based on the extent of
     your activity of conveying the work, and under which the third
     party grants, to any of the parties who would receive the covered
     work from you, a discriminatory patent license (a) in connection
     with copies of the covered work conveyed by you (or copies made
     from those copies), or (b) primarily for and in connection with
     specific products or compilations that contain the covered work,
     unless you entered into that arrangement, or that patent license
     was granted, prior to 28 March 2007.

     Nothing in this License shall be construed as excluding or limiting
     any implied license or other defenses to infringement that may
     otherwise be available to you under applicable patent law.

 12. No Surrender of Others' Freedom.

     If conditions are imposed on you (whether by court order,
     agreement or otherwise) that contradict the conditions of this
     License, they do not excuse you from the conditions of this
     License.  If you cannot convey a covered work so as to satisfy
     simultaneously your obligations under this License and any other
     pertinent obligations, then as a consequence you may not convey it
     at all.  For example, if you agree to terms that obligate you to
     collect a royalty for further conveying from those to whom you
     convey the Program, the only way you could satisfy both those
     terms and this License would be to refrain entirely from conveying
     the Program.

 13. Use with the GNU Affero General Public License.

     Notwithstanding any other provision of this License, you have
     permission to link or combine any covered work with a work licensed
     under version 3 of the GNU Affero General Public License into a
     single combined work, and to convey the resulting work.  The terms
     of this License will continue to apply to the part which is the
     covered work, but the special requirements of the GNU Affero
     General Public License, section 13, concerning interaction through
     a network will apply to the combination as such.

 14. Revised Versions of this License.

     The Free Software Foundation may publish revised and/or new
     versions of the GNU General Public License from time to time.
     Such new versions will be similar in spirit to the present
     version, but may differ in detail to address new problems or
     concerns.

     Each version is given a distinguishing version number.  If the
     Program specifies that a certain numbered version of the GNU
     General Public License "or any later version" applies to it, you
     have the option of following the terms and conditions either of
     that numbered version or of any later version published by the
     Free Software Foundation.  If the Program does not specify a
     version number of the GNU General Public License, you may choose
     any version ever published by the Free Software Foundation.

     If the Program specifies that a proxy can decide which future
     versions of the GNU General Public License can be used, that
     proxy's public statement of acceptance of a version permanently
     authorizes you to choose that version for the Program.

     Later license versions may give you additional or different
     permissions.  However, no additional obligations are imposed on any
     author or copyright holder as a result of your choosing to follow a
     later version.

 15. Disclaimer of Warranty.

     THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
     APPLICABLE LAW.  EXCEPT WHEN OTHERWISE STATED IN WRITING THE
     COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS"
     WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED,
     INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
     MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.  THE ENTIRE
     RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU.
     SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL
     NECESSARY SERVICING, REPAIR OR CORRECTION.

 16. Limitation of Liability.

     IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN
     WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES
     AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU
     FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR
     CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE
     THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA
     BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
     PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER
     PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF
     THE POSSIBILITY OF SUCH DAMAGES.

 17. Interpretation of Sections 15 and 16.

     If the disclaimer of warranty and limitation of liability provided
     above cannot be given local legal effect according to their terms,
     reviewing courts shall apply local law that most closely
     approximates an absolute waiver of all civil liability in
     connection with the Program, unless a warranty or assumption of
     liability accompanies a copy of the Program in return for a fee.


END OF TERMS AND CONDITIONS
===========================

How to Apply These Terms to Your New Programs
=============================================

If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these
terms.

   To do so, attach the following notices to the program.  It is safest
to attach them to the start of each source file to most effectively
state the exclusion of warranty; and each file should have at least the
"copyright" line and a pointer to where the full notice is found.

     one line to give the program's name and a brief idea
     of whAT IT DOES.
     Copyright (C) YEAR NAME OF AUTHOR

     This program is free software: you can redistribute it and/or modify
     it under the terms of the GNU General Public License as published by
     the Free Software Foundation, either version 3 of the License, or (at
     your option) any later version.

     This program is distributed in the hope that it will be useful, but
     WITHOUT ANY WARRANTY; without even the implied warranty of
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
     General Public License for more details.

     You should have received a copy of the GNU General Public License
     along with this program.  If not, see `http://www.gnu.org/licenses/'.

   Also add information on how to contact you by electronic and paper
mail.

   If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:

     PROGRAM Copyright (C) YEAR NAME OF AUTHOR
     This program 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, your
program's commands might be different; for a GUI interface, you would
use an "about box".

   You should also get your employer (if you work as a programmer) or
school, if any, to sign a "copyright disclaimer" for the program, if
necessary.  For more information on this, and how to apply and follow
the GNU GPL, see `http://www.gnu.org/licenses/'.

   The GNU 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 Lesser General Public License instead of this License.  But first,
please read `http://www.gnu.org/philosophy/why-not-lgpl.html'.


File: gsl-ref.info,  Node: GNU Free Documentation License,  Next: Function Index,  Prev: GNU General Public License,  Up: Top

GNU Free Documentation License
******************************

                     Version 1.3, 3 November 2008
     Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
     `http://fsf.org/'

     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
     author and publisher a way to get credit for their work, while not
     being considered responsible for modifications made by others.

     This License is a kind of "copyleft", which means that derivative
     works of the document must themselves be free in the same sense.
     It complements the GNU General Public License, which is a copyleft
     license designed for free software.

     We have designed this License in order to use it for manuals for
     free software, because free software needs free documentation: a
     free program should come with manuals providing the same freedoms
     that the software does.  But this License is not limited to
     software manuals; it can be used for any textual work, regardless
     of subject matter or whether it is published as a printed book.
     We recommend this License principally for works whose purpose is
     instruction or reference.

  1. APPLICABILITY AND DEFINITIONS

     This License applies to any manual or other work, in any medium,
     that contains a notice placed by the copyright holder saying it
     can be distributed under the terms of this License.  Such a notice
     grants a world-wide, royalty-free license, unlimited in duration,
     to use that work under the conditions stated herein.  The
     "Document", below, refers to any such manual or work.  Any member
     of the public is a licensee, and is addressed as "you".  You
     accept the license if you copy, modify or distribute the work in a
     way requiring permission under copyright law.

     A "Modified Version" of the Document means any work containing the
     Document or a portion of it, either copied verbatim, or with
     modifications and/or translated into another language.

     A "Secondary Section" is a named appendix or a front-matter section
     of the Document that deals exclusively with the relationship of the
     publishers or authors of the Document to the Document's overall
     subject (or to related matters) and contains nothing that could
     fall directly within that overall subject.  (Thus, if the Document
     is in part a textbook of mathematics, a Secondary Section may not
     explain any mathematics.)  The relationship could be a matter of
     historical connection with the subject or with related matters, or
     of legal, commercial, philosophical, ethical or political position
     regarding them.

     The "Invariant Sections" are certain Secondary Sections whose
     titles are designated, as being those of Invariant Sections, in
     the notice that says that the Document is released under this
     License.  If a section does not fit the above definition of
     Secondary then it is not allowed to be designated as Invariant.
     The Document may contain zero Invariant Sections.  If the Document
     does not identify any Invariant Sections then there are none.

     The "Cover Texts" are certain short passages of text that are
     listed, as Front-Cover Texts or Back-Cover Texts, in the notice
     that says that the Document is released under this License.  A
     Front-Cover Text may be at most 5 words, and a Back-Cover Text may
     be at most 25 words.

     A "Transparent" copy of the Document means a machine-readable copy,
     represented in a format whose specification is available to the
     general public, that is suitable for revising the document
     straightforwardly with generic text editors or (for images
     composed of pixels) generic paint programs or (for drawings) some
     widely available drawing editor, and that is suitable for input to
     text formatters or for automatic translation to a variety of
     formats suitable for input to text formatters.  A copy made in an
     otherwise Transparent file format whose markup, or absence of
     markup, has been arranged to thwart or discourage subsequent
     modification by readers is not Transparent.  An image format is
     not Transparent if used for any substantial amount of text.  A
     copy that is not "Transparent" is called "Opaque".

     Examples of suitable formats for Transparent copies include plain
     ASCII without markup, Texinfo input format, LaTeX input format,
     SGML or XML using a publicly available DTD, and
     standard-conforming simple HTML, PostScript or PDF designed for
     human modification.  Examples of transparent image formats include
     PNG, XCF and JPG.  Opaque formats include proprietary formats that
     can be read and edited only by proprietary word processors, SGML or
     XML for which the DTD and/or processing tools are not generally
     available, and the machine-generated HTML, PostScript or PDF
     produced by some word processors for output purposes only.

     The "Title Page" means, for a printed book, the title page itself,
     plus such following pages as are needed to hold, legibly, the
     material this License requires to appear in the title page.  For
     works in formats which do not have any title page as such, "Title
     Page" means the text near the most prominent appearance of the
     work's title, preceding the beginning of the body of the text.

     The "publisher" means any person or entity that distributes copies
     of the Document to the public.

     A section "Entitled XYZ" means a named subunit of the Document
     whose title either is precisely XYZ or contains XYZ in parentheses
     following text that translates XYZ in another language.  (Here XYZ
     stands for a specific section name mentioned below, such as
     "Acknowledgements", "Dedications", "Endorsements", or "History".)
     To "Preserve the Title" of such a section when you modify the
     Document means that it remains a section "Entitled XYZ" according
     to this definition.

     The Document may include Warranty Disclaimers next to the notice
     which states that this License applies to the Document.  These
     Warranty Disclaimers are considered to be included by reference in
     this License, but only as regards disclaiming warranties: any other
     implication that these Warranty Disclaimers may have is void and
     has no effect on the meaning of this License.

  2. VERBATIM COPYING

     You may copy and distribute the Document in any medium, either
     commercially or noncommercially, provided that this License, the
     copyright notices, and the license notice saying this License
     applies to the Document are reproduced in all copies, and that you
     add no other conditions whatsoever to those of this License.  You
     may not use technical measures to obstruct or control the reading
     or further copying of the copies you make or distribute.  However,
     you may accept compensation in exchange for copies.  If you
     distribute a large enough number of copies you must also follow
     the conditions in section 3.

     You may also lend copies, under the same conditions stated above,
     and you may publicly display copies.

  3. COPYING IN QUANTITY

     If you publish printed copies (or copies in media that commonly
     have printed covers) of the Document, numbering more than 100, and
     the Document's license notice requires Cover Texts, you must
     enclose the copies in covers that carry, clearly and legibly, all
     these Cover Texts: Front-Cover Texts on the front cover, and
     Back-Cover Texts on the back cover.  Both covers must also clearly
     and legibly identify you as the publisher of these copies.  The
     front cover must present the full title with all words of the
     title equally prominent and visible.  You may add other material
     on the covers in addition.  Copying with changes limited to the
     covers, as long as they preserve the title of the Document and
     satisfy these conditions, can be treated as verbatim copying in
     other respects.

     If the required texts for either cover are too voluminous to fit
     legibly, you should put the first ones listed (as many as fit
     reasonably) on the actual cover, and continue the rest onto
     adjacent pages.

     If you publish or distribute Opaque copies of the Document
     numbering more than 100, you must either include a
     machine-readable Transparent copy along with each Opaque copy, or
     state in or with each Opaque copy a computer-network location from
     which the general network-using public has access to download
     using public-standard network protocols a complete Transparent
     copy of the Document, free of added material.  If you use the
     latter option, you must take reasonably prudent steps, when you
     begin distribution of Opaque copies in quantity, to ensure that
     this Transparent copy will remain thus accessible at the stated
     location until at least one year after the last time you
     distribute an Opaque copy (directly or through your agents or
     retailers) of that edition to the public.

     It is requested, but not required, that you contact the authors of
     the Document well before redistributing any large number of
     copies, to give them a chance to provide you with an updated
     version of the Document.

  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
     the Modified Version filling the role of the Document, thus
     licensing distribution and modification of the Modified Version to
     whoever possesses a copy of it.  In addition, you must do these
     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
          previous versions (which should, if there were any, be listed
          in the History section of the Document).  You may use the
          same title as a previous version if the original publisher of
          that version gives permission.

       B. List on the Title Page, as authors, one or more persons or
          entities responsible for authorship of the modifications in
          the Modified Version, together with at least five of the
          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
          Modified Version, as the publisher.

       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,
          and add to it an item stating at least the title, year, new
          authors, and publisher of the Modified Version as given on
          the Title Page.  If there is no section Entitled "History" in
          the Document, create one stating the title, year, authors,
          and publisher of the Document as given on its Title Page,
          then add an item describing the Modified Version as stated in
          the previous sentence.

       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
     appendices that qualify as Secondary Sections and contain no
     material copied from the Document, you may at your option
     designate some or all of these sections as invariant.  To do this,
     add their titles to the list of Invariant Sections in the Modified
     Version's license notice.  These titles must be distinct from any
     other section titles.

     You may add a section Entitled "Endorsements", provided it contains
     nothing but endorsements of your Modified Version by various
     parties--for example, statements of peer review or that the text
     has been approved by an organization as the authoritative
     definition of a standard.

     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
     of the list of Cover Texts in the Modified Version.  Only one
     passage of Front-Cover Text and one of Back-Cover Text may be
     added by (or through arrangements made by) any one entity.  If the
     Document already includes a cover text for the same cover,
     previously added by you or by arrangement made by the same entity
     you are acting on behalf of, you may not add another; but you may
     replace the old one, on explicit permission from the previous
     publisher that added the old one.

     The author(s) and publisher(s) of the Document do not by this
     License give permission to use their names for publicity for or to
     assert or imply endorsement of any Modified Version.

  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
     but different contents, make the title of each such section unique
     by adding at the end of it, in parentheses, the name of the
     original author or publisher of that section if known, or else a
     unique number.  Make the same adjustment to the section titles in
     the list of Invariant Sections in the license notice of the
     combined work.

     In the combination, you must combine any sections Entitled
     "History" in the various original documents, forming one section
     Entitled "History"; likewise combine any sections Entitled
     "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 under this License.  Any attempt
     otherwise to copy, modify, sublicense, or distribute it is void,
     and will automatically terminate your rights under this License.

     However, if you cease all violation of this License, then your
     license from a particular copyright holder is reinstated (a)
     provisionally, unless and until the copyright holder explicitly
     and finally terminates your license, and (b) permanently, if the
     copyright holder fails to notify you of the violation by some
     reasonable means prior to 60 days after the cessation.

     Moreover, your license from a particular copyright holder is
     reinstated permanently if the copyright holder notifies you of the
     violation by some reasonable means, this is the first time you have
     received notice of violation of this License (for any work) from
     that copyright holder, and you cure the violation prior to 30 days
     after your receipt of the notice.

     Termination of your rights under this section does not terminate
     the licenses of parties who have received copies or rights from
     you under this License.  If your rights have been terminated and
     not permanently reinstated, receipt of a copy of some or all of
     the same material does not give you any rights to use it.

 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
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====================================================

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       Copyright (C)  YEAR  YOUR NAME.
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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  33)
* gsl_asinh:                             Elementary Functions.
                                                              (line  37)
* gsl_atanh:                             Elementary Functions.
                                                              (line  41)
* 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 177)
* gsl_blas_cherk:                        Level 3 GSL BLAS Interface.
                                                              (line 137)
* 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 160)
* gsl_blas_csyrk:                        Level 3 GSL BLAS Interface.
                                                              (line 123)
* 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 155)
* gsl_blas_dsyrk:                        Level 3 GSL BLAS Interface.
                                                              (line 119)
* 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 152)
* gsl_blas_ssyrk:                        Level 3 GSL BLAS Interface.
                                                              (line 116)
* 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 181)
* gsl_blas_zherk:                        Level 3 GSL BLAS Interface.
                                                              (line 140)
* 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 164)
* gsl_blas_zsyrk:                        Level 3 GSL BLAS Interface.
                                                              (line 126)
* 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_bspline_alloc:                     Initializing the B-splines solver.
                                                              (line  12)
* gsl_bspline_deriv_alloc:               Initializing the B-splines solver.
                                                              (line  22)
* gsl_bspline_deriv_eval:                Evaluation of B-spline basis function derivatives.
                                                              (line   9)
* gsl_bspline_deriv_eval_nonzero:        Evaluation of B-spline basis function derivatives.
                                                              (line  24)
* gsl_bspline_deriv_free:                Initializing the B-splines solver.
                                                              (line  28)
* gsl_bspline_eval:                      Evaluation of B-spline basis functions.
                                                              (line   8)
* gsl_bspline_eval_nonzero:              Evaluation of B-spline basis functions.
                                                              (line  19)
* gsl_bspline_free:                      Initializing the B-splines solver.
                                                              (line  18)
* gsl_bspline_greville_abscissa:         Working with the Greville abscissae.
                                                              (line  16)
* gsl_bspline_knots:                     Constructing the knots vector.
                                                              (line   8)
* gsl_bspline_knots_uniform:             Constructing the knots vector.
                                                              (line  13)
* gsl_bspline_ncoeffs:                   Evaluation of B-spline basis functions.
                                                              (line  29)
* 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  35)
* gsl_cdf_gamma_Pinv:                    The Gamma Distribution.
                                                              (line  37)
* gsl_cdf_gamma_Q:                       The Gamma Distribution.
                                                              (line  36)
* gsl_cdf_gamma_Qinv:                    The Gamma Distribution.
                                                              (line  38)
* 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_coeffs:                       Auxiliary Functions for Chebyshev Series.
                                                              (line  14)
* 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_cheb_order:                        Auxiliary Functions for Chebyshev Series.
                                                              (line  10)
* gsl_cheb_size:                         Auxiliary Functions for Chebyshev Series.
                                                              (line  13)
* 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  19)
* gsl_complex_arccot:                    Inverse Complex Trigonometric Functions.
                                                              (line  53)
* gsl_complex_arccoth:                   Inverse Complex Hyperbolic Functions.
                                                              (line  40)
* 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  36)
* 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  32)
* 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  23)
* gsl_complex_arctanh_real:              Inverse Complex Hyperbolic Functions.
                                                              (line  28)
* 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:                     Representation of complex numbers.
                                                              (line  30)
* gsl_complex_poly_complex_eval:         Polynomial Evaluation.
                                                              (line  23)
* gsl_complex_pow:                       Elementary Complex Functions.
                                                              (line  16)
* gsl_complex_pow_real:                  Elementary Complex Functions.
                                                              (line  21)
* gsl_complex_rect:                      Representation of 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  37)
* gsl_dht_new:                           Discrete Hankel Transform Functions.
                                                              (line  16)
* gsl_dht_x_sample:                      Discrete Hankel Transform Functions.
                                                              (line  32)
* gsl_eigen_gen:                         Real Generalized Nonsymmetric Eigensystems.
                                                              (line  75)
* gsl_eigen_gen_alloc:                   Real Generalized Nonsymmetric Eigensystems.
                                                              (line  43)
* gsl_eigen_gen_free:                    Real Generalized Nonsymmetric Eigensystems.
                                                              (line  48)
* gsl_eigen_gen_params:                  Real Generalized Nonsymmetric Eigensystems.
                                                              (line  52)
* gsl_eigen_gen_QZ:                      Real Generalized Nonsymmetric Eigensystems.
                                                              (line  92)
* gsl_eigen_genherm:                     Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  30)
* gsl_eigen_genherm_alloc:               Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  19)
* gsl_eigen_genherm_free:                Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  25)
* gsl_eigen_genhermv:                    Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  48)
* gsl_eigen_genhermv_alloc:              Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  37)
* gsl_eigen_genhermv_free:               Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  43)
* gsl_eigen_genhermv_sort:               Sorting Eigenvalues and Eigenvectors.
                                                              (line  51)
* gsl_eigen_gensymm:                     Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  34)
* gsl_eigen_gensymm_alloc:               Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  24)
* gsl_eigen_gensymm_free:                Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  30)
* gsl_eigen_gensymmv:                    Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  52)
* gsl_eigen_gensymmv_alloc:              Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  41)
* gsl_eigen_gensymmv_free:               Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  47)
* gsl_eigen_gensymmv_sort:               Sorting Eigenvalues and Eigenvectors.
                                                              (line  44)
* gsl_eigen_genv:                        Real Generalized Nonsymmetric Eigensystems.
                                                              (line 108)
* gsl_eigen_genv_alloc:                  Real Generalized Nonsymmetric Eigensystems.
                                                              (line  98)
* gsl_eigen_genv_free:                   Real Generalized Nonsymmetric Eigensystems.
                                                              (line 103)
* gsl_eigen_genv_QZ:                     Real Generalized Nonsymmetric Eigensystems.
                                                              (line 124)
* gsl_eigen_genv_sort:                   Sorting Eigenvalues and Eigenvectors.
                                                              (line  59)
* gsl_eigen_herm:                        Complex Hermitian Matrices.
                                                              (line  20)
* gsl_eigen_herm_alloc:                  Complex Hermitian Matrices.
                                                              (line  11)
* gsl_eigen_herm_free:                   Complex Hermitian Matrices.
                                                              (line  16)
* gsl_eigen_hermv:                       Complex Hermitian Matrices.
                                                              (line  40)
* gsl_eigen_hermv_alloc:                 Complex Hermitian Matrices.
                                                              (line  30)
* gsl_eigen_hermv_free:                  Complex Hermitian Matrices.
                                                              (line  35)
* gsl_eigen_hermv_sort:                  Sorting Eigenvalues and Eigenvectors.
                                                              (line  28)
* gsl_eigen_nonsymm:                     Real Nonsymmetric Matrices.
                                                              (line  62)
* gsl_eigen_nonsymm_alloc:               Real Nonsymmetric Matrices.
                                                              (line  19)
* gsl_eigen_nonsymm_free:                Real Nonsymmetric Matrices.
                                                              (line  25)
* gsl_eigen_nonsymm_params:              Real Nonsymmetric Matrices.
                                                              (line  29)
* gsl_eigen_nonsymm_Z:                   Real Nonsymmetric Matrices.
                                                              (line  75)
* gsl_eigen_nonsymmv:                    Real Nonsymmetric Matrices.
                                                              (line 100)
* gsl_eigen_nonsymmv_alloc:              Real Nonsymmetric Matrices.
                                                              (line  80)
* gsl_eigen_nonsymmv_free:               Real Nonsymmetric Matrices.
                                                              (line  86)
* gsl_eigen_nonsymmv_params:             Real Nonsymmetric Matrices.
                                                              (line  90)
* gsl_eigen_nonsymmv_sort:               Sorting Eigenvalues and Eigenvectors.
                                                              (line  35)
* gsl_eigen_nonsymmv_Z:                  Real Nonsymmetric Matrices.
                                                              (line 114)
* gsl_eigen_symm:                        Real Symmetric Matrices.
                                                              (line  23)
* gsl_eigen_symm_alloc:                  Real Symmetric Matrices.
                                                              (line  14)
* gsl_eigen_symm_free:                   Real Symmetric Matrices.
                                                              (line  19)
* gsl_eigen_symmv:                       Real Symmetric Matrices.
                                                              (line  41)
* gsl_eigen_symmv_alloc:                 Real Symmetric Matrices.
                                                              (line  32)
* gsl_eigen_symmv_free:                  Real Symmetric Matrices.
                                                              (line  37)
* gsl_eigen_symmv_sort:                  Sorting Eigenvalues and Eigenvectors.
                                                              (line   8)
* 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  59)
* gsl_fft_halfcomplex_radix2_inverse:    Radix-2 FFT routines for real data.
                                                              (line  57)
* gsl_fft_halfcomplex_radix2_unpack:     Radix-2 FFT routines for real data.
                                                              (line  68)
* 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  45)
* 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  30)
* gsl_fit_wmul:                          Linear fitting without a constant term.
                                                              (line  25)
* gsl_frexp:                             Elementary Functions.
                                                              (line  49)
* 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_hypot3:                            Elementary Functions.
                                                              (line  29)
* 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:                              Representation of complex numbers.
                                                              (line  36)
* gsl_integration_cquad:                 CQUAD doubly-adaptive integration.
                                                              (line  37)
* gsl_integration_cquad_workspace_alloc: CQUAD doubly-adaptive integration.
                                                              (line  22)
* gsl_integration_cquad_workspace_free:  CQUAD doubly-adaptive integration.
                                                              (line  31)
* gsl_integration_glfixed:               Fixed order Gauss-Legendre integration.
                                                              (line  24)
* gsl_integration_glfixed_point:         Fixed order Gauss-Legendre integration.
                                                              (line  30)
* gsl_integration_glfixed_table_alloc:   Fixed order Gauss-Legendre integration.
                                                              (line  16)
* gsl_integration_glfixed_table_free:    Fixed order Gauss-Legendre integration.
                                                              (line  37)
* gsl_integration_qag:                   QAG adaptive integration.
                                                              (line  31)
* 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  25)
* gsl_interp_accel_alloc:                Index Look-up and Acceleration.
                                                              (line  20)
* gsl_interp_accel_find:                 Index Look-up and Acceleration.
                                                              (line  27)
* gsl_interp_accel_free:                 Index Look-up and Acceleration.
                                                              (line  40)
* gsl_interp_accel_reset:                Index Look-up and Acceleration.
                                                              (line  35)
* 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  21)
* gsl_interp_eval_deriv2:                Evaluation of Interpolating Functions.
                                                              (line  31)
* gsl_interp_eval_deriv2_e:              Evaluation of Interpolating Functions.
                                                              (line  34)
* gsl_interp_eval_deriv_e:               Evaluation of Interpolating Functions.
                                                              (line  24)
* gsl_interp_eval_e:                     Evaluation of Interpolating Functions.
                                                              (line  12)
* gsl_interp_eval_integ:                 Evaluation of Interpolating Functions.
                                                              (line  41)
* gsl_interp_eval_integ_e:               Evaluation of Interpolating Functions.
                                                              (line  44)
* gsl_interp_free:                       Interpolation Functions.
                                                              (line  25)
* 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_interp_type_min_size:              Interpolation Types. (line  60)
* 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  45)
* gsl_linalg_balance_matrix:             Balancing.           (line  19)
* 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_invert:            Cholesky Decomposition.
                                                              (line  56)
* gsl_linalg_cholesky_solve:             Cholesky Decomposition.
                                                              (line  37)
* gsl_linalg_cholesky_svx:               Cholesky Decomposition.
                                                              (line  47)
* gsl_linalg_complex_cholesky_decomp:    Cholesky Decomposition.
                                                              (line  21)
* gsl_linalg_complex_cholesky_invert:    Cholesky Decomposition.
                                                              (line  58)
* gsl_linalg_complex_cholesky_solve:     Cholesky Decomposition.
                                                              (line  40)
* gsl_linalg_complex_cholesky_svx:       Cholesky Decomposition.
                                                              (line  49)
* gsl_linalg_complex_householder_hm:     Householder Transformations.
                                                              (line  29)
* gsl_linalg_complex_householder_hv:     Householder Transformations.
                                                              (line  45)
* gsl_linalg_complex_householder_mh:     Householder Transformations.
                                                              (line  37)
* gsl_linalg_complex_householder_transform: Householder Transformations.
                                                              (line  20)
* gsl_linalg_complex_LU_decomp:          LU Decomposition.    (line  22)
* gsl_linalg_complex_LU_det:             LU Decomposition.    (line  86)
* gsl_linalg_complex_LU_invert:          LU Decomposition.    (line  74)
* gsl_linalg_complex_LU_lndet:           LU Decomposition.    (line  94)
* gsl_linalg_complex_LU_refine:          LU Decomposition.    (line  64)
* gsl_linalg_complex_LU_sgndet:          LU Decomposition.    (line 102)
* 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_hessenberg_decomp:          Hessenberg Decomposition of Real Matrices.
                                                              (line  18)
* gsl_linalg_hessenberg_set_zero:        Hessenberg Decomposition of Real Matrices.
                                                              (line  44)
* gsl_linalg_hessenberg_unpack:          Hessenberg Decomposition of Real Matrices.
                                                              (line  29)
* gsl_linalg_hessenberg_unpack_accum:    Hessenberg Decomposition of Real Matrices.
                                                              (line  36)
* gsl_linalg_hesstri_decomp:             Hessenberg-Triangular Decomposition of Real Matrices.
                                                              (line  19)
* 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  27)
* gsl_linalg_householder_hv:             Householder Transformations.
                                                              (line  43)
* gsl_linalg_householder_mh:             Householder Transformations.
                                                              (line  35)
* gsl_linalg_householder_transform:      Householder Transformations.
                                                              (line  18)
* gsl_linalg_LU_decomp:                  LU Decomposition.    (line  20)
* gsl_linalg_LU_det:                     LU Decomposition.    (line  84)
* gsl_linalg_LU_invert:                  LU Decomposition.    (line  71)
* gsl_linalg_LU_lndet:                   LU Decomposition.    (line  92)
* gsl_linalg_LU_refine:                  LU Decomposition.    (line  60)
* gsl_linalg_LU_sgndet:                  LU Decomposition.    (line 100)
* 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  55)
* gsl_linalg_QR_QRsolve:                 QR Decomposition.    (line 106)
* gsl_linalg_QR_QTmat:                   QR Decomposition.    (line  80)
* gsl_linalg_QR_QTvec:                   QR Decomposition.    (line  65)
* gsl_linalg_QR_Qvec:                    QR Decomposition.    (line  73)
* gsl_linalg_QR_Rsolve:                  QR Decomposition.    (line  88)
* gsl_linalg_QR_Rsvx:                    QR Decomposition.    (line  94)
* gsl_linalg_QR_solve:                   QR Decomposition.    (line  38)
* gsl_linalg_QR_svx:                     QR Decomposition.    (line  46)
* gsl_linalg_QR_unpack:                  QR Decomposition.    (line 101)
* gsl_linalg_QR_update:                  QR Decomposition.    (line 112)
* 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 119)
* gsl_linalg_R_svx:                      QR Decomposition.    (line 124)
* gsl_linalg_solve_cyc_tridiag:          Tridiagonal Systems. (line  43)
* gsl_linalg_solve_symm_cyc_tridiag:     Tridiagonal Systems. (line  57)
* gsl_linalg_solve_symm_tridiag:         Tridiagonal Systems. (line  30)
* gsl_linalg_solve_tridiag:              Tridiagonal Systems. (line  16)
* gsl_linalg_SV_decomp:                  Singular Value Decomposition.
                                                              (line  39)
* gsl_linalg_SV_decomp_jacobi:           Singular Value Decomposition.
                                                              (line  58)
* gsl_linalg_SV_decomp_mod:              Singular Value Decomposition.
                                                              (line  52)
* gsl_linalg_SV_leverage:                Singular Value Decomposition.
                                                              (line  81)
* gsl_linalg_SV_solve:                   Singular Value Decomposition.
                                                              (line  66)
* 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  40)
* 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  65)
* gsl_matrix_const_ptr:                  Accessing matrix elements.
                                                              (line  38)
* gsl_matrix_const_row:                  Creating row and column views.
                                                              (line  16)
* gsl_matrix_const_subcolumn:            Creating row and column views.
                                                              (line  53)
* gsl_matrix_const_subdiagonal:          Creating row and column views.
                                                              (line  78)
* gsl_matrix_const_submatrix:            Matrix views.        (line  22)
* gsl_matrix_const_subrow:               Creating row and column views.
                                                              (line  40)
* gsl_matrix_const_superdiagonal:        Creating row and column views.
                                                              (line  90)
* 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  63)
* gsl_matrix_div_elements:               Matrix operations.   (line  29)
* gsl_matrix_equal:                      Matrix properties.   (line  22)
* 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_isneg:                      Matrix properties.   (line  13)
* gsl_matrix_isnonneg:                   Matrix properties.   (line  14)
* gsl_matrix_isnull:                     Matrix properties.   (line  11)
* gsl_matrix_ispos:                      Matrix properties.   (line  12)
* 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  22)
* gsl_matrix_ptr:                        Accessing matrix elements.
                                                              (line  36)
* gsl_matrix_row:                        Creating row and column views.
                                                              (line  14)
* gsl_matrix_scale:                      Matrix operations.   (line  35)
* gsl_matrix_set:                        Accessing matrix elements.
                                                              (line  29)
* 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  15)
* gsl_matrix_subcolumn:                  Creating row and column views.
                                                              (line  51)
* gsl_matrix_subdiagonal:                Creating row and column views.
                                                              (line  76)
* gsl_matrix_submatrix:                  Matrix views.        (line  20)
* gsl_matrix_subrow:                     Creating row and column views.
                                                              (line  38)
* gsl_matrix_superdiagonal:              Creating row and column views.
                                                              (line  88)
* 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  11)
* GSL_MAX_DBL:                           Maximum and Minimum functions.
                                                              (line  19)
* GSL_MAX_INT:                           Maximum and Minimum functions.
                                                              (line  33)
* GSL_MAX_LDBL:                          Maximum and Minimum functions.
                                                              (line  41)
* GSL_MIN:                               Maximum and Minimum functions.
                                                              (line  15)
* GSL_MIN_DBL:                           Maximum and Minimum functions.
                                                              (line  26)
* 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_quad_golden:        Minimization Algorithms.
                                                              (line  48)
* 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  34)
* GSL_MIN_LDBL:                          Maximum and Minimum functions.
                                                              (line  43)
* 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_miser_params_get:            MISER.               (line  78)
* gsl_monte_miser_params_set:            MISER.               (line  83)
* 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_chisq:                 VEGAS.               (line 112)
* gsl_monte_vegas_free:                  VEGAS.               (line  78)
* gsl_monte_vegas_init:                  VEGAS.               (line  56)
* gsl_monte_vegas_integrate:             VEGAS.               (line  64)
* gsl_monte_vegas_params_get:            VEGAS.               (line 130)
* gsl_monte_vegas_params_set:            VEGAS.               (line 135)
* gsl_monte_vegas_runval:                VEGAS.               (line 121)
* 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_dif_df:         Finite Difference Jacobian.
                                                              (line  18)
* gsl_multifit_fdfsolver_dif_fdf:        Finite Difference Jacobian.
                                                              (line  30)
* gsl_multifit_fdfsolver_driver:         High Level Driver.   (line  15)
* 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  65)
* 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_driver:           High Level Driver.   (line  12)
* 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  53)
* gsl_multifit_linear_alloc:             Multi-parameter fitting.
                                                              (line  43)
* gsl_multifit_linear_est:               Multi-parameter fitting.
                                                              (line 111)
* gsl_multifit_linear_free:              Multi-parameter fitting.
                                                              (line  48)
* gsl_multifit_linear_residuals:         Multi-parameter fitting.
                                                              (line 118)
* gsl_multifit_linear_svd:               Multi-parameter fitting.
                                                              (line  89)
* gsl_multifit_linear_usvd:              Multi-parameter fitting.
                                                              (line 101)
* gsl_multifit_robust:                   Robust linear regression.
                                                              (line 167)
* gsl_multifit_robust_alloc:             Robust linear regression.
                                                              (line  77)
* gsl_multifit_robust_bisquare:          Robust linear regression.
                                                              (line  87)
* gsl_multifit_robust_cauchy:            Robust linear regression.
                                                              (line  96)
* gsl_multifit_robust_default:           Robust linear regression.
                                                              (line  82)
* gsl_multifit_robust_est:               Robust linear regression.
                                                              (line 181)
* gsl_multifit_robust_fair:              Robust linear regression.
                                                              (line 107)
* gsl_multifit_robust_free:              Robust linear regression.
                                                              (line 149)
* gsl_multifit_robust_huber:             Robust linear regression.
                                                              (line 116)
* gsl_multifit_robust_name:              Robust linear regression.
                                                              (line 153)
* gsl_multifit_robust_ols:               Robust linear regression.
                                                              (line 129)
* gsl_multifit_robust_statistics:        Robust linear regression.
                                                              (line 188)
* gsl_multifit_robust_tune:              Robust linear regression.
                                                              (line 158)
* gsl_multifit_robust_welsch:            Robust linear regression.
                                                              (line 139)
* 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  74)
* gsl_multifit_wlinear_svd:              Multi-parameter fitting.
                                                              (line  93)
* gsl_multifit_wlinear_usvd:             Multi-parameter fitting.
                                                              (line 105)
* gsl_multimin_fdfminimizer_alloc:       Initializing the Multidimensional Minimizer.
                                                              (line  13)
* gsl_multimin_fdfminimizer_conjugate_fr: Multimin Algorithms with Derivatives.
                                                              (line  12)
* gsl_multimin_fdfminimizer_conjugate_pr: Multimin Algorithms with Derivatives.
                                                              (line  29)
* gsl_multimin_fdfminimizer_free:        Initializing the Multidimensional Minimizer.
                                                              (line  48)
* gsl_multimin_fdfminimizer_gradient:    Multimin Iteration.  (line  36)
* gsl_multimin_fdfminimizer_iterate:     Multimin Iteration.  (line  13)
* gsl_multimin_fdfminimizer_minimum:     Multimin Iteration.  (line  32)
* gsl_multimin_fdfminimizer_name:        Initializing the Multidimensional Minimizer.
                                                              (line  54)
* gsl_multimin_fdfminimizer_restart:     Multimin Iteration.  (line  45)
* gsl_multimin_fdfminimizer_set:         Initializing the Multidimensional Minimizer.
                                                              (line  24)
* gsl_multimin_fdfminimizer_steepest_descent: Multimin Algorithms with Derivatives.
                                                              (line  57)
* gsl_multimin_fdfminimizer_vector_bfgs: Multimin Algorithms with Derivatives.
                                                              (line  37)
* gsl_multimin_fdfminimizer_vector_bfgs2: Multimin Algorithms with Derivatives.
                                                              (line  36)
* gsl_multimin_fdfminimizer_x:           Multimin Iteration.  (line  28)
* gsl_multimin_fminimizer_alloc:         Initializing the Multidimensional Minimizer.
                                                              (line  15)
* gsl_multimin_fminimizer_free:          Initializing the Multidimensional Minimizer.
                                                              (line  50)
* gsl_multimin_fminimizer_iterate:       Multimin Iteration.  (line  15)
* gsl_multimin_fminimizer_minimum:       Multimin Iteration.  (line  34)
* gsl_multimin_fminimizer_name:          Initializing the Multidimensional Minimizer.
                                                              (line  56)
* gsl_multimin_fminimizer_nmsimplex:     Multimin Algorithms without Derivatives.
                                                              (line  11)
* gsl_multimin_fminimizer_nmsimplex2:    Multimin Algorithms without Derivatives.
                                                              (line  10)
* gsl_multimin_fminimizer_nmsimplex2rand: Multimin Algorithms without Derivatives.
                                                              (line  50)
* gsl_multimin_fminimizer_set:           Initializing the Multidimensional Minimizer.
                                                              (line  27)
* gsl_multimin_fminimizer_size:          Multimin Iteration.  (line  38)
* gsl_multimin_fminimizer_x:             Multimin Iteration.  (line  30)
* 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  56)
* gsl_multiroot_fdfsolver_gnewton:       Algorithms using Derivatives.
                                                              (line  92)
* 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  62)
* 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  47)
* gsl_multiroot_fsolver_alloc:           Initializing the Multidimensional Solver.
                                                              (line  13)
* gsl_multiroot_fsolver_broyden:         Algorithms without Derivatives.
                                                              (line  45)
* gsl_multiroot_fsolver_dnewton:         Algorithms without Derivatives.
                                                              (line  26)
* 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  54)
* gsl_multiroot_fsolver_hybrid:          Algorithms without Derivatives.
                                                              (line  22)
* 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  60)
* 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_multiset_alloc:                    Multiset allocation. (line   7)
* gsl_multiset_calloc:                   Multiset allocation. (line  15)
* gsl_multiset_data:                     Multiset properties. (line  13)
* gsl_multiset_fprintf:                  Reading and writing multisets.
                                                              (line  28)
* gsl_multiset_fread:                    Reading and writing multisets.
                                                              (line  18)
* gsl_multiset_free:                     Multiset allocation. (line  29)
* gsl_multiset_fscanf:                   Reading and writing multisets.
                                                              (line  36)
* gsl_multiset_fwrite:                   Reading and writing multisets.
                                                              (line  11)
* gsl_multiset_get:                      Accessing multiset elements.
                                                              (line  10)
* gsl_multiset_init_first:               Multiset allocation. (line  21)
* gsl_multiset_init_last:                Multiset allocation. (line  25)
* gsl_multiset_k:                        Multiset properties. (line  10)
* gsl_multiset_memcpy:                   Multiset allocation. (line  33)
* gsl_multiset_n:                        Multiset properties. (line   7)
* gsl_multiset_next:                     Multiset functions.  (line   7)
* gsl_multiset_prev:                     Multiset functions.  (line  15)
* gsl_multiset_valid:                    Multiset properties. (line  17)
* 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  38)
* gsl_ntuple_read:                       Reading ntuples.     (line   7)
* gsl_ntuple_write:                      Writing ntuples.     (line   7)
* gsl_odeiv2_control_alloc:              Adaptive Step-size Control.
                                                              (line  75)
* gsl_odeiv2_control_errlevel:           Adaptive Step-size Control.
                                                              (line 118)
* gsl_odeiv2_control_free:               Adaptive Step-size Control.
                                                              (line  87)
* gsl_odeiv2_control_hadjust:            Adaptive Step-size Control.
                                                              (line  93)
* gsl_odeiv2_control_init:               Adaptive Step-size Control.
                                                              (line  82)
* gsl_odeiv2_control_name:               Adaptive Step-size Control.
                                                              (line 107)
* gsl_odeiv2_control_scaled_new:         Adaptive Step-size Control.
                                                              (line  63)
* gsl_odeiv2_control_set_driver:         Adaptive Step-size Control.
                                                              (line 124)
* gsl_odeiv2_control_standard_new:       Adaptive Step-size Control.
                                                              (line  12)
* gsl_odeiv2_control_y_new:              Adaptive Step-size Control.
                                                              (line  46)
* gsl_odeiv2_control_yp_new:             Adaptive Step-size Control.
                                                              (line  54)
* gsl_odeiv2_driver_alloc_scaled_new:    Driver.              (line  24)
* gsl_odeiv2_driver_alloc_standard_new:  Driver.              (line  19)
* gsl_odeiv2_driver_alloc_y_new:         Driver.              (line  12)
* gsl_odeiv2_driver_alloc_yp_new:        Driver.              (line  15)
* gsl_odeiv2_driver_apply:               Driver.              (line  48)
* gsl_odeiv2_driver_apply_fixed_step:    Driver.              (line  65)
* gsl_odeiv2_driver_free:                Driver.              (line  80)
* gsl_odeiv2_driver_reset:               Driver.              (line  71)
* gsl_odeiv2_driver_reset_hstart:        Driver.              (line  75)
* gsl_odeiv2_driver_set_hmax:            Driver.              (line  38)
* gsl_odeiv2_driver_set_hmin:            Driver.              (line  33)
* gsl_odeiv2_driver_set_nmax:            Driver.              (line  43)
* gsl_odeiv2_evolve_alloc:               Evolution.           (line  11)
* gsl_odeiv2_evolve_apply:               Evolution.           (line  18)
* gsl_odeiv2_evolve_apply_fixed_step:    Evolution.           (line  56)
* gsl_odeiv2_evolve_free:                Evolution.           (line  69)
* gsl_odeiv2_evolve_reset:               Evolution.           (line  64)
* gsl_odeiv2_evolve_set_driver:          Evolution.           (line  74)
* gsl_odeiv2_step_alloc:                 Stepping Functions.  (line  12)
* gsl_odeiv2_step_apply:                 Stepping Functions.  (line  54)
* gsl_odeiv2_step_bsimp:                 Stepping Functions.  (line 124)
* gsl_odeiv2_step_free:                  Stepping Functions.  (line  24)
* gsl_odeiv2_step_msadams:               Stepping Functions.  (line 129)
* gsl_odeiv2_step_msbdf:                 Stepping Functions.  (line 137)
* gsl_odeiv2_step_name:                  Stepping Functions.  (line  29)
* gsl_odeiv2_step_order:                 Stepping Functions.  (line  39)
* gsl_odeiv2_step_reset:                 Stepping Functions.  (line  19)
* gsl_odeiv2_step_rk1imp:                Stepping Functions.  (line 106)
* gsl_odeiv2_step_rk2:                   Stepping Functions.  (line  88)
* gsl_odeiv2_step_rk2imp:                Stepping Functions.  (line 112)
* gsl_odeiv2_step_rk4:                   Stepping Functions.  (line  91)
* gsl_odeiv2_step_rk4imp:                Stepping Functions.  (line 118)
* gsl_odeiv2_step_rk8pd:                 Stepping Functions.  (line 103)
* gsl_odeiv2_step_rkck:                  Stepping Functions.  (line 100)
* gsl_odeiv2_step_rkf45:                 Stepping Functions.  (line  96)
* gsl_odeiv2_step_set_driver:            Stepping Functions.  (line  45)
* 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  18)
* 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_eval:                 Polynomial Evaluation.
                                                              (line  18)
* gsl_poly_complex_solve:                General Polynomial Equations.
                                                              (line  27)
* gsl_poly_complex_solve_cubic:          Cubic Equations.     (line  26)
* 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  40)
* gsl_poly_dd_hermite_init:              Divided Difference Representation of Polynomials.
                                                              (line  57)
* gsl_poly_dd_init:                      Divided Difference Representation of Polynomials.
                                                              (line  32)
* gsl_poly_dd_taylor:                    Divided Difference Representation of Polynomials.
                                                              (line  47)
* gsl_poly_eval:                         Polynomial Evaluation.
                                                              (line  13)
* gsl_poly_eval_derivs:                  Polynomial Evaluation.
                                                              (line  28)
* gsl_poly_solve_cubic:                  Cubic Equations.     (line   8)
* gsl_poly_solve_quadratic:              Quadratic Equations. (line   8)
* gsl_pow_2:                             Small integer powers.
                                                              (line  20)
* gsl_pow_3:                             Small integer powers.
                                                              (line  21)
* gsl_pow_4:                             Small integer powers.
                                                              (line  22)
* gsl_pow_5:                             Small integer powers.
                                                              (line  23)
* gsl_pow_6:                             Small integer powers.
                                                              (line  24)
* gsl_pow_7:                             Small integer powers.
                                                              (line  25)
* gsl_pow_8:                             Small integer powers.
                                                              (line  26)
* gsl_pow_9:                             Small integer powers.
                                                              (line  27)
* gsl_pow_int:                           Small integer powers.
                                                              (line  12)
* gsl_pow_uint:                          Small integer powers.
                                                              (line  13)
* 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_halton:                       Quasi-random number generator algorithms.
                                                              (line  19)
* 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_reversehalton:                Quasi-random number generator algorithms.
                                                              (line  20)
* 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_knuth:                   The Gamma Distribution.
                                                              (line  25)
* gsl_ran_gamma_pdf:                     The Gamma Distribution.
                                                              (line  29)
* 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  35)
* gsl_ran_multinomial_pdf:               The Multinomial Distribution.
                                                              (line  29)
* 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:                              Representation of complex numbers.
                                                              (line  35)
* gsl_rng_alloc:                         Random number generator initialization.
                                                              (line   7)
* gsl_rng_borosh13:                      Other random number generators.
                                                              (line 175)
* gsl_rng_clone:                         Copying random number generator state.
                                                              (line  17)
* gsl_rng_cmrg:                          Random number generator algorithms.
                                                              (line  97)
* gsl_rng_coveyou:                       Other random number generators.
                                                              (line 203)
* gsl_rng_env_setup:                     Random number environment variables.
                                                              (line  12)
* gsl_rng_fishman18:                     Other random number generators.
                                                              (line 176)
* gsl_rng_fishman20:                     Other random number generators.
                                                              (line 177)
* gsl_rng_fishman2x:                     Other random number generators.
                                                              (line 192)
* gsl_rng_fread:                         Reading and writing random number generator state.
                                                              (line  18)
* gsl_rng_free:                          Random number generator initialization.
                                                              (line  46)
* gsl_rng_fwrite:                        Reading and writing random number generator state.
                                                              (line  10)
* gsl_rng_get:                           Sampling from a random number generator.
                                                              (line  12)
* gsl_rng_gfsr4:                         Random number generator algorithms.
                                                              (line 172)
* gsl_rng_knuthran:                      Other random number generators.
                                                              (line 168)
* gsl_rng_knuthran2:                     Other random number generators.
                                                              (line 158)
* gsl_rng_knuthran2002:                  Other random number generators.
                                                              (line 167)
* gsl_rng_lecuyer21:                     Other random number generators.
                                                              (line 178)
* 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 119)
* gsl_rng_mt19937:                       Random number generator algorithms.
                                                              (line  20)
* 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  58)
* 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  70)
* gsl_rng_ranlux389:                     Random number generator algorithms.
                                                              (line  71)
* gsl_rng_ranlxd1:                       Random number generator algorithms.
                                                              (line  64)
* gsl_rng_ranlxd2:                       Random number generator algorithms.
                                                              (line  65)
* gsl_rng_ranlxs0:                       Random number generator algorithms.
                                                              (line  45)
* gsl_rng_ranlxs1:                       Random number generator algorithms.
                                                              (line  46)
* gsl_rng_ranlxs2:                       Random number generator algorithms.
                                                              (line  47)
* 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 136)
* gsl_rng_taus2:                         Random number generator algorithms.
                                                              (line 137)
* 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  19)
* gsl_rng_uniform_int:                   Sampling from a random number generator.
                                                              (line  38)
* gsl_rng_uniform_pos:                   Sampling from a random number generator.
                                                              (line  29)
* gsl_rng_vax:                           Other random number generators.
                                                              (line  87)
* gsl_rng_waterman14:                    Other random number generators.
                                                              (line 179)
* 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:                       Representation of complex numbers.
                                                              (line  40)
* gsl_set_error_handler:                 Error Handlers.      (line  44)
* gsl_set_error_handler_off:             Error Handlers.      (line  68)
* GSL_SET_IMAG:                          Representation of complex numbers.
                                                              (line  50)
* GSL_SET_REAL:                          Representation of complex numbers.
                                                              (line  49)
* 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  16)
* gsl_sf_angle_restrict_pos_e:           Restriction Functions.
                                                              (line  17)
* 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 Spherical Bessel Functions.
                                                              (line  11)
* gsl_sf_bessel_I0_scaled:               Regular Modified Cylindrical Bessel Functions.
                                                              (line  32)
* 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 Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_I1_scaled:               Regular Modified Cylindrical Bessel Functions.
                                                              (line  38)
* gsl_sf_bessel_i1_scaled_e:             Regular Modified Spherical Bessel Functions.
                                                              (line  19)
* gsl_sf_bessel_I1_scaled_e:             Regular Modified Cylindrical Bessel Functions.
                                                              (line  40)
* 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 Cylindrical Bessel Functions.
                                                              (line  39)
* gsl_sf_bessel_k1_scaled:               Irregular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_k1_scaled_e:             Irregular Modified Spherical Bessel Functions.
                                                              (line  19)
* gsl_sf_bessel_K1_scaled_e:             Irregular Modified Cylindrical Bessel Functions.
                                                              (line  41)
* 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  36)
* gsl_sf_ellint_D_e:                     Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  38)
* gsl_sf_ellint_E:                       Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  17)
* gsl_sf_ellint_E_e:                     Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  19)
* gsl_sf_ellint_Ecomp:                   Legendre Form of Complete Elliptic Integrals.
                                                              (line  15)
* gsl_sf_ellint_Ecomp_e:                 Legendre Form of Complete Elliptic Integrals.
                                                              (line  17)
* 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  26)
* gsl_sf_ellint_P_e:                     Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  28)
* gsl_sf_ellint_Pcomp:                   Legendre Form of Complete Elliptic Integrals.
                                                              (line  24)
* gsl_sf_ellint_Pcomp_e:                 Legendre Form of Complete Elliptic Integrals.
                                                              (line  26)
* 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_expint_En:                      Exponential Integral.
                                                              (line  24)
* gsl_sf_expint_En_e:                    Exponential Integral.
                                                              (line  26)
* 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  20)
* gsl_sf_fermi_dirac_0_e:                Complete Fermi-Dirac Integrals.
                                                              (line  22)
* gsl_sf_fermi_dirac_1:                  Complete Fermi-Dirac Integrals.
                                                              (line  26)
* gsl_sf_fermi_dirac_1_e:                Complete Fermi-Dirac Integrals.
                                                              (line  28)
* gsl_sf_fermi_dirac_2:                  Complete Fermi-Dirac Integrals.
                                                              (line  32)
* gsl_sf_fermi_dirac_2_e:                Complete Fermi-Dirac Integrals.
                                                              (line  34)
* gsl_sf_fermi_dirac_3half:              Complete Fermi-Dirac Integrals.
                                                              (line  57)
* gsl_sf_fermi_dirac_3half_e:            Complete Fermi-Dirac Integrals.
                                                              (line  59)
* gsl_sf_fermi_dirac_half:               Complete Fermi-Dirac Integrals.
                                                              (line  51)
* gsl_sf_fermi_dirac_half_e:             Complete Fermi-Dirac Integrals.
                                                              (line  53)
* 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  38)
* gsl_sf_fermi_dirac_int_e:              Complete Fermi-Dirac Integrals.
                                                              (line  40)
* gsl_sf_fermi_dirac_m1:                 Complete Fermi-Dirac Integrals.
                                                              (line  13)
* gsl_sf_fermi_dirac_m1_e:               Complete Fermi-Dirac Integrals.
                                                              (line  15)
* gsl_sf_fermi_dirac_mhalf:              Complete Fermi-Dirac Integrals.
                                                              (line  45)
* gsl_sf_fermi_dirac_mhalf_e:            Complete Fermi-Dirac Integrals.
                                                              (line  47)
* gsl_sf_gamma:                          Gamma Functions.     (line  15)
* gsl_sf_gamma_e:                        Gamma Functions.     (line  16)
* 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  50)
* gsl_sf_gammainv_e:                     Gamma Functions.     (line  51)
* gsl_sf_gammastar:                      Gamma Functions.     (line  41)
* gsl_sf_gammastar_e:                    Gamma Functions.     (line  42)
* 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  53)
* gsl_sf_hyperg_2F1_conj:                Hypergeometric Functions.
                                                              (line  65)
* gsl_sf_hyperg_2F1_conj_e:              Hypergeometric Functions.
                                                              (line  67)
* 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  55)
* 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  42)
* gsl_sf_hyperg_U_e10_e:                 Hypergeometric Functions.
                                                              (line  47)
* 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  50)
* 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  32)
* gsl_sf_legendre_Q0_e:                  Legendre Polynomials.
                                                              (line  34)
* gsl_sf_legendre_Q1:                    Legendre Polynomials.
                                                              (line  38)
* gsl_sf_legendre_Q1_e:                  Legendre Polynomials.
                                                              (line  40)
* gsl_sf_legendre_Ql:                    Legendre Polynomials.
                                                              (line  44)
* gsl_sf_legendre_Ql_e:                  Legendre Polynomials.
                                                              (line  46)
* 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  14)
* gsl_sf_lnbeta_e:                       Beta Functions.      (line  16)
* 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  23)
* gsl_sf_lngamma_complex_e:              Gamma Functions.     (line  56)
* gsl_sf_lngamma_e:                      Gamma Functions.     (line  24)
* gsl_sf_lngamma_sgn_e:                  Gamma Functions.     (line  32)
* gsl_sf_lnpoch:                         Pochhammer Symbol.   (line  16)
* gsl_sf_lnpoch_e:                       Pochhammer Symbol.   (line  18)
* gsl_sf_lnpoch_sgn_e:                   Pochhammer Symbol.   (line  23)
* 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_mathieu_a:                      Mathieu Function Characteristic Values.
                                                              (line   8)
* gsl_sf_mathieu_a_array:                Mathieu Function Characteristic Values.
                                                              (line  16)
* gsl_sf_mathieu_alloc:                  Mathieu Function Workspace.
                                                              (line  12)
* gsl_sf_mathieu_b:                      Mathieu Function Characteristic Values.
                                                              (line  10)
* gsl_sf_mathieu_b_array:                Mathieu Function Characteristic Values.
                                                              (line  19)
* gsl_sf_mathieu_ce:                     Angular Mathieu Functions.
                                                              (line   8)
* gsl_sf_mathieu_ce_array:               Angular Mathieu Functions.
                                                              (line  16)
* gsl_sf_mathieu_free:                   Mathieu Function Workspace.
                                                              (line  19)
* gsl_sf_mathieu_Mc:                     Radial Mathieu Functions.
                                                              (line   8)
* gsl_sf_mathieu_Mc_array:               Radial Mathieu Functions.
                                                              (line  21)
* gsl_sf_mathieu_Ms:                     Radial Mathieu Functions.
                                                              (line  10)
* gsl_sf_mathieu_Ms_array:               Radial Mathieu Functions.
                                                              (line  24)
* gsl_sf_mathieu_se:                     Angular Mathieu Functions.
                                                              (line  10)
* gsl_sf_mathieu_se_array:               Angular Mathieu Functions.
                                                              (line  19)
* 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  29)
* gsl_sf_pochrel_e:                      Pochhammer Symbol.   (line  31)
* 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  25)
* gsl_sort2:                             Sorting vectors.     (line  30)
* gsl_sort_index:                        Sorting vectors.     (line  46)
* 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  36)
* gsl_sort_vector2:                      Sorting vectors.     (line  40)
* gsl_sort_vector_index:                 Sorting vectors.     (line  56)
* 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_correlation:                 Correlation.         (line   9)
* gsl_stats_covariance:                  Covariance.          (line   9)
* gsl_stats_covariance_m:                Covariance.          (line  18)
* 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  75)
* gsl_stats_skew:                        Higher moments (skewness and kurtosis).
                                                              (line   8)
* gsl_stats_skew_m_sd:                   Higher moments (skewness and kurtosis).
                                                              (line  21)
* gsl_stats_spearman:                    Correlation.         (line  21)
* gsl_stats_tss:                         Mean and standard deviation and variance.
                                                              (line  54)
* gsl_stats_tss_m:                       Mean and standard deviation and variance.
                                                              (line  56)
* 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  65)
* gsl_stats_wabsdev:                     Weighted Samples.    (line  85)
* gsl_stats_wabsdev_m:                   Weighted Samples.    (line  94)
* gsl_stats_wkurtosis:                   Weighted Samples.    (line 112)
* gsl_stats_wkurtosis_m_sd:              Weighted Samples.    (line 119)
* 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  99)
* gsl_stats_wskew_m_sd:                  Weighted Samples.    (line 106)
* gsl_stats_wtss:                        Weighted Samples.    (line  73)
* gsl_stats_wtss_m:                      Weighted Samples.    (line  76)
* 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  38)
* 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   7)
* gsl_vector_add_constant:               Vector operations.   (line  33)
* 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  54)
* 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  23)
* gsl_vector_equal:                      Vector properties.   (line  20)
* 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  40)
* gsl_vector_isneg:                      Vector properties.   (line  13)
* gsl_vector_isnonneg:                   Vector properties.   (line  14)
* gsl_vector_isnull:                     Vector properties.   (line  11)
* gsl_vector_ispos:                      Vector properties.   (line  12)
* gsl_vector_max:                        Finding maximum and minimum elements of vectors.
                                                              (line   9)
* gsl_vector_max_index:                  Finding maximum and minimum elements of vectors.
                                                              (line  20)
* gsl_vector_memcpy:                     Copying vectors.     (line  14)
* gsl_vector_min:                        Finding maximum and minimum elements of vectors.
                                                              (line  12)
* gsl_vector_min_index:                  Finding maximum and minimum elements of vectors.
                                                              (line  25)
* gsl_vector_minmax:                     Finding maximum and minimum elements of vectors.
                                                              (line  16)
* gsl_vector_minmax_index:               Finding maximum and minimum elements of vectors.
                                                              (line  31)
* gsl_vector_mul:                        Vector operations.   (line  18)
* gsl_vector_ptr:                        Accessing vector elements.
                                                              (line  52)
* gsl_vector_reverse:                    Exchanging elements. (line  15)
* gsl_vector_scale:                      Vector operations.   (line  28)
* gsl_vector_set:                        Accessing vector elements.
                                                              (line  46)
* 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  12)
* 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  66)
* gsl_wavelet2d_nstransform_forward:     DWT in two dimension.
                                                              (line  69)
* gsl_wavelet2d_nstransform_inverse:     DWT in two dimension.
                                                              (line  72)
* gsl_wavelet2d_nstransform_matrix:      DWT in two dimension.
                                                              (line  78)
* gsl_wavelet2d_nstransform_matrix_forward: DWT in two dimension.
                                                              (line  80)
* gsl_wavelet2d_nstransform_matrix_inverse: DWT in two dimension.
                                                              (line  82)
* 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  56)
* gsl_wavelet2d_transform_matrix_forward: DWT in two dimension.
                                                              (line  58)
* gsl_wavelet2d_transform_matrix_inverse: DWT in two dimension.
                                                              (line  60)
* 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  63)


File: gsl-ref.info,  Node: Variable Index,  Next: Type Index,  Prev: Function Index,  Up: Top

Variable Index
**************

[index]
* Menu:

* alpha <1>:                             VEGAS.               (line 146)
* alpha:                                 MISER.               (line 115)
* dither:                                MISER.               (line 131)
* estimate_frac:                         MISER.               (line  94)
* GSL_C99_INLINE <1>:                    Inline functions.    (line   6)
* GSL_C99_INLINE:                        Accessing vector elements.
                                                              (line  26)
* gsl_check_range:                       Accessing vector elements.
                                                              (line  31)
* GSL_EDOM:                              Error Codes.         (line  14)
* GSL_EINVAL:                            Error Codes.         (line  30)
* GSL_ENOMEM:                            Error Codes.         (line  24)
* GSL_ERANGE:                            Error Codes.         (line  19)
* GSL_IEEE_MODE:                         Setting up your IEEE environment.
                                                              (line  24)
* GSL_NAN:                               Infinities and Not-a-number.
                                                              (line  15)
* GSL_NEGINF:                            Infinities and Not-a-number.
                                                              (line  11)
* GSL_POSINF:                            Infinities and Not-a-number.
                                                              (line   7)
* GSL_RANGE_CHECK_OFF:                   Accessing vector elements.
                                                              (line  17)
* gsl_rng_default:                       Random number environment variables.
                                                              (line  12)
* gsl_rng_default_seed <1>:              Random number generator initialization.
                                                              (line  17)
* gsl_rng_default_seed:                  Random number environment variables.
                                                              (line  12)
* GSL_RNG_SEED <1>:                      Random number environment variables.
                                                              (line  12)
* GSL_RNG_SEED:                          Random number generator initialization.
                                                              (line  17)
* GSL_RNG_TYPE:                          Random number environment variables.
                                                              (line  12)
* HAVE_INLINE:                           Inline functions.    (line   6)
* iterations:                            VEGAS.               (line 151)
* min_calls:                             MISER.               (line  99)
* min_calls_per_bisection:               MISER.               (line 107)
* mode:                                  VEGAS.               (line 169)
* ostream:                               VEGAS.               (line 179)
* stage:                                 VEGAS.               (line 155)
* verbose:                               VEGAS.               (line 178)


File: gsl-ref.info,  Node: Type Index,  Next: Concept Index,  Prev: Variable Index,  Up: Top

Type Index
**********

[index]
* Menu:

* gsl_block:                             Blocks.              (line   6)
* gsl_bspline_deriv_workspace:           Initializing the B-splines solver.
                                                              (line  22)
* gsl_bspline_workspace:                 Initializing the B-splines solver.
                                                              (line  12)
* gsl_cheb_series:                       Chebyshev Definitions.
                                                              (line   6)
* gsl_combination:                       The Combination struct.
                                                              (line   6)
* gsl_complex:                           Representation of complex numbers.
                                                              (line   6)
* gsl_dht:                               Discrete Hankel Transform Functions.
                                                              (line   7)
* gsl_eigen_gen_workspace:               Real Generalized Nonsymmetric Eigensystems.
                                                              (line  43)
* gsl_eigen_genherm_workspace:           Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  19)
* gsl_eigen_genhermv_workspace:          Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  37)
* gsl_eigen_gensymm_workspace:           Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  24)
* gsl_eigen_gensymmv_workspace:          Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  41)
* gsl_eigen_genv_workspace:              Real Generalized Nonsymmetric Eigensystems.
                                                              (line  98)
* gsl_eigen_herm_workspace:              Complex Hermitian Matrices.
                                                              (line  11)
* gsl_eigen_hermv_workspace:             Complex Hermitian Matrices.
                                                              (line  30)
* gsl_eigen_nonsymm_workspace:           Real Nonsymmetric Matrices.
                                                              (line  19)
* gsl_eigen_nonsymmv_workspace:          Real Nonsymmetric Matrices.
                                                              (line  80)
* gsl_eigen_symm_workspace:              Real Symmetric Matrices.
                                                              (line  14)
* gsl_eigen_symmv_workspace:             Real Symmetric Matrices.
                                                              (line  32)
* gsl_error_handler_t:                   Error Handlers.      (line  24)
* gsl_fft_complex_wavetable:             Mixed-radix FFT routines for complex data.
                                                              (line  79)
* gsl_fft_complex_workspace:             Mixed-radix FFT routines for complex data.
                                                              (line 107)
* gsl_fft_halfcomplex_wavetable:         Mixed-radix FFT routines for real data.
                                                              (line  76)
* gsl_fft_real_wavetable:                Mixed-radix FFT routines for real data.
                                                              (line  76)
* gsl_fft_real_workspace:                Mixed-radix FFT routines for real data.
                                                              (line 106)
* gsl_function:                          Providing the function to solve.
                                                              (line  12)
* gsl_function_fdf:                      Providing the function to solve.
                                                              (line  51)
* 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_integration_cquad_workspace:       CQUAD doubly-adaptive integration.
                                                              (line  22)
* gsl_integration_glfixed_table:         Fixed order Gauss-Legendre integration.
                                                              (line  16)
* gsl_integration_qawo_table:            QAWO adaptive integration for oscillatory functions.
                                                              (line  14)
* gsl_integration_qaws_table:            QAWS adaptive integration for singular functions.
                                                              (line  14)
* gsl_integration_workspace:             QAG adaptive integration.
                                                              (line  17)
* gsl_interp:                            Interpolation Functions.
                                                              (line  11)
* gsl_interp_accel:                      Index Look-up and Acceleration.
                                                              (line  20)
* gsl_interp_type:                       Interpolation Types. (line   6)
* gsl_matrix:                            Matrices.            (line  10)
* gsl_matrix_const_view:                 Matrix views.        (line   6)
* gsl_matrix_view:                       Matrix views.        (line   6)
* gsl_min_fminimizer:                    Initializing the Minimizer.
                                                              (line   8)
* gsl_min_fminimizer_type:               Initializing the Minimizer.
                                                              (line   8)
* gsl_monte_function:                    Monte Carlo Interface.
                                                              (line  28)
* gsl_monte_miser_state:                 MISER.               (line  47)
* gsl_monte_plain_state:                 PLAIN Monte Carlo.   (line  30)
* gsl_monte_vegas_state:                 VEGAS.               (line  51)
* gsl_multifit_fdfsolver:                Initializing the Nonlinear Least-Squares Solver.
                                                              (line  18)
* gsl_multifit_fdfsolver_type:           Initializing the Nonlinear Least-Squares Solver.
                                                              (line  18)
* gsl_multifit_fsolver:                  Initializing the Nonlinear Least-Squares Solver.
                                                              (line   8)
* gsl_multifit_fsolver_type:             Initializing the Nonlinear Least-Squares Solver.
                                                              (line   8)
* gsl_multifit_function:                 Providing the Function to be Minimized.
                                                              (line  11)
* gsl_multifit_function_fdf:             Providing the Function to be Minimized.
                                                              (line  31)
* gsl_multifit_linear_workspace:         Multi-parameter fitting.
                                                              (line  43)
* gsl_multifit_robust_workspace:         Robust linear regression.
                                                              (line  77)
* gsl_multimin_fdfminimizer:             Initializing the Multidimensional Minimizer.
                                                              (line  15)
* gsl_multimin_fdfminimizer_type:        Initializing the Multidimensional Minimizer.
                                                              (line  15)
* gsl_multimin_fminimizer:               Initializing the Multidimensional Minimizer.
                                                              (line  15)
* gsl_multimin_fminimizer_type:          Initializing the Multidimensional Minimizer.
                                                              (line  15)
* gsl_multimin_function:                 Providing a function to minimize.
                                                              (line  43)
* gsl_multimin_function_fdf:             Providing a function to minimize.
                                                              (line  14)
* gsl_multiroot_fdfsolver:               Initializing the Multidimensional Solver.
                                                              (line  29)
* gsl_multiroot_fdfsolver_type:          Initializing the Multidimensional Solver.
                                                              (line  29)
* gsl_multiroot_fsolver:                 Initializing the Multidimensional Solver.
                                                              (line  13)
* gsl_multiroot_fsolver_type:            Initializing the Multidimensional Solver.
                                                              (line  13)
* 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_multiset:                          The Multiset struct. (line   6)
* gsl_ntuple:                            The ntuple struct.   (line   6)
* gsl_ntuple_select_fn:                  Histogramming ntuple values.
                                                              (line  13)
* gsl_ntuple_value_fn:                   Histogramming ntuple values.
                                                              (line  24)
* gsl_odeiv2_control:                    Adaptive Step-size Control.
                                                              (line  12)
* gsl_odeiv2_control_type:               Adaptive Step-size Control.
                                                              (line  12)
* gsl_odeiv2_evolve:                     Evolution.           (line  11)
* gsl_odeiv2_step:                       Stepping Functions.  (line  12)
* gsl_odeiv2_step_type:                  Stepping Functions.  (line  12)
* gsl_odeiv2_system:                     Defining the ODE System.
                                                              (line  15)
* gsl_permutation:                       The Permutation struct.
                                                              (line   6)
* gsl_poly_complex_workspace:            General Polynomial Equations.
                                                              (line  13)
* gsl_qrng:                              Quasi-random number generator initialization.
                                                              (line   8)
* gsl_qrng_type:                         Quasi-random number generator initialization.
                                                              (line   8)
* gsl_ran_discrete_t:                    General Discrete Distributions.
                                                              (line  52)
* gsl_rng:                               Random number generator initialization.
                                                              (line   7)
* gsl_rng_type:                          The Random Number Generator Interface.
                                                              (line  18)
* gsl_root_fdfsolver:                    Initializing the Solver.
                                                              (line  23)
* gsl_root_fdfsolver_type:               Initializing the Solver.
                                                              (line  23)
* gsl_root_fsolver:                      Initializing the Solver.
                                                              (line   8)
* gsl_root_fsolver_type:                 Initializing the Solver.
                                                              (line   8)
* gsl_sf_mathieu_workspace:              Mathieu Function Workspace.
                                                              (line  12)
* gsl_sf_result:                         The gsl_sf_result struct.
                                                              (line   6)
* gsl_sf_result_e10:                     The gsl_sf_result struct.
                                                              (line   6)
* 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)
* gsl_spline:                            Higher-level Interface.
                                                              (line  16)
* gsl_sum_levin_u_workspace:             Acceleration functions.
                                                              (line  25)
* gsl_sum_levin_utrunc_workspace:        Acceleration functions without error estimation.
                                                              (line  24)
* gsl_vector:                            Vectors.             (line  11)
* gsl_vector_const_view:                 Vector views.        (line  12)
* gsl_vector_view:                       Vector views.        (line  12)
* gsl_wavelet:                           DWT Initialization.  (line  11)
* gsl_wavelet_type:                      DWT Initialization.  (line   6)
* gsl_wavelet_workspace:                 DWT Initialization.  (line  53)


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  33)
* Adams method:                          Stepping Functions.  (line 129)
* 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 Mathieu Functions:             Angular Mathieu Functions.
                                                              (line   6)
* angular reduction:                     Restriction Functions.
                                                              (line   6)
* ANSI C, use of:                        Using the library.   (line   6)
* Apell symbol, see Pochhammer 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  37)
* astronomical constants:                Astronomy and Astrophysics.
                                                              (line   6)
* ASUM, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line  47)
* atanh:                                 Elementary Functions.
                                                              (line  41)
* 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 124)
* balancing matrices:                    Balancing.           (line   6)
* Basic Linear Algebra Subroutines (BLAS) <1>: BLAS Support.  (line   6)
* Basic Linear Algebra Subroutines (BLAS): GSL CBLAS Library. (line   6)
* basis splines, B-splines:              Basis Splines.       (line   6)
* basis splines, derivatives:            Evaluation of B-spline basis function derivatives.
                                                              (line   6)
* basis splines, evaluation:             Evaluation of B-spline basis functions.
                                                              (line   6)
* basis splines, examples:               Example programs for B-splines.
                                                              (line   6)
* basis splines, Greville abscissae:     Working with the Greville abscissae.
                                                              (line   6)
* basis splines, initializing:           Initializing the B-splines solver.
                                                              (line   6)
* basis splines, Marsden-Schoenberg points: Working with the Greville abscissae.
                                                              (line   6)
* basis splines, overview:               Overview of B-splines.
                                                              (line   6)
* BDF method:                            Stepping Functions.  (line 137)
* 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 algorithm, minimization:          Multimin Algorithms with Derivatives.
                                                              (line  37)
* 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  45)
* 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 124)
* C extensions, compatible use of:       Using the library.   (line   6)
* C++, compatibility:                    Compatibility with C++.
                                                              (line   6)
* C99, inline keyword:                   Inline functions.    (line   6)
* Carlson forms of Elliptic integrals:   Definition of Carlson Forms.
                                                              (line   6)
* Cash-Karp, Runge-Kutta method:         Stepping Functions.  (line 100)
* 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)
* ce(q,x), Mathieu function:             Angular Mathieu Functions.
                                                              (line   6)
* Chebyshev series:                      Chebyshev Approximations.
                                                              (line   6)
* checking combination for validity:     Combination properties.
                                                              (line  18)
* checking multiset for validity:        Multiset properties. (line  17)
* 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  97)
* 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  56)
* complex hermitian matrix, eigensystem: Complex Hermitian Matrices.
                                                              (line   9)
* 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 with Derivatives.
                                                              (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)
* correlation, of two datasets:          Correlation.         (line   6)
* 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)
* cquad, doubly-adaptive integration:    CQUAD doubly-adaptive integration.
                                                              (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)
* DAXPY, Level-1 BLAS:                   Level 1 GSL BLAS Interface.
                                                              (line  96)
* 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  83)
* Deuflhard and Bader, Bulirsch-Stoer method.: Stepping Functions.
                                                              (line 124)
* DFTs, see FFT:                         Fast Fourier Transforms.
                                                              (line   6)
* diagonal, of a matrix:                 Creating row and column views.
                                                              (line  62)
* 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)
* discontinuities, in ODE systems:       Evolution.           (line  77)
* 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  26)
* Discrete random numbers:               General Discrete Distributions.
                                                              (line  62)
* 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  13)
* error codes, reserved:                 Error Codes.         (line   6)
* 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)
* Eta Function:                          Eta Function.        (line   6)
* euclidean distance function, hypot:    Elementary Functions.
                                                              (line  29)
* 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, C++:                       Compatibility with C++.
                                                              (line   6)
* exceptions, floating point:            Handling floating point exceptions.
                                                              (line   6)
* exceptions, IEEE arithmetic:           Setting up your IEEE environment.
                                                              (line   6)
* exchanging permutation elements:       Accessing permutation elements.
                                                              (line  18)
* 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)
* extern inline:                         Inline functions.    (line   6)
* F-distribution:                        The F-distribution.  (line  16)
* factorial:                             Factorials.          (line   6)
* 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)
* Fehlberg method, differential equations: Stepping Functions.
                                                              (line  96)
* 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 with Derivatives.
                                                              (line  12)
* floating point exceptions:             Handling floating point exceptions.
                                                              (line   6)
* floating point numbers, approximate comparison: Approximate Comparison of Floating Point Numbers.
                                                              (line  13)
* floating point registers:              Examining floating point registers.
                                                              (line   6)
* 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 172)
* 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 software, explanation of:         GSL is Free Software.
                                                              (line   6)
* frexp:                                 Elementary Functions.
                                                              (line  49)
* 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)
* 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)
* generalized eigensystems:              Real Generalized Nonsymmetric Eigensystems.
                                                              (line   6)
* generalized hermitian definite eigensystems: Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line   6)
* generalized symmetric eigensystems:    Real Generalized Symmetric-Definite Eigensystems.
                                                              (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_C99_INLINE:                        Inline functions.    (line   6)
* GSL_RNG_SEED:                          Random number generator initialization.
                                                              (line  17)
* 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 181)
* HERK, Level-3 BLAS:                    Level 3 GSL BLAS Interface.
                                                              (line 140)
* hermitian matrix, complex, eigensystem: Complex Hermitian Matrices.
                                                              (line   9)
* Hessenberg decomposition:              Hessenberg Decomposition of Real Matrices.
                                                              (line   6)
* Hessenberg triangular decomposition:   Hessenberg-Triangular Decomposition of Real Matrices.
                                                              (line   6)
* histogram statistics:                  Histogram Statistics.
                                                              (line   6)
* histogram, from ntuple:                Histogramming ntuple values.
                                                              (line  35)
* 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)
* Hurwitz Zeta Function:                 Hurwitz Zeta Function.
                                                              (line   6)
* HYBRID algorithm, unscaled without derivatives: Algorithms without Derivatives.
                                                              (line  22)
* 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  33)
* hyperbolic functions, complex numbers: Complex Hyperbolic Functions.
                                                              (line   6)
* hyperbolic integrals:                  Hyperbolic Integrals.
                                                              (line   6)
* hyperbolic sine, inverse:              Elementary Functions.
                                                              (line  37)
* hyperbolic space:                      Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* hyperbolic tangent, inverse:           Elementary Functions.
                                                              (line  41)
* 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)
* i(x), Bessel Functions:                Regular Modified Spherical Bessel Functions.
                                                              (line   6)
* I(x), Bessel Functions:                Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* 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)
* Implicit Euler method:                 Stepping Functions.  (line 106)
* Implicit Runge-Kutta method:           Stepping Functions.  (line 112)
* 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  44)
* 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  33)
* inverse hyperbolic functions, complex numbers: Inverse Complex Hyperbolic Functions.
                                                              (line   6)
* inverse hyperbolic sine:               Elementary Functions.
                                                              (line  37)
* inverse hyperbolic tangent:            Elementary Functions.
                                                              (line  41)
* inverse of a matrix, by LU decomposition: LU Decomposition. (line  69)
* 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 multisets:           Multiset functions.  (line   7)
* iterating through permutations:        Permutation functions.
                                                              (line  15)
* iterative refinement of solutions in linear systems: LU Decomposition.
                                                              (line  57)
* j(x), Bessel Functions:                Regular Spherical Bessel Functions.
                                                              (line   6)
* J(x), Bessel Functions:                Regular Cylindrical Bessel Functions.
                                                              (line   6)
* Jacobi elliptic functions:             Elliptic Functions (Jacobi).
                                                              (line   6)
* Jacobi orthogonalization:              Singular Value Decomposition.
                                                              (line  58)
* Jacobian matrix, fitting:              Overview of Nonlinear Least-Squares Fitting.
                                                              (line  34)
* Jacobian matrix, ODEs:                 Defining the ODE System.
                                                              (line  34)
* Jacobian matrix, root finding:         Overview of Multidimensional Root Finding.
                                                              (line  42)
* K(x), Bessel Functions:                Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* k(x), Bessel Functions:                Irregular Modified Spherical Bessel Functions.
                                                              (line   6)
* knots, basis splines:                  Constructing the knots vector.
                                                              (line   6)
* kurtosis:                              Higher moments (skewness and kurtosis).
                                                              (line   6)
* Laguerre functions:                    Laguerre Functions.  (line   6)
* Lambert function:                      Lambert W Functions. (line   6)
* Landau distribution:                   The Landau Distribution.
                                                              (line   8)
* LAPACK:                                Eigenvalue and Eigenvector References.
                                                              (line  18)
* Laplace distribution:                  The Laplace Distribution.
                                                              (line   7)
* LD_LIBRARY_PATH:                       Shared Libraries.    (line   6)
* ldexp:                                 Elementary Functions.
                                                              (line  45)
* leading dimension, matrices:           Matrices.            (line   6)
* least squares fit:                     Least-Squares Fitting.
                                                              (line   6)
* least squares fitting, nonlinear:      Nonlinear Least-Squares Fitting.
                                                              (line   6)
* least squares troubleshooting:         Troubleshooting.     (line   6)
* 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)
* Legendre polynomials:                  Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* length, computed accurately using hypot: Elementary Functions.
                                                              (line  29)
* 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  57)
* 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  16)
* 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  24)
* logarithm of Pochhammer symbol:        Pochhammer Symbol.   (line  18)
* logarithm of sinh function, special functions: Hyperbolic Trigonometric Functions.
                                                              (line   8)
* logarithm of the determinant of a matrix: LU Decomposition. (line  91)
* 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)
* Mathieu Function Characteristic Values: Mathieu Function Characteristic Values.
                                                              (line   6)
* Mathieu functions:                     Mathieu 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  83)
* matrix diagonal:                       Creating row and column views.
                                                              (line  62)
* matrix factorization:                  Linear Algebra.      (line   6)
* matrix inverse:                        LU Decomposition.    (line  69)
* matrix square root, Cholesky decomposition: Cholesky Decomposition.
                                                              (line   6)
* matrix subdiagonal:                    Creating row and column views.
                                                              (line  74)
* matrix superdiagonal:                  Creating row and column views.
                                                              (line  86)
* 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  11)
* maximum value, from histogram:         Histogram Statistics.
                                                              (line   6)
* mean:                                  Statistics.          (line   6)
* mean value, from histogram:            Histogram Statistics.
                                                              (line  24)
* Mills' ratio, inverse:                 Probability functions.
                                                              (line  19)
* min:                                   Statistics.          (line   6)
* minimization, BFGS algorithm:          Multimin Algorithms with Derivatives.
                                                              (line  37)
* minimization, caveats:                 Minimization Caveats.
                                                              (line   6)
* minimization, conjugate gradient algorithm: Multimin Algorithms with Derivatives.
                                                              (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 with Derivatives.
                                                              (line  29)
* minimization, providing a function to minimize: Providing the function to minimize.
                                                              (line   6)
* minimization, simplex algorithm:       Multimin Algorithms without Derivatives.
                                                              (line  11)
* minimization, steepest descent algorithm: Multimin Algorithms with Derivatives.
                                                              (line  57)
* 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  15)
* minimum value, from histogram:         Histogram Statistics.
                                                              (line   6)
* MINPACK, minimization algorithms <1>:  Algorithms using Derivatives.
                                                              (line  14)
* MINPACK, minimization algorithms:      Minimization Algorithms using Derivatives.
                                                              (line  13)
* 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  92)
* 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 119)
* MT19937 random number generator:       Random number generator algorithms.
                                                              (line  20)
* multi-parameter regression:            Multi-parameter fitting.
                                                              (line   6)
* multidimensional integration:          Monte Carlo Integration.
                                                              (line   6)
* multidimensional root finding, Broyden algorithm: Algorithms without Derivatives.
                                                              (line  45)
* 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)
* multisets:                             Multisets.           (line   6)
* multistep methods, ODEs:               Stepping Functions.  (line 129)
* 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 without Derivatives.
                                                              (line  11)
* Newton algorithm, discrete:            Algorithms without Derivatives.
                                                              (line  26)
* Newton algorithm, globally convergent: Algorithms using Derivatives.
                                                              (line  92)
* 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   6)
* nonlinear least squares fitting, overview: Overview of Nonlinear Least-Squares Fitting.
                                                              (line   6)
* nonlinear systems of equations, solution of: Multidimensional Root-Finding.
                                                              (line   6)
* nonsymmetric matrix, real, eigensystem: Real Nonsymmetric Matrices.
                                                              (line   6)
* Nordsieck form:                        Stepping Functions.  (line 129)
* 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)
* ordering, matrix elements:             Matrices.            (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 with Derivatives.
                                                              (line  29)
* polar form of complex numbers:         Representation of 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)
* predictor-corrector method, ODEs:      Stepping Functions.  (line 129)
* prefixes:                              Prefixes.            (line   6)
* pressure, units of:                    Pressure.            (line   6)
* Prince-Dormand, Runge-Kutta method:    Stepping Functions.  (line 103)
* 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  35)
* psi function:                          Psi (Digamma) Function.
                                                              (line   6)
* QAG quadrature algorithm:              QAG adaptive integration.
                                                              (line   6)
* QAGI quadrature algorithm:             QAGI adaptive integration on infinite intervals.
                                                              (line   6)
* QAGP quadrature algorithm:             QAGP adaptive integration with known singular points.
                                                              (line   6)
* QAGS quadrature algorithm:             QAGS adaptive integration with singularities.
                                                              (line   6)
* QAWC quadrature algorithm:             QAWC adaptive integration for Cauchy principal values.
                                                              (line   6)
* QAWF quadrature algorithm:             QAWF adaptive integration for Fourier integrals.
                                                              (line   6)
* QAWO quadrature algorithm:             QAWO adaptive integration for oscillatory functions.
                                                              (line   6)
* QAWS quadrature algorithm:             QAWS adaptive integration for singular functions.
                                                              (line   6)
* QNG quadrature algorithm:              QNG non-adaptive Gauss-Kronrod integration.
                                                              (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)
* Radial Mathieu Functions:              Radial Mathieu Functions.
                                                              (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  58)
* 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  71)
* RANLXD random number generator:        Random number generator algorithms.
                                                              (line  65)
* RANLXS random number generator:        Random number generator algorithms.
                                                              (line  47)
* RANMAR random number generator:        Other random number generators.
                                                              (line  54)
* Rayleigh distribution:                 The Rayleigh Distribution.
                                                              (line   7)
* Rayleigh Tail distribution:            The Rayleigh Tail Distribution.
                                                              (line   8)
* real nonsymmetric matrix, eigensystem: Real Nonsymmetric Matrices.
                                                              (line   6)
* real symmetric matrix, eigensystem:    Real Symmetric Matrices.
                                                              (line   6)
* Reciprocal Gamma function:             Gamma Functions.     (line  51)
* 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  57)
* 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  42)
* relative Pochhammer symbol:            Pochhammer Symbol.   (line  31)
* reporting bugs in GSL:                 Reporting Bugs.      (line   6)
* representations of complex numbers:    Representation of complex numbers.
                                                              (line   6)
* resampling from histograms:            Resampling from histograms.
                                                              (line   6)
* residual, in nonlinear systems of equations <1>: Search Stopping Parameters for Minimization Algorithms.
                                                              (line  30)
* residual, in nonlinear systems of equations: Search Stopping Parameters for the multidimensional solver.
                                                              (line  31)
* reversing a permutation:               Permutation functions.
                                                              (line   7)
* Riemann Zeta Function:                 Riemann Zeta Function.
                                                              (line   6)
* RK2, Runge-Kutta method:               Stepping Functions.  (line  88)
* RK4, Runge-Kutta method:               Stepping Functions.  (line  91)
* RKF45, Runge-Kutta-Fehlberg method:    Stepping Functions.  (line  96)
* robust regression:                     Robust linear regression.
                                                              (line   6)
* 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 for the multidimensional solver.
                                                              (line   6)
* root finding, stopping parameters:     Search Stopping Parameters.
                                                              (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 100)
* Runge-Kutta methods, ordinary differential equations: Stepping Functions.
                                                              (line  88)
* Runge-Kutta Prince-Dormand method:     Stepping Functions.  (line 103)
* safe comparison of floating point numbers: Approximate Comparison of Floating Point Numbers.
                                                              (line  13)
* safeguarded step-length algorithm:     Minimization Algorithms.
                                                              (line  48)
* sampling from histograms <1>:          Resampling from histograms.
                                                              (line   6)
* sampling from histograms:              The histogram probability distribution struct.
                                                              (line   6)
* SAXPY, Level-1 BLAS:                   Level 1 GSL BLAS Interface.
                                                              (line  96)
* SCAL, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line 109)
* schedule, cooling:                     Simulated Annealing algorithm.
                                                              (line  23)
* se(q,x), Mathieu function:             Angular Mathieu Functions.
                                                              (line   6)
* 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  99)
* simplex algorithm, minimization:       Multimin Algorithms without Derivatives.
                                                              (line  11)
* 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)
* skewness:                              Higher moments (skewness and kurtosis).
                                                              (line   6)
* 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)
* splines, basis:                        Basis Splines.       (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 with Derivatives.
                                                              (line  57)
* 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  74)
* summation, acceleration:               Series Acceleration. (line   6)
* superdiagonal, matrix:                 Creating row and column views.
                                                              (line  86)
* SVD:                                   Singular Value Decomposition.
                                                              (line   6)
* SWAP, Level-1 BLAS:                    Level 1 GSL BLAS Interface.
                                                              (line  76)
* swapping permutation elements:         Accessing permutation elements.
                                                              (line  18)
* 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 164)
* SYRK, Level-3 BLAS:                    Level 3 GSL BLAS Interface.
                                                              (line 126)
* 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 137)
* Taylor coefficients, computation of:   Factorials.          (line  54)
* testing combination for validity:      Combination properties.
                                                              (line  18)
* testing multiset for validity:         Multiset properties. (line  17)
* 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  24)
* 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  47)
* 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 and Matrices.
                                                              (line   6)
* vectors:                               Vectors.             (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)
* y(x), Bessel Functions:                Irregular Spherical Bessel Functions.
                                                              (line   6)
* Y(x), Bessel Functions:                Irregular Cylindrical Bessel Functions.
                                                              (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)
* Ziggurat method:                       The Gaussian Distribution.
                                                              (line  29)



Tag Table:
Node: Top715
Node: Introduction3167
Node: Routines available in GSL3819
Node: GSL is Free Software5734
Node: Obtaining GSL8126
Node: No Warranty9153
Node: Reporting Bugs9659
Ref: Reporting Bugs-Footnote-110615
Node: Further Information10665
Node: Conventions used in this manual11678
Node: Using the library12442
Node: An Example Program13028
Ref: An Example Program-Footnote-113771
Node: Compiling and Linking13878
Node: Linking programs with the library14948
Ref: Linking programs with the library-Footnote-116350
Ref: Linking programs with the library-Footnote-216387
Node: Linking with an alternative BLAS library16440
Node: Shared Libraries17522
Ref: Shared Libraries-Footnote-118997
Node: ANSI C Compliance19045
Node: Inline functions20138
Node: Long double21676
Node: Portability functions23264
Node: Alternative optimized functions24661
Node: Support for different numeric types26119
Node: Compatibility with C++29182
Node: Aliasing of arrays29754
Node: Thread-safety30505
Node: Deprecated Functions31592
Node: Code Reuse32217
Node: Error Handling32861
Node: Error Reporting33611
Node: Error Codes35503
Node: Error Handlers37359
Node: Using GSL error reporting in your own functions40996
Node: Error Reporting Examples42954
Node: Mathematical Functions44166
Node: Mathematical Constants44927
Node: Infinities and Not-a-number46057
Ref: Infinities and Not-a-number-Footnote-147164
Node: Elementary Functions47574
Node: Small integer powers50021
Node: Testing the Sign of Numbers51668
Node: Testing for Odd and Even Numbers52099
Node: Maximum and Minimum functions52655
Node: Approximate Comparison of Floating Point Numbers54930
Node: Complex Numbers56289
Ref: Complex Numbers-Footnote-157615
Node: Representation of complex numbers57679
Node: Properties of complex numbers59616
Node: Complex arithmetic operators60621
Node: Elementary Complex Functions63367
Node: Complex Trigonometric Functions65187
Node: Inverse Complex Trigonometric Functions66408
Node: Complex Hyperbolic Functions69014
Node: Inverse Complex Hyperbolic Functions70310
Node: Complex Number References and Further Reading72298
Node: Polynomials73694
Node: Polynomial Evaluation74461
Node: Divided Difference Representation of Polynomials75871
Node: Quadratic Equations79711
Node: Cubic Equations81651
Node: General Polynomial Equations83233
Node: Roots of Polynomials Examples85578
Node: Roots of Polynomials References and Further Reading86967
Node: Special Functions88013
Node: Special Function Usage90007
Node: The gsl_sf_result struct91190
Node: Special Function Modes92454
Node: Airy Functions and Derivatives93390
Node: Airy Functions94091
Node: Derivatives of Airy Functions95451
Node: Zeros of Airy Functions96963
Node: Zeros of Derivatives of Airy Functions97682
Node: Bessel Functions98441
Node: Regular Cylindrical Bessel Functions99633
Node: Irregular Cylindrical Bessel Functions100957
Node: Regular Modified Cylindrical Bessel Functions102415
Node: Irregular Modified Cylindrical Bessel Functions105242
Node: Regular Spherical Bessel Functions108210
Node: Irregular Spherical Bessel Functions110424
Node: Regular Modified Spherical Bessel Functions112111
Node: Irregular Modified Spherical Bessel Functions114114
Node: Regular Bessel Function - Fractional Order116177
Node: Irregular Bessel Functions - Fractional Order117175
Node: Regular Modified Bessel Functions - Fractional Order117757
Node: Irregular Modified Bessel Functions - Fractional Order118682
Node: Zeros of Regular Bessel Functions119918
Node: Clausen Functions121021
Node: Coulomb Functions121625
Node: Normalized Hydrogenic Bound States122074
Node: Coulomb Wave Functions123187
Node: Coulomb Wave Function Normalization Constant126597
Node: Coupling Coefficients127350
Node: 3-j Symbols128073
Node: 6-j Symbols128677
Node: 9-j Symbols129301
Node: Dawson Function130007
Node: Debye Functions130574
Node: Dilogarithm132348
Node: Real Argument132643
Node: Complex Argument133336
Node: Elementary Operations133805
Node: Elliptic Integrals134629
Node: Definition of Legendre Forms135201
Node: Definition of Carlson Forms136021
Node: Legendre Form of Complete Elliptic Integrals136718
Node: Legendre Form of Incomplete Elliptic Integrals138286
Node: Carlson Forms140474
Node: Elliptic Functions (Jacobi)142021
Node: Error Functions142602
Node: Error Function143053
Node: Complementary Error Function143426
Node: Log Complementary Error Function143886
Node: Probability functions144338
Node: Exponential Functions145591
Node: Exponential Function145974
Node: Relative Exponential Functions147197
Node: Exponentiation With Error Estimate148869
Node: Exponential Integrals150078
Node: Exponential Integral150560
Node: Ei(x)151487
Node: Hyperbolic Integrals151935
Node: Ei_3(x)152591
Node: Trigonometric Integrals152983
Node: Arctangent Integral153582
Node: Fermi-Dirac Function153982
Node: Complete Fermi-Dirac Integrals154352
Node: Incomplete Fermi-Dirac Integrals156920
Node: Gamma and Beta Functions157563
Node: Gamma Functions158136
Node: Factorials161245
Node: Pochhammer Symbol163798
Node: Incomplete Gamma Functions165304
Node: Beta Functions166553
Node: Incomplete Beta Function167305
Node: Gegenbauer Functions168016
Node: Hypergeometric Functions169666
Node: Laguerre Functions173929
Node: Lambert W Functions175492
Node: Legendre Functions and Spherical Harmonics176530
Node: Legendre Polynomials177119
Node: Associated Legendre Polynomials and Spherical Harmonics179174
Node: Conical Functions182007
Node: Radial Functions for Hyperbolic Space184130
Node: Logarithm and Related Functions186307
Node: Mathieu Functions187931
Node: Mathieu Function Workspace189280
Node: Mathieu Function Characteristic Values190094
Node: Angular Mathieu Functions191166
Node: Radial Mathieu Functions192202
Node: Power Function193444
Node: Psi (Digamma) Function194371
Node: Digamma Function194891
Node: Trigamma Function195721
Node: Polygamma Function196298
Node: Synchrotron Functions196682
Node: Transport Functions197445
Node: Trigonometric Functions198592
Node: Circular Trigonometric Functions199215
Node: Trigonometric Functions for Complex Arguments200224
Node: Hyperbolic Trigonometric Functions201260
Node: Conversion Functions201863
Node: Restriction Functions202646
Node: Trigonometric Functions With Error Estimates203578
Node: Zeta Functions204488
Node: Riemann Zeta Function204946
Node: Riemann Zeta Function Minus One205625
Node: Hurwitz Zeta Function206396
Node: Eta Function206901
Node: Special Functions Examples207460
Node: Special Functions References and Further Reading209154
Node: Vectors and Matrices210424
Node: Data types211183
Node: Blocks212386
Node: Block allocation213302
Node: Reading and writing blocks214706
Node: Example programs for blocks216770
Node: Vectors217397
Node: Vector allocation219283
Node: Accessing vector elements220819
Ref: Accessing vector elements-Footnote-1223961
Node: Initializing vector elements224200
Node: Reading and writing vectors224894
Node: Vector views226984
Node: Copying vectors234426
Node: Exchanging elements235286
Node: Vector operations235844
Node: Finding maximum and minimum elements of vectors237610
Node: Vector properties239196
Node: Example programs for vectors240170
Node: Matrices242459
Node: Matrix allocation245317
Node: Accessing matrix elements246902
Node: Initializing matrix elements248851
Node: Reading and writing matrices249647
Node: Matrix views251750
Node: Creating row and column views258890
Node: Copying matrices263360
Node: Copying rows and columns263956
Node: Exchanging rows and columns265633
Node: Matrix operations267111
Node: Finding maximum and minimum elements of matrices269078
Node: Matrix properties270982
Node: Example programs for matrices272084
Node: Vector and Matrix References and Further Reading276155
Node: Permutations276641
Node: The Permutation struct277929
Node: Permutation allocation278433
Node: Accessing permutation elements279867
Node: Permutation properties280710
Node: Permutation functions281415
Node: Applying Permutations282672
Node: Reading and writing permutations284598
Ref: Reading and writing permutations-Footnote-1286813
Node: Permutations in cyclic form286925
Node: Permutation Examples290350
Node: Permutation References and Further Reading292893
Node: Combinations293567
Node: The Combination struct294414
Node: Combination allocation294966
Node: Accessing combination elements296751
Node: Combination properties297410
Node: Combination functions298296
Node: Reading and writing combinations299236
Ref: Reading and writing combinations-Footnote-1301470
Node: Combination Examples301582
Node: Combination References and Further Reading302890
Node: Multisets303294
Node: The Multiset struct304220
Node: Multiset allocation304742
Node: Accessing multiset elements306433
Node: Multiset properties307062
Node: Multiset functions307875
Node: Reading and writing multisets308785
Ref: Reading and writing multisets-Footnote-1310936
Node: Multiset Examples311048
Node: Sorting313156
Node: Sorting objects314103
Node: Sorting vectors317014
Node: Selecting the k smallest or largest elements320174
Node: Computing the rank323508
Node: Sorting Examples324709
Node: Sorting References and Further Reading326353
Node: BLAS Support326899
Ref: BLAS Support-Footnote-1330274
Node: GSL BLAS Interface330442
Node: Level 1 GSL BLAS Interface330929
Node: Level 2 GSL BLAS Interface337668
Node: Level 3 GSL BLAS Interface346964
Node: BLAS Examples358104
Node: BLAS References and Further Reading359525
Node: Linear Algebra360872
Node: LU Decomposition361997
Node: QR Decomposition367242
Node: QR Decomposition with Column Pivoting373551
Node: Singular Value Decomposition378006
Node: Cholesky Decomposition382417
Node: Tridiagonal Decomposition of Real Symmetric Matrices385657
Node: Tridiagonal Decomposition of Hermitian Matrices387505
Node: Hessenberg Decomposition of Real Matrices389457
Node: Hessenberg-Triangular Decomposition of Real Matrices391867
Node: Bidiagonalization392987
Node: Householder Transformations395412
Node: Householder solver for linear systems397764
Node: Tridiagonal Systems398669
Node: Balancing401541
Node: Linear Algebra Examples402342
Node: Linear Algebra References and Further Reading404330
Node: Eigensystems405977
Node: Real Symmetric Matrices407191
Node: Complex Hermitian Matrices409705
Node: Real Nonsymmetric Matrices412332
Node: Real Generalized Symmetric-Definite Eigensystems418045
Node: Complex Generalized Hermitian-Definite Eigensystems420917
Node: Real Generalized Nonsymmetric Eigensystems423624
Node: Sorting Eigenvalues and Eigenvectors430208
Node: Eigenvalue and Eigenvector Examples433485
Node: Eigenvalue and Eigenvector References438875
Node: Fast Fourier Transforms439900
Node: Mathematical Definitions441109
Node: Overview of complex data FFTs443514
Node: Radix-2 FFT routines for complex data446457
Node: Mixed-radix FFT routines for complex data450669
Node: Overview of real data FFTs459998
Node: Radix-2 FFT routines for real data462432
Node: Mixed-radix FFT routines for real data466861
Node: FFT References and Further Reading477536
Node: Numerical Integration480405
Node: Numerical Integration Introduction482171
Node: Integrands without weight functions484440
Node: Integrands with weight functions485269
Node: Integrands with singular weight functions485980
Node: QNG non-adaptive Gauss-Kronrod integration486898
Node: QAG adaptive integration488179
Node: QAGS adaptive integration with singularities490956
Node: QAGP adaptive integration with known singular points492756
Node: QAGI adaptive integration on infinite intervals494070
Node: QAWC adaptive integration for Cauchy principal values496363
Node: QAWS adaptive integration for singular functions497538
Node: QAWO adaptive integration for oscillatory functions500760
Node: QAWF adaptive integration for Fourier integrals504539
Node: CQUAD doubly-adaptive integration507164
Node: Fixed order Gauss-Legendre integration510402
Node: Numerical integration error codes512380
Node: Numerical integration examples513139
Node: Numerical integration References and Further Reading515185
Node: Random Number Generation516176
Node: General comments on random numbers517678
Node: The Random Number Generator Interface519628
Node: Random number generator initialization520986
Node: Sampling from a random number generator523359
Node: Auxiliary random number generator functions526603
Node: Random number environment variables528923
Node: Copying random number generator state531449
Node: Reading and writing random number generator state532424
Node: Random number generator algorithms533818
Node: Unix random number generators543753
Node: Other random number generators547471
Node: Random Number Generator Performance556118
Node: Random Number Generator Examples557239
Node: Random Number References and Further Reading558795
Node: Random Number Acknowledgements560092
Node: Quasi-Random Sequences560578
Node: Quasi-random number generator initialization561685
Node: Sampling from a quasi-random number generator562700
Node: Auxiliary quasi-random number generator functions563397
Node: Saving and resorting quasi-random number generator state564343
Node: Quasi-random number generator algorithms565151
Node: Quasi-random number generator examples566287
Node: Quasi-random number references567272
Node: Random Number Distributions567798
Node: Random Number Distribution Introduction570997
Node: The Gaussian Distribution572792
Node: The Gaussian Tail Distribution575453
Node: The Bivariate Gaussian Distribution577122
Node: The Exponential Distribution578445
Node: The Laplace Distribution579584
Node: The Exponential Power Distribution580681
Node: The Cauchy Distribution581949
Node: The Rayleigh Distribution583198
Node: The Rayleigh Tail Distribution584370
Node: The Landau Distribution585242
Node: The Levy alpha-Stable Distributions586197
Node: The Levy skew alpha-Stable Distribution587255
Node: The Gamma Distribution588867
Node: The Flat (Uniform) Distribution590524
Node: The Lognormal Distribution591676
Node: The Chi-squared Distribution593024
Node: The F-distribution594428
Node: The t-distribution596064
Node: The Beta Distribution597458
Node: The Logistic Distribution598617
Node: The Pareto Distribution599746
Node: Spherical Vector Distributions600916
Node: The Weibull Distribution603750
Node: The Type-1 Gumbel Distribution604944
Node: The Type-2 Gumbel Distribution606181
Node: The Dirichlet Distribution607412
Node: General Discrete Distributions609079
Node: The Poisson Distribution612946
Node: The Bernoulli Distribution613954
Node: The Binomial Distribution614705
Node: The Multinomial Distribution615910
Node: The Negative Binomial Distribution617679
Node: The Pascal Distribution619044
Node: The Geometric Distribution620202
Node: The Hypergeometric Distribution621450
Node: The Logarithmic Distribution623103
Node: Shuffling and Sampling623894
Node: Random Number Distribution Examples626713
Node: Random Number Distribution References and Further Reading629909
Node: Statistics632052
Node: Mean and standard deviation and variance633452
Node: Absolute deviation636960
Node: Higher moments (skewness and kurtosis)638252
Node: Autocorrelation640385
Node: Covariance641199
Node: Correlation642175
Node: Weighted Samples643543
Node: Maximum and Minimum values649393
Node: Median and Percentiles652132
Node: Example statistical programs654547
Node: Statistics References and Further Reading657207
Node: Histograms658415
Node: The histogram struct660175
Node: Histogram allocation661978
Node: Copying Histograms664939
Node: Updating and accessing histogram elements665615
Node: Searching histogram ranges668882
Node: Histogram Statistics669881
Node: Histogram Operations671741
Node: Reading and writing histograms673813
Node: Resampling from histograms676855
Node: The histogram probability distribution struct677653
Node: Example programs for histograms680684
Node: Two dimensional histograms682747
Node: The 2D histogram struct683468
Node: 2D Histogram allocation685276
Node: Copying 2D Histograms687347
Node: Updating and accessing 2D histogram elements688052
Node: Searching 2D histogram ranges691707
Node: 2D Histogram Statistics692719
Node: 2D Histogram Operations695578
Node: Reading and writing 2D histograms697752
Node: Resampling from 2D histograms701381
Node: Example programs for 2D histograms704400
Node: N-tuples706228
Node: The ntuple struct707484
Node: Creating ntuples707962
Node: Opening an existing ntuple file708629
Node: Writing ntuples709257
Node: Reading ntuples709718
Node: Closing an ntuple file710049
Node: Histogramming ntuple values710389
Node: Example ntuple programs712397
Node: Ntuple References and Further Reading715726
Node: Monte Carlo Integration716047
Node: Monte Carlo Interface717292
Node: PLAIN Monte Carlo719915
Node: MISER722369
Ref: MISER-Footnote-1729045
Node: VEGAS729170
Node: Monte Carlo Examples738721
Node: Monte Carlo Integration References and Further Reading744706
Node: Simulated Annealing745488
Node: Simulated Annealing algorithm746695
Node: Simulated Annealing functions747849
Node: Examples with Simulated Annealing752384
Node: Trivial example752938
Node: Traveling Salesman Problem755589
Node: Simulated Annealing References and Further Reading758902
Node: Ordinary Differential Equations759313
Node: Defining the ODE System760504
Node: Stepping Functions763330
Node: Adaptive Step-size Control770272
Node: Evolution776345
Node: Driver780462
Node: ODE Example programs784687
Node: ODE References and Further Reading789729
Node: Interpolation791489
Node: Introduction to Interpolation792660
Node: Interpolation Functions793101
Node: Interpolation Types794308
Node: Index Look-up and Acceleration797031
Node: Evaluation of Interpolating Functions799083
Node: Higher-level Interface801584
Node: Interpolation Example programs803623
Node: Interpolation References and Further Reading806839
Node: Numerical Differentiation807412
Node: Numerical Differentiation functions808001
Node: Numerical Differentiation Examples810859
Node: Numerical Differentiation References812272
Node: Chebyshev Approximations812823
Node: Chebyshev Definitions813884
Node: Creation and Calculation of Chebyshev Series814674
Node: Auxiliary Functions for Chebyshev Series815659
Node: Chebyshev Series Evaluation816399
Node: Derivatives and Integrals817779
Node: Chebyshev Approximation Examples819021
Node: Chebyshev Approximation References and Further Reading820517
Node: Series Acceleration820966
Node: Acceleration functions821731
Node: Acceleration functions without error estimation824056
Node: Example of accelerating a series826666
Node: Series Acceleration References829012
Node: Wavelet Transforms829900
Node: DWT Definitions830441
Node: DWT Initialization831393
Node: DWT Transform Functions834042
Node: DWT in one dimension834577
Node: DWT in two dimension836596
Node: DWT Examples841174
Node: DWT References842991
Node: Discrete Hankel Transforms845154
Node: Discrete Hankel Transform Definition845622
Node: Discrete Hankel Transform Functions847821
Node: Discrete Hankel Transform References849485
Node: One dimensional Root-Finding849889
Node: Root Finding Overview851149
Node: Root Finding Caveats853007
Node: Initializing the Solver854774
Node: Providing the function to solve857407
Node: Search Bounds and Guesses860955
Node: Root Finding Iteration861818
Node: Search Stopping Parameters863667
Node: Root Bracketing Algorithms866181
Node: Root Finding Algorithms using Derivatives869479
Ref: Root Finding Algorithms using Derivatives-Footnote-1873021
Node: Root Finding Examples873176
Node: Root Finding References and Further Reading880464
Node: One dimensional Minimization881103
Node: Minimization Overview882405
Node: Minimization Caveats884111
Node: Initializing the Minimizer885448
Node: Providing the function to minimize887689
Node: Minimization Iteration888167
Node: Minimization Stopping Parameters890303
Node: Minimization Algorithms891912
Node: Minimization Examples894516
Node: Minimization References and Further Reading897443
Node: Multidimensional Root-Finding897899
Node: Overview of Multidimensional Root Finding899388
Node: Initializing the Multidimensional Solver901575
Node: Providing the multidimensional system of equations to solve904806
Node: Iteration of the multidimensional solver909708
Node: Search Stopping Parameters for the multidimensional solver911987
Node: Algorithms using Derivatives913724
Node: Algorithms without Derivatives918553
Node: Example programs for Multidimensional Root finding921641
Node: References and Further Reading for Multidimensional Root Finding930254
Node: Multidimensional Minimization931493
Node: Multimin Overview932846
Node: Multimin Caveats934923
Node: Initializing the Multidimensional Minimizer935672
Node: Providing a function to minimize938893
Node: Multimin Iteration942945
Node: Multimin Stopping Criteria945144
Node: Multimin Algorithms with Derivatives946713
Node: Multimin Algorithms without Derivatives950241
Node: Multimin Examples953241
Node: Multimin References and Further Reading959807
Node: Least-Squares Fitting960672
Node: Fitting Overview961720
Node: Linear regression964054
Node: Linear fitting without a constant term966656
Node: Multi-parameter fitting968849
Node: Robust linear regression974700
Node: Troubleshooting984919
Node: Fitting Examples985541
Node: Fitting References and Further Reading996063
Node: Nonlinear Least-Squares Fitting997434
Node: Overview of Nonlinear Least-Squares Fitting998936
Node: Initializing the Nonlinear Least-Squares Solver1000403
Node: Providing the Function to be Minimized1003268
Node: Finite Difference Jacobian1006720
Node: Iteration of the Minimization Algorithm1008588
Node: Search Stopping Parameters for Minimization Algorithms1010426
Node: High Level Driver1012497
Node: Minimization Algorithms using Derivatives1013402
Ref: Minimization Algorithms using Derivatives-Footnote-11016846
Node: Minimization Algorithms without Derivatives1016935
Node: Computing the covariance matrix of best fit parameters1017327
Node: Example programs for Nonlinear Least-Squares Fitting1019363
Node: References and Further Reading for Nonlinear Least-Squares Fitting1027093
Node: Basis Splines1027829
Node: Overview of B-splines1028707
Node: Initializing the B-splines solver1030016
Node: Constructing the knots vector1031392
Node: Evaluation of B-spline basis functions1032179
Node: Evaluation of B-spline basis function derivatives1033836
Node: Working with the Greville abscissae1035806
Node: Example programs for B-splines1036800
Node: References and Further Reading1040790
Node: Physical Constants1041595
Node: Fundamental Constants1043005
Node: Astronomy and Astrophysics1044144
Node: Atomic and Nuclear Physics1044809
Node: Measurement of Time1046460
Node: Imperial Units1046890
Node: Speed and Nautical Units1047332
Node: Printers Units1047836
Node: Volume Area and Length1048159
Node: Mass and Weight1048845
Node: Thermal Energy and Power1049664
Node: Pressure1050087
Node: Viscosity1050700
Node: Light and Illumination1050976
Node: Radioactivity1051568
Node: Force and Energy1051903
Node: Prefixes1052307
Node: Physical Constant Examples1053050
Node: Physical Constant References and Further Reading1054839
Node: IEEE floating-point arithmetic1055569
Node: Representation of floating point numbers1056155
Node: Setting up your IEEE environment1060636
Node: IEEE References and Further Reading1067628
Node: Debugging Numerical Programs1068783
Node: Using gdb1069267
Node: Examining floating point registers1072610
Node: Handling floating point exceptions1073895
Node: GCC warning options for numerical programs1075307
Node: Debugging References1079269
Node: Contributors to GSL1079981
Node: Autoconf Macros1084476
Node: GSL CBLAS Library1088505
Node: Level 1 CBLAS Functions1089020
Node: Level 2 CBLAS Functions1094322
Node: Level 3 CBLAS Functions1110992
Node: GSL CBLAS Examples1120654
Node: GNU General Public License1122216
Node: GNU Free Documentation License1159807
Node: Function Index1184975
Node: Variable Index1437314
Node: Type Index1440617
Node: Concept Index1455532

End Tag Table
