     GSL 2.5, Jun 03, 2018

     The GSL Team

     Copyright © 1996-2018 The GSL Team

   Generated by Sphinx 1.7.5.

GNU Scientific Library
**********************

     GSL 2.5, Jun 03, 2018

     The GSL Team

     Copyright © 1996-2018 The GSL Team

 — The Detailed Node Listing —

Introduction

Using the Library

Compiling and Linking

Error Handling

Mathematical Functions

Complex Numbers

Polynomials

Special Functions

Airy Functions and Derivatives

Bessel Functions

Coulomb Functions

Coupling Coefficients

Dilogarithm

Elliptic Integrals

Error Functions

Exponential Functions

Exponential Integrals

Fermi-Dirac Function

Gamma and Beta Functions

Hermite Polynomials and Functions

Legendre Functions and Spherical Harmonics

Mathieu Functions

Psi (Digamma) Function

Trigonometric Functions

Zeta Functions

Vectors and Matrices

Blocks

Vectors

Matrices

Permutations

Combinations

Multisets

Sorting

BLAS Support

GSL BLAS Interface

Linear Algebra

Eigensystems

Fast Fourier Transforms (FFTs)

Numerical Integration

Introduction

Examples

Random Number Generation

Quasi-Random Sequences

Random Number Distributions

Statistics

Robust Location Estimates

Robust Scale Estimates

Running Statistics

Moving Window Statistics

Robust Scale Estimation

Examples

Digital Filtering

Linear Digital Filters

Nonlinear Digital Filters

Examples

Histograms

N-tuples

Monte Carlo Integration

Simulated Annealing

Examples

Ordinary Differential Equations

Interpolation

Numerical Differentiation

Chebyshev Approximations

Series Acceleration

Wavelet Transforms

Transform Functions

Discrete Hankel Transforms

One Dimensional Root-Finding

One Dimensional Minimization

Multidimensional Root-Finding

Multidimensional Minimization

Linear Least-Squares Fitting

Linear regression

Large dense linear systems

Examples

Nonlinear Least-Squares Fitting

Solving the Trust Region Subproblem (TRS)

Examples

Basis Splines

Sparse Matrices

Sparse BLAS Support

Sparse Linear Algebra

Sparse Iterative Solvers

Physical Constants

IEEE floating-point arithmetic

Debugging Numerical Programs

GSL CBLAS Library

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.

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                        Sparse BLAS Support             Sparse Linear Algebra

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.

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(1).
Further information about this license is available from the GNU Project
webpage Frequently Asked Questions about the GNU GPL(2).

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

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

   (1)
https://www.gnu.org/software/gsl/manual/html_node/GNU-General-Public-License.html#GNU-General-Public-License

   (2) http://www.gnu.org/copyleft/gpl-faq.html

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.

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(1) for further details.

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

   (1)
https://www.gnu.org/software/gsl/manual/html_node/GNU-General-Public-License.html#GNU-General-Public-License

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’

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/’”.

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’).

2 Using the Library
*******************

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

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) = -1.775967713143382642e-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

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.

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/’

2.2.2 Linking with an alternative BLAS library
----------------------------------------------

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

     $ gcc example.o -lgsl -lcblas -lm

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

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

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

For more information about BLAS functions see BLAS Support.

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

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_’.

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

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

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

     $ gcc -Wall -c -DHAVE_INLINE example.c

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

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

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

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.

2.7 Portability functions
=========================

To help in writing portable applications GSL provides some
implementations of functions that are found in other libraries, such as
the BSD math library.  You can write your application to use the native
versions of these functions, and substitute the GSL versions via a
preprocessor macro if they are unavailable on another platform.

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

     /* Substitute gsl_hypot for missing system hypot */

     #ifndef HAVE_HYPOT
     #define hypot gsl_hypot
     #endif

The application source files can then use the include command ‘#include
<config.h>’ to replace each occurrence of ‘hypot()’ by
gsl_hypot() when ‘hypot()’ is not available.  This substitution can
be made automatically if you use ‘autoconf’, see Autoconf Macros.

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

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.

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

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’.

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.

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.

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.

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.

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’.

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.

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,

 -- Variable: 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)

 -- Variable: 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)

 -- Variable: 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()’.

 -- Variable: 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.

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’,

 -- 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’.

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);
       }

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_forward() 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.

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’.

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

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().

4.3 Elementary Functions
========================

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

 -- Function: double gsl_log1p (const double x)

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

 -- Function: double gsl_expm1 (const double x)

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

 -- Function: double gsl_hypot (const double x, const double y)

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

 -- Function: double gsl_hypot3 (const double x, const double y, const
          double z)

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

 -- Function: double gsl_acosh (const double x)

     This function computes the value of \arccosh{(x)}.  It provides an
     alternative to the standard math function ‘acosh(x)’.

 -- Function: double gsl_asinh (const double x)

     This function computes the value of \arcsinh{(x)}.  It provides an
     alternative to the standard math function ‘asinh(x)’.

 -- Function: double gsl_atanh (const double x)

     This function computes the value of \arctanh{(x)}.  It provides an
     alternative to the standard math function ‘atanh(x)’.

 -- Function: double gsl_ldexp (double x, int e)

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

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

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

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 */

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).

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.  It evaluates
     to 1 if ‘n’ is even and 0 if ‘n’ is odd.  The argument ‘n’ must be
     of integer type.

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.

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.

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.  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’.

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

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

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.

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.

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).

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).

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).

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).

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).

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) = \arcsinh(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).

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

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’.

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

The functions described here evaluate the polynomial

P(x) = c[0] + c[1] x + c[2] x^2 + … + 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(x)/d x^k for the specified value of
     ‘x’ starting with k = 0.

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’.

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’.

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.

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.

 -- Type: gsl_poly_complex_workspace

     This workspace contains parameters used for finding roots of
     general 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).

6.6 Examples
============

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

     z0 = -0.809016994374947673 +0.587785252292473359
     z1 = -0.809016994374947673 -0.587785252292473359
     z2 = +0.309016994374947507 +0.951056516295152976
     z3 = +0.309016994374947507 -0.951056516295152976
     z4 = +0.999999999999999889 +0.000000000000000000

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

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.

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, Hermite polynomials and 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’.

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’.

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 following struct contains value and error fields.

 -- Type: gsl_sf_result

          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 following struct
contains value and error fields as well as an exponent field such that
the actual result is obtained as ‘result * 10^(e10)’.

 -- Type: gsl_sf_result_e10

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

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, of type
gsl_mode_t allows the accuracy of the function to be reduced in
order to improve performance.  The following precision levels are
available for the mode argument,

 -- Type: gsl_mode_t

      -- Macro: GSL_PREC_DOUBLE

          Double-precision, a relative accuracy of approximately 2 *
          10^{-16}.

      -- Macro: GSL_PREC_SINGLE

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

      -- Macro: GSL_PREC_APPROX

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

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

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’.

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.

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.

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).

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).

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’.

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.

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.

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.

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.

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 \geq 0 and x \geq 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 \geq 0
     and x \geq 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 \geq 0 and x \geq 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.

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 \geq 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 \geq 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.

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
     spherical Bessel functions \exp(-|x|) i_l(x) for l from 0 to ‘lmax’
     inclusive for lmax \geq 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.

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 \geq 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.

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).

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).

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.

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.

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’.

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).

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.

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

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.

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.

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’.

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.

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.

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.

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’.

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).

 -- 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).

 -- 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).

 -- 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).

 -- 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).

 -- 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).

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

The dilogarithm is defined as

     Li_2(z) = - \int_0^z ds log(1-s) / s

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

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 \le 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).

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’.

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: double gsl_sf_multiply (double x, double y)
 -- 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 \pm xy \sqrt{(dx/x)^2
     +(dy/y)^2} is stored in ‘result’.

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.

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.

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)

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.

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,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_D_e (double phi, double k,
          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) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).

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.

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.

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’.

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).

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)

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)).

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.

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

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

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.

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)

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.

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’.

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.

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.

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’).

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^x dt \exp(-t^3)`

     for x \ge 0.

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

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

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

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

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).

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)

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

The 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’.

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.

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 \ge 0,
     n \ge 0

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).

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
     \ge 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 \ge 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 \ge 0.

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

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.

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 \le x \le 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)

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 \ge 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 \ge 0.

7.21 Hermite Polynomials and Functions
======================================

The Hermite polynomials exist in two variants: the probabilists’ version
He_n(x) and the physicists’version H_n(x).  The are defined by the
derivatives

     He_n(x) = (-1)^n e^{x^2/2} (d / dx)^n e^{-x^2/2}
     H_n(x) = (-1)^n e^{x^2} (d / dx)^n e^{-x^2}

They are connected via

     He_n(x) = 2^{-n/2} H_n(x / \sqrt{2})
     H_n(x) = 2^{n/2} He_n(\sqrt{2} x)

and satisfy the ordinary differential equations

     He_n^{''}(x) - x He_n^{'}(x) + n He_n(x) = 0
     H_n^{''}(x) - 2x H_n^{'}(x) + 2n H_n(x) = 0

The closely related Hermite functions are defined by

     psi_n = (n! sqrt(\pi))^{-1/2} e^{-x^2/2} He_n({sqrt(2) x})

and satisfy the Schrödinger equation for a quantum mechanical harmonic
oscillator

     psi_n^{''}(x) + (2n + 1 - x^2) psi_n(x) = 0

Maybe most importantly, the Hermite functions \psi_n are eigenfunctions
of the (continuous) Fourier transform.

For further information see Abramowitz & Stegun, Chapter 22 and Szego,
Gabor (1939, 1958, 1967), Orthogonal Polynomials, American Mathematical
Society.  The Hermite polynomials and functions are defined in the
header file ‘gsl_sf_hermite.h’.

7.21.1 Hermite Polynomials
--------------------------

 -- Function: double gsl_sf_hermite_prob (const int n, const double x)
 -- Function: int gsl_sf_hermite_prob_e (const int n, const double x,
          gsl_sf_result * result)

     These routines evaluate the probabilists’ Hermite polynomial
     He_n(x) of order ‘n’ at position ‘x’.

 -- Function: int gsl_sf_hermite_prob_array (const int nmax, const
          double x, double * result_array)

     This routine evaluates all probabilists’ Hermite polynomials
     He_n(x) up to order ‘nmax’ at position ‘x’.  The results are stored
     in ‘result_array’.

 -- Function: double gsl_sf_hermite_prob_series (const int n, const
          double x, const double * a)
 -- Function: int gsl_sf_hermite_prob_series_e (const int n, const
          double x, const double * a, gsl_sf_result * result)

     These routines evaluate the series \sum_{j=0}^n a_j He_j(x) with
     He_j being the j-th probabilists’ Hermite polynomial using the
     Clenshaw algorithm.

 -- Function: double gsl_sf_hermite_phys (const int n, const double x)
 -- Function: int gsl_sf_hermite_phys_e (const int n, const double x,
          gsl_sf_result * result)

     These routines evaluate the physicists’ Hermite polynomial H_n(x)
     of order ‘n’ at position ‘x’.

 -- Function: int gsl_sf_hermite_phys_array (const int nmax, const
          double x, double * result_array)

     This routine evaluates all physicists’ Hermite polynomials H_n up
     to order ‘nmax’ at position ‘x’.  The results are stored in
     ‘result_array’.

 -- Function: double gsl_sf_hermite_phys_series (const int n, const
          double x, const double * a)
 -- Function: int gsl_sf_hermite_phys_series_e (const int n, const
          double x, const double * a, gsl_sf_result * result)

     These routines evaluate the series \sum_{j=0}^n a_j H_j(x) with H_j
     being the j-th physicists’ Hermite polynomial using the Clenshaw
     algorithm.

7.21.2 Hermite Functions
------------------------

 -- Function: double gsl_sf_hermite_func (const int n, const double x)
 -- Function: int gsl_sf_hermite_func_e (const int n, const double x,
          gsl_sf_result * result)

     These routines evaluate the Hermite function \psi_n(x) of order ‘n’
     at position ‘x’.

 -- Function: int gsl_sf_hermite_func_array (const int nmax, const
          double x, double * result_array)

     This routine evaluates all Hermite functions \psi_n(x) up to order
     ‘nmax’ at position ‘x’.  The results are stored in ‘result_array’.

 -- Function: double gsl_sf_hermite_func_series (const int n, const
          double x, const double * a)
 -- Function: int gsl_sf_hermite_func_series_e (const int n, const
          double x, const double * a, gsl_sf_result * result)

     These routines evaluate the series \sum_{j=0}^n a_j \psi_j(x) with
     \psi_j being the j-th Hermite function using the Clenshaw
     algorithm.

7.21.3 Derivatives of Hermite Polynomials
-----------------------------------------

 -- Function: double gsl_sf_hermite_prob_der (const int m, const int n,
          const double x)
 -- Function: int gsl_sf_hermite_prob_der_e (const int m, const int n,
          const double x, gsl_sf_result * result)

     These routines evaluate the ‘m’-th derivative of the probabilists’
     Hermite polynomial He_n(x) of order ‘n’ at position ‘x’.

 -- Function: int gsl_sf_hermite_prob_array_der (const int m, const
          int nmax, const double x, double * result_array)

     This routine evaluates the ‘m’-th derivative of all probabilists’
     Hermite polynomials He_n(x) up to order ‘nmax’ at position ‘x’.
     The results are stored in ‘result_array’.

 -- Function: int gsl_sf_hermite_prob_der_array (const int mmax, const
          int n, const double x, double * result_array)

     This routine evaluates all derivatives (starting from 0) up to the
     ‘mmax’-th derivative of the probabilists’ Hermite polynomial of
     order ‘n’ He_n(x) at position ‘x’.  The results are stored in
     ‘result_array’.

 -- Function: double gsl_sf_hermite_phys_der (const int m, const int n,
          const double x)
 -- Function: int gsl_sf_hermite_phys_der_e (const int m, const int n,
          const double x, gsl_sf_result * result)

     These routines evaluate the ‘m’-th derivative of the physicists’
     Hermite polynomial H_n(x) of order ‘n’ at position ‘x’.

 -- Function: int gsl_sf_hermite_phys_array_der (const int m, const
          int nmax, const double x, double * result_array)

     This routine evaluates the ‘m’-th derivative of all physicists’
     Hermite polynomials H_n up to order ‘nmax’ at position ‘x’.  The
     results are stored in ‘result_array’.

 -- Function: int gsl_sf_hermite_phys_der_array (const int mmax, const
          int n, const double x, double * result_array)

     This routine evaluates all derivatives (starting from 0) up to the
     ‘mmax’-th derivative of the physicists’ Hermite polynomial of order
     ‘n’ H_n at position ‘x’.  The results are stored in ‘result_array’.

7.21.4 Derivatives of Hermite Functions
---------------------------------------

 -- Function: double gsl_sf_hermite_func_der (const int m, const int n,
          const double x)
 -- Function: int gsl_sf_hermite_func_der_e (const int m, const int n,
          const double x, gsl_sf_result * result)

     These routines evaluate the ‘m’-th derivative of the Hermite
     function \psi_n(x) of order ‘n’ at position ‘x’.

7.21.5 Zeros of Hermite Polynomials and Hermite Functions
---------------------------------------------------------

These routines calculate the s-th zero of the Hermite
Polynomial/Function of order n.  Since the zeros are symmetrical around
zero, only positive zeros are calculated, ordered from smallest to
largest, starting from index 1.  Only for odd polynomial orders a zeroth
zero exists, its value always being zero.

 -- Function: double gsl_sf_hermite_prob_zero (const int n, const int s)
 -- Function: int gsl_sf_hermite_prob_zero_e (const int n, const int s,
          gsl_sf_result * result)

     These routines evaluate the ‘s’-th zero of the probabilists’
     Hermite polynomial He_n(x) of order ‘n’.

 -- Function: double gsl_sf_hermite_phys_zero (const int n, const int s)
 -- Function: int gsl_sf_hermite_phys_zero_e (const int n, const int s,
          gsl_sf_result * result)

     These routines evaluate the ‘s’-th zero of the physicists’ Hermite
     polynomial H_n(x) of order ‘n’.

 -- Function: double gsl_sf_hermite_func_zero (const int n, const int s)
 -- Function: int gsl_sf_hermite_func_zero_e (const int n, const int s,
          gsl_sf_result * result)

     These routines evaluate the ‘s’-th zero of the Hermite function
     \psi_n(x) of order ‘n’.

7.22 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, aR - i aI, 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)

7.23 Laguerre Functions
=======================

The generalized Laguerre polynomials, sometimes referred to as
associated Laguerre polynomials, are defined in terms of confluent
hypergeometric functions as

L^a_n(x) = ((a+1)_n / n!)  1F1(-n,a+1,x)

where (a)_n is the Pochhammer symbol. (rising factorial).
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 \ge 0.

7.24 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).

7.25 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’.

7.25.1 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 \ge 0 and |x| \le 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 and |x| \le 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 and
     x \ne 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 and
     x \ne 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
     \ne 1 and l \ge 0.

7.25.2 Associated Legendre Polynomials and Spherical Harmonics
--------------------------------------------------------------

The following functions compute the associated Legendre polynomials
P_l^m(x) which are solutions of the differential equation

     (1 - x^2) d^2 P_l^m(x) / dx^2 P_l^m(x) - 2x d/dx P_l^m(x) +
     ( l(l+1) - m^2 / (1 - x^2) ) P_l^m(x) = 0

where the degree l and order m satisfy 0 \le l and 0 \le m \le l.  The
functions P_l^m(x) grow combinatorially with l and can overflow for l
larger than about 150.  Alternatively, one may calculate normalized
associated Legendre polynomials.  There are a number of different
normalization conventions, and these functions can be stably computed up
to degree and order 2700.  The following normalizations are provided:

   * Schmidt semi-normalization

     Schmidt semi-normalized associated Legendre polynomials are often
     used in the magnetics community and are defined as

          S_l^0(x) = P_l^0(x)
          S_l^m(x) = (-1)^m \sqrt((2(l-m)! / (l+m)!)) P_l^m(x), m > 0

     The factor of (-1)^m is called the Condon-Shortley phase factor and
     can be excluded if desired by setting the parameter ‘csphase = 1’
     in the functions below.

   * Spherical Harmonic Normalization

     The associated Legendre polynomials suitable for calculating
     spherical harmonics are defined as

          Y_l^m(x) = (-1)^m \sqrt((2l + 1) * (l-m)! / (4 \pi) / (l+m)!) P_l^m(x)

     where again the phase factor (-1)^m can be included or excluded if
     desired.

   * Full Normalization

     The fully normalized associated Legendre polynomials are defined as

          N_l^m(x) = (-1)^m \sqrt((l + 1/2) (l-m)! / (l+m)!) P_l^m(x)

     and have the property

          \int_{-1}^1 N_l^m(x)^2 dx = 1

The normalized associated Legendre routines below use a recurrence
relation which is stable up to a degree and order of about 2700.  Beyond
this, the computed functions could suffer from underflow leading to
incorrect results.  Routines are provided to compute first and second
derivatives dP_l^m(x)/dx and d^2 P_l^m(x)/dx^2 as well as their
alternate versions d P_l^m(\cos{\theta})/d\theta and d^2
P_l^m(\cos{\theta})/d\theta^2.  While there is a simple scaling
relationship between the two forms, the derivatives involving \theta are
heavily used in spherical harmonic expansions and so these routines are
also provided.

In the functions below, a parameter of type gsl_sf_legendre_t
specifies the type of normalization to use.  The possible values
are

 -- Type: gsl_sf_legendre_t

     Value                                  Description

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

     ‘GSL_SF_LEGENDRE_NONE’                 The unnormalized associated Legendre polynomials P_l^m(x)

     ‘GSL_SF_LEGENDRE_SCHMIDT’              The Schmidt semi-normalized associated Legendre polynomials S_l^m(x)

     ‘GSL_SF_LEGENDRE_SPHARM’               The spherical harmonic associated Legendre polynomials Y_l^m(x)

     ‘GSL_SF_LEGENDRE_FULL’                 The fully normalized associated Legendre polynomials N_l^m(x)

 -- Function: int gsl_sf_legendre_array (const gsl_sf_legendre_t norm,
          const size_t lmax, const double x, double result_array[])
 -- Function: int gsl_sf_legendre_array_e (const gsl_sf_legendre_t norm,
          const size_t lmax, const double x, const double csphase,
          double result_array[])

     These functions calculate all normalized associated Legendre
     polynomials for 0 \le l \le lmax and 0 \le m \le l for |x| \le 1.
     The ‘norm’ parameter specifies which normalization is used.  The
     normalized P_l^m(x) values are stored in ‘result_array’, whose
     minimum size can be obtained from calling
     gsl_sf_legendre_array_n().  The array index of P_l^m(x) is
     obtained from calling ‘gsl_sf_legendre_array_index(l, m)’.  To
     include or exclude the Condon-Shortley phase factor of (-1)^m, set
     the parameter ‘csphase’ to either -1 or 1 respectively in the ‘_e’
     function.  This factor is excluded by default.

 -- Function: int gsl_sf_legendre_deriv_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[])
 -- Function: int gsl_sf_legendre_deriv_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[],
          double result_deriv_array[])

     These functions calculate all normalized associated Legendre
     functions and their first derivatives up to degree ‘lmax’ for |x| <
     1.  The parameter ‘norm’ specifies the normalization used.  The
     normalized P_l^m(x) values and their derivatives dP_l^m(x)/dx are
     stored in ‘result_array’ and ‘result_deriv_array’ respectively.  To
     include or exclude the Condon-Shortley phase factor of (-1)^m, set
     the parameter ‘csphase’ to either -1 or 1 respectively in the ‘_e’
     function.  This factor is excluded by default.

 -- Function: int gsl_sf_legendre_deriv_alt_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[])
 -- Function: int gsl_sf_legendre_deriv_alt_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[],
          double result_deriv_array[])

     These functions calculate all normalized associated Legendre
     functions and their (alternate) first derivatives up to degree
     ‘lmax’ for |x| < 1.  The normalized P_l^m(x) values and their
     derivatives dP_l^m(\cos{\theta})/d\theta are stored in
     ‘result_array’ and ‘result_deriv_array’ respectively.  To include
     or exclude the Condon-Shortley phase factor of (-1)^m, set the
     parameter ‘csphase’ to either -1 or 1 respectively in the ‘_e’
     function.  This factor is excluded by default.

 -- Function: int gsl_sf_legendre_deriv2_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[],
          double result_deriv2_array[])
 -- Function: int gsl_sf_legendre_deriv2_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[],
          double result_deriv_array[], double result_deriv2_array[])

     These functions calculate all normalized associated Legendre
     functions and their first and second derivatives up to degree
     ‘lmax’ for |x| < 1.  The parameter ‘norm’ specifies the
     normalization used.  The normalized P_l^m(x), their first
     derivatives dP_l^m(x)/dx, and their second derivatives d^2
     P_l^m(x)/dx^2 are stored in ‘result_array’, ‘result_deriv_array’,
     and ‘result_deriv2_array’ respectively.  To include or exclude the
     Condon-Shortley phase factor of (-1)^m, set the parameter ‘csphase’
     to either -1 or 1 respectively in the ‘_e’ function.  This factor
     is excluded by default.

 -- Function: int gsl_sf_legendre_deriv2_alt_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[],
          double result_deriv2_array[])
 -- Function: int gsl_sf_legendre_deriv2_alt_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[],
          double result_deriv_array[], double result_deriv2_array[])

     These functions calculate all normalized associated Legendre
     functions and their (alternate) first and second derivatives up to
     degree ‘lmax’ for |x| < 1.  The parameter ‘norm’ specifies the
     normalization used.  The normalized P_l^m(x), their first
     derivatives dP_l^m(\cos{\theta})/d\theta, and their second
     derivatives d^2 P_l^m(\cos{\theta})/d\theta^2 are stored in
     ‘result_array’, ‘result_deriv_array’, and ‘result_deriv2_array’
     respectively.  To include or exclude the Condon-Shortley phase
     factor of (-1)^m, set the parameter ‘csphase’ to either -1 or 1
     respectively in the ‘_e’ function.  This factor is excluded by
     default.

 -- Function: size_t gsl_sf_legendre_array_n (const size_t lmax)

     This function returns the minimum array size for maximum degree
     ‘lmax’ needed for the array versions of the associated Legendre
     functions.  Size is calculated as the total number of P_l^m(x)
     functions, plus extra space for precomputing multiplicative factors
     used in the recurrence relations.

 -- Function: size_t gsl_sf_legendre_array_index (const size_t l, const
          size_t m)

     This function returns the index into ‘result_array’,
     ‘result_deriv_array’, or ‘result_deriv2_array’ corresponding to
     P_l^m(x), P_l^{'m}(x), or P_l^{''m}(x).  The index is given by
     l(l+1)/2 + m.

 -- 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 \ge 0, l \ge m, and |x| \le 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 \ge 0, l \ge m, and |x| \le 1.  These routines avoid the
     overflows that occur for the standard normalization of P_l^m(x).

 -- 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 are now deprecated and will be removed in a future
     release; see gsl_sf_legendre_array() and
     gsl_sf_legendre_deriv_array().

 -- 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 are now deprecated and will be removed in a future
     release; see gsl_sf_legendre_array() and
     gsl_sf_legendre_deriv_array().

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

     This function is now deprecated and will be removed in a future
     release.

7.25.3 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 and l \ge -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 and m \ge -1.

7.25.4 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 H^3.  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) \over \lambda\sinh(\eta)}

     for \eta \ge 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\over\sqrt{\lambda^2 + 1}} {\left(\sin(\lambda \eta)\over \lambda \sinh(\eta)\right)} \left(\coth(\eta) - \lambda \cot(\lambda\eta)\right)

     for \eta \ge 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 \ge 0 and l
     \ge 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 \le l \le lmax.

7.26 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’.

7.27 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) and 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’.

7.27.1 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.

 -- Type: gsl_sf_mathieu_workspace

     Workspace required for array-based routines

 -- 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’.

7.27.2 Mathieu Function Characteristic Values
---------------------------------------------

 -- Function: int gsl_sf_mathieu_a (int n, double q)
 -- Function: int gsl_sf_mathieu_a_e (int n, double q, gsl_sf_result
          * result)
 -- Function: int gsl_sf_mathieu_b (int n, double q)
 -- Function: int gsl_sf_mathieu_b_e (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’.

7.27.3 Angular Mathieu Functions
--------------------------------

 -- Function: int gsl_sf_mathieu_ce (int n, double q, double x)
 -- Function: int gsl_sf_mathieu_ce_e (int n, double q, double x,
          gsl_sf_result * result)
 -- Function: int gsl_sf_mathieu_se (int n, double q, double x)
 -- Function: int gsl_sf_mathieu_se_e (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’.

7.27.4 Radial Mathieu Functions
-------------------------------

 -- Function: int gsl_sf_mathieu_Mc (int j, int n, double q, double x)
 -- Function: int gsl_sf_mathieu_Mc_e (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)
 -- Function: int gsl_sf_mathieu_Ms_e (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’.

7.28 Power Function
===================

The following functions are equivalent to the function
gsl_pow_int() 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.  The following is a simple
     example:

          #include <gsl/gsl_sf_pow_int.h>
          /* compute 3.0**12 */
          double y = gsl_sf_pow_int(3.0, 12);

7.29 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’.

7.29.1 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)].

7.29.2 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’.

7.29.3 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
     \ge 0, x > 0.

7.30 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 \ge 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 \ge 0.

7.31 Transport Functions
========================

The transport functions J(n,x) are defined by the integral
representations

     J(n,x) = \int_0^x t^n e^t /(e^t - 1)^2 dt

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).

7.32 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’.

7.32.1 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’.

7.32.2 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’.

7.32.3 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’.

7.32.4 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].

7.32.5 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.

7.32.6 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.

7.33 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’.

7.33.1 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.

7.33.2 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.

7.33.3 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.

7.33.4 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’.

7.34 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,

     status  = success
     J0(5.0) = -0.177596771314338264
           +/-  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,

     J0(5.0) = -0.177596771314338264
     exact   = -0.177596771314338292

7.35 References and Further Reading
===================================

The library follows the conventions of the following book where
possible,

   * Handbook of Mathematical Functions, edited by Abramowitz & Stegun,
     Dover, ISBN 0486612724.

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).

   * S. A. Holmes and W. E. Featherstone, A unified approach to the
     Clenshaw summation and the recursive computation of very high
     degree and order normalised associated Legendre functions, Journal
     of Geodesy, 76, pg.  279-299, 2002.

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.

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,

Prefix                              Type

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

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.

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,

 -- Type: gsl_block

          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’.

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.

     Zero-sized requests are valid and return a non-null result.  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().

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.

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 = %zu\n", b->size);
       printf ("block data address = %p\n", b->data);

       gsl_block_free (b);
       return 0;
     }

Here is the output from the program,

     length of block = 100
     block data address = 0x804b0d8

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,

 -- Type: gsl_vector

          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’.

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.  Zero-sized requests
     are valid and return a non-null result.

 -- 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.

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.

 -- Macro: GSL_RANGE_CHECK_OFF

     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.

 -- Macro: GSL_C99_INLINE

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

 -- Variable: gsl_check_range

     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, const
          size_t i)

     This function returns the ‘i’-th element of a vector ‘v’.  If ‘i’
     lies outside the allowed range of 0 to ‘size - 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, const 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 ‘size - 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 ‘size - 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.

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.

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.

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.

 -- Type: gsl_vector_view
 -- Type: gsl_vector_const_view

     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’.

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

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

 -- Function: int gsl_vector_memcpy (gsl_vector * dest, const gsl_vector
          * src)

     This function copies the elements of the vector ‘src’ into the
     vector ‘dest’.  The two vectors must have the same length.

 -- Function: int gsl_vector_swap (gsl_vector * v, gsl_vector * w)

     This function exchanges the elements of the vectors ‘v’ and ‘w’ by
     copying.  The two vectors must have the same length.

8.3.7 Exchanging elements
-------------------------

The following functions 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’.

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’.

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.

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.

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;
     }

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.

 -- Type: gsl_matrix

     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’.

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.  Requesting zero for ‘n1’ or ‘n2’ is valid and returns
     a non-null result.

 -- 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.

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, const
          size_t i, const 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, const size_t i, const
          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.

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.

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.

8.4.5 Matrix views
------------------

 -- Type: gsl_matrix_view
 -- Type: gsl_matrix_const_view

     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’.

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_matrix_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_matrix_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_matrix_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_matrix_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’.

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.

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.

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.

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’.

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.

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.

 -- 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.

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 %zu, norm = %g\n", j, d);
         }

       gsl_matrix_free (m);

       return 0;
     }

Here is the output of the program,

     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

8.4.14 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.

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’.

9.1 The Permutation struct
==========================

 -- Type: gsl_permutation

     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;

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, \dots, 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.

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’.

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.

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.

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
     :data:`p`_j-th column of the identity matrix.  The permutation ‘p’
     and the vector ‘v’ must have the same length.

 -- Function: int gsl_permute_matrix (const gsl_permutation * p,
          gsl_matrix * A)

     This function applies the permutation ‘p’ to the matrix ‘A’ from
     the right, A' = A P. The j-th column of the permutation matrix P is
     given by the p_j-th column of the identity matrix.  This
     effectively permutes the columns of ‘A’ according to the
     permutation ‘p’, and so the number of columns of ‘A’ must equal the
     size of the permutation ‘p’.

 -- 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’.

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.

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.

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().

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

The subject of permutations is covered extensively in the following,

   * 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.

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’.

10.1 The Combination struct
===========================

 -- Type: gsl_combination

     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;

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, \dots, 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, \dots,
     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.

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.

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.

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.

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.

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,

     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.

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

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’.

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

 -- Type: gsl_multiset

     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;

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.

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.

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.

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.

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.

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,

     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.

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.

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

      -- Type: gsl_comparison_fn_t

               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.

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.

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’.

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);
     }

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

       for (i = 0; i < k; i++)
         {
           printf ("%zu: %.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,

     5 smallest values from 100000
     0: 0.000003489200025797
     1: 0.000008199829608202
     2: 0.000008953968062997
     3: 0.000010712770745158
     4: 0.000033531803637743

12.6 References and Further Reading
===================================

The subject of sorting is covered extensively in the following,

   * 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.

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.

There are three levels of BLAS operations,

`Level 1'       Vector operations, e.g.  y = \alpha x + y

`Level 2'       Matrix-vector operations, e.g.  y = \alpha A x + \beta y

`Level 3'       Matrix-matrix operations, e.g.  C = \alpha A B + C

Each routine has a name which specifies the operation, the type of
matrices involved and their precisions.  Some of the most common
operations and their names are given below,

`DOT'        scalar product, x^T y

`AXPY'       vector sum, \alpha x + y

`MV'         matrix-vector product, A x

`SV'         matrix-vector solve, inv(A) x

`MM'         matrix-matrix product, A B

`SM'         matrix-matrix solve, inv(A) B

The types of matrices are,

`GE'       general

`GB'       general band

`SY'       symmetric

`SB'       symmetric band

`SP'       symmetric packed

`HE'       hermitian

`HB'       hermitian band

`HP'       hermitian packed

`TR'       triangular

`TB'       triangular band

`TP'       triangular packed

Each operation is defined for four precisions,

`S'       single real

`D'       double real

`C'       single complex

`Z'       double complex

Thus, for example, the name SGEMM stands for “single-precision general
matrix-matrix multiply” and ZGEMM stands for “double-precision complex
matrix-matrix multiply”.

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

   ---------- 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.

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’.

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 \left( |\Re(x_i)| +
     |\Im(x_i)| \right).

 -- 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.

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.

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.

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,

     [ 367.76, 368.12
       674.06, 674.72 ]

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.

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’.

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

A general N-by-N 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’ on
     output.  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 * work)
 -- 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 * work)

     These functions apply an iterative improvement to ‘x’, the solution
     of A x = b, from the precomputed LU decomposition of A into (‘LU’,
     ‘p’).  Additional workspace of length ‘N’ is required in ‘work’.

 -- 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’.

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.

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

The QR decomposition of an M-by-N matrix A 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. When A is rank deficient with r = {\rm rank}(A), the matrix R can
be partitioned as

     R = [ R11 R12 ] =~ [ R11 R12 ]
         [  0  R22 ]    [  0   0  ]

where R_{11} is r-by-r and nonsingular.  In this case, a `basic' least
squares solution for the overdetermined system A x = b can be obtained
as

     x = P [ R11^-1 c1 ]
           [     0     ]

where c_1 consists of the first r elements of Q^T b.  This basic
solution is not guaranteed to be the minimum norm solution unless R_{12}
= 0 (see Complete Orthogonal Decomposition).

 -- 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_lssolve (const gsl_matrix * QR, const
          gsl_vector * tau, const gsl_permutation * p, 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 and is assumed to have full rank.  The least squares
     solution minimizes the Euclidean norm of the residual, ||b - A x||.
     The routine requires as input the QR decomposition of A into (‘QR’,
     ‘tau’, ‘p’) given by gsl_linalg_QRPT_decomp().  The
     solution is returned in ‘x’.  The residual is computed as a
     by-product and stored in ‘residual’.  For rank deficient matrices,
     see gsl_linalg_QRPT_lssolve2() should be used instead.

 -- Function: int gsl_linalg_QRPT_lssolve2 (const gsl_matrix * QR, const
          gsl_vector * tau, const gsl_permutation * p, const gsl_vector
          * b, const size_t rank, 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 and has rank given by the input ‘rank’.  If the user
     does not know the rank of A, the routine
     gsl_linalg_QRPT_rank() can be called to estimate it.  The
     least squares solution is the so-called “basic” solution discussed
     above and may not be the minimum norm solution.  The routine
     requires as input the QR decomposition of A into (‘QR’, ‘tau’, ‘p’)
     given by gsl_linalg_QRPT_decomp().  The solution is
     returned in ‘x’.  The residual is computed as a by-product and
     stored in ‘residual’.

 -- 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.

 -- Function: size_t gsl_linalg_QRPT_rank (const gsl_matrix * QR, const
          double tol)

     This function estimates the rank of the triangular matrix R
     contained in ‘QR’.  The algorithm simply counts the number of
     diagonal elements of R whose absolute value is greater than the
     specified tolerance ‘tol’.  If the input ‘tol’ is negative, a
     default value of 20 (M + N) eps(max(|diag(R)|)) is used.

 -- Function: int gsl_linalg_QRPT_rcond (const gsl_matrix * QR, double
          * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the R factor, stored in the upper triangle of ‘QR’.  The
     reciprocal condition number estimate, defined as 1 / (||R||_1 \cdot
     ||R^{-1}||_1), is stored in ‘rcond’.  Additional workspace of size
     3 N is required in ‘work’.

14.4 Complete Orthogonal Decomposition
======================================

The complete orthogonal decomposition of a M-by-N matrix A is a
generalization of the QR decomposition with column pivoting, given by

     A P = Q [ R11 0 ] Z^T
             [  0  0 ]

where P is a N-by-N permutation matrix, Q is M-by-M orthogonal, R_{11}
is r-by-r upper triangular, with r = {\rm rank}(A), and Z is N-by-N
orthogonal.  If A has full rank, then R_{11} = R, Z = I and this reduces
to the QR decomposition with column pivoting.

For a rank deficient least squares problem, \min_x{|| b - Ax||^2}, the
solution vector x is not unique.  However if we further require that
||x||^2 is minimized, then the complete orthogonal decomposition gives
us the ability to compute the unique minimum norm solution, which is
given by

     x = P Z [ R11^-1 c1 ]
             [     0     ]

and the vector c_1 is the first r elements of Q^T b.

The COD also enables a straightforward solution of regularized least
squares problems in Tikhonov standard form, written as

     \min_x ||b - A x||^2 + \lambda^2 ||x||^2

where \lambda > 0 is a regularization parameter which represents a
tradeoff between minimizing the residual norm ||b - A x|| and the
solution norm ||x||.  For this system, the solution is given by

     x = P Z [ y1 ]
             [ 0  ]

where y_1 is a vector of length r which is found by solving

     [     R11     ] y_1 = [ c_1 ]
     [ \lambda I_r ]       [  0  ]

and c_1 is defined above.  The equation above can be solved efficiently
for different values of \lambda using QR factorizations of the left hand
side matrix.

 -- Function: int gsl_linalg_COD_decomp (gsl_matrix * A, gsl_vector
          * tau_Q, gsl_vector * tau_Z, gsl_permutation * p, size_t
          * rank, gsl_vector * work)
 -- Function: int gsl_linalg_COD_decomp_e (gsl_matrix * A, gsl_vector
          * tau_Q, gsl_vector * tau_Z, gsl_permutation * p, double tol,
          size_t * rank, gsl_vector * work)

     These functions factor the M-by-N matrix ‘A’ into the decomposition
     A = Q R Z P^T. The rank of ‘A’ is computed as the number of
     diagonal elements of R greater than the tolerance ‘tol’ and output
     in ‘rank’.  If ‘tol’ is not specified, a default value is used (see
     gsl_linalg_QRPT_rank()).  On output, the permutation
     matrix P is stored in ‘p’.  The matrix R_{11} is stored in the
     upper ‘rank’-by-‘rank’ block of ‘A’.  The matrices Q and Z are
     encoded in packed storage in ‘A’ on output.  The vectors ‘tau_Q’
     and ‘tau_Z’ contain the Householder scalars corresponding to the
     matrices Q and Z respectively and must be of length k = \min(M,N).
     The vector ‘work’ is additional workspace of length N.

 -- Function: int gsl_linalg_COD_lssolve (const gsl_matrix * QRZT, const
          gsl_vector * tau_Q, const gsl_vector * tau_Z, const
          gsl_permutation * p, const size_t rank, const gsl_vector * b,
          gsl_vector * x, gsl_vector * residual)

     This function finds the unique minimum norm 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, ||b - A x|| as well as the norm of
     the solution ||x||.  The routine requires as input the QRZT
     decomposition of A into (‘QRZT’, ‘tau_Q’, ‘tau_Z’, ‘p’, ‘rank’)
     given by gsl_linalg_COD_decomp().  The solution is
     returned in ‘x’.  The residual, b - Ax, is computed as a by-product
     and stored in ‘residual’.

 -- Function: int gsl_linalg_COD_lssolve2 (const double lambda, const
          gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector
          * tau_Z, const gsl_permutation * p, const size_t rank, const
          gsl_vector * b, gsl_vector * x, gsl_vector * residual,
          gsl_matrix * S, gsl_vector * work)

     This function finds the solution to the regularized least squares
     problem in Tikhonov standard form, \min_x ||b - Ax||^2 + \lambda^2
     ||x||^2.  The routine requires as input the QRZT decomposition of A
     into (‘QRZT’, ‘tau_Q’, ‘tau_Z’, ‘p’, ‘rank’) given by
     gsl_linalg_COD_decomp().  The parameter \lambda is supplied in
     ‘lambda’.  The solution is returned in ‘x’.  The residual, b - Ax,
     is stored in ‘residual’ on output.  ‘S’ is additional workspace of
     size ‘rank’-by-‘rank’.  ‘work’ is additional workspace of length
     ‘rank’.

 -- Function: int gsl_linalg_COD_unpack (const gsl_matrix * QRZT, const
          gsl_vector * tau_Q, const gsl_vector * tau_Z, const
          size_t rank, gsl_matrix * Q, gsl_matrix * R, gsl_matrix * Z)

     This function unpacks the encoded QRZT decomposition (‘QRZT’,
     ‘tau_Q’, ‘tau_Z’, ‘rank’) into the matrices ‘Q’, ‘R’, and ‘Z’,
     where ‘Q’ is M-by-M, ‘R’ is M-by-N, and ‘Z’ is N-by-N.

 -- Function: int gsl_linalg_COD_matZ (const gsl_matrix * QRZT, const
          gsl_vector * tau_Z, const size_t rank, gsl_matrix * A,
          gsl_vector * work)

     This function multiplies the input matrix ‘A’ on the right by ‘Z’,
     A' = A Z using the encoded QRZT decomposition (‘QRZT’, ‘tau_Z’,
     ‘rank’).  ‘A’ must have N columns but may have any number of rows.
     Additional workspace of length M is provided in ‘work’.

14.5 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 \ge 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 \gg 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 \ge 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.

14.6 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.

If the matrix A is near singular, it is sometimes possible to reduce the
condition number and recover a more accurate solution vector x by
scaling as

     ( S A S ) ( S^(-1) x ) = S b

where S is a diagonal matrix whose elements are given by S_{ii} =
1/\sqrt{A_{ii}}.  This scaling is also known as Jacobi preconditioning.
There are routines below to solve both the scaled and unscaled systems.

 -- Function: int gsl_linalg_cholesky_decomp1 (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^{\dagger} 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 is unmodified.  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_decomp (gsl_matrix * A)

     This function is now deprecated and is provided only for backward
     compatibility.

 -- 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’.

 -- Function: int gsl_linalg_cholesky_decomp2 (gsl_matrix * A,
          gsl_vector * S)

     This function calculates a diagonal scaling transformation S for
     the symmetric, positive-definite square matrix ‘A’, and then
     computes the Cholesky decomposition S A S = L L^T. 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.
     The diagonal scale factors are stored in ‘S’ on output.

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

 -- Function: int gsl_linalg_cholesky_solve2 (const gsl_matrix
          * cholesky, const gsl_vector * S, const gsl_vector * b,
          gsl_vector * x)

     This function solves the system (S A S) (S^{-1} x) = S b using the
     Cholesky decomposition of S A S held in the matrix ‘cholesky’ which
     must have been previously computed by
     gsl_linalg_cholesky_decomp2().

 -- Function: int gsl_linalg_cholesky_svx2 (const gsl_matrix * cholesky,
          const gsl_vector * S, gsl_vector * x)

     This function solves the system (S A S) (S^{-1} x) = S b in-place
     using the Cholesky decomposition of S A S held in the matrix
     ‘cholesky’ which must have been previously computed by
     gsl_linalg_cholesky_decomp2().  On input ‘x’ should contain
     the right-hand side b, which is replaced by the solution on output.

 -- Function: int gsl_linalg_cholesky_scale (const gsl_matrix * A,
          gsl_vector * S)

     This function calculates a diagonal scaling transformation of the
     symmetric, positive definite matrix ‘A’, such that S A S has a
     condition number within a factor of N of the matrix of smallest
     possible condition number over all possible diagonal scalings.  On
     output, ‘S’ contains the scale factors, given by S_i =
     1/\sqrt{A_{ii}}.  For any A_{ii} \le 0, the corresponding scale
     factor S_i is set to 1.

 -- Function: int gsl_linalg_cholesky_scale_apply (gsl_matrix * A, const
          gsl_vector * S)

     This function applies the scaling transformation ‘S’ to the matrix
     ‘A’.  On output, ‘A’ is replaced by S A S.

 -- Function: int gsl_linalg_cholesky_rcond (const gsl_matrix
          * cholesky, double * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric positive definite matrix A, using its
     Cholesky decomposition provided in ‘cholesky’.  The reciprocal
     condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in ‘rcond’.  Additional workspace of size
     3 N is required in ‘work’.

14.7 Pivoted Cholesky Decomposition
===================================

A symmetric, positive definite square matrix A has an alternate Cholesky
decomposition into a product of a lower unit triangular matrix L, a
diagonal matrix D and L^T, given by L D L^T. This is equivalent to the
Cholesky formulation discussed above, with the standard Cholesky lower
triangular factor given by L D^{1 \over 2}.  For ill-conditioned
matrices, it can help to use a pivoting strategy to prevent the entries
of D and L from growing too large, and also ensure D_1 \ge D_2 \ge
\cdots \ge D_n > 0, where D_i are the diagonal entries of D. The final
decomposition is given by

     P A P^T = L D L^T

where P is a permutation matrix.

 -- Function: int gsl_linalg_pcholesky_decomp (gsl_matrix * A,
          gsl_permutation * p)

     This function factors the symmetric, positive-definite square
     matrix ‘A’ into the Pivoted Cholesky decomposition P A P^T = L D
     L^T. On input, the values from the diagonal and lower-triangular
     part of the matrix ‘A’ are used to construct the factorization.  On
     output the diagonal of the input matrix ‘A’ stores the diagonal
     elements of D, and the lower triangular portion of ‘A’ contains the
     matrix L. Since L has ones on its diagonal these do not need to be
     explicitely stored.  The upper triangular portion of ‘A’ is
     unmodified.  The permutation matrix P is stored in ‘p’ on output.

 -- Function: int gsl_linalg_pcholesky_solve (const gsl_matrix * LDLT,
          const gsl_permutation * p, const gsl_vector * b, gsl_vector
          * x)

     This function solves the system A x = b using the Pivoted Cholesky
     decomposition of A held in the matrix ‘LDLT’ and permutation ‘p’
     which must have been previously computed by
     gsl_linalg_pcholesky_decomp().

 -- Function: int gsl_linalg_pcholesky_svx (const gsl_matrix * LDLT,
          const gsl_permutation * p, gsl_vector * x)

     This function solves the system A x = b in-place using the Pivoted
     Cholesky decomposition of A held in the matrix ‘LDLT’ and
     permutation ‘p’ which must have been previously computed by
     gsl_linalg_pcholesky_decomp().  On input, ‘x’ contains the
     right hand side vector b which is replaced by the solution vector
     on output.

 -- Function: int gsl_linalg_pcholesky_decomp2 (gsl_matrix * A,
          gsl_permutation * p, gsl_vector * S)

     This function computes the pivoted Cholesky factorization of the
     matrix S A S, where the input matrix ‘A’ is symmetric and positive
     definite, and the diagonal scaling matrix ‘S’ is computed to reduce
     the condition number of ‘A’ as much as possible.  See
     Cholesky Decomposition for more information on the matrix
     ‘S’.  The Pivoted Cholesky decomposition satisfies P S A S P^T = L
     D L^T. On input, the values from the diagonal and lower-triangular
     part of the matrix ‘A’ are used to construct the factorization.  On
     output the diagonal of the input matrix ‘A’ stores the diagonal
     elements of D, and the lower triangular portion of ‘A’ contains the
     matrix L. Since L has ones on its diagonal these do not need to be
     explicitely stored.  The upper triangular portion of ‘A’ is
     unmodified.  The permutation matrix P is stored in ‘p’ on output.
     The diagonal scaling transformation is stored in ‘S’ on output.

 -- Function: int gsl_linalg_pcholesky_solve2 (const gsl_matrix * LDLT,
          const gsl_permutation * p, const gsl_vector * S, const
          gsl_vector * b, gsl_vector * x)

     This function solves the system (S A S) (S^{-1} x) = S b using the
     Pivoted Cholesky decomposition of S A S held in the matrix ‘LDLT’,
     permutation ‘p’, and vector ‘S’, which must have been previously
     computed by gsl_linalg_pcholesky_decomp2().

 -- Function: int gsl_linalg_pcholesky_svx2 (const gsl_matrix * LDLT,
          const gsl_permutation * p, const gsl_vector * S, gsl_vector
          * x)

     This function solves the system (S A S) (S^{-1} x) = S b in-place
     using the Pivoted Cholesky decomposition of S A S held in the
     matrix ‘LDLT’, permutation ‘p’ and vector ‘S’, which must have been
     previously computed by gsl_linalg_pcholesky_decomp2().
     On input, ‘x’ contains the right hand side vector b which is
     replaced by the solution vector on output.

 -- Function: int gsl_linalg_pcholesky_invert (const gsl_matrix * LDLT,
          const gsl_permutation * p, gsl_matrix * Ainv)

     This function computes the inverse of the matrix A, using the
     Pivoted Cholesky decomposition stored in ‘LDLT’ and ‘p’.  On
     output, the matrix ‘Ainv’ contains A^{-1}.

 -- Function: int gsl_linalg_pcholesky_rcond (const gsl_matrix * LDLT,
          const gsl_permutation * p, double * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric positive definite matrix A, using its
     pivoted Cholesky decomposition provided in ‘LDLT’.  The reciprocal
     condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in ‘rcond’.  Additional workspace of size
     3 N is required in ‘work’.

14.8 Modified Cholesky Decomposition
====================================

The modified Cholesky decomposition is suitable for solving systems A x
= b where A is a symmetric indefinite matrix.  Such matrices arise in
nonlinear optimization algorithms.  The standard Cholesky decomposition
requires a positive definite matrix and would fail in this case.
Instead of resorting to a method like QR or SVD, which do not take into
account the symmetry of the matrix, we can instead introduce a small
perturbation to the matrix A to make it positive definite, and then use
a Cholesky decomposition on the perturbed matrix.  The resulting
decomposition satisfies

     P (A + E) P^T = L D L^T

where P is a permutation matrix, E is a diagonal perturbation matrix, L
is unit lower triangular, and D is diagonal.  If A is sufficiently
positive definite, then the perturbation matrix E will be zero and this
method is equivalent to the pivoted Cholesky algorithm.  For indefinite
matrices, the perturbation matrix E is computed to ensure that A + E is
positive definite and well conditioned.

 -- Function: int gsl_linalg_mcholesky_decomp (gsl_matrix * A,
          gsl_permutation * p, gsl_vector * E)

     This function factors the symmetric, indefinite square matrix ‘A’
     into the Modified Cholesky decomposition P (A + E) P^T = L D L^T.
     On input, the values from the diagonal and lower-triangular part of
     the matrix ‘A’ are used to construct the factorization.  On output
     the diagonal of the input matrix ‘A’ stores the diagonal elements
     of D, and the lower triangular portion of ‘A’ contains the matrix
     L. Since L has ones on its diagonal these do not need to be
     explicitely stored.  The upper triangular portion of ‘A’ is
     unmodified.  The permutation matrix P is stored in ‘p’ on output.
     The diagonal perturbation matrix is stored in ‘E’ on output.  The
     parameter ‘E’ may be set to NULL if it is not required.

 -- Function: int gsl_linalg_mcholesky_solve (const gsl_matrix * LDLT,
          const gsl_permutation * p, const gsl_vector * b, gsl_vector
          * x)

     This function solves the perturbed system (A + E) x = b using the
     Cholesky decomposition of A + E held in the matrix ‘LDLT’ and
     permutation ‘p’ which must have been previously computed by
     gsl_linalg_mcholesky_decomp().

 -- Function: int gsl_linalg_mcholesky_svx (const gsl_matrix * LDLT,
          const gsl_permutation * p, gsl_vector * x)

     This function solves the perturbed system (A + E) x = b in-place
     using the Cholesky decomposition of A + E held in the matrix ‘LDLT’
     and permutation ‘p’ which must have been previously computed by
     gsl_linalg_mcholesky_decomp().  On input, ‘x’ contains
     the right hand side vector b which is replaced by the solution
     vector on output.

 -- Function: int gsl_linalg_mcholesky_rcond (const gsl_matrix * LDLT,
          const gsl_permutation * p, double * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the perturbed matrix A + E, using its pivoted Cholesky
     decomposition provided in ‘LDLT’.  The reciprocal condition number
     estimate, defined as 1 / (||A + E||_1 \cdot ||(A + E)^{-1}||_1), is
     stored in ‘rcond’.  Additional workspace of size 3 N is required in
     ‘work’.

14.9 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’.

14.10 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’.

14.11 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().

14.12 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’.

14.13 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’.

14.14 Givens Rotations
======================

A Givens rotation is a rotation in the plane acting on two elements of a
given vector.  It can be represented in matrix form as

where the \cos{\theta} and \sin{\theta} appear at the intersection of
the i-th and j-th rows and columns.  When acting on a vector x,
G(i,j,\theta) x performs a rotation of the (i,j) elements of x.  Givens
rotations are typically used to introduce zeros in vectors, such as
during the QR decomposition of a matrix.  In this case, it is typically
desired to find c and s such that

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

with r = \sqrt{a^2 + b^2}.

 -- Function: void gsl_linalg_givens (const double a, const double b,
          double * c, double * s)

     This function computes c = \cos{\theta} and s = \sin{\theta} so
     that the Givens matrix G(\theta) acting on the vector (a,b)
     produces (r, 0), with r = \sqrt{a^2 + b^2}.

 -- Function: void gsl_linalg_givens_gv (gsl_vector * v, const size_t i,
          const size_t j, const double c, const double s)

     This function applies the Givens rotation defined by c =
     \cos{\theta} and s = \sin{\theta} to the ‘i’ and ‘j’ elements of
     ‘v’.  On output, (v(i),v(j)) \leftarrow G(\theta) (v(i),v(j)).

14.15 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 * w)
 -- Function: gsl_complex gsl_linalg_complex_householder_transform
          (gsl_vector_complex * w)

     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
     ‘w’ except the first.  On output the Householder vector ‘v’ is
     stored in ‘w’ and the scalar \tau is returned.  The householder
     vector ‘v’ is normalized so that ‘v[0] = 1’, however this 1 is not
     stored in the output vector.  Instead, ‘w[0]’ is set to the first
     element of the transformed vector, so that if u = P w, ‘w[0] =
     u[0]’ on output and the remainder of u is zero.

 -- 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’.

14.16 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.

14.17 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 \geq 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 \geq 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 \geq 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 \geq 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 )

14.18 Triangular Systems
========================

 -- Function: int gsl_linalg_tri_upper_invert (gsl_matrix * T)
 -- Function: int gsl_linalg_tri_lower_invert (gsl_matrix * T)
 -- Function: int gsl_linalg_tri_upper_unit_invert (gsl_matrix * T)
 -- Function: int gsl_linalg_tri_lower_unit_invert (gsl_matrix * T)

     These functions calculate the in-place inverse of the triangular
     matrix ‘T’.  When the ‘upper’ prefix is specified, then the upper
     triangle of ‘T’ is used, and when the ‘lower’ prefix is specified,
     the lower triangle is used.  If the ‘unit’ prefix is specified,
     then the diagonal elements of the matrix ‘T’ are taken as unity and
     are not referenced.  Otherwise the diagonal elements are used in
     the inversion.

 -- Function: int gsl_linalg_tri_upper_rcond (const gsl_matrix * T,
          double * rcond, gsl_vector * work)
 -- Function: int gsl_linalg_tri_lower_rcond (const gsl_matrix * T,
          double * rcond, gsl_vector * work)

     These functions estimate the reciprocal condition number, in the
     1-norm, of the upper or lower N-by-N triangular matrix ‘T’.  The
     reciprocal condition number is stored in ‘rcond’ on output, and is
     defined by 1 / (||T||_1 \cdot ||T^{-1}||_1).  Additional workspace
     of size 3 N is required in ‘work’.

14.19 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’.

14.20 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.

14.21 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

The LAPACK source code can be found at ‘http://www.netlib.org/lapack’,
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.

   * J. Demmel, K. 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.

The algorithm for estimating a matrix condition number is described in
the following paper,

   * N. J. Higham, “FORTRAN codes for estimating the one-norm of a real
     or complex matrix, with applications to condition estimation”, ACM
     Trans.  Math.  Soft., vol.  14, no.  4, pp.  381-396, December
     1988.

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’.

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.

 -- Type: gsl_eigen_symm_workspace

     This workspace contains internal parameters used for solving
     symmetric eigenvalue problems.

 -- 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.

 -- Type: gsl_eigen_symmv_workspace

     This workspace contains internal parameters used for solving
     symmetric eigenvalue and eigenvector problems.

 -- 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.

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

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

 -- Type: gsl_eigen_herm_workspace

     This workspace contains internal parameters used for solving
     hermitian eigenvalue problems.

 -- 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.

 -- Type: gsl_eigen_hermv_workspace

     This workspace contains internal parameters used for solving
     hermitian eigenvalue and eigenvector problems.

 -- 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.

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.

 -- Type: gsl_eigen_nonsymm_workspace

     This workspace contains internal parameters used for solving
     nonsymmetric eigenvalue problems.

 -- Function: gsl_eigen_nonsymm_workspace * gsl_eigen_nonsymm_alloc
          (const size_t n)

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

 -- Function: void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace
          * w)

     This function frees the memory associated with the workspace ‘w’.

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

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

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

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

          T = Q^T D^{-1} A D Q

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

          T = Z^{-1} A Z

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

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

     This function computes the eigenvalues of the real nonsymmetric
     matrix ‘A’ and stores them in the vector ‘eval’.  If T is desired,
     it is stored in the upper portion of ‘A’ on output.  Otherwise, on
     output, the diagonal of ‘A’ will contain the 1-by-1 real
     eigenvalues and 2-by-2 complex conjugate eigenvalue systems, and
     the rest of ‘A’ is destroyed.  In rare cases, this function may
     fail to find all eigenvalues.  If this happens, an error code is
     returned and the number of converged eigenvalues is stored in
     ‘w->n_evals’.  The converged eigenvalues are stored in the
     beginning of ‘eval’.

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

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

 -- Type: gsl_eigen_nonsymmv_workspace

     This workspace contains internal parameters used for solving
     nonsymmetric eigenvalue and eigenvector problems.

 -- 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’.

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.

 -- Type: gsl_eigen_gensymm_workspace

     This workspace contains internal parameters used for solving
     generalized symmetric eigenvalue problems.

 -- 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.

 -- Type: gsl_eigen_gensymmv_workspace

     This workspace contains internal parameters used for solving
     generalized symmetric eigenvalue and eigenvector problems.

 -- 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.

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^{-\dagger} is hermitian, and y = L^{\dagger} 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.

 -- Type: gsl_eigen_genherm_workspace

     This workspace contains internal parameters used for solving
     generalized hermitian eigenvalue problems.

 -- 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.

 -- Type: gsl_eigen_genhermv_workspace

     This workspace contains internal parameters used for solving
     generalized hermitian eigenvalue and eigenvector problems.

 -- 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.

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).

 -- Type: gsl_eigen_gen_workspace

     This workspace contains internal parameters used for solving
     generalized eigenvalue problems.

 -- 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
     (see 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
     (see 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.

 -- Type: gsl_eigen_genv_workspace

     This workspace contains internal parameters used for solving
     generalized eigenvalue and eigenvector problems.

 -- 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 (see 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.

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’,

      -- Type: gsl_eigen_sort_t

          ‘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 (see 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.

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.

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.

The LAPACK source code can be found at the website
‘http://www.netlib.org/lapack’ along with an online copy of the users
guide.

16 Fast Fourier Transforms (FFTs)
*********************************

This chapter describes functions for performing Fast Fourier Transforms
(FFTs).  The library includes radix-2 routines (for lengths which are a
power of two) and mixed-radix routines (which work for any length).  For
efficiency there are separate versions of the routines for real data and
for complex data.  The mixed-radix routines are a reimplementation of
the FFTPACK library of Paul Swarztrauber.  Fortran code for FFTPACK is
available on Netlib (FFTPACK also includes some routines for sine and
cosine transforms but these are currently not available in GSL). For
details and derivations of the underlying algorithms consult the
document “GSL FFT Algorithms” (see References and Further Reading)

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 \ldots 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 = \hbox{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 =
\hbox{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.

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.

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 \dots 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\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.

     \int_{-a}^{+a} e^{-2 \pi i k x} dx = {\sin(2\pi k a) \over\pi k}

The output of the example program is plotted in Fig.  %s.

Figure: A pulse and its discrete Fourier transform, output from the
example program.

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’.

 -- 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.

 -- Type: gsl_fft_complex_workspace

     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).

     See 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 < (int) wavetable->nf; i++)
         {
            printf ("# factor %d: %zu\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.

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).

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;
            }

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’.

 -- Type: gsl_fft_real_wavetable
 -- Type: gsl_fft_halfcomplex_wavetable

     These data structures contain lookup tables for an FFT of a fixed
     size.

 -- 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,

 -- Type: gsl_fft_real_workspace

     This workspace contains parameters needed to compute a real FFT.

 -- 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;
     }

The program output is shown in Fig.  %s.

Figure: Low-pass filtered version of a real pulse, output from the
example program.

16.8 References and Further Reading
===================================

A good starting point for learning more about the FFT is the following
review article,

   * 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,

   *
       5. 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
‘http://www.netlib.org/fftpack/’

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, as
well as fixed-order quadrature rules for a variety of weighting
functions from IQPACK.

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

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.

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.

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.

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.

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.

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 the following struct,

 -- Type: gsl_integration_workspace

     This workspace 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,

        *  -- Macro: GSL_INTEG_GAUSS15 (key = 1)

        *  -- Macro: GSL_INTEG_GAUSS21 (key = 2)

        *  -- Macro: GSL_INTEG_GAUSS31 (key = 3)

        *  -- Macro: GSL_INTEG_GAUSS41 (key = 4)

        *  -- Macro: GSL_INTEG_GAUSS51 (key = 5)

        *  -- Macro: 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.

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.

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
     gsl_integration_qags().

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.

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.

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.

 -- Type: gsl_integration_qaws_table

     This structure contains precomputed quantities for the QAWS
     algorithm.

 -- 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,

     Weight function w(x)                                             (\mu,\nu)

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

     (x - a)^\alpha (b - x)^\beta                                     (0,0)

     (x - a)^\alpha (b - x)^\beta \log{(x-a)}                         (1,0)

     (x - a)^\alpha (b - x)^\beta \log{(b-x)}                         (0,1)

     (x - a)^\alpha (b - x)^\beta \log{(x-a)} \log{(b-x)}             (1,1)

     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.

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:

      -- Macro: GSL_INTEG_COSINE

      -- Macro: 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.

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.

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’.

17.12 Romberg integration
=========================

The Romberg integration method estimates the definite integral

     I = \int_a^b f(x) dx

by applying Richardson extrapolation on the trapezoidal rule, using
equally spaced points with spacing

     h_k = (b - a) 2^{-k}

for k = 1, \dots, n.  For each k, Richardson extrapolation is used k-1
times on previous approximations to improve the order of accuracy as
much as possible.  Romberg integration typically works well (and
converges quickly) for smooth integrands with no singularities in the
interval or at the end points.

 -- Function: gsl_integration_romberg_workspace *
          gsl_integration_romberg_alloc (const size_t n)

     This function allocates a workspace for Romberg integration,
     specifying a maximum of n iterations, or divisions of the interval.
     Since the number of divisions is 2^n + 1, n can be kept relatively
     small (i.e.  10 or 20).  It is capped at a maximum value of 30 to
     prevent overflow.  The size of the workspace is O(2n).

 -- Function: void gsl_integration_romberg_free
          (gsl_integration_romberg_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_integration_romberg (const gsl_function * f, const
          double a, const double b, const double epsabs, const
          double epsrel, double * result, size_t * neval,
          gsl_integration_romberg_workspace * w)

     This function integrates f(x), specified by ‘f’, from ‘a’ to ‘b’,
     storing the answer in ‘result’.  At each step in the iteration,
     convergence is tested by checking:

          | I_k - I_{k-1} | \le \textrm{max} \left( epsabs, epsrel \times |I_k| \right)

     where I_k is the current approximation and I_{k-1} is the
     approximation of the previous iteration.  If the method does not
     converge within the previously specified n iterations, the function
     stores the best current estimate in ‘result’ and returns
     ‘GSL_EMAXITER’.  If the method converges, the function returns
     ‘GSL_SUCCESS’.  The total number of function evaluations is
     returned in ‘neval’.

17.13 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 2n - 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, \dots, 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’.

17.14 Fixed point quadratures
=============================

The routines in this section approximate an integral by the sum

     \int_a^b w(x) f(x) dx = \sum_{i=1}^n w_i f(x_i)

where f(x) is the function to be integrated and w(x) is a weighting
function.  The n weights w_i and nodes x_i are carefully chosen so that
the result is exact when f(x) is a polynomial of degree 2n - 1 or less.
Once the user chooses the order n and weighting function w(x), the
weights w_i and nodes x_i can be precomputed and used to efficiently
evaluate integrals for any number of functions f(x).

This method works best when f(x) is well approximated by a polynomial on
the interval (a,b), and so is not suitable for functions with
singularities.  Since the user specifies ahead of time how many
quadrature nodes will be used, these routines do not accept absolute or
relative error bounds.  The table below lists the weighting functions
currently supported.

Name                 Interval                     Weighting function w(x)                         Constraints

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

Legendre             (a,b)                        1                                               b > a

Chebyshev Type 1     (a,b)                        1 / \sqrt{(b - x) (x - a)}                      b > a

Gegenbauer           (a,b)                        ((b - x) (x - a))^{\alpha}                      \alpha > -1, b > a

Jacobi               (a,b)                        (b - x)^{\alpha} (x - a)^{\beta}                \alpha,\beta > -1, b > a

Laguerre             (a,\infty)                   (x-a)^{\alpha} \exp{( -b (x - a) )}             \alpha > -1, b > 0

Hermite              (-\infty,\infty)             |x-a|^{\alpha} \exp{( -b (x-a)^2 )}             \alpha > -1, b > 0

Exponential          (a,b)                        |x - (a + b)/2|^{\alpha}                        \alpha > -1, b > a

Rational             (a,\infty)                   (x - a)^{\alpha} (x + b)^{\beta}                \alpha > -1, \alpha + \beta + 2n < 0, a + b > 0

Chebyshev Type 2     (a,b)                        \sqrt{(b - x) (x - a)}                          b > a

The fixed point quadrature routines use the following workspace to store
the nodes and weights, as well as additional variables for intermediate
calculations:

 -- Type: gsl_integration_fixed_workspace

     This workspace is used for fixed point quadrature rules and looks
     like this:

          typedef struct
          {
            size_t n;        /* number of nodes/weights */
            double *weights; /* quadrature weights */
            double *x;       /* quadrature nodes */
            double *diag;    /* diagonal of Jacobi matrix */
            double *subdiag; /* subdiagonal of Jacobi matrix */
            const gsl_integration_fixed_type * type;
          } gsl_integration_fixed_workspace;

 -- Function: gsl_integration_fixed_workspace *
          gsl_integration_fixed_alloc (const gsl_integration_fixed_type
          * T, const size_t n, const double a, const double b, const
          double alpha, const double beta)

     This function allocates a workspace for computing integrals with
     interpolating quadratures using ‘n’ quadrature nodes.  The
     parameters ‘a’, ‘b’, see alpha, and ‘beta’ specify the
     integration interval and/or weighting function for the various
     quadrature types.  See the table above for constraints
     on these parameters.  The size of the workspace is O(4n).

      -- Type: gsl_integration_fixed_type

          The type of quadrature used is specified by ‘T’ which can be
          set to the following choices:

           -- Variable: gsl_integration_fixed_legendre

               This specifies Legendre quadrature integration.  The
               parameters alpha and ‘beta’ are ignored for
               this type.

           -- Variable: gsl_integration_fixed_chebyshev

               This specifies Chebyshev type 1 quadrature integration.
               The parameters alpha and ‘beta’ are ignored
               for this type.

           -- Variable: gsl_integration_fixed_gegenbauer

               This specifies Gegenbauer quadrature integration.  The
               parameter ‘beta’ is ignored for this type.

           -- Variable: gsl_integration_fixed_jacobi

               This specifies Jacobi quadrature integration.

           -- Variable: gsl_integration_fixed_laguerre

               This specifies Laguerre quadrature integration.  The
               parameter ‘beta’ is ignored for this type.

           -- Variable: gsl_integration_fixed_hermite

               This specifies Hermite quadrature integration.  The
               parameter ‘beta’ is ignored for this type.

           -- Variable: gsl_integration_fixed_exponential

               This specifies exponential quadrature integration.  The
               parameter ‘beta’ is ignored for this type.

           -- Variable: gsl_integration_fixed_rational

               This specifies rational quadrature integration.

           -- Variable: gsl_integration_fixed_chebyshev2

               This specifies Chebyshev type 2 quadrature integration.
               The parameters alpha and ‘beta’ are ignored
               for this type.

 -- Function: void gsl_integration_fixed_free
          (gsl_integration_fixed_workspace * w)

     This function frees the memory assocated with the workspace ‘w’

 -- Function: size_t gsl_integration_fixed_n (const
          gsl_integration_fixed_workspace * w)

     This function returns the number of quadrature nodes and weights.

 -- Function: double * gsl_integration_fixed_nodes (const
          gsl_integration_fixed_workspace * w)

     This function returns a pointer to an array of size ‘n’ containing
     the quadrature nodes x_i.

 -- Function: double * gsl_integration_fixed_weights (const
          gsl_integration_fixed_workspace * w)

     This function returns a pointer to an array of size ‘n’ containing
     the quadrature weights w_i.

 -- Function: int gsl_integration_fixed (const gsl_function * func,
          double * result, const gsl_integration_fixed_workspace * w)

     This function integrates the function f(x) provided in ‘func’ using
     previously computed fixed quadrature rules.  The integral is
     approximated as

          \sum_{i=1}^n w_i f(x_i)

     where w_i are the quadrature weights and x_i are the quadrature
     nodes computed previously by gsl_integration_fixed_alloc().
     The sum is stored in ‘result’ on output.

17.15 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.

See GSL_EDOM.       error in the values of the input arguments

17.16 Examples
==============

17.16.1 Adaptive integration example
------------------------------------

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       = %zu\n", w->size);

       gsl_integration_workspace_free (w);

       return 0;
     }

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

     result          = -4.000000000000085265
     exact result    = -4.000000000000000000
     estimated error =  0.000000000000135447
     actual error    = -0.000000000000085265
     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.

17.16.2 Fixed-point quadrature example
--------------------------------------

In this example, we use a fixed-point quadrature rule to integrate the
integral

     \int_{-\infty}^{\infty} e^{-x^2} \left( x^m + 1 \right) dx = \left\{ \begin{array}{cc} \sqrt{\pi} + \Gamma{\left( \frac{m+1}{2} \right)}, & m \textrm{ even} \\ \sqrt{\pi}, & m \textrm{ odd} \end{array} \right.

for integer m.  Consulting our table of fixed point
quadratures, we see that this integral can be evaluated with a Hermite
quadrature rule, setting \alpha = 0, a = 0, b = 1.  Since we are
integrating a polynomial of degree m, we need to choose the number of
nodes n \ge (m+1)/2 to achieve the best results.

First we will try integrating for m = 10, n = 5, which does not satisfy
our criteria above:

     $ ./integration2 10 5

The output is,

     m               = 10
     intervals       = 5
     result          =  47.468529694563351029
     exact result    =  54.115231635459025483
     actual error    = -6.646701940895674454

So, we find a large error.  Now we try integrating for m = 10, n = 6
which does satisfy the criteria above:

     $ ./integration2 10 6

The output is,

     m               = 10
     intervals       = 6
     result          =  54.115231635459096537
     exact result    =  54.115231635459025483
     actual error    =  0.000000000000071054

The program is given below.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_integration.h>
     #include <gsl/gsl_sf_gamma.h>

     double
     f(double x, void * params)
     {
       int m = *(int *) params;
       double f = gsl_pow_int(x, m) + 1.0;
       return f;
     }

     int
     main (int argc, char *argv[])
     {
       gsl_integration_fixed_workspace * w;
       const gsl_integration_fixed_type * T = gsl_integration_fixed_hermite;
       int m = 10;
       int n = 6;
       double expected, result;
       gsl_function F;

       if (argc > 1)
         m = atoi(argv[1]);

       if (argc > 2)
         n = atoi(argv[2]);

       w = gsl_integration_fixed_alloc(T, n, 0.0, 1.0, 0.0, 0.0);

       F.function = &f;
       F.params = &m;

       gsl_integration_fixed(&F, &result, w);

       if (m % 2 == 0)
         expected = M_SQRTPI + gsl_sf_gamma(0.5*(1.0 + m));
       else
         expected = M_SQRTPI;

       printf ("m               = %d\n", m);
       printf ("intervals       = %zu\n", gsl_integration_fixed_n(w));
       printf ("result          = % .18f\n", result);
       printf ("exact result    = % .18f\n", expected);
       printf ("actual error    = % .18f\n", result - expected);

       gsl_integration_fixed_free (w);

       return 0;
     }

17.17 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.

The fixed-point quadrature routines are based on IQPACK, described in
the following papers:

   * S. Elhay, J. Kautsky, Algorithm 655: IQPACK, FORTRAN Subroutines
     for the Weights of Interpolatory Quadrature, ACM Transactions on
     Mathematical Software, Volume 13, Number 4, December 1987, pages
     399-415.

   * J. Kautsky, S. Elhay, Calculation of the Weights of Interpolatory
     Quadratures, Numerische Mathematik, Volume 40, Number 3, October
     1982, pages 407-422.

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’.

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”.

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.

 -- Type: gsl_rng_type
 -- Type: gsl_rng

     The random number generator library uses two special structs, see
     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’.

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,
     see gsl_rng_default_seed.  This is zero by default but can
     be changed either directly or by using the environment variable
     GSL_RNG_SEED.

     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 \geq 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’.

18.4 Sampling from a random number generator
============================================

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

 -- Function: unsigned long int gsl_rng_get (const gsl_rng * r)

     This function returns a random integer from the generator ‘r’.  The
     minimum and maximum values depend on the algorithm used, but all
     integers in the range [‘min’, ‘max’] are equally likely.  The
     values of ‘min’ and ‘max’ can be determined using the auxiliary
     functions gsl_rng_max() and gsl_rng_min().

 -- 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.

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)

     This function returns the largest value that gsl_rng_get()
     can return.

 -- Function: unsigned long int gsl_rng_min (const gsl_rng * r)

     This function 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);
            }

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.

 -- Macro: GSL_RNG_TYPE

     This environment variable specifies the default random number
     generator.  It should be the name of a generator, such as ‘taus’ or
     ‘mt19937’.

 -- Macro: GSL_RNG_SEED

     This environment variable specifies the default seed for the random
     number generator

 -- Variable: gsl_rng_default

     This global library variable specifies the default random number
     generator, and can be initialized from GSL_RNG_TYPE.
     using gsl_rng_env_setup().  It is defined as follows:

          extern const gsl_rng_type *gsl_rng_default

 -- Variable: gsl_rng_default_seed

     This global library variable specifies the seed for the default
     random number generator, and can be initialized from
     GSL_RNG_SEED using gsl_rng_env_setup().  It is set
     to zero by default and is defined as follows:

          extern unsigned long int gsl_rng_default_seed

 -- 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.

     The value of GSL_RNG_SEED 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,

     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

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’.

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.

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.

 -- Variable: 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’.

 -- Variable: gsl_rng_ranlxs0
 -- Variable: gsl_rng_ranlxs1
 -- Variable: gsl_rng_ranlxs2

     The generator ‘ranlxs0’ is a second-generation version of the
     RANLUX algorithm of Luscher, 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}.

 -- Variable: gsl_rng_ranlxd1
 -- Variable: 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.

 -- Variable: gsl_rng_ranlux
 -- Variable: gsl_rng_ranlux389

     The ‘ranlux’ generator is an implementation of the original
     algorithm developed by Luscher.  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 Luscher 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. Luscher, “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 Luscher”,
          Computer Physics Communications, 79 (1994) 111–114

 -- Variable: 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 \hbox{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.

 -- Variable: 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).

 -- Variable: gsl_rng_taus
 -- Variable: 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 \oplus 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.

 -- Variable: 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.

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.

 -- Variable: 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.

 -- Variable: gsl_rng_random_bsd
 -- Variable: gsl_rng_random_libc5
 -- Variable: 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.

 -- Variable: 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).

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.

 -- Variable: 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}.

 -- Variable: 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.

 -- Variable: 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 \oplus 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)

 -- Variable: 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.

 -- Variable: 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.

 -- Variable: 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.

 -- Variable: 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.

 -- Variable: 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.

 -- Variable: gsl_rng_uni
 -- Variable: 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.

 -- Variable: gsl_rng_slatec

     This is the SLATEC random number generator RAND. It is ancient.
     The original source code is available from NETLIB.

 -- Variable: 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).

 -- Variable: 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.

 -- Variable: gsl_rng_knuthran2002
 -- Variable: 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.

 -- Variable: gsl_rng_borosh13
 -- Variable: gsl_rng_fishman18
 -- Variable: gsl_rng_fishman20
 -- Variable: gsl_rng_lecuyer21
 -- Variable: 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}.

 -- Variable: 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.

 -- Variable: 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.

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

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,

     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

     0.33050
     0.86631
     0.32982
     0.67620
     0.53391
     0.06457
     0.16847
     0.70229
     0.04371
     0.86374

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/’

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
Luscher for providing notes and source code for the RANLXS and RANLXD
generators.

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’.

19.1 Quasi-random number generator initialization
=================================================

 -- Type: gsl_qrng

     This is a workspace for computing quasi-random sequences.

 -- 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.

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.

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);

19.4 Saving and restoring 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’.

19.5 Quasi-random number generator algorithms
=============================================

The following quasi-random sequence algorithms are available,

 -- Type: gsl_qrng_type

      -- Variable: 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.

      -- Variable: 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.

      -- Variable: gsl_qrng_halton
      -- Variable: 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.

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.

Fig.  %s shows the distribution in the x-y plane of the first 1024
points from the Sobol sequence,

Figure: Distribution of the first 1024 points from the quasi-random
Sobol sequence

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.

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.

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(x) 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 \le P(x) \le 1, 0 \le Q(x) \le 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 \le 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).

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.

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 :data:x,
          double :data: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.

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.

20.5 The Multivariate Gaussian Distribution
===========================================

 -- Function: int gsl_ran_multivariate_gaussian (const gsl_rng * r,
          const gsl_vector * mu, const gsl_matrix * L, gsl_vector
          * result)

     This function generates a random vector satisfying the
     k-dimensional multivariate Gaussian distribution with mean \mu and
     variance-covariance matrix \Sigma.  On input, the k-vector \mu is
     given in ‘mu’, and the Cholesky factor of the k-by-k matrix \Sigma
     = L L^T is given in the lower triangle of ‘L’, as output from
     gsl_linalg_cholesky_decomp().  The random vector is stored in
     ‘result’ on output.  The probability distribution for multivariate
     Gaussian random variates is

          p(x_1,...,x_k) dx_1 ... dx_k = 1 / ( \sqrt{(2 \pi)^k |\Sigma| ) \exp (-1/2 (x - \mu)^T \Sigma^{-1} (x - \mu)) dx_1 ... dx_k

 -- Function: int gsl_ran_multivariate_gaussian_pdf (const gsl_vector
          * x, const gsl_vector * mu, const gsl_matrix * L, double
          * result, gsl_vector * work)
 -- Function: int gsl_ran_multivariate_gaussian_log_pdf (const
          gsl_vector * x, const gsl_vector * mu, const gsl_matrix * L,
          double * result, gsl_vector * work)

     These functions compute p(x) or \log{p(x)} at the point ‘x’, using
     mean vector ‘mu’ and variance-covariance matrix specified by its
     Cholesky factor ‘L’ using the formula above.  Additional workspace
     of length k is required in ‘work’.

 -- Function: int gsl_ran_multivariate_gaussian_mean (const gsl_matrix
          * X, gsl_vector * mu_hat)

     Given a set of n samples X_j from a k-dimensional multivariate
     Gaussian distribution, this function computes the maximum
     likelihood estimate of the mean of the distribution, given by

          \Hat{\mu} = {1 \over n} \sum_{j=1}^n X_j

     The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix ‘X’,
     and the maximum likelihood estimate of the mean is stored in
     ‘mu_hat’ on output.

 -- Function: int gsl_ran_multivariate_gaussian_vcov (const gsl_matrix
          * X, gsl_matrix * sigma_hat)

     Given a set of n samples X_j from a k-dimensional multivariate
     Gaussian distribution, this function computes the maximum
     likelihood estimate of the variance-covariance matrix of the
     distribution, given by

          \Hat{\Sigma} = (1 / n) \sum_{j=1}^n ( X_j - \Hat{\mu} ) ( X_j - \Hat{\mu} )^T

     The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix ‘X’
     and the maximum likelihood estimate of the variance-covariance
     matrix is stored in ‘sigma_hat’ on output.

20.6 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 \ge 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’.

20.7 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’.

20.8 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 \ge 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’.

20.9 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’.

20.10 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’.

20.11 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.

20.12 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.

20.13 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 / (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 \le 2.

20.14 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 / (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 \le 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).

20.15 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’.

20.16 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 \le 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’.

20.17 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’.

20.18 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 \ge 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.

20.19 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 \ge 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.

20.20 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.

20.21 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 \le x \le 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’.

20.22 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’.

20.23 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 \ge 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’.

20.24 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,\ldots,x_n) in ‘n’ dimensions.  The vector is normalized
     such that |v|^2 = x_1^2 + x_2^2 + \cdots + 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).

20.25 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 \ge 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’.

20.26 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’.

20.27 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’.

20.28 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 \ge 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) \over \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, \ldots ,
     \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, \ldots , \theta_K) for a Dirichlet distribution with
     parameters ‘alpha[K]’.

20.29 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] \le 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.

 -- Type: gsl_ran_discrete_t

     This structure contains the lookup table for the discrete random
     number generator.

 -- 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’.

20.30 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 \ge 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’.

20.31 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.

20.32 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 \le k \le 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’.

20.33 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, \ldots, n_K) are nonnegative integers with
     \sum_{k=1}^{K} n_k = N, and (p_1, p_2, \ldots, 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, \ldots, 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, \ldots, n_K) with parameters
     ‘p[K]’.

20.34 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’.

20.35 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 \ge 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’.

20.36 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 \ge 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’.

20.37 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 \leq n_1 + n_2.  The domain of k
     is \max(0, t - n_2), \ldots, \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’.

20.38 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 \ge 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.

20.39 The Wishart Distribution
==============================

 -- Function: int gsl_ran_wishart (const gsl_rng * r, const double n,
          const gsl_matrix * L, gsl_matrix * result, gsl_matrix * work)

     This function computes a random symmetric p-by-p matrix from the
     Wishart distribution.  The probability distribution for Wishart
     random variates is,

          p(X) = \frac{|X|^{(n-p-1)/2} e^{-tr( V^{-1} X)/2}}{2^{(np)/2} |V|^{n/2} \Gamma_p(n/2)}

     Here, n > p - 1 is the number of degrees of freedom, V is a
     symmetric positive definite p-by-p scale matrix, whose Cholesky
     factor is specified by ‘L’, and ‘work’ is p-by-p workspace.  The
     p-by-p Wishart distributed matrix X is stored in ‘result’ on
     output.

 -- Function: int gsl_ran_wishart_pdf (const gsl_matrix * X, const
          gsl_matrix * L_X, const double n, const gsl_matrix * L, double
          * result, gsl_matrix * work)
 -- Function: int gsl_ran_wishart_log_pdf (const gsl_matrix * X, const
          gsl_matrix * L_X, const double n, const gsl_matrix * L, double
          * result, gsl_matrix * work)

     These functions compute p(X) or \log{p(X)} for the p-by-p matrix
     ‘X’, whose Cholesky factor is specified in ‘L_X’.  The degrees of
     freedom is given by ‘n’, the Cholesky factor of the scale matrix V
     is specified in ‘L’, and ‘work’ is p-by-p workspace.  The probably
     density value is returned in ‘result’.

20.40 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’.  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.

20.41 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,

      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

giving output

      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;
     }

Fig.  %s shows the output from the program.

Figure: 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

20.42 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).

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.

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 \over 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.

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).

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.

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 = 2}^{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’.

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.

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.

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’.

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 \ge 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 \le 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 \ge 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 \ge 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.

21.9 Median and Percentiles
===========================

The median and percentile functions described in this section operate on
sorted data in O(1) time.  There is also a routine for computing the
median of an unsorted input array in average O(n) time using the
quickselect algorithm.  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[], const size_t stride, const 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_median (double data[], const
          size_t stride, const size_t n)

     This function returns the median value of ‘data’, a dataset of
     length ‘n’ with stride ‘stride’.  The median is found using the
     quickselect algorithm.  The input array does not need to be sorted,
     but note that the algorithm rearranges the array and so the input
     is not preserved on output.

 -- 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.

21.10 Order Statistics
======================

The k-th `order statistic' of a sample is equal to its k-th smallest
value.  The k-th order statistic of a set of n values x = \left\{ x_i
\right\}, 1 \le i \le n is denoted x_{(k)}.  The median of the set x is
equal to x_{\left( \frac{n}{2} \right)} if n is odd, or the average of
x_{\left( \frac{n}{2} \right)} and x_{\left( \frac{n}{2} + 1 \right)} if
n is even.  The k-th smallest element of a length n vector can be found
in average O(n) time using the quickselect algorithm.

 -- Function: gsl_stats_select (double data[], const size_t stride,
          const size_t n, const size_t k)

     This function finds the ‘k’-th smallest element of the input array
     ‘data’ of length ‘n’ and stride ‘stride’ using the quickselect
     method.  The algorithm rearranges the elements of ‘data’ and so the
     input array is not preserved on output.

21.11 Robust Location Estimates
===============================

A `location estimate' refers to a typical or central value which best
describes a given dataset.  The mean and median are both examples of
location estimators.  However, the mean has a severe sensitivity to data
outliers and can give erroneous values when even a small number of
outliers are present.  The median on the other hand, has a strong
insensitivity to data outliers, but due to its non-smoothness it can
behave unexpectedly in certain situations.  GSL offers the following
alternative location estimators, which are robust to the presence of
outliers.

21.11.1 Trimmed Mean
--------------------

The trimmed mean, or `truncated mean', discards a certain number of
smallest and largest samples from the input vector before computing the
mean of the remaining samples.  The amount of trimming is specified by a
factor \alpha \in [0,0.5].  Then the number of samples discarded from
both ends of the input vector is \left\lfloor \alpha n \right\rfloor,
where n is the length of the input.  So to discard 25% of the samples
from each end, one would set \alpha = 0.25.

 -- Function: double gsl_stats_trmean_from_sorted_data (const
          double alpha, const double sorted_data[], const size_t stride,
          const size_t n)

     This function returns the trimmed mean 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.  The trimming factor \alpha is given
     in alpha.  If \alpha \ge 0.5, then the median of the
     input is returned.

21.11.2 Gastwirth Estimator
---------------------------

Gastwirth’s location estimator is a weighted sum of three order
statistics,

     gastwirth = 0.3 * Q_{1/3} + 0.4 * Q_{1/2} + 0.3 * Q_{2/3}

where Q_{\frac{1}{3}} is the one-third quantile, Q_{\frac{1}{2}} is the
one-half quantile (i.e.  median), and Q_{\frac{2}{3}} is the two-thirds
quantile.

 -- Function: double gsl_stats_gastwirth_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n)

     This function returns the Gastwirth location estimator 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.

21.12 Robust Scale Estimates
============================

A `robust scale estimate', also known as a robust measure of scale,
attempts to quantify the statistical dispersion (variability, scatter,
spread) in a set of data which may contain outliers.  In such datasets,
the usual variance or standard deviation scale estimate can be rendered
useless by even a single outlier.

21.12.1 Median Absolute Deviation (MAD)
---------------------------------------

The median absolute deviation (MAD) is defined as

     MAD = 1.4826 median { | x_i - median(x) | }

In words, first the median of all samples is computed.  Then the median
is subtracted from all samples in the input to find the deviation of
each sample from the median.  The median of all absolute deviations is
then the MAD. The factor 1.4826 makes the MAD an unbiased estimator of
the standard deviation for Gaussian data.  The median absolute deviation
has an asymptotic efficiency of 37%.

 -- Function: double gsl_stats_mad0 (const double data[], const
          size_t stride, const size_t n, double work[])

 -- Function: double gsl_stats_mad (const double data[], const
          size_t stride, const size_t n, double work[])

     These functions return the median absolute deviation of ‘data’, a
     dataset of length ‘n’ and stride ‘stride’.  The ‘mad0’ function
     calculates \textrm{median} \left\{ \left| x_i - \textrm{median}
     \left( x \right) \right| \right\} (i.e.  the MAD statistic without
     the bias correction scale factor).  These functions require
     additional workspace of size ‘n’ provided in ‘work’.

21.12.2 S_n Statistic
---------------------

The S_n statistic developed by Croux and Rousseeuw is defined as

     S_n = 1.1926 * c_n * median_i { median_j ( | x_i - x_j | ) }

For each sample x_i, 1 \le i \le n, the median of the values \left| x_i
- x_j \right| is computed for all x_j, 1 \le j \le n.  This yields n
values, whose median then gives the final S_n.  The factor 1.1926 makes
S_n an unbiased estimate of the standard deviation for Gaussian data.
The factor c_n is a correction factor to correct bias in small sample
sizes.  S_n has an asymptotic efficiency of 58%.

 -- Function: double gsl_stats_Sn0_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n,
          double work[])

 -- Function: double gsl_stats_Sn_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n,
          double work[])

     These functions return the S_n statistic 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.  The ‘Sn0’ function calculates
     \textrm{median}_i \left\{ \textrm{median}_j \left( \left| x_i - x_j
     \right| \right) \right\} (i.e.  the S_n statistic without the bias
     correction scale factors).  These functions require additional
     workspace of size ‘n’ provided in ‘work’.

21.12.3 Q_n Statistic
---------------------

The Q_n statistic developed by Croux and Rousseeuw is defined as

     Q_n = 2.21914 * d_n * { | x_i - x_j |, i < j }_{(k)}

The factor 2.21914 makes Q_n an unbiased estimate of the standard
deviation for Gaussian data.  The factor d_n is a correction factor to
correct bias in small sample sizes.  The order statistic is

     k = ( floor(n/2) + 1 )
         (       2        )

Q_n has an asymptotic efficiency of 82%.

 -- Function: double gsl_stats_Qn0_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n,
          double work[], int work_int[])
 -- Function: double gsl_stats_Qn_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n,
          double work[], int work_int[])

     These functions return the Q_n statistic 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.  The ‘Qn0’ function calculates
     \left\{ \left| x_i - x_j \right|, i < j \right\}_{(k)} (i.e.  Q_n
     without the bias correction scale factors).  These functions
     require additional workspace of size ‘3n’ provided in ‘work’ and
     integer workspace of size ‘5n’ provided in ‘work_int’.

21.13 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

21.14 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/’.

The following papers describe robust scale estimation,

   * C. Croux and P. J. Rousseeuw, `Time-Efficient algorithms for two
     highly robust estimators of scale', Comp.  Stat., Physica,
     Heidelberg, 1992.

   * P. J. Rousseeuw and C. Croux, `Explicit scale estimators with high
     breakdown point', L1-Statistical Analysis and Related Methods, pp.
     77-92, 1992.

22 Running Statistics
*********************

This chapter describes routines for computing running statistics, also
known as online statistics, of data.  These routines are suitable for
handling large datasets for which it may be inconvenient or impractical
to store in memory all at once.  The data can be processed in a single
pass, one point at a time.  Each time a data point is added to the
accumulator, internal parameters are updated in order to compute the
current mean, variance, standard deviation, skewness, and kurtosis.
These statistics are exact, and are updated with numerically stable
single-pass algorithms.  The median and arbitrary quantiles are also
available, however these calculations use algorithms which provide
approximations, and grow more accurate as more data is added to the
accumulator.

The functions described in this chapter are declared in the header file
‘gsl_rstat.h’.

22.1 Initializing the Accumulator
=================================

 -- Type: gsl_rstat_workspace

     This workspace contains parameters used to calculate various
     statistics and are updated after each data point is added to the
     accumulator.

 -- Function: gsl_rstat_workspace * gsl_rstat_alloc (void)

     This function allocates a workspace for computing running
     statistics.  The size of the workspace is O(1).

 -- Function: void gsl_rstat_free (gsl_rstat_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_rstat_reset (gsl_rstat_workspace * w)

     This function resets the workspace ‘w’ to its initial state, so it
     can begin working on a new set of data.

22.2 Adding Data to the Accumulator
===================================

 -- Function: int gsl_rstat_add (const double x, gsl_rstat_workspace
          * w)

     This function adds the data point ‘x’ to the statistical
     accumulator, updating calculations of the mean, variance, standard
     deviation, skewness, kurtosis, and median.

 -- Function: size_t gsl_rstat_n (const gsl_rstat_workspace * w)

     This function returns the number of data so far added to the
     accumulator.

22.3 Current Statistics
=======================

 -- Function: double gsl_rstat_min (const gsl_rstat_workspace * w)

     This function returns the minimum value added to the accumulator.

 -- Function: double gsl_rstat_max (const gsl_rstat_workspace * w)

     This function returns the maximum value added to the accumulator.

 -- Function: double gsl_rstat_mean (const gsl_rstat_workspace * w)

     This function returns the mean of all data added to the
     accumulator, defined as

          \Hat\mu = (1/N) \sum x_i

 -- Function: double gsl_rstat_variance (const gsl_rstat_workspace * w)

     This function returns the variance of all data added to the
     accumulator, defined as

          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2

 -- Function: double gsl_rstat_sd (const gsl_rstat_workspace * w)

     This function returns the standard deviation of all data added to
     the accumulator, defined as the square root of the variance given
     above.

 -- Function: double gsl_rstat_sd_mean (const gsl_rstat_workspace * w)

     This function returns the standard deviation of the mean, defined
     as

          sd_mean = \Hat\sigma / \sqrt{N}

 -- Function: double gsl_rstat_rms (const gsl_rstat_workspace * w)

     This function returns the root mean square of all data added to the
     accumulator, defined as

          rms = \sqrt{{1 \over N} \sum x_i^2}

 -- Function: double gsl_rstat_skew (const gsl_rstat_workspace * w)

     This function returns the skewness of all data added to the
     accumulator, defined as

          skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3

 -- Function: double gsl_rstat_kurtosis (const gsl_rstat_workspace * w)

     This function returns the kurtosis of all data added to the
     accumulator, defined as

          kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4)  - 3

 -- Function: double gsl_rstat_median (gsl_rstat_workspace * w)

     This function returns an estimate of the median of the data added
     to the accumulator.

22.4 Quantiles
==============

The functions in this section estimate quantiles dynamically without
storing the entire dataset, using the algorithm of Jain and Chlamtec,
1985.  Only five points (markers) are stored which represent the minimum
and maximum of the data, as well as current estimates of the p/2-, p-,
and (1+p)/2-quantiles.  Each time a new data point is added, the marker
positions and heights are updated.

 -- Type: gsl_rstat_quantile_workspace

     This workspace contains parameters for estimating quantiles of the
     current dataset

 -- Function: gsl_rstat_quantile_workspace * gsl_rstat_quantile_alloc
          (const double p)

     This function allocates a workspace for the dynamic estimation of
     ‘p’-quantiles, where ‘p’ is between 0 and 1.  The median
     corresponds to p = 0.5.  The size of the workspace is O(1).

 -- Function: void gsl_rstat_quantile_free (gsl_rstat_quantile_workspace
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_rstat_quantile_reset (gsl_rstat_quantile_workspace
          * w)

     This function resets the workspace ‘w’ to its initial state, so it
     can begin working on a new set of data.

 -- Function: int gsl_rstat_quantile_add (const double x,
          gsl_rstat_quantile_workspace * w)

     This function updates the estimate of the p-quantile with the new
     data point ‘x’.

 -- Function: double gsl_rstat_quantile_get
          (gsl_rstat_quantile_workspace * w)

     This function returns the current estimate of the p-quantile.

22.5 Examples
=============

Here is a basic example of how to use the statistical functions:

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

     int
     main(void)
     {
       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
       double mean, variance, largest, smallest, sd,
              rms, sd_mean, median, skew, kurtosis;
       gsl_rstat_workspace *rstat_p = gsl_rstat_alloc();
       size_t i, n;

       /* add data to rstat accumulator */
       for (i = 0; i < 5; ++i)
         gsl_rstat_add(data[i], rstat_p);

       mean     = gsl_rstat_mean(rstat_p);
       variance = gsl_rstat_variance(rstat_p);
       largest  = gsl_rstat_max(rstat_p);
       smallest = gsl_rstat_min(rstat_p);
       median   = gsl_rstat_median(rstat_p);
       sd       = gsl_rstat_sd(rstat_p);
       sd_mean  = gsl_rstat_sd_mean(rstat_p);
       skew     = gsl_rstat_skew(rstat_p);
       rms      = gsl_rstat_rms(rstat_p);
       kurtosis = gsl_rstat_kurtosis(rstat_p);
       n        = gsl_rstat_n(rstat_p);

       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);
       printf( "The median is %g\n", median);
       printf( "The standard deviation is %g\n", sd);
       printf( "The root mean square is %g\n", rms);
       printf( "The standard devation of the mean is %g\n", sd_mean);
       printf( "The skew is %g\n", skew);
       printf( "The kurtosis %g\n", kurtosis);
       printf( "There are %zu items in the accumulator\n", n);

       gsl_rstat_reset(rstat_p);
       n = gsl_rstat_n(rstat_p);
       printf( "There are %zu items in the accumulator\n", n);

       gsl_rstat_free(rstat_p);

       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
     The median is 16.5
     The standard deviation is 2.31797
     The root mean square is 16.6694
     The standard devation of the mean is 1.03663
     The skew is -0.829058
     The kurtosis -1.2217
     There are 5 items in the accumulator
     There are 0 items in the accumulator

This next program estimates the lower quartile, median and upper
quartile from 10,000 samples of a random Rayleigh distribution, using
the P^2 algorithm of Jain and Chlamtec.  For comparison, the exact
values are also computed from the sorted dataset.

     #include <stdio.h>
     #include <stdlib.h>
     #include <gsl/gsl_rstat.h>
     #include <gsl/gsl_statistics.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_sort.h>

     int
     main(void)
     {
       const size_t N = 10000;
       double *data = malloc(N * sizeof(double));
       gsl_rstat_quantile_workspace *work_25 = gsl_rstat_quantile_alloc(0.25);
       gsl_rstat_quantile_workspace *work_50 = gsl_rstat_quantile_alloc(0.5);
       gsl_rstat_quantile_workspace *work_75 = gsl_rstat_quantile_alloc(0.75);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       double exact_p25, exact_p50, exact_p75;
       double val_p25, val_p50, val_p75;
       size_t i;

       /* add data to quantile accumulators; also store data for exact
        * comparisons */
       for (i = 0; i < N; ++i)
         {
           data[i] = gsl_ran_rayleigh(r, 1.0);
           gsl_rstat_quantile_add(data[i], work_25);
           gsl_rstat_quantile_add(data[i], work_50);
           gsl_rstat_quantile_add(data[i], work_75);
         }

       /* exact values */
       gsl_sort(data, 1, N);
       exact_p25 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.25);
       exact_p50 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.5);
       exact_p75 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.75);

       /* estimated values */
       val_p25 = gsl_rstat_quantile_get(work_25);
       val_p50 = gsl_rstat_quantile_get(work_50);
       val_p75 = gsl_rstat_quantile_get(work_75);

       printf ("The dataset is %g, %g, %g, %g, %g, ...\n",
              data[0], data[1], data[2], data[3], data[4]);

       printf ("0.25 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
               exact_p25, val_p25, (val_p25 - exact_p25) / exact_p25);
       printf ("0.50 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
               exact_p50, val_p50, (val_p50 - exact_p50) / exact_p50);
       printf ("0.75 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
               exact_p75, val_p75, (val_p75 - exact_p75) / exact_p75);

       gsl_rstat_quantile_free(work_25);
       gsl_rstat_quantile_free(work_50);
       gsl_rstat_quantile_free(work_75);
       gsl_rng_free(r);
       free(data);

       return 0;
     }

The program should produce the following output,

     The dataset is 0.00645272, 0.0074002, 0.0120706, 0.0207256, 0.0227282, ...
     0.25 quartile: exact = 0.75766, estimated = 0.75580, error = -2.450209e-03
     0.50 quartile: exact = 1.17508, estimated = 1.17438, error = -5.995912e-04
     0.75 quartile: exact = 1.65347, estimated = 1.65696, error = 2.110571e-03

22.6 References and Further Reading
===================================

The algorithm used to dynamically estimate p-quantiles is described in
the paper,

   * R. Jain and I. Chlamtac.  `The P^2 algorithm for dynamic
     calculation of quantiles and histograms without storing
     observations', Communications of the ACM, Volume 28 (October),
     Number 10, 1985, p.  1076-1085.

23 Moving Window Statistics
***************************

This chapter describes routines for computing `moving window statistics'
(also called rolling statistics and running statistics), using a window
around a sample which is used to calculate various local statistical
properties of an input data stream.  The window is then slid forward by
one sample to process the next data point and so on.

The functions described in this chapter are declared in the header file
‘gsl_movstat.h’.

23.1 Introduction
=================

This chapter is concerned with calculating various statistics from
subsets of a given dataset.  The main idea is to compute statistics in
the vicinity of a given data sample by defining a `window' which
includes the sample itself as well as some specified number of samples
before and after the sample in question.  For a sample x_i, we define a
window W_i^{H,J} as

     W_i^{H,J} = {x_{i-H},...,x_i,...,x_{i+J}}

The parameters H and J are non-negative integers specifying the number
of samples to include before and after the sample x_i.  Statistics such
as the mean and standard deviation of the window W_i^{H,J} may be
computed, and then the window is shifted forward by one sample to focus
on x_{i+1}.  The total number of samples in the window is K = H + J + 1.
To define a symmetric window centered on x_i, one would set H = J =
\left\lfloor K / 2 \right\rfloor.

23.2 Handling Endpoints
=======================

When processing samples near the ends of the input signal, there will
not be enough samples to fill the window W_i^{H,J} defined above.
Therefore the user must specify how to construct the windows near the
end points.  This is done by passing an input argument of type
gsl_movstat_end_t.:

 -- Type: gsl_movstat_end_t

     This data type specifies how to construct windows near end points
     and can be selected from the following choices:

      -- Macro: GSL_MOVSTAT_END_PADZERO

          With this option, a full window of length K will be
          constructed by inserting zeros into the window near the signal
          end points.  Effectively, the input signal is modified to

               x~ = {0, ..., 0, x_1, x_2, ..., x_{n-1}, x_n, 0, ..., 0}

          to ensure a well-defined window for all x_i.

      -- Macro: GSL_MOVSTAT_END_PADVALUE

          With this option, a full window of length K will be
          constructed by padding the window with the first and last
          sample in the input signal.  Effectively, the input signal is
          modified to

               x~ = {x_1, ..., x_1, x_1, x_2, ..., x_{n-1}, x_n, x_n, ..., x_n}

      -- Macro: GSL_MOVSTAT_END_TRUNCATE

          With this option, no padding is performed, and the windows are
          simply truncated as the end points are approached.

23.3 Allocation for Moving Window Statistics
============================================

 -- Type: gsl_movstat_workspace

     The moving window statistical routines use a common workspace.

 -- Function: gsl_movstat_workspace * gsl_movstat_alloc (const size_t K)

     This function allocates a workspace for computing symmetric,
     centered moving statistics with a window length of K samples.  In
     this case, H = J = \left\lfloor K/2 \right\rfloor.  The size of the
     workspace is O(7K).

 -- Function: gsl_movstat_workspace * gsl_movstat_alloc2 (const
          size_t H, const size_t J)

     This function allocates a workspace for computing moving statistics
     using a window with H samples prior to the current sample, and J
     samples after the current sample.  The total window size is K = H +
     J + 1.  The size of the workspace is O(7K).

 -- Function: void * gsl_movstat_free (gsl_movstat_workspace * w)

     This function frees the memory associated with ‘w’.

23.4 Moving Mean
================

The moving window mean calculates the mean of the values of each window
W_i^{H,J}.

     \hat{\mu}_i = 1/| W_i^{H,J} | \sum_{x_m \in W_i^{H,J}} x_m

Here, \left| W_i^{H,J} \right| represents the number of elements in the
window W_i^{H,J}.  This will normally be K, unless the
GSL_MOVSTAT_END_TRUNCATE option is selected, in which case it
could be less than K near the signal end points.

 -- Function: int gsl_movstat_mean (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving window mean of the input vector
     ‘x’, storing the output in ‘y’.  The parameter ‘endtype’ specifies
     how windows near the ends of the input should be handled.  It is
     allowed to have ‘x’ = ‘y’ for an in-place moving mean.

23.5 Moving Variance and Standard Deviation
===========================================

The moving window variance calculates the `sample variance' of the
values of each window W_i^{H,J}, defined by

     \hat{\sigma}_i^2 = 1/(|W_i^{H,J}| - 1) \sum_{x_m \in W_i^{H,J}} ( x_m - \hat{\mu}_i )^2

where \hat{\mu}_i is the mean of W_i^{H,J} defined above.  The standard
deviation \hat{\sigma}_i is the square root of the variance.

 -- Function: int gsl_movstat_variance (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving window variance of the input
     vector ‘x’, storing the output in ‘y’.  The parameter ‘endtype’
     specifies how windows near the ends of the input should be handled.
     It is allowed to have ‘x’ = ‘y’ for an in-place moving variance.

 -- Function: int gsl_movstat_sd (const gsl_movstat_end_t endtype, const
          gsl_vector * x, gsl_vector * y, gsl_movstat_workspace * w)

     This function computes the moving window standard deviation of the
     input vector ‘x’, storing the output in ‘y’.  The parameter
     ‘endtype’ specifies how windows near the ends of the input should
     be handled.  It is allowed to have ‘x’ = ‘y’ for an in-place moving
     standard deviation.

23.6 Moving Minimum and Maximum
===============================

The moving minimum/maximum calculates the minimum and maximum values of
each window W_i^{H,J}.

     y_i^{min} = \min W_i^{H,J}
     y_i^{max} = \max W_i^{H,J}

 -- Function: int gsl_movstat_min (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving minimum of the input vector ‘x’,
     storing the result in ‘y’.  The parameter ‘endtype’ specifies how
     windows near the ends of the input should be handled.  It is
     allowed to have ‘x’ = ‘y’ for an in-place moving minimum.

 -- Function: int gsl_movstat_max (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving maximum of the input vector ‘x’,
     storing the result in ‘y’.  The parameter ‘endtype’ specifies how
     windows near the ends of the input should be handled.  It is
     allowed to have ‘x’ = ‘y’ for an in-place moving maximum.

 -- Function: int gsl_movstat_minmax (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y_min, gsl_vector * y_max,
          gsl_movstat_workspace * w)

     This function computes the moving minimum and maximum of the input
     vector ‘x’, storing the window minimums in ‘y_min’ and the window
     maximums in ‘y_max’.  The parameter ‘endtype’ specifies how windows
     near the ends of the input should be handled.

23.7 Moving Sum
===============

The moving window sum calculates the sum of the values of each window
W_i^{H,J}.

     y_i = \sum_{x_m \in W_i^{H,J}} x_m

 -- Function: int gsl_movstat_sum (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving window sum of the input vector
     ‘x’, storing the output in ‘y’.  The parameter ‘endtype’ specifies
     how windows near the ends of the input should be handled.  It is
     allowed to have ‘x’ = ‘y’ for an in-place moving sum.

23.8 Moving Median
==================

The moving median calculates the median of the window W_i^{H,J} for each
sample x_i:

     y_i = median(W_i^{H,J})

 -- Function: int gsl_movstat_median (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving median of the input vector ‘x’,
     storing the output in ‘y’.  The parameter ‘endtype’ specifies how
     windows near the ends of the input should be handled.  It is
     allowed for ‘x’ = ‘y’ for an in-place moving window median.

23.9 Robust Scale Estimation
============================

A common problem in statistics is to quantify the dispersion (also known
as the variability, scatter, and spread) of a set of data.  Often this
is done by calculating the variance or standard deviation.  However
these statistics are strongly influenced by outliers, and can often
provide erroneous results when even a small number of outliers are
present.

Several useful statistics have emerged to provide robust estimates of
scale which are not as susceptible to data outliers.  A few of these
statistical scale estimators are described below.

23.9.1 Moving MAD
-----------------

The median absolute deviation (MAD) for the window W_i^{H,J} is defined
to be the median of the absolute deviations from the window’s median:

     MAD_i = 1.4826 * median[ |W_i^{H,J} - median(W_i^{H,J})| ]

The factor of 1.4826 makes the MAD an unbiased estimator of the standard
deviation for Gaussian data.  The MAD has an efficiency of 37%.  See
here for more information.

 -- Function: int gsl_movstat_mad0 (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * xmedian, gsl_vector * xmad,
          gsl_movstat_workspace * w)
 -- Function: int gsl_movstat_mad (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * xmedian, gsl_vector * xmad,
          gsl_movstat_workspace * w)

     These functions compute the moving MAD of the input vector ‘x’ and
     store the result in ‘xmad’.  The medians of each window W_i^{H,J}
     are stored in ‘xmedian’ on output.  The inputs ‘x’, ‘xmedian’, and
     ‘xmad’ must all be the same length.  The parameter ‘endtype’
     specifies how windows near the ends of the input should be handled.
     The function ‘mad0’ does not include the scale factor of 1.4826,
     while the function ‘mad’ does include this factor.

23.9.2 Moving QQR
-----------------

The q-quantile range (QQR) is the difference between the (1-q) and q
quantiles of a set of data,

     QQR = Q_{1-q} - Q_q

The case q = 0.25 corresponds to the well-known `interquartile range
(IQR)', which is the difference between the 75th and 25th percentiles of
a set of data.  The QQR is a `trimmed estimator', the main idea being to
discard the largest and smallest values in a data window and compute a
scale estimate from the remaining middle values.  In the case of the
IQR, the largest and smallest 25% of the data are discarded and the
scale is estimated from the remaining (middle) 50%.

 -- Function: int gsl_movstat_qqr (const gsl_movstat_end_t endtype,
          const gsl_vector * x, const double q, gsl_vector * xqqr,
          gsl_movstat_workspace * w)

     This function computes the moving QQR of the input vector ‘x’ and
     stores the q-quantile ranges of each window W_i^{H,J} in ‘xqqr’.
     The quantile parameter ‘q’ must be between 0 and 0.5.  The input q
     = 0.25 corresponds to the IQR. The inputs ‘x’ and ‘xqqr’ must be
     the same length.  The parameter ‘endtype’ specifies how windows
     near the ends of the input should be handled.

23.9.3 Moving S_n
-----------------

The S_n statistic proposed by Croux and Rousseeuw is based on pairwise
differences between all samples in the window.  It has an efficiency of
58%, significantly higher than the MAD. See here for more
information.

 -- Function: int gsl_movstat_Sn (const gsl_movstat_end_t endtype, const
          gsl_vector * x, gsl_vector * xscale, gsl_movstat_workspace
          * w)

     This function computes the moving S_n of the input vector ‘x’ and
     stores the output in ‘xscale’.  The inputs ‘x’ and ‘xscale’ must be
     the same length.  The parameter ‘endtype’ specifies how windows
     near the ends of the input should be handled.  It is allowed for
     ‘x’ = ‘xscale’ for an in-place moving window S_n.

23.9.4 Moving Q_n
-----------------

The Q_n statistic proposed by Croux and Rousseeuw is loosely based on
the Hodges-Lehmann location estimator.  It has a relatively high
efficiency of 82%.  See here for more information.

 -- Function: int gsl_movstat_Qn (const gsl_movstat_end_t endtype, const
          gsl_vector * x, gsl_vector * xscale, gsl_movstat_workspace
          * w)

     This function computes the moving Q_n of the input vector ‘x’ and
     stores the output in ‘xscale’.  The inputs ‘x’ and ‘xscale’ must be
     the same length.  The parameter ‘endtype’ specifies how windows
     near the ends of the input should be handled.  It is allowed for
     ‘x’ = ‘xscale’ for an in-place moving window Q_n.

23.10 User-defined Moving Statistics
====================================

GSL offers an interface for users to define their own moving window
statistics functions, without needing to implement the edge-handling and
accumulator machinery.  This can be done by explicitly constructing the
windows W_i^{H,J} for a given input signal (see gsl_movstat_fill()),
or by calculating a user-defined function for each window
automatically.  In order to apply a user-defined function to each
window, users must define a variable of type gsl_movstat_function
to pass into gsl_movstat_apply().  This structure is
defined as follows.

 -- Type: gsl_movstat_function

     Structure specifying user-defined moving window statistical
     function:

          typedef struct
          {
            double (* function) (const size_t n, double x[], void * params);
            void * params;
          } gsl_movstat_function;

     This structure contains a pointer to the user-defined function as
     well as possible parameters to pass to the function.

      -- Member: double (* function) (const size_t n, double x[], void
               * params)

          This function returns the user-defined statistic of the array
          ‘x’ of length ‘n’.  User-specified parameters are passed in
          via ‘params’.  It is allowed to modify the array ‘x’.

      -- Member: void * params

          User-specified parameters to be passed into the function.

 -- Function: int gsl_movstat_apply (const gsl_movstat_end_t endtype,
          const gsl_movstat_function * F, const gsl_vector * x,
          gsl_vector * y, gsl_movstat_workspace * w)

     This function applies the user-defined moving window statistic
     specified in ‘F’ to the input vector ‘x’, storing the output in
     ‘y’.  The parameter ‘endtype’ specifies how windows near the ends
     of the input should be handled.  It is allowed for ‘x’ = ‘y’ for an
     in-place moving window calculation.

 -- Function: size_t gsl_movstat_fill (const gsl_movstat_end_t endtype,
          const gsl_vector * x, const size_t idx, const size_t H, const
          size_t J, double * window)

     This function explicitly constructs the sliding window for the
     input vector ‘x’ which is centered on the sample ‘idx’.  On output,
     the array ‘window’ will contain W_{idx}^{H,J}.  The number of
     samples to the left and right of the sample ‘idx’ are specified by
     ‘H’ and ‘J’ respectively.  The parameter ‘endtype’ specifies how
     windows near the ends of the input should be handled.  The function
     returns the size of the window.

23.11 Accumulators
==================

Many of the algorithms of this chapter are based on an accumulator
design, which process the input vector one sample at a time, updating
calculations of the desired statistic for the current window.  Each
accumulator is stored in the following structure:

 -- Type: gsl_movstat_accum

     Structure specifying accumulator for moving window statistics:

          typedef struct
          {
            size_t (* size) (const size_t n);
            int (* init) (const size_t n, void * vstate);
            int (* insert) (const double x, void * vstate);
            int (* delete) (void * vstate);
            int (* get) (void * params, double * result, const void * vstate);
          } gsl_movstat_accum;

     The structure contains function pointers responsible for performing
     different tasks for the accumulator.

      -- Member: size_t (* size) (const size_t n)

          This function returns the size of the workspace (in bytes)
          needed by the accumulator for a moving window of length ‘n’.

      -- Member: int (* init) (const size_t n, void * vstate)

          This function initializes the workspace ‘vstate’ for a moving
          window of length ‘n’.

      -- Member: int (* insert) (const double x, void * vstate)

          This function inserts a single sample ‘x’ into the
          accumulator, updating internal calculations of the desired
          statistic.  If the accumulator is full (i.e.  n samples have
          already been inserted), then the oldest sample is deleted from
          the accumulator.

      -- Member: int (* delete) (void * vstate)

          This function deletes the oldest sample from the accumulator,
          updating internal calculations of the desired statistic.

      -- Member: int (* get) (void * params, double * result, const void
               * vstate)

          This function stores the desired statistic for the current
          window in ‘result’.  The input ‘params’ specifies optional
          parameters for calculating the statistic.

The following accumulators of type gsl_movstat_accum are
defined by GSL to perform moving window statistics calculations.

 -- Variable: gsl_movstat_accum_min
 -- Variable: gsl_movstat_accum_max
 -- Variable: gsl_movstat_accum_minmax

     These accumulators calculate moving window minimum/maximums
     efficiently, using the algorithm of D. Lemire.

 -- Variable: gsl_movstat_accum_mean
 -- Variable: gsl_movstat_accum_sd
 -- Variable: gsl_movstat_accum_variance

     These accumulators calculate the moving window mean, standard
     deviation, and variance, using the algorithm of B. P. Welford.

 -- Variable: gsl_movstat_median

     This accumulator calculates the moving window median using the
     min/max heap algorithm of Härdle and Steiger.

 -- Variable: gsl_movstat_accum_Sn
 -- Variable: gsl_movstat_accum_Qn

     These accumulators calculate the moving window S_n and Q_n
     statistics developed by Croux and Rousseeuw.

 -- Variable: gsl_movstat_accum_sum

     This accumulator calculates the moving window sum.

 -- Variable: gsl_movstat_accum_qqr

     This accumulator calculates the moving window q-quantile range.

23.12 Examples
==============

23.12.1 Example 1
-----------------

The following example program computes the moving mean, minimum and
maximum of a noisy sinusoid signal of length N = 500 with a symmetric
moving window of size K = 11.

Figure: Original signal time series (gray) with moving mean (green),
moving minimum (blue), and moving maximum (orange).

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_movstat.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 500;                      /* length of time series */
       const size_t K = 11;                       /* window size */
       gsl_movstat_workspace * w = gsl_movstat_alloc(K);
       gsl_vector *x = gsl_vector_alloc(N);
       gsl_vector *xmean = gsl_vector_alloc(N);
       gsl_vector *xmin = gsl_vector_alloc(N);
       gsl_vector *xmax = gsl_vector_alloc(N);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       size_t i;

       for (i = 0; i < N; ++i)
         {
           double xi = cos(2.0 * M_PI * i / (double) N);
           double ei = gsl_ran_gaussian(r, 0.1);

           gsl_vector_set(x, i, xi + ei);
         }

       /* compute moving statistics */
       gsl_movstat_mean(GSL_MOVSTAT_END_PADVALUE, x, xmean, w);
       gsl_movstat_minmax(GSL_MOVSTAT_END_PADVALUE, x, xmin, xmax, w);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           printf("%zu %f %f %f %f\n",
                  i,
                  gsl_vector_get(x, i),
                  gsl_vector_get(xmean, i),
                  gsl_vector_get(xmin, i),
                  gsl_vector_get(xmax, i));
         }

       gsl_vector_free(x);
       gsl_vector_free(xmean);
       gsl_rng_free(r);
       gsl_movstat_free(w);

       return 0;
     }

23.12.2 Example 2: Robust Scale
-------------------------------

The following example program analyzes a time series of length N = 1000
composed of Gaussian random variates with zero mean whose standard
deviation changes in a piecewise constant fashion as shown in the table
below.

Sample Range     \sigma

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

1-200            1.0

201-450          5.0

451-600          1.0

601-850          3.0

851-1000         5.0

Additionally, about 1% of the samples are perturbed to represent
outliers by adding \pm 15 to the random Gaussian variate.  The program
calculates the moving statistics MAD, IQR, S_n, Q_n, and the standard
deviation using a symmetric moving window of length K = 41.  The results
are shown in Fig.  %s.

Figure: Top: time series of piecewise constant variance.  Bottom: scale
estimates using a moving window; the true sigma value is in light blue,
MAD in green, IQR in red, S_n in yellow, and Q_n in dark blue.  The
moving standard deviation is shown in gray.

The robust statistics follow the true standard deviation piecewise
changes well, without being influenced by the outliers.  The moving
standard deviation (gray curve) is heavily influenced by the presence of
the outliers.  The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_movstat.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                                 /* length of time series */
       const double sigma[] = { 1.0, 5.0, 1.0, 3.0, 5.0 };    /* variances */
       const size_t N_sigma[] = { 200, 450, 600, 850, 1000 }; /* samples where variance changes */
       const size_t K = 41;                                   /* window size */
       gsl_vector *x = gsl_vector_alloc(N);
       gsl_vector *xmedian = gsl_vector_alloc(N);
       gsl_vector *xmad = gsl_vector_alloc(N);
       gsl_vector *xiqr = gsl_vector_alloc(N);
       gsl_vector *xSn = gsl_vector_alloc(N);
       gsl_vector *xQn = gsl_vector_alloc(N);
       gsl_vector *xsd = gsl_vector_alloc(N);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_movstat_workspace * w = gsl_movstat_alloc(K);
       size_t idx = 0;
       size_t i;

       for (i = 0; i < N; ++i)
         {
           double gi = gsl_ran_gaussian(r, sigma[idx]);
           double u = gsl_rng_uniform(r);
           double outlier = (u < 0.01) ? 15.0*GSL_SIGN(gi) : 0.0;
           double xi = gi + outlier;

           gsl_vector_set(x, i, xi);

           if (i == N_sigma[idx] - 1)
             ++idx;
         }

       /* compute moving statistics */
       gsl_movstat_mad(GSL_MOVSTAT_END_TRUNCATE, x, xmedian, xmad, w);
       gsl_movstat_qqr(GSL_MOVSTAT_END_TRUNCATE, x, 0.25, xiqr, w);
       gsl_movstat_Sn(GSL_MOVSTAT_END_TRUNCATE, x, xSn, w);
       gsl_movstat_Qn(GSL_MOVSTAT_END_TRUNCATE, x, xQn, w);
       gsl_movstat_sd(GSL_MOVSTAT_END_TRUNCATE, x, xsd, w);

       /* scale IQR by factor to approximate standard deviation */
       gsl_vector_scale(xiqr, 0.7413);

       /* print results */
       idx = 0;
       for (i = 0; i < N; ++i)
         {
           printf("%zu %f %f %f %f %f %f %f\n",
                  i,
                  gsl_vector_get(x, i),
                  sigma[idx],
                  gsl_vector_get(xmad, i),
                  gsl_vector_get(xiqr, i),
                  gsl_vector_get(xSn, i),
                  gsl_vector_get(xQn, i),
                  gsl_vector_get(xsd, i));

           if (i == N_sigma[idx] - 1)
             ++idx;
         }

       gsl_vector_free(x);
       gsl_vector_free(xmedian);
       gsl_vector_free(xmad);
       gsl_vector_free(xiqr);
       gsl_vector_free(xSn);
       gsl_vector_free(xQn);
       gsl_vector_free(xsd);
       gsl_rng_free(r);
       gsl_movstat_free(w);

       return 0;
     }

23.12.3 Example 3: User-defined Moving Window
---------------------------------------------

This example program illustrates how a user can define their own moving
window function to apply to an input vector.  It constructs a random
noisy time series of length N = 1000 with some outliers added.  Then it
applies a moving window trimmed mean to the time series with trim
parameter \alpha = 0.1.  The length of the moving window is K = 11, so
the smallest and largest sample of each window is discarded prior to
computing the mean.  The results are shown in Fig.  %s.

Figure: Noisy time series data (black) with moving window trimmed mean
(red)

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_movstat.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_sort.h>
     #include <gsl/gsl_statistics.h>

     double
     func(const size_t n, double x[], void * params)
     {
       const double alpha = *(double *) params;

       gsl_sort(x, 1, n);

       return gsl_stats_trmean_from_sorted_data(alpha, x, 1, n);
     }

     int
     main(void)
     {
       const size_t N = 1000;                /* length of time series */
       const size_t K = 11;                  /* window size */
       double alpha = 0.1;                   /* trimmed mean parameter */
       gsl_vector *x = gsl_vector_alloc(N);  /* input vector */
       gsl_vector *y = gsl_vector_alloc(N);  /* filtered output vector for alpha1 */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_movstat_workspace *w = gsl_movstat_alloc(K);
       gsl_movstat_function F;
       size_t i;
       double sum = 0.0;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double ui = gsl_ran_gaussian(r, 1.0);
           double outlier = (gsl_rng_uniform(r) < 0.01) ? 10.0*GSL_SIGN(ui) : 0.0;
           sum += ui;
           gsl_vector_set(x, i, sum + outlier);
         }

       /* apply moving window function */
       F.function = func;
       F.params = &alpha;
       gsl_movstat_apply(GSL_MOVSTAT_END_PADVALUE, &F, x, y, w);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);

           printf("%f %f\n", xi, yi);
         }

       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_rng_free(r);
       gsl_movstat_free(w);

       return 0;
     }

23.13 References and Further Reading
====================================

The following publications are relevant to the algorithms described in
this chapter,

   * W.Hardle and W. Steiger, `Optimal Median Smoothing', Appl.
     Statist., 44 (2), 1995.

   * D. Lemire, `Streaming Maximum-Minimum Filter Using No More than
     Three Comparisons per Element', Nordic Journal of Computing, 13
     (4), 2006 (‘https://arxiv.org/abs/cs/0610046’).

   * B. P. Welford, `Note on a method for calculating corrected sums of
     squares and products', Technometrics, 4 (3), 1962.

24 Digital Filtering
********************

24.1 Introduction
=================

The filters discussed in this chapter are based on the following moving
data window which is centered on i-th sample:

     W_i^H = { x_{i-H}, ..., x_i, ..., x_{i+H} }

Here, H is a non-negative integer called the `window half-length', which
represents the number of samples before and after sample i.  The total
window length is K = 2 H + 1.

24.2 Handling Endpoints
=======================

When processing samples near the ends of the input signal, there will
not be enough samples to fill the window W_i^H defined above.  Therefore
the user must specify how to construct the windows near the end points.
This is done by passing an input argument of type
gsl_filter_end_t.:

 -- Type: gsl_filter_end_t

     This data type specifies how to construct windows near end points
     and can be selected from the following choices:

      -- Macro: GSL_FILTER_END_PADZERO

          With this option, a full window of length K will be
          constructed by inserting zeros into the window near the signal
          end points.  Effectively, the input signal is modified to

               x~ = { 0, ..., 0, x_1, x_2, ..., x_{n-1}, x_n, 0, ..., 0 }

          to ensure a well-defined window for all x_i.

      -- Macro: GSL_FILTER_END_PADVALUE

          With this option, a full window of length K will be
          constructed by padding the window with the first and last
          sample in the input signal.  Effectively, the input signal is
          modified to

               x~ = { x_1, ..., x_1, x_1, x_2, ..., x_{n-1}, x_n, x_n, ..., x_n }

      -- Macro: GSL_FILTER_END_TRUNCATE

          With this option, no padding is performed, and the windows are
          simply truncated as the end points are approached.

24.3 Linear Digital Filters
===========================

24.3.1 Gaussian Filter
----------------------

The Gaussian filter convolves the input signal with a Gaussian kernel or
window.  This filter is often used as a smoothing or noise reduction
filter.  The Gaussian kernel is defined by

     G(k) = e^{-1/2 ( \alpha k/((K-1)/2) )^2} = e^{-k^2/2\sigma^2}

for -(K-1)/2 \le k \le (K-1)/2, and K is the size of the kernel.  The
parameter \alpha specifies the number of standard deviations \sigma
desired in the kernel.  So for example setting \alpha = 3 would define a
Gaussian window of length K which spans \pm 3 \sigma.  It is often more
convenient to specify the parameter \alpha rather than the standard
deviation \sigma when constructing the kernel, since a fixed value of
\alpha would correspond to the same shape of Gaussian regardless of the
size K. The appropriate value of the standard deviation depends on K and
is related to \alpha as

     \sigma = (K - 1)/(2 \alpha)

The routines below accept \alpha as an input argument instead of \sigma.

The Gaussian filter offers a convenient way of differentiating and
smoothing an input signal in a single pass.  Using the derivative
property of a convolution,

     d/dt ( G * x ) = dG/dt * x

the input signal x(t) can be smoothed and differentiated at the same
time by convolution with a derivative Gaussian kernel, which can be
readily computed from the analytic expression above.  The same principle
applies to higher order derivatives.

 -- Function: gsl_filter_gaussian_workspace * gsl_filter_gaussian_alloc
          (const size_t K)

     This function initializes a workspace for Gaussian filtering using
     a kernel of size ‘K’.  Here, H = K / 2.  If K is even, it is
     rounded up to the next odd integer to ensure a symmetric window.
     The size of the workspace is O(K).

 -- Function: void gsl_filter_gaussian_free
          (gsl_filter_gaussian_workspace * w)

     This function frees the memory associated with ‘w’.

 -- Function: int gsl_filter_gaussian (const gsl_filter_end_t endtype,
          const double alpha, const size_t order, const gsl_vector * x,
          gsl_vector * y, gsl_filter_gaussian_workspace * w)

     This function applies a Gaussian filter parameterized by
     alpha to the input vector ‘x’, storing the output in ‘y’.
     The derivative order is specified by ‘order’, with ‘0’
     corresponding to a Gaussian, ‘1’ corresponding to a first
     derivative Gaussian, and so on.  The parameter ‘endtype’ specifies
     how the signal end points are handled.  It is allowed for ‘x’ = ‘y’
     for an in-place filter.

 -- Function: int gsl_filter_gaussian_kernel (const double alpha, const
          size_t order, const int normalize, gsl_vector * kernel)

     This function constructs a Gaussian kernel parameterized by
     alpha and stores the output in ‘kernel’.  The parameter
     ‘order’ specifies the derivative order, with ‘0’ corresponding to a
     Gaussian, ‘1’ corresponding to a first derivative Gaussian, and so
     on.  If ‘normalize’ is set to ‘1’, then the kernel will be
     normalized to sum to one on output.  If ‘normalize’ is set to ‘0’,
     no normalization is performed.

24.4 Nonlinear Digital Filters
==============================

The nonlinear digital filters described below are based on the window
median, which is given by

     m_i = median { W_i^H } = median { x_{i-H}, ..., x_i, ..., x_{i+H} }

The median is considered robust to local outliers, unlike the mean.
Median filters can preserve sharp edges while at the same removing
signal noise, and are used in a wide range of applications.

24.4.1 Standard Median Filter
-----------------------------

The `standard median filter' (SMF) simply replaces the sample x_i by the
median m_i of the window W_i^H: This filter has one tuning parameter
given by H. The standard median filter is considered highly resistant to
local outliers and local noise in the data sequence \{x_i\}.

 -- Function: gsl_filter_median_workspace * gsl_filter_median_alloc
          (const size_t K)

     This function initializes a workspace for standard median filtering
     using a symmetric centered moving window of size ‘K’.  Here, H = K
     / 2.  If K is even, it is rounded up to the next odd integer to
     ensure a symmetric window.  The size of the workspace is O(7K).

 -- Function: void gsl_filter_median_free (gsl_filter_median_workspace
          * w)

     This function frees the memory associated with ‘w’.

 -- Function: int gsl_filter_median (const gsl_filter_end_t endtype,
          const gsl_vector * x, gsl_vector * y,
          gsl_filter_median_workspace * w)

     This function applies a standard median filter to the input ‘x’,
     storing the output in ‘y’.  The parameter ‘endtype’ specifies how
     the signal end points are handled.  It is allowed to have ‘x’ = ‘y’
     for an in-place filter.

24.4.2 Recursive Median Filter
------------------------------

The `recursive median filter' (RMF) is a modification of the SMF to
include previous filter outputs in the window before computing the
median.  The filter’s response is

     y_i = median ( y_{i-H}, ..., y_{i-1}, x_i, x_{i+1}, ..., x_{i+H} )

Sometimes, the SMF must be applied several times in a row to achieve
adequate smoothing (i.e.  a cascade filter).  The RMF, on the other
hand, converges to a `root sequence' in one pass, and can sometimes
provide a smoother result than several passes of the SMF. A root
sequence is an input which is left unchanged by the filter.  So there is
no need to apply a recursive median filter twice to an input vector.

 -- Function: gsl_filter_rmedian_workspace * gsl_filter_rmedian_alloc
          (const size_t K)

     This function initializes a workspace for recursive median
     filtering using a symmetric centered moving window of size ‘K’.
     Here, H = K / 2.  If K is even, it is rounded up to the next odd
     integer to ensure a symmetric window.  The size of the workspace is
     O(K).

 -- Function: void gsl_filter_rmedian_free (gsl_filter_rmedian_workspace
          * w)

     This function frees the memory associated with ‘w’.

 -- Function: int gsl_filter_rmedian (const gsl_filter_end_t endtype,
          const gsl_vector * x, gsl_vector * y,
          gsl_filter_rmedian_workspace * w)

     This function applies a recursive median filter to the input ‘x’,
     storing the output in ‘y’.  The parameter ‘endtype’ specifies how
     the signal end points are handled.  It is allowed to have ‘x’ = ‘y’
     for an in-place filter.

24.4.3 Impulse Detection Filter
-------------------------------

Impulsive noise is characterized by short sequences of data points
distinct from those in the surrounding neighborhood.  This section
describes a powerful class of filters, also known as `impulse rejection
filters' and `decision-based filters', designed to detect and remove
such outliers from data.  The filter’s response is given by

     y_i = { x_i, |x_i - m_i| <= t * S_i
           { m_i, |x_i - m_i| > t * S_i

where m_i is the median value of the window W_i^H, S_i is a robust
estimate of the scatter or dispersion for the window W_i^H, and t is a
tuning parameter specifying the number of scale factors needed to
determine that a point is an outlier.  The main idea is that the median
m_i will be unaffected by a small number of outliers in the window, and
so a given sample x_i is tested to determine how far away it is from the
median in terms of the local scale estimate S_i.  Samples which are more
than t scale estimates away from the median are labeled as outliers and
replaced by the window median m_i.  Samples which are less than t scale
estimates from the median are left unchanged by the filter.

Note that when t = 0, the impulse detection filter is equivalent to the
standard median filter.  When t \rightarrow \infty, it becomes the
identity filter.  This means the impulse detection filter can be viewed
as a “less aggressive” version of the standard median filter, becoming
less aggressive as t is increased.  Note that this filter modifies only
samples identified as outliers, while the standard median filter changes
all samples to the local median, regardless of whether they are
outliers.  This fact, plus the additional flexibility offered by the
additional tuning parameter t can make the impulse detection filter a
better choice for some applications.

It is important to have a robust and accurate scale estimate S_i in
order to detect impulse outliers even in the presence of noise.  The
window standard deviation is not typically a good choice, as it can be
significantly perturbed by the presence of even one outlier.  GSL offers
the following choices (specified by a parameter of type
gsl_filter_scale_t) for computing the scale estimate S_i, all of
which are robust to the presence of impulse outliers.

 -- Type: gsl_filter_scale_t

     This type specifies how the scale estimate S_i of the window W_i^H
     is calculated.

      -- Macro: GSL_FILTER_SCALE_MAD

          This option specifies the median absolute deviation (MAD)
          scale estimate, defined by

               S_i = 1.4826 median { | W_i^H - m_i | }

          This choice of scale estimate is also known as the `Hampel
          filter' in the statistical literature.  See here.
          for more information.

      -- Macro: GSL_FILTER_SCALE_IQR

          This option specifies the interquartile range (IQR) scale
          estimate, defined as the difference between the 75th and 25th
          percentiles of the window W_i^H,

               S_i = 0.7413 ( Q_{0.75} - Q_{0.25} )

          where Q_p is the p-quantile of the window W_i^H. The idea is
          to throw away the largest and smallest 25% of the window
          samples (where the outliers would be), and estimate a scale
          from the middle 50%.  The factor 0.7413 provides an unbiased
          estimate of the standard deviation for Gaussian data.

      -- Macro: GSL_FILTER_SCALE_SN

          This option specifies the so-called S_n statistic proposed by
          Croux and Rousseeuw.  See here for more
          information.

      -- Macro: GSL_FILTER_SCALE_QN

          This option specifies the so-called Q_n statistic proposed by
          Croux and Rousseeuw.  See here for more
          information.

     Warning: While the scale estimates defined above are much less
     sensitive to outliers than the standard deviation, they can suffer
     from an effect called `implosion'.  The standard deviation of a
     window W_i^H will be zero if and only if all samples in the window
     are equal.  However, it is possible for the MAD of a window to be
     zero even if all the samples in the window are not equal.  For
     example, if K/2 + 1 or more of the K samples in the window are
     equal to some value x^{*}, then the window median will be equal to
     x^{*}.  Consequently, at least K/2 + 1 of the absolute deviations
     |x_j - x^{*}| will be zero, and so the MAD will be zero.  In such a
     case, the Hampel filter will act like the standard median filter
     regardless of the value of t.  Caution should also be exercised if
     dividing by S_i.

 -- Function: gsl_filter_impulse_workspace * gsl_filter_impulse_alloc
          (const size_t K)

     This function initializes a workspace for impulse detection
     filtering using a symmetric moving window of size ‘K’.  Here, H = K
     / 2.  If K is even, it is rounded up to the next odd integer to
     ensure a symmetric window.  The size of the workspace is O(6K).

 -- Function: void gsl_filter_impulse_free (gsl_filter_impulse_workspace
          * w)

     This function frees the memory associated with ‘w’.

 -- Function: int gsl_filter_impulse (const gsl_filter_end_t endtype,
          const gsl_filter_scale_t scale_type, const double t, const
          gsl_vector * x, gsl_vector * y, gsl_vector * xmedian,
          gsl_vector * xsigma, size_t * noutlier, gsl_vector_int
          * ioutlier, gsl_filter_impulse_workspace * w)

     These functions apply an impulse detection filter to the input
     vector ‘x’, storing the filtered output in ‘y’.  The tuning
     parameter t is provided in ‘t’.  The window medians m_i are stored
     in ‘xmedian’ and the S_i are stored in ‘xsigma’ on output.  The
     number of outliers detected is stored in ‘noutlier’ on output,
     while the locations of flagged outliers are stored in the boolean
     array ‘ioutlier’.  The input ‘ioutlier’ may be ‘NULL’ if not
     desired.  It is allowed to have ‘x’ = ‘y’ for an in-place filter.

24.5 Examples
=============

24.5.1 Gaussian Example 1
-------------------------

This example program illustrates the Gaussian filter applied to
smoothing a time series of length N = 500 with a kernel size of K = 51.
Three filters are applied with parameters \alpha = 0.5, 3, 10.  The
results are shown in Fig.  %s.

Figure: Top panel: Gaussian kernels (unnormalized) for \alpha = 0.5, 3,
10.  Bottom panel: Time series (gray) with Gaussian filter output for
same \alpha values.

We see that the filter corresponding to \alpha = 0.5 applies the most
smoothing, while \alpha = 10 corresponds to the least amount of
smoothing.  The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 500;                        /* length of time series */
       const size_t K = 51;                         /* window size */
       const double alpha[3] = { 0.5, 3.0, 10.0 };  /* alpha values */
       gsl_vector *x = gsl_vector_alloc(N);         /* input vector */
       gsl_vector *y1 = gsl_vector_alloc(N);        /* filtered output vector for alpha1 */
       gsl_vector *y2 = gsl_vector_alloc(N);        /* filtered output vector for alpha2 */
       gsl_vector *y3 = gsl_vector_alloc(N);        /* filtered output vector for alpha3 */
       gsl_vector *k1 = gsl_vector_alloc(K);        /* Gaussian kernel for alpha1 */
       gsl_vector *k2 = gsl_vector_alloc(K);        /* Gaussian kernel for alpha2 */
       gsl_vector *k3 = gsl_vector_alloc(K);        /* Gaussian kernel for alpha3 */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_filter_gaussian_workspace *gauss_p = gsl_filter_gaussian_alloc(K);
       size_t i;
       double sum = 0.0;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double ui = gsl_ran_gaussian(r, 1.0);
           sum += ui;
           gsl_vector_set(x, i, sum);
         }

       /* compute kernels without normalization */
       gsl_filter_gaussian_kernel(alpha[0], 0, 0, k1);
       gsl_filter_gaussian_kernel(alpha[1], 0, 0, k2);
       gsl_filter_gaussian_kernel(alpha[2], 0, 0, k3);

       /* apply filters */
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[0], 0, x, y1, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[1], 0, x, y2, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[2], 0, x, y3, gauss_p);

       /* print kernels */
       for (i = 0; i < K; ++i)
         {
           double k1i = gsl_vector_get(k1, i);
           double k2i = gsl_vector_get(k2, i);
           double k3i = gsl_vector_get(k3, i);

           printf("%e %e %e\n", k1i, k2i, k3i);
         }

       printf("\n\n");

       /* print filter results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double y1i = gsl_vector_get(y1, i);
           double y2i = gsl_vector_get(y2, i);
           double y3i = gsl_vector_get(y3, i);

           printf("%.12e %.12e %.12e %.12e\n", xi, y1i, y2i, y3i);
         }

       gsl_vector_free(x);
       gsl_vector_free(y1);
       gsl_vector_free(y2);
       gsl_vector_free(y3);
       gsl_vector_free(k1);
       gsl_vector_free(k2);
       gsl_vector_free(k3);
       gsl_rng_free(r);
       gsl_filter_gaussian_free(gauss_p);

       return 0;
     }

24.5.2 Gaussian Example 2
-------------------------

A common application of the Gaussian filter is to detect edges, or
sudden jumps, in a noisy input signal.  It is used both for 1D edge
detection in time series, as well as 2D edge detection in images.  Here
we will examine a noisy time series of length N = 1000 with a single
edge.  The input signal is defined as

     x(n) = e(n) + { 0,   n <= N/2
                   { 0.5, n >  N/2

where e(n) is Gaussian random noise.  The program smooths the input
signal with order 0,1, and 2 Gaussian filters of length K = 61 with
\alpha = 3.  For comparison, the program also computes finite
differences of the input signal without smoothing.  The results are
shown in Fig.  %s.

Figure: Top row: original input signal x(n) (black) with Gaussian
smoothed signal in red.  Second row: First finite differences of input
signal.  Third row: Input signal smoothed with a first order Gaussian
filter.  Fourth row: Input signal smoothed with a second order Gaussian
filter.

The finite difference approximation of the first derivative (second row)
shows the common problem with differentiating a noisy signal.  The noise
is amplified and makes it extremely difficult to detect the sharp
gradient at sample 500.  The third row shows the first order Gaussian
smoothed signal with a clear peak at the location of the edge.
Alternatively, one could examine the second order Gaussian smoothed
signal (fourth row) and look for zero crossings to determine the edge
location.

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                       /* length of time series */
       const size_t K = 61;                         /* window size */
       const double alpha = 3.0;                    /* Gaussian kernel has +/- 3 standard deviations */
       gsl_vector *x = gsl_vector_alloc(N);         /* input vector */
       gsl_vector *y = gsl_vector_alloc(N);         /* filtered output vector */
       gsl_vector *dy = gsl_vector_alloc(N);        /* first derivative filtered vector */
       gsl_vector *d2y = gsl_vector_alloc(N);       /* second derivative filtered vector */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_filter_gaussian_workspace *gauss_p = gsl_filter_gaussian_alloc(K);
       size_t i;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double xi = (i > N / 2) ? 0.5 : 0.0;
           double ei = gsl_ran_gaussian(r, 0.1);

           gsl_vector_set(x, i, xi + ei);
         }

       /* apply filters */
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 0, x, y, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 1, x, dy, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 2, x, d2y, gauss_p);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);
           double dyi = gsl_vector_get(dy, i);
           double d2yi = gsl_vector_get(d2y, i);
           double dxi;

           /* compute finite difference of x vector */
           if (i == 0)
             dxi = gsl_vector_get(x, i + 1) - xi;
           else if (i == N - 1)
             dxi = gsl_vector_get(x, i) - gsl_vector_get(x, i - 1);
           else
             dxi = 0.5 * (gsl_vector_get(x, i + 1) - gsl_vector_get(x, i - 1));

           printf("%.12e %.12e %.12e %.12e %.12e\n",
                  xi,
                  yi,
                  dxi,
                  dyi,
                  d2yi);
         }

       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_vector_free(dy);
       gsl_vector_free(d2y);
       gsl_rng_free(r);
       gsl_filter_gaussian_free(gauss_p);

       return 0;
     }

24.5.3 Square Wave Signal Example
---------------------------------

The following example program illustrates the median filters on a noisy
square wave signal.  Median filters are well known for preserving sharp
edges in the input signal while reducing noise.  The program constructs
a 5 Hz square wave signal with Gaussian noise added.  Then the signal is
filtered with a standard median filter and recursive median filter using
a symmetric window of length K = 7.  The results are shown in Fig.  %s.

Figure: Original time series is in gray.  The standard median filter
output is in green and the recursive median filter output is in red.

Both filters preserve the sharp signal edges while reducing the noise.
The recursive median filter achieves a smoother result than the standard
median filter.  The “blocky” nature of the output is characteristic of
all median filters.  The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                               /* length of time series */
       const size_t K = 7;                                  /* window size */
       const double f = 5.0;                                /* frequency of square wave in Hz */
       gsl_filter_median_workspace *median_p = gsl_filter_median_alloc(K);
       gsl_filter_rmedian_workspace *rmedian_p = gsl_filter_rmedian_alloc(K);
       gsl_vector *t = gsl_vector_alloc(N);                 /* time */
       gsl_vector *x = gsl_vector_alloc(N);                 /* input vector */
       gsl_vector *y_median = gsl_vector_alloc(N);          /* median filtered output */
       gsl_vector *y_rmedian = gsl_vector_alloc(N);         /* recursive median filtered output */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       size_t i;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double ti = (double) i / (N - 1.0);
           double tmp = sin(2.0 * M_PI * f * ti);
           double xi = (tmp >= 0.0) ? 1.0 : -1.0;
           double ei = gsl_ran_gaussian(r, 0.1);

           gsl_vector_set(t, i, ti);
           gsl_vector_set(x, i, xi + ei);
         }

       gsl_filter_median(GSL_FILTER_END_PADVALUE, x, y_median, median_p);
       gsl_filter_rmedian(GSL_FILTER_END_PADVALUE, x, y_rmedian, rmedian_p);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double ti = gsl_vector_get(t, i);
           double xi = gsl_vector_get(x, i);
           double medi = gsl_vector_get(y_median, i);
           double rmedi = gsl_vector_get(y_rmedian, i);

           printf("%f %f %f %f\n",
                  ti,
                  xi,
                  medi,
                  rmedi);
         }

       gsl_vector_free(t);
       gsl_vector_free(x);
       gsl_vector_free(y_median);
       gsl_vector_free(y_rmedian);
       gsl_rng_free(r);
       gsl_filter_median_free(median_p);

       return 0;
     }

24.5.4 Impulse Detection Example
--------------------------------

The following example program illustrates the impulse detection filter.
First, it constructs a sinusoid signal of length N = 1000 with Gaussian
noise added.  Then, about 1% of the data are perturbed to represent
large outliers.  An impulse detecting filter is applied with a window
size K = 25 and tuning parameter t = 4, using the Q_n statistic as the
robust measure of scale.  The results are plotted in Fig.  %s.

Figure: Original time series is in blue, filter output is in green,
upper and lower intervals for detecting outliers are in red and yellow
respectively.  Detected outliers are marked with squares.

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                               /* length of time series */
       const size_t K = 25;                                 /* window size */
       const double t = 4.0;                                /* number of scale factors for outlier detection */
       gsl_vector *x = gsl_vector_alloc(N);                 /* input vector */
       gsl_vector *y = gsl_vector_alloc(N);                 /* output (filtered) vector */
       gsl_vector *xmedian = gsl_vector_alloc(N);           /* window medians */
       gsl_vector *xsigma = gsl_vector_alloc(N);            /* window scale estimates */
       gsl_vector_int *ioutlier = gsl_vector_int_alloc(N);  /* outlier detected? */
       gsl_filter_impulse_workspace * w = gsl_filter_impulse_alloc(K);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       size_t noutlier;
       size_t i;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double xi = 10.0 * sin(2.0 * M_PI * i / (double) N);
           double ei = gsl_ran_gaussian(r, 2.0);
           double u = gsl_rng_uniform(r);
           double outlier = (u < 0.01) ? 15.0*GSL_SIGN(ei) : 0.0;

           gsl_vector_set(x, i, xi + ei + outlier);
         }

       /* apply impulse detection filter */
       gsl_filter_impulse(GSL_FILTER_END_TRUNCATE, GSL_FILTER_SCALE_QN, t, x, y,
                          xmedian, xsigma, &noutlier, ioutlier, w);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);
           double xmedi = gsl_vector_get(xmedian, i);
           double xsigmai = gsl_vector_get(xsigma, i);
           int outlier = gsl_vector_int_get(ioutlier, i);

           printf("%zu %f %f %f %f %d\n",
                  i,
                  xi,
                  yi,
                  xmedi + t * xsigmai,
                  xmedi - t * xsigmai,
                  outlier);
         }

       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_vector_free(xmedian);
       gsl_vector_free(xsigma);
       gsl_vector_int_free(ioutlier);
       gsl_filter_impulse_free(w);
       gsl_rng_free(r);

       return 0;
     }

25 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’.

25.1 The histogram struct
=========================

A histogram is defined by the following struct,

 -- 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 \le 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’.

25.2 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.

25.3 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’.

25.4 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.

25.5 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.

25.6 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.

25.7 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

25.8 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.

25.9 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.

25.10 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.

 -- 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] \le r < sum[i+1] and
     delta is (r - sum[i])/(sum[i+1] - sum[i]).

25.11 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

Fig.  %s shows the familiar shape of the Cauchy distribution and the
fluctuations caused by the finite sample size.

Figure: Histogram output from example program

25.12 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).

25.13 The 2D histogram struct
=============================

Two dimensional histograms are defined by the following struct,

 -- 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’.

25.14 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.

25.15 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’.

25.16 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.

25.17 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.

25.18 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.

25.19 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

25.20 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.

25.21 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,:math:‘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.

 -- 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’.

25.22 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;
     }

The following plot shows the distribution of the simulated events.
Using a higher resolution grid we can see the original underlying
histogram and also the statistical fluctuations caused by the events
being uniformly distributed over the area of the original bins.

Figure: Distribution of simulated events from example program

26 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’.

26.1 The ntuple struct
======================

 -- Type: gsl_ntuple

     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;

26.2 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.

26.3 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.

26.4 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().

26.5 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’.

26.6 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.

26.7 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.

 -- Type: gsl_ntuple_select_fn

     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.

 -- Type: gsl_ntuple_value_fn

     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.

26.8 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)
     {
       (void)(params); /* avoid unused parameter warning */
       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;
     }

Fig.  %s shows the distribution of the selected events.  Note the
cut-off at the lower bound.

Figure: Distribution of selected events

26.9 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).

27 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_{x_l}^{x_u} dx \int_{y_l}^{y_u} 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’.

27.1 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’.

 -- 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)

27.2 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’.

 -- Type: gsl_monte_plain_state

     This is a workspace for plain Monte Carlo integration

 -- 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’.

27.3 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\over 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’.

 -- Type: gsl_monte_miser_state

     This workspace is used for MISER Monte Carlo integration

 -- 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:

 -- Type: gsl_monte_miser_params

      -- 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.

27.4 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, \ldots) = g_1(x_1) g_2(x_2) \ldots 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).

 -- Type: gsl_monte_vegas_state

     This workspace is used for VEGAS Monte Carlo integration

 -- 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:

 -- Type: gsl_monte_vegas_params

      -- 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.

27.5 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)
     {
       (void)(dim); /* avoid unused parameter warnings */
       (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.

27.6 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

28 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’.

28.1 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} \le 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 \rightarrow
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.

28.2 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.

 -- Type: gsl_siman_Efunc_t

     This function type should return the energy of a configuration
     ‘xp’:

          double (*gsl_siman_Efunc_t) (void *xp)

 -- 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)

 -- 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)

 -- Type: gsl_siman_print_t

     This function type should print the contents of the configuration
     ‘xp’:

          void (*gsl_siman_print_t) (void *xp)

 -- Type: gsl_siman_copy_t

     This function type should copy the configuration ‘source’ into
     ‘dest’:

          void (*gsl_siman_copy_t) (void *source, void *dest)

 -- 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)

 -- Type: gsl_siman_destroy_t

     This function type should destroy the configuration ‘xp’, freeing
     its memory:

          void (*gsl_siman_destroy_t) (void *xp)

 -- 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.

28.3 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.

28.3.1 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(void)
     {
       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;
     }

Fig.  %s is 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

Fig.  %s is generated by running ‘siman_test’ in the following way:

     $ ./siman_test | awk '!/^#/ {print $1, $5}'
      | graph -y -0.88 -0.83 -W0 -X generation -Y energy
      | plot -Tps > siman-energy.eps

Figure: Example of a simulated annealing run: at higher temperatures
(early in the plot) you see that the solution can fluctuate, but at
lower temperatures it converges.

Figure: Simulated annealing energy vs generation

28.3.2 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 is given below.

     /* siman/siman_tsp.c
      *
      * Copyright (C) 1996, 1997, 1998, 1999, 2000 Mark Galassi
      *
      * 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, write to the Free Software
      * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
      */

     #include <config.h>
     #include <math.h>
     #include <string.h>
     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_siman.h>
     #include <gsl/gsl_ieee_utils.h>

     /* set up parameters for this simulated annealing run */

     #define N_TRIES 200             /* how many points do we try before stepping */
     #define ITERS_FIXED_T 2000      /* how many iterations for each T? */
     #define STEP_SIZE 1.0           /* max step size in random walk */
     #define K 1.0                   /* Boltzmann constant */
     #define T_INITIAL 5000.0        /* initial temperature */
     #define MU_T 1.002              /* damping factor for temperature */
     #define T_MIN 5.0e-1

     gsl_siman_params_t params = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
                                  K, T_INITIAL, MU_T, T_MIN};

     struct s_tsp_city {
       const char * name;
       double lat, longitude;        /* coordinates */
     };
     typedef struct s_tsp_city Stsp_city;

     void prepare_distance_matrix(void);
     void exhaustive_search(void);
     void print_distance_matrix(void);
     double city_distance(Stsp_city c1, Stsp_city c2);
     double Etsp(void *xp);
     double Mtsp(void *xp, void *yp);
     void Stsp(const gsl_rng * r, void *xp, double step_size);
     void Ptsp(void *xp);

     /* in this table, latitude and longitude are obtained from the US
        Census Bureau, at http://www.census.gov/cgi-bin/gazetteer */

     Stsp_city cities[] = {{"Santa Fe",    35.68,   105.95},
                           {"Phoenix",     33.54,   112.07},
                           {"Albuquerque", 35.12,   106.62},
                           {"Clovis",      34.41,   103.20},
                           {"Durango",     37.29,   107.87},
                           {"Dallas",      32.79,    96.77},
                           {"Tesuque",     35.77,   105.92},
                           {"Grants",      35.15,   107.84},
                           {"Los Alamos",  35.89,   106.28},
                           {"Las Cruces",  32.34,   106.76},
                           {"Cortez",      37.35,   108.58},
                           {"Gallup",      35.52,   108.74}};

     #define N_CITIES (sizeof(cities)/sizeof(Stsp_city))

     double distance_matrix[N_CITIES][N_CITIES];

     /* distance between two cities */
     double city_distance(Stsp_city c1, Stsp_city c2)
     {
       const double earth_radius = 6375.000; /* 6000KM approximately */
       /* sin and cos of lat and long; must convert to radians */
       double sla1 = sin(c1.lat*M_PI/180), cla1 = cos(c1.lat*M_PI/180),
         slo1 = sin(c1.longitude*M_PI/180), clo1 = cos(c1.longitude*M_PI/180);
       double sla2 = sin(c2.lat*M_PI/180), cla2 = cos(c2.lat*M_PI/180),
         slo2 = sin(c2.longitude*M_PI/180), clo2 = cos(c2.longitude*M_PI/180);

       double x1 = cla1*clo1;
       double x2 = cla2*clo2;

       double y1 = cla1*slo1;
       double y2 = cla2*slo2;

       double z1 = sla1;
       double z2 = sla2;

       double dot_product = x1*x2 + y1*y2 + z1*z2;

       double angle = acos(dot_product);

       /* distance is the angle (in radians) times the earth radius */
       return angle*earth_radius;
     }

     /* energy for the travelling salesman problem */
     double Etsp(void *xp)
     {
       /* an array of N_CITIES integers describing the order */
       int *route = (int *) xp;
       double E = 0;
       unsigned int i;

       for (i = 0; i < N_CITIES; ++i) {
         /* use the distance_matrix to optimize this calculation; it had
            better be allocated!! */
         E += distance_matrix[route[i]][route[(i + 1) % N_CITIES]];
       }

       return E;
     }

     double Mtsp(void *xp, void *yp)
     {
       int *route1 = (int *) xp, *route2 = (int *) yp;
       double distance = 0;
       unsigned int i;

       for (i = 0; i < N_CITIES; ++i) {
         distance += ((route1[i] == route2[i]) ? 0 : 1);
       }

       return distance;
     }

     /* take a step through the TSP space */
     void Stsp(const gsl_rng * r, void *xp, double step_size)
     {
       int x1, x2, dummy;
       int *route = (int *) xp;

       step_size = 0 ; /* prevent warnings about unused parameter */

       /* pick the two cities to swap in the matrix; we leave the first
          city fixed */
       x1 = (gsl_rng_get (r) % (N_CITIES-1)) + 1;
       do {
         x2 = (gsl_rng_get (r) % (N_CITIES-1)) + 1;
       } while (x2 == x1);

       dummy = route[x1];
       route[x1] = route[x2];
       route[x2] = dummy;
     }

     void Ptsp(void *xp)
     {
       unsigned int i;
       int *route = (int *) xp;
       printf("  [");
       for (i = 0; i < N_CITIES; ++i) {
         printf(" %d ", route[i]);
       }
       printf("]  ");
     }

     int main(void)
     {
       int x_initial[N_CITIES];
       unsigned int i;

       const gsl_rng * r = gsl_rng_alloc (gsl_rng_env_setup()) ;

       gsl_ieee_env_setup ();

       prepare_distance_matrix();

       /* set up a trivial initial route */
       printf("# initial order of cities:\n");
       for (i = 0; i < N_CITIES; ++i) {
         printf("# \"%s\"\n", cities[i].name);
         x_initial[i] = i;
       }

       printf("# distance matrix is:\n");
       print_distance_matrix();

       printf("# initial coordinates of cities (longitude and latitude)\n");
       /* this can be plotted with */
       /* ./siman_tsp > hhh ; grep city_coord hhh | awk '{print $2 "   " $3}' | xyplot -ps -d "xy" > c.eps */
       for (i = 0; i < N_CITIES+1; ++i) {
         printf("###initial_city_coord: %g %g \"%s\"\n",
                -cities[x_initial[i % N_CITIES]].longitude,
                cities[x_initial[i % N_CITIES]].lat,
                cities[x_initial[i % N_CITIES]].name);
       }

     /*   exhaustive_search(); */

       gsl_siman_solve(r, x_initial, Etsp, Stsp, Mtsp, Ptsp, NULL, NULL, NULL,
                       N_CITIES*sizeof(int), params);

       printf("# final order of cities:\n");
       for (i = 0; i < N_CITIES; ++i) {
         printf("# \"%s\"\n", cities[x_initial[i]].name);
       }

       printf("# final coordinates of cities (longitude and latitude)\n");
       /* this can be plotted with */
       /* ./siman_tsp > hhh ; grep city_coord hhh | awk '{print $2 "   " $3}' | xyplot -ps -d "xy" > c.eps */
       for (i = 0; i < N_CITIES+1; ++i) {
         printf("###final_city_coord: %g %g %s\n",
                -cities[x_initial[i % N_CITIES]].longitude,
                cities[x_initial[i % N_CITIES]].lat,
                cities[x_initial[i % N_CITIES]].name);
       }

       printf("# ");
       fflush(stdout);
     #if 0
       system("date");
     #endif /* 0 */
       fflush(stdout);

       return 0;
     }

     void prepare_distance_matrix()
     {
       unsigned int i, j;
       double dist;

       for (i = 0; i < N_CITIES; ++i) {
         for (j = 0; j < N_CITIES; ++j) {
           if (i == j) {
             dist = 0;
           } else {
             dist = city_distance(cities[i], cities[j]);
           }
           distance_matrix[i][j] = dist;
         }
       }
     }

     void print_distance_matrix()
     {
       unsigned int i, j;

       for (i = 0; i < N_CITIES; ++i) {
         printf("# ");
         for (j = 0; j < N_CITIES; ++j) {
           printf("%15.8f   ", distance_matrix[i][j]);
         }
         printf("\n");
       }
     }

     /* [only works for 12] search the entire space for solutions */
     static double best_E = 1.0e100, second_E = 1.0e100, third_E = 1.0e100;
     static int best_route[N_CITIES];
     static int second_route[N_CITIES];
     static int third_route[N_CITIES];
     static void do_all_perms(int *route, int n);

     void exhaustive_search()
     {
       static int initial_route[N_CITIES] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11};
       printf("\n# ");
       fflush(stdout);
     #if 0
       system("date");
     #endif
       fflush(stdout);
       do_all_perms(initial_route, 1);
       printf("\n# ");
       fflush(stdout);
     #if 0
       system("date");
     #endif /* 0 */
       fflush(stdout);

       printf("# exhaustive best route: ");
       Ptsp(best_route);
       printf("\n# its energy is: %g\n", best_E);

       printf("# exhaustive second_best route: ");
       Ptsp(second_route);
       printf("\n# its energy is: %g\n", second_E);

       printf("# exhaustive third_best route: ");
       Ptsp(third_route);
       printf("\n# its energy is: %g\n", third_E);
     }

     /* James Theiler's recursive algorithm for generating all routes */
     static void do_all_perms(int *route, int n)
     {
       if (n == (N_CITIES-1)) {
         /* do it! calculate the energy/cost for that route */
         double E;
         E = Etsp(route);            /* TSP energy function */
         /* now save the best 3 energies and routes */
         if (E < best_E) {
           third_E = second_E;
           memcpy(third_route, second_route, N_CITIES*sizeof(*route));
           second_E = best_E;
           memcpy(second_route, best_route, N_CITIES*sizeof(*route));
           best_E = E;
           memcpy(best_route, route, N_CITIES*sizeof(*route));
         } else if (E < second_E) {
           third_E = second_E;
           memcpy(third_route, second_route, N_CITIES*sizeof(*route));
           second_E = E;
           memcpy(second_route, route, N_CITIES*sizeof(*route));
         } else if (E < third_E) {
           third_E = E;
           memcpy(route, third_route, N_CITIES*sizeof(*route));
         }
       } else {
         int new_route[N_CITIES];
         unsigned int j;
         int swap_tmp;
         memcpy(new_route, route, N_CITIES*sizeof(*route));
         for (j = n; j < N_CITIES; ++j) {
           swap_tmp = new_route[j];
           new_route[j] = new_route[n];
           new_route[n] = swap_tmp;
           do_all_perms(new_route, n+1);
         }
       }
     }

Below are 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

Figure: Initial route for the 12 southwestern cities Flying Salesman
Problem.

Figure: Final (optimal) route for the 12 southwestern cities Flying
Salesman Problem.

Here’s a plot of the cost function (energy) versus generation (point in
the calculation at which a new temperature is set) for this problem:

Figure: Example of a simulated annealing run for the 12 southwestern
cities Flying Salesman Problem.

28.4 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).

29 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.

29.1 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.

 -- 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.  If
          ‘function’ is modified (for example contents of ‘params’), the
          user must call an appropriate reset function (see
          gsl_odeiv2_driver_reset(),
          gsl_odeiv2_evolve_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

               df_i(t,y,params)/dt

          in the array ‘dfdt’ and the Jacobian matrix J_{ij} in the
          array ‘dfdy’, regarded as a row-ordered matrix ‘J(i,j) =
          dfdy[i * dimension + j]’ where ‘dimension’ is the dimension of
          the system.

          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.  If
          ‘jacobian’ is modified (for example contents of ‘params’), the
          user must call an appropriate reset function (see
          gsl_odeiv2_driver_reset(),
          gsl_odeiv2_evolve_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.

29.2 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.

 -- Type: gsl_odeiv2_step

     This contains internal parameters for a stepping function.

 -- 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,

 -- Type: gsl_odeiv2_step_type

      -- Variable: gsl_odeiv2_step_rk2

          Explicit embedded Runge-Kutta (2, 3) method.

      -- Variable: 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.

      -- Variable: gsl_odeiv2_step_rkf45

          Explicit embedded Runge-Kutta-Fehlberg (4, 5) method.  This
          method is a good general-purpose integrator.

      -- Variable: gsl_odeiv2_step_rkck

          Explicit embedded Runge-Kutta Cash-Karp (4, 5) method.

      -- Variable: gsl_odeiv2_step_rk8pd

          Explicit embedded Runge-Kutta Prince-Dormand (8, 9) method.

      -- Variable: 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().

      -- Variable: 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().

      -- Variable: 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().

      -- Variable: 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.

      -- Variable: 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().

      -- Variable: 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().

29.3 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.

 -- Type: gsl_odeiv2_control

     This is a workspace for controlling step size.

 -- Type: gsl_odeiv2_control_type

     This specifies the control type.

 -- 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’.

29.4 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.

 -- Type: gsl_odeiv2_evolve

     This workspace contains parameters for controlling the evolution
     function

 -- 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).

29.5 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.

29.6 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)
     {
       (void)(t); /* avoid unused parameter warning */
       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)
     {
       (void)(t); /* avoid unused parameter warning */
       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.

Figure: Numerical solution of the Van der Pol oscillator equation using
Prince-Dormand 8th order Runge-Kutta.

It is also possible to work with a non-adaptive integrator, using only
the stepping function itself, see
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;
     }

29.7 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.

30 Interpolation
****************

This chapter describes functions for performing interpolation.  The
library provides a variety of interpolation methods, including Cubic,
Akima, and Steffen 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.  Routines are
provided for interpolating both one and two dimensional datasets.

These interpolation methods produce curves that pass through each
datapoint.  To interpolate noisy data with a smoothing curve see
Basis Splines.

The functions described in this section are declared in the header files
‘gsl_interp.h’ and ‘gsl_spline.h’.

30.1 Introduction to 1D Interpolation
=====================================

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.

30.2 1D Interpolation Functions
===============================

The interpolation function for a given dataset is stored in a
gsl_interp object.  These are created by the following functions.

 -- Type: gsl_interp

     Workspace for 1D interpolation

 -- 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 (see 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’.

30.3 1D Interpolation Types
===========================

The interpolation library provides the following interpolation types:

 -- Type: gsl_interp_type

      -- Variable: gsl_interp_linear

          Linear interpolation.  This interpolation method does not
          require any additional memory.

      -- Variable: 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.

      -- Variable: 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.

      -- Variable: 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.

      -- Variable: gsl_interp_akima

          Non-rounded Akima spline with natural boundary conditions.
          This method uses the non-rounded corner algorithm of Wodicka.

      -- Variable: gsl_interp_akima_periodic

          Non-rounded Akima spline with periodic boundary conditions.
          This method uses the non-rounded corner algorithm of Wodicka.

      -- Variable: gsl_interp_steffen

          Steffen’s method guarantees the monotonicity of the
          interpolating function between the given data points.
          Therefore, minima and maxima can only occur exactly at the
          data points, and there can never be spurious oscillations
          between data points.  The interpolated function is piecewise
          cubic in each interval.  The resulting curve and its first
          derivative are guaranteed to be continuous, but the second
          derivative may be discontinuous.

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.

30.4 1D 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.

 -- Type: gsl_interp_accel

     This workspace stores state variables 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’.

30.5 1D 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’.

30.6 1D 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’.

 -- Type: gsl_spline

     This workspace provides a higher level interface for the
     gsl_interp object

 -- 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)

30.7 1D Interpolation Example Programs
======================================

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=17\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

Figure: Cubic spline interpolation

Fig.  %s 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

Figure: Periodic cubic spline interpolation

Fig.  %s 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.

The next program illustrates the difference between the cubic spline,
Akima, and Steffen interpolation types on a difficult dataset.

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_spline.h>

     int
     main(void)
     {
       size_t i;
       const size_t N = 9;

       /* this dataset is taken from
        * J. M. Hyman, Accurate Monotonicity preserving cubic interpolation,
        * SIAM J. Sci. Stat. Comput. 4, 4, 1983. */
       const double x[] = { 7.99, 8.09, 8.19, 8.7, 9.2,
                            10.0, 12.0, 15.0, 20.0 };
       const double y[] = { 0.0, 2.76429e-5, 4.37498e-2,
                            0.169183, 0.469428, 0.943740,
                            0.998636, 0.999919, 0.999994 };

       gsl_interp_accel *acc = gsl_interp_accel_alloc();
       gsl_spline *spline_cubic = gsl_spline_alloc(gsl_interp_cspline, N);
       gsl_spline *spline_akima = gsl_spline_alloc(gsl_interp_akima, N);
       gsl_spline *spline_steffen = gsl_spline_alloc(gsl_interp_steffen, N);

       gsl_spline_init(spline_cubic, x, y, N);
       gsl_spline_init(spline_akima, x, y, N);
       gsl_spline_init(spline_steffen, x, y, N);

       for (i = 0; i < N; ++i)
         printf("%g %g\n", x[i], y[i]);

       printf("\n\n");

       for (i = 0; i <= 100; ++i)
         {
           double xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
           double yi_cubic = gsl_spline_eval(spline_cubic, xi, acc);
           double yi_akima = gsl_spline_eval(spline_akima, xi, acc);
           double yi_steffen = gsl_spline_eval(spline_steffen, xi, acc);

           printf("%g %g %g %g\n", xi, yi_cubic, yi_akima, yi_steffen);
         }

       gsl_spline_free(spline_cubic);
       gsl_spline_free(spline_akima);
       gsl_spline_free(spline_steffen);
       gsl_interp_accel_free(acc);

       return 0;
     }

Figure: Comparison of different 1D interpolation methods

The output is shown in Fig.  %s.  The cubic method exhibits a local
maxima between the 6th and 7th data points and continues oscillating for
the rest of the data.  Akima also shows a local maxima but recovers and
follows the data well after the 7th grid point.  Steffen preserves
monotonicity in all intervals and does not exhibit oscillations, at the
expense of having a discontinuous second derivative.

30.8 Introduction to 2D Interpolation
=====================================

Given a set of x coordinates x_1,...,x_m and a set of y coordinates
y_1,...,y_n, each in increasing order, plus a set of function values
z_{ij} for each grid point (x_i,y_j), the routines described in this
section compute a continuous interpolation function z(x,y) such that
z(x_i,y_j) = z_{ij}.

30.9 2D Interpolation Functions
===============================

The interpolation function for a given dataset is stored in a
gsl_interp2d object.  These are created by the following
functions.

 -- Type: gsl_interp2d

     Workspace for 2D interpolation

 -- Function: gsl_interp2d * gsl_interp2d_alloc (const gsl_interp2d_type
          * T, const size_t xsize, const size_t ysize)

     This function returns a pointer to a newly allocated interpolation
     object of type ‘T’ for ‘xsize’ grid points in the x direction and
     ‘ysize’ grid points in the y direction.

 -- Function: int gsl_interp2d_init (gsl_interp2d * interp, const
          double xa[], const double ya[], const double za[], const
          size_t xsize, const size_t ysize)

     This function initializes the interpolation object ‘interp’ for the
     data (‘xa’, ‘ya’, ‘za’) where ‘xa’ and ‘ya’ are arrays of the x and
     y grid points of size ‘xsize’ and ‘ysize’ respectively, and ‘za’ is
     an array of function values of size ‘xsize’ * ‘ysize’.  The
     interpolation object (see gsl_interp2d) does not save the
     data arrays ‘xa’, ‘ya’, and ‘za’ and only stores the static state
     computed from the data.  The ‘xa’ and ‘ya’ data arrays are always
     assumed to be strictly ordered, with increasing x,y values; the
     behavior for other arrangements is not defined.

 -- Function: void gsl_interp2d_free (gsl_interp2d * interp)

     This function frees the interpolation object ‘interp’.

30.10 2D Interpolation Grids
============================

The 2D interpolation routines access the function values z_{ij} with the
following ordering:

     z_{ij} = za[j*xsize + i]

with i = 0,...,xsize-1 and j = 0,...,ysize-1.  However, for ease of use,
the following functions are provided to add and retrieve elements from
the function grid without requiring knowledge of the internal ordering.

 -- Function: int gsl_interp2d_set (const gsl_interp2d * interp,
          double za[], const size_t i, const size_t j, const double z)

     This function sets the value z_{ij} for grid point (‘i’, ‘j’) of
     the array ‘za’ to ‘z’.

 -- Function: double gsl_interp2d_get (const gsl_interp2d * interp,
          const double za[], const size_t i, const size_t j)

     This function returns the value z_{ij} for grid point (‘i’, ‘j’)
     stored in the array ‘za’.

 -- Function: size_t gsl_interp2d_idx (const gsl_interp2d * interp,
          const size_t i, const size_t j)

     This function returns the index corresponding to the grid point
     (‘i’, ‘j’).  The index is given by j*xsize + i.

30.11 2D Interpolation Types
============================

 -- Type: gsl_interp2d_type

     The interpolation library provides the following 2D interpolation
     types:

      -- Variable: gsl_interp2d_bilinear

          Bilinear interpolation.  This interpolation method does not
          require any additional memory.

      -- Variable: gsl_interp2d_bicubic

          Bicubic interpolation.

 -- Function: const char * gsl_interp2d_name (const gsl_interp2d
          * interp)

     This function returns the name of the interpolation type used by
     ‘interp’.  For example:

          printf ("interp uses '%s' interpolation.\n", gsl_interp2d_name (interp));

     would print something like:

          interp uses 'bilinear' interpolation.

 -- Function: unsigned int gsl_interp2d_min_size (const gsl_interp2d
          * interp)
 -- Function: unsigned int gsl_interp2d_type_min_size (const
          gsl_interp2d_type * T)

     These functions return the minimum number of points required by the
     interpolation object ‘interp’ or interpolation type ‘T’.  For
     example, bicubic interpolation requires a minimum of 4 points.

30.12 2D Evaluation of Interpolating Functions
==============================================

 -- Function: double gsl_interp2d_eval (const gsl_interp2d * interp,
          const double xa[], const double ya[], const double za[], const
          double x, const double y, gsl_interp_accel * xacc,
          gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_e (const gsl_interp2d * interp,
          const double xa[], const double ya[], const double za[], const
          double x, const double y, gsl_interp_accel * xacc,
          gsl_interp_accel * yacc, double * z)

     These functions return the interpolated value of ‘z’ for a given
     point (‘x’, ‘y’), using the interpolation object ‘interp’, data
     arrays ‘xa’, ‘ya’, and ‘za’ and the accelerators ‘xacc’ and ‘yacc’.
     When ‘x’ is outside the range of ‘xa’ or ‘y’ is outside the range
     of ‘ya’, the error code GSL_EDOM is returned.

 -- Function: double gsl_interp2d_eval_extrap (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_extrap_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * z)

     These functions return the interpolated value of ‘z’ for a given
     point (‘x’, ‘y’), using the interpolation object ‘interp’, data
     arrays ‘xa’, ‘ya’, and ‘za’ and the accelerators ‘xacc’ and ‘yacc’.
     The functions perform no bounds checking, so when ‘x’ is outside
     the range of ‘xa’ or ‘y’ is outside the range of ‘ya’,
     extrapolation is performed.

 -- Function: double gsl_interp2d_eval_deriv_x (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_x_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial z /
     \partial x for a given point (‘x’, ‘y’), using the interpolation
     object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and the
     accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range of
     ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code
     GSL_EDOM is returned.

 -- Function: double gsl_interp2d_eval_deriv_y (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_y_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial z /
     \partial y for a given point (‘x’, ‘y’), using the interpolation
     object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and the
     accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range of
     ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code
     GSL_EDOM is returned.

 -- Function: double gsl_interp2d_eval_deriv_xx (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_xx_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial^2 z /
     \partial x^2 for a given point (‘x’, ‘y’), using the interpolation
     object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and the
     accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range of
     ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code
     GSL_EDOM is returned.

 -- Function: double gsl_interp2d_eval_deriv_yy (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_yy_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial^2 z /
     \partial y^2 for a given point (‘x’, ‘y’), using the interpolation
     object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and the
     accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range of
     ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code
     GSL_EDOM is returned.

 -- Function: double gsl_interp2d_eval_deriv_xy (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_xy_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial^2 z /
     \partial x \partial y for a given point (‘x’, ‘y’), using the
     interpolation object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and
     the accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range
     of ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code
     GSL_EDOM is returned.

30.13 2D Higher-level Interface
===============================

The functions described in the previous sections required the user to
supply pointers to the x, y, and z arrays on each call.  The following
functions are equivalent to the corresponding ‘gsl_interp2d’ functions
but maintain a copy of this data in the gsl_spline2d object.
This removes the need to pass ‘xa’, ‘ya’, and ‘za’ as arguments on each
evaluation.  These functions are defined in the header file
‘gsl_spline2d.h’.

 -- Type: gsl_spline2d

     This workspace provides a higher level interface for the
     gsl_interp2d object

 -- Function: gsl_spline2d * gsl_spline2d_alloc (const gsl_interp2d_type
          * T, size_t xsize, size_t ysize)

 -- Function: int gsl_spline2d_init (gsl_spline2d * spline, const
          double xa[], const double ya[], const double za[],
          size_t xsize, size_t ysize)

 -- Function: void gsl_spline2d_free (gsl_spline2d * spline)

 -- Function: const char * gsl_spline2d_name (const gsl_spline2d
          * spline)

 -- Function: unsigned int gsl_spline2d_min_size (const gsl_spline2d
          * spline)

 -- Function: double gsl_spline2d_eval (const gsl_spline2d * spline,
          const double x, const double y, gsl_interp_accel * xacc,
          gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_e (const gsl_spline2d * spline,
          const double x, const double y, gsl_interp_accel * xacc,
          gsl_interp_accel * yacc, double * z)

 -- Function: double gsl_spline2d_eval_deriv_x (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_x_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: double gsl_spline2d_eval_deriv_y (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_y_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: double gsl_spline2d_eval_deriv_xx (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_xx_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: double gsl_spline2d_eval_deriv_yy (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_yy_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: double gsl_spline2d_eval_deriv_xy (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_xy_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: int gsl_spline2d_set (const gsl_spline2d * spline,
          double za[], const size_t i, const size_t j, const double z)

 -- Function: double gsl_spline2d_get (const gsl_spline2d * spline,
          const double za[], const size_t i, const size_t j)

     This function returns the value z_{ij} for grid point (‘i’, ‘j’)
     stored in the array ‘za’.

30.14 2D Interpolation Example programs
=======================================

The following example performs bilinear interpolation on the unit
square, using z values of (0,1,0.5,1) going clockwise around the square.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_interp2d.h>
     #include <gsl/gsl_spline2d.h>

     int
     main()
     {
       const gsl_interp2d_type *T = gsl_interp2d_bilinear;
       const size_t N = 100;             /* number of points to interpolate */
       const double xa[] = { 0.0, 1.0 }; /* define unit square */
       const double ya[] = { 0.0, 1.0 };
       const size_t nx = sizeof(xa) / sizeof(double); /* x grid points */
       const size_t ny = sizeof(ya) / sizeof(double); /* y grid points */
       double *za = malloc(nx * ny * sizeof(double));
       gsl_spline2d *spline = gsl_spline2d_alloc(T, nx, ny);
       gsl_interp_accel *xacc = gsl_interp_accel_alloc();
       gsl_interp_accel *yacc = gsl_interp_accel_alloc();
       size_t i, j;

       /* set z grid values */
       gsl_spline2d_set(spline, za, 0, 0, 0.0);
       gsl_spline2d_set(spline, za, 0, 1, 1.0);
       gsl_spline2d_set(spline, za, 1, 1, 0.5);
       gsl_spline2d_set(spline, za, 1, 0, 1.0);

       /* initialize interpolation */
       gsl_spline2d_init(spline, xa, ya, za, nx, ny);

       /* interpolate N values in x and y and print out grid for plotting */
       for (i = 0; i < N; ++i)
         {
           double xi = i / (N - 1.0);

           for (j = 0; j < N; ++j)
             {
               double yj = j / (N - 1.0);
               double zij = gsl_spline2d_eval(spline, xi, yj, xacc, yacc);

               printf("%f %f %f\n", xi, yj, zij);
             }
           printf("\n");
         }

       gsl_spline2d_free(spline);
       gsl_interp_accel_free(xacc);
       gsl_interp_accel_free(yacc);
       free(za);

       return 0;
     }

The results of the interpolation are shown in Fig.  %s, where the
corners are labeled with their fixed z values.

Figure: 2D interpolation example

30.15 References and Further Reading
====================================

Descriptions of the interpolation algorithms and further references can
be found in the following publications:

   * 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.

   * M. Steffen, `A simple method for monotonic interpolation in one
     dimension', Astron.  Astrophys.  239, 443-450, 1990.

31 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’.

31.1 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.

31.2 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)
     {
       (void)(params); /* avoid unused parameter warning */
       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,

     f(x) = x^(3/2)
     x = 2.0
     f'(x) = 2.1213203120 +/- 0.0000005006
     exact = 2.1213203436

     x = 0.0
     f'(x) = 0.0000000160 +/- 0.0000000339
     exact = 0.0000000000

31.3 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.

32 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’.

32.1 Definitions
================

 -- Type: gsl_cheb_series

     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.

32.2 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.

32.3 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’.

32.4 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.

32.5 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’.

32.6 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)
     {
       (void)(p); /* avoid unused parameter warning */

       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;
     }

Fig.  %s shows output from the program with the original function, 10-th
order approximation and 40-th order approximation, all sampled at
intervals of 0.001 in x.

Figure: Chebyshev approximations to a step function

32.7 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)

33 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’.

33.1 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.

 -- Type: gsl_sum_levin_u_workspace

     Workspace for a Leven u-transform.

 -- 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.

33.2 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.

 -- Type: gsl_sum_levin_utrunc_workspace

     Workspace for a Levin u-transform without error estimation

 -- 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.

33.3 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 + \dots

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 %zu terms\n",
               w->sum_plain, w->terms_used);

       printf ("exact value      = % .16f\n", zeta_2);
       printf ("accelerated sum  = % .16f using %zu 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.

     term-by-term sum =  1.5961632439130233 using 20 terms
     term-by-term sum =  1.5759958390005426 using 13 terms
     exact value      =  1.6449340668482264
     accelerated sum  =  1.6449340669228176 using 13 terms
     estimated error  =  0.0000000000888360
     actual error     =  0.0000000000745912

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.

33.4 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’

34 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’.

34.1 Definitions
================

The continuous wavelet transform and its inverse are defined by the
relations,

     w(s, \tau) = \int_{-\infty}^\infty f(t) * \psi^*_{s,\tau}(t) dt

and,

     f(t) = \int_0^\infty ds \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau

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'.

34.2 Initialization
===================

 -- Type: gsl_wavelet

     This 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:

 -- Type: gsl_wavelet_type

      -- Variable: gsl_wavelet_daubechies
      -- Variable: 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,
          \dots, 20, with ‘k’ even.

      -- Variable: gsl_wavelet_haar
      -- Variable: gsl_wavelet_haar_centered

          This is the Haar wavelet.  The only valid choice of k for the
          Haar wavelet is k=2.

      -- Variable: gsl_wavelet_bspline
      -- Variable: 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’.

 -- Type: gsl_wavelet_workspace

     This 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’.

34.3 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.

34.3.1 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 \rightarrow w_{j,k}
     in a packed triangular storage layout, where ‘j’ is the index of
     the level j = 0 \dots J-1 and ‘k’ is the index of the coefficient
     within each level, k = 0 \dots 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},\cdots, d_{j,k},\cdots, 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.

34.3.2 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 ‘m’.

 -- 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 ‘m’.

34.4 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)
     {
       (void)(argc); /* avoid unused parameter warning */
       int i, n = 256, nc = 20;
       double *orig_data = malloc (n * sizeof (double));
       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", &orig_data[i]);
           data[i] = orig_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 %g\n", orig_data[i], data[i]);
         }

       gsl_wavelet_free (w);
       gsl_wavelet_workspace_free (work);

       free (data);
       free (orig_data);
       free (abscoeff);
       free (p);
       return 0;
     }

The output can be used with the GNU plotutils ‘graph’ program:

     $ ./a.out ecg.dat > dwt.txt
     $ graph -T ps -x 0 256 32 -h 0.3 -a dwt.txt > dwt.ps

Fig.  %s shows an original and compressed version of a sample ECG
recording from the MIT-BIH Arrhythmia Database, part of the PhysioNet
archive of public-domain of medical datasets.

Figure: Original (upper) and wavelet-compressed (lower) ECG signals,
using the 20 largest components of the Daubechies(4) discrete wavelet
transform.

34.5 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.

35 Discrete Hankel Transforms
*****************************

This chapter describes functions for performing Discrete Hankel
Transforms (DHTs).  The functions are declared in the header file
‘gsl_dht.h’.

35.1 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
zeros 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.

Starting with its definition, the Hankel transform (or Bessel transform)
of order \nu of a function f with \nu > -1/2 is defined as (see Johnson,
1987 and Lemoine, 1994)

     F_\nu(u) = \int_0^\infty f(t) J_\nu(u t) t dt

If the integral exists, F_\nu is called the Hankel transformation of f.
The reverse transform is given by

     f(t) = \int_0^\infty F_\nu(u) J_\nu(u t) u du

where \int_0^\infty f(t) t^{1/2} dt must exist and be absolutely
convergent, and where f(t) satisfies Dirichlet’s conditions (of limited
total fluctuations) in the interval [0,\infty].

Now the discrete Hankel transform works on a discrete function f, which
is sampled on points n=1...M located at positions
t_n=(j_{\nu,n}/j_{\nu,M}) X in real space and at u_n=j_{\nu,n}/X in
reciprocal space.  Here, j_{\nu,m} are the m-th zeros of the Bessel
function J_\nu(x) arranged in ascending order.  Moreover, the discrete
functions are assumed to be band limited, so f(t_n)=0 and F(u_n)=0 for
n>M. Accordingly, the function f is defined on the interval [0,X].

Following the work of Johnson, 1987 and Lemoine, 1994, the discrete
Hankel transform is given by

     F_\nu(u_m) = (2 X^2 / j_(\nu,M)^2)
           \sum_{k=1}^{M-1} f(j_(\nu,k) X/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 calculated by GSL. In GSL, forward and backward transforms are
defined equally and calculate F_\nu(u_m).  Following Johnson, the
backward transform reads

     f(t_k) = (2 / X^2)
           \sum_{m=1}^{M-1} F(j_(\nu,m)/X)
               (J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,m))^2).

Obviously, using the forward transform instead of the backward transform
gives an additional factor X^4/j_{\nu,M}^2=t_m^2/u_m^2.

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 to define the length X of the fundamental interval, for
convenience, while discrete Hankel transforms are often defined on the
unit interval instead of [0,X].

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.

35.2 Functions
==============

 -- Type: gsl_dht

     Workspace for computing discrete Hankel transforms

 -- 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 (X^2/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}.

35.3 References and Further Reading
===================================

The algorithms used by these functions are described in the following
papers,

   *
       8. Fisk Johnson, Comp.: Phys.: Comm.: 43, 181 (1987).

   *
       4. Lemoine, J. Chem.: Phys.: 101, 3936 (1994).

36 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.

36.1 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.

36.2 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.

36.3 Initializing the Solver
============================

 -- Type: gsl_root_fsolver

     This is a workspace for finding roots using methods which do not
     require derivatives.

 -- Type: gsl_root_fdfsolver

     This is a workspace for finding roots using methods which require
     derivatives.

 -- 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’.

36.4 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:

 -- 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’.

 -- 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) = \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’.

36.5 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.

36.6 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’.

36.7 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.

36.8 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.

 -- Type: gsl_root_fsolver_type

      -- Variable: 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.

      -- Variable: 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.

      -- Variable: 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.

36.9 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.

 -- Type: gsl_root_fdfsolver_type

      -- Variable: 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.

      -- Variable: 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.

      -- Variable: 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.

36.10 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;

       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.

36.11 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

37 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.

37.1 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, as shown in
Fig.  %s.

Figure: Function with lower and upper bounds with an estimate of the
minimum.

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).

37.2 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.

37.3 Initializing the Minimizer
===============================

 -- Type: gsl_min_fminimizer

     This is a workspace for minimizing functions.

 -- 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’.

37.4 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 (see
Providing the function to solve).

37.5 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’.

37.6 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.

37.7 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.

 -- Type: gsl_min_fminimizer_type

      -- Variable: 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 \approx 0.3819660 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.

      -- Variable: 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.

      -- Variable: gsl_min_fminimizer_quad_golden

          This is a variant of Brent’s algorithm which uses the
          safeguarded step-length algorithm of Gill and Murray.

37.8 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)
     {
       (void)(params); /* avoid unused parameter warning */
       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.

     using brent method
      iter [    lower,     upper]       min        err  err(est)
         0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
         1 [2.0000000, 6.0000000] 3.5278640 +0.3862713 4.0000000
         2 [2.0000000, 3.5278640] 3.1748217 +0.0332290 1.5278640
         3 [2.0000000, 3.1748217] 3.1264576 -0.0151351 1.1748217
         4 [3.1264576, 3.1748217] 3.1414743 -0.0001183 0.0483641
         5 [3.1414743, 3.1748217] 3.1415930 +0.0000004 0.0333474
     Converged:
         6 [3.1414743, 3.1415930] 3.1415927 +0.0000000 0.0001187

37.9 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.

38 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.

38.1 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} =
\partial f_i / \partial 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.

38.2 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.

 -- Type: gsl_multiroot_fsolver

     This is a workspace for multidimensional root-finding without
     derivatives.

 -- Type: gsl_multiroot_fdfsolver

     This is a workspace for multidimensional root-finding with
     derivatives.

 -- 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’.

38.3 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:

 -- 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;

 -- 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.

38.4 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’.

38.5 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.

38.6 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.

 -- Type: gsl_multiroot_fdfsolver_type

     The following are available algorithms for minimizing functions
     using derivatives.

      -- Variable: 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. Moré, 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.

      -- Variable: 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.

      -- Variable: 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.

      -- Variable: 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.

38.7 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.

 -- Type: gsl_multiroot_fsolver_type

     The following are available algorithms for minimizing functions
     without derivatives.

      -- Variable: 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.

      -- Variable: gsl_multiroot_fsolver_hybrid

          This is a finite difference version of the Hybrid algorithm
          without internal scaling.

      -- Variable: 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.

      -- Variable: 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.

38.8 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 gsl_multiroot_fsolver_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
gsl_multiroot_fsolver_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
gsl_multiroot_fdfsolver_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
gsl_multiroot_fdfsolver_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.

38.9 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. Moré, 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. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained
     Optimization Software”, ACM Transactions on Mathematical Software,
     Vol 7, No 1 (1981), p 17–41

39 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.

39.1 Overview
=============

The problem of multidimensional minimization requires finding a point x
such that the scalar function,

     f(x_1, \dots, 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.

39.2 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.

39.3 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.

 -- Type: gsl_multimin_fdfminimizer

     This is a workspace for minimizing functions using derivatives.

 -- Type: gsl_multimin_fminimizer

     This is a workspace for minimizing functions without derivatives.

 -- 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 p \cdot 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’.

39.4 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:

 -- 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.

 -- 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;

39.5 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: gsl_vector * gsl_multimin_fdfminimizer_dx (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,
     the last step increment of the estimate, 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.

39.6 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.

39.7 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.

 -- Type: gsl_multimin_fdfminimizer_type

     This type specifies a minimization algorithm using gradients.

      -- Variable: 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 p\cdot 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.

      -- Variable: 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.

      -- Variable: gsl_multimin_fdfminimizer_vector_bfgs2
      -- Variable: 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).

      -- Variable: 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.

39.8 Algorithms without Derivatives
===================================

The algorithms described in this section use only the value of the
function at each evaluation point.

 -- Type: gsl_multimin_fminimizer_type

     This type specifies minimization algorithms which do not use
     gradients.

      -- Variable: gsl_multimin_fminimizer_nmsimplex2
      -- Variable: gsl_multimin_fminimizer_nmsimplex

          These methods use the Simplex algorithm of Nelder and Mead.
          Starting from the initial vector :data:`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 gsl_multimin_fminimizer_nmsimplex2 version of
          this minimiser is a new O(N) operations implementation of the
          earlier O(N^2) operations
          gsl_multimin_fminimizer_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.

      -- Variable: gsl_multimin_fminimizer_nmsimplex2rand

          This method is a variant of
          gsl_multimin_fminimizer_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.

39.9 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 in Fig.  %s.

Figure: Function contours with path taken by minimization algorithm

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.

39.10 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.

40 Linear 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.

40.1 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, \dots\}.  The weight
factors w_i are given by w_i = 1/\sigma_i^2 where \sigma_i is the
experimental error on the data-point y_i.  The errors are assumed to be
Gaussian and uncorrelated.  For unweighted data the chi-squared sum is
computed without any weight factors.

The fitting routines return the best-fit parameters c and their p \times
p covariance matrix.  The covariance matrix measures the statistical
errors on the best-fit parameters resulting from the errors on the data,
\sigma_i, and is defined as

     C_{ab} = <\delta c_a \delta c_b>

where \langle \, \rangle denotes an average over the Gaussian error
distributions of the underlying datapoints.

The covariance matrix is calculated by error propagation from the data
errors \sigma_i.  The change in a fitted parameter \delta c_a caused by
a small change in the data \delta y_i is given by

     \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}}.

40.2 Linear regression
======================

The functions in this section are used to fit simple one or two
parameter linear regression models.  The functions are declared in the
header file ‘gsl_fit.h’.

40.2.1 Linear regression with a constant term
---------------------------------------------

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(), 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’.

40.2.2 Linear regression 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’.

40.3 Multi-parameter regression
===============================

This section describes routines which perform least squares fits to a
linear model by minimizing the cost function

     \chi^2 = \sum_i w_i (y_i - \sum_j X_{ij} c_j)^2 = || y - Xc ||_W^2

where y is a vector of n observations, X is an n-by-p matrix of
predictor variables, c is a vector of the p unknown best-fit parameters
to be estimated, and ||r||_W^2 = r^T W r.  The matrix W =
\diag(w_1,w_2,...,w_n) defines the weights or uncertainties of the
observation vector.

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, \ldots, \omega_p, use,

     X_{ij} = \sin(\omega_j x_i)

To fit to p independent variables x_1, x_2, \ldots, x_p, use,

     X_{ij} = x_j(i)

where x_j(i) is the i-th value of the predictor variable x_j.

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.

These functions are declared in the header file ‘gsl_multifit.h’.

 -- Type: gsl_multifit_linear_workspace

     This workspace contains internal variables for fitting
     multi-parameter models.

 -- Function: gsl_multifit_linear_workspace * gsl_multifit_linear_alloc
          (const size_t n, const size_t p)

     This function allocates a workspace for fitting a model to a
     maximum of ‘n’ observations using a maximum of ‘p’ parameters.  The
     user may later supply a smaller least squares system if desired.
     The size of the workspace is O(np + p^2).

 -- 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_svd (const gsl_matrix * X,
          gsl_multifit_linear_workspace * work)

     This function performs a singular value decomposition of the matrix
     ‘X’ and stores the SVD factors internally in ‘work’.

 -- Function: int gsl_multifit_linear_bsvd (const gsl_matrix * X,
          gsl_multifit_linear_workspace * work)

     This function performs a singular value decomposition of the matrix
     ‘X’ and stores the SVD factors internally in ‘work’.  The matrix
     ‘X’ is first balanced by applying column scaling factors to improve
     the accuracy of the singular values.

 -- 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_linear_tsvd (const gsl_matrix * X, const
          gsl_vector * y, const double tol, gsl_vector * c, gsl_matrix
          * cov, double * chisq, size_t * rank,
          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 a truncated SVD expansion.  Singular values which
     satisfy s_i \le tol \times s_0 are discarded from the fit, where
     s_0 is the largest singular value.  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’.  The effective rank (number of
     singular values used in solution) is returned in ‘rank’.  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().

 -- 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_wlinear_tsvd (const gsl_matrix * X, const
          gsl_vector * w, const gsl_vector * y, const double tol,
          gsl_vector * c, gsl_matrix * cov, double * chisq, size_t
          * rank, 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 a truncated SVD expansion.
     Singular values which satisfy s_i \le tol \times s_0 are discarded
     from the fit, where s_0 is the largest singular value.  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’.  The effective rank of
     the system (number of singular values used in the solution) is
     returned in ‘rank’.  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_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’.

 -- Function: size_t gsl_multifit_linear_rank (const double tol, const
          gsl_multifit_linear_workspace * work)

     This function returns the rank of the matrix X which must first
     have its singular value decomposition computed.  The rank is
     computed by counting the number of singular values \sigma_j which
     satisfy \sigma_j > tol \times \sigma_0, where \sigma_0 is the
     largest singular value.

40.4 Regularized regression
===========================

Ordinary weighted least squares models seek a solution vector c which
minimizes the residual

     \chi^2 = || y - Xc ||_W^2

where y is the n-by-1 observation vector, X is the n-by-p design matrix,
c is the p-by-1 solution vector, W = \diag(w_1,...,w_n) is the data
weighting matrix, and ||r||_W^2 = r^T W r.  In cases where the least
squares matrix X is ill-conditioned, small perturbations (ie: noise) in
the observation vector could lead to widely different solution vectors
c.  One way of dealing with ill-conditioned matrices is to use a
“truncated SVD” in which small singular values, below some given
tolerance, are discarded from the solution.  The truncated SVD method is
available using the functions gsl_multifit_linear_tsvd() and
gsl_multifit_wlinear_tsvd().  Another way to help solve
ill-posed problems is to include a regularization term in the least
squares minimization

     \chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2

for a suitably chosen regularization parameter \lambda and matrix L.
This type of regularization is known as Tikhonov, or ridge, regression.
In some applications, L is chosen as the identity matrix, giving
preference to solution vectors c with smaller norms.  Including this
regularization term leads to the explicit “normal equations” solution

     c = ( X^T W X + \lambda^2 L^T L )^-1 X^T W y

which reduces to the ordinary least squares solution when L = 0.  In
practice, it is often advantageous to transform a regularized least
squares system into the form

     \chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2

This is known as the Tikhonov “standard form” and has the normal
equations solution

     \tilde{c} = ( \tilde{X}^T \tilde{X} + \lambda^2 I )^{-1} \tilde{X}^T \tilde{y}

For an m-by-p matrix L which is full rank and has m >= p (ie: L is
square or has more rows than columns), we can calculate the “thin” QR
decomposition of L, and note that ||L c|| = ||R c|| since the Q factor
will not change the norm.  Since R is p-by-p, we can then use the
transformation

     X~ = sqrt(W) X R^-1
     y~ = sqrt(W) y
     c~ = R c

to achieve the standard form.  For a rectangular matrix L with m < p, a
more sophisticated approach is needed (see Hansen 1998, chapter 2.3).
In practice, the normal equations solution above is not desirable due to
numerical instabilities, and so the system is solved using the singular
value decomposition of the matrix \tilde{X}.  The matrix L is often
chosen as the identity matrix, or as a first or second finite difference
operator, to ensure a smoothly varying coefficient vector c, or as a
diagonal matrix to selectively damp each model parameter differently.
If L \ne I, the user must first convert the least squares problem to
standard form using gsl_multifit_linear_stdform1() or
gsl_multifit_linear_stdform2(), solve the system, and then
backtransform the solution vector to recover the solution of the
original problem (see gsl_multifit_linear_genform1() and
gsl_multifit_linear_genform2()).

In many regularization problems, care must be taken when choosing the
regularization parameter \lambda.  Since both the residual norm ||y - X
c|| and solution norm ||L c|| are being minimized, the parameter \lambda
represents a tradeoff between minimizing either the residuals or the
solution vector.  A common tool for visualizing the comprimise between
the minimization of these two quantities is known as the L-curve.  The
L-curve is a log-log plot of the residual norm ||y - X c|| on the
horizontal axis and the solution norm ||L c|| on the vertical axis.
This curve nearly always as an L shaped appearance, with a distinct
corner separating the horizontal and vertical sections of the curve.
The regularization parameter corresponding to this corner is often
chosen as the optimal value.  GSL provides routines to calculate the
L-curve for all relevant regularization parameters as well as locating
the corner.

Another method of choosing the regularization parameter is known as
Generalized Cross Validation (GCV). This method is based on the idea
that if an arbitrary element y_i is left out of the right hand side, the
resulting regularized solution should predict this element accurately.
This leads to choosing the parameter \lambda which minimizes the GCV
function

     G(\lambda) = (||y - X c_{\lambda}||^2) / Tr(I_n - X X^I)^2

where X_{\lambda}^I is the matrix which relates the solution c_{\lambda}
to the right hand side y, ie: c_{\lambda} = X_{\lambda}^I y.  GSL
provides routines to compute the GCV curve and its minimum.

For most applications, the steps required to solve a regularized least
squares problem are as follows:

  1. Construct the least squares system (X, y, W, L)

  2. Transform the system to standard form (\tilde{X}, \tilde{y}).  This
     step can be skipped if L = I_p and W = I_n.

  3. Calculate the SVD of \tilde{X}.

  4. Determine an appropriate regularization parameter \lambda (using
     for example L-curve or GCV analysis).

  5. Solve the standard form system using the chosen \lambda and the SVD
     of \tilde{X}.

  6. Backtransform the standard form solution \tilde{c} to recover the
     original solution vector c.

 -- Function: int gsl_multifit_linear_stdform1 (const gsl_vector * L,
          const gsl_matrix * X, const gsl_vector * y, gsl_matrix * Xs,
          gsl_vector * ys, gsl_multifit_linear_workspace * work)
 -- Function: int gsl_multifit_linear_wstdform1 (const gsl_vector * L,
          const gsl_matrix * X, const gsl_vector * w, const gsl_vector
          * y, gsl_matrix * Xs, gsl_vector * ys,
          gsl_multifit_linear_workspace * work)

     These functions define a regularization matrix L =
     \diag(l_0,l_1,...,l_{p-1}).  The diagonal matrix element l_i is
     provided by the i-th element of the input vector ‘L’.  The n-by-p
     least squares matrix ‘X’ and vector ‘y’ of length n are then
     converted to standard form as described above and the parameters
     (\tilde{X}, \tilde{y}) are stored in ‘Xs’ and ‘ys’ on output.  ‘Xs’
     and ‘ys’ have the same dimensions as ‘X’ and ‘y’.  Optional data
     weights may be supplied in the vector ‘w’ of length n.  In order to
     apply this transformation, L^{-1} must exist and so none of the l_i
     may be zero.  After the standard form system has been solved, use
     gsl_multifit_linear_genform1() to recover the original
     solution vector.  It is allowed to have ‘X’ = ‘Xs’ and ‘y’ = ‘ys’
     for an in-place transform.  In order to perform a weighted
     regularized fit with L = I, the user may call
     gsl_multifit_linear_applyW() to convert to standard form.

 -- Function: int gsl_multifit_linear_L_decomp (gsl_matrix * L,
          gsl_vector * tau)

     This function factors the m-by-p regularization matrix ‘L’ into a
     form needed for the later transformation to standard form.  ‘L’ may
     have any number of rows m.  If m \ge p the QR decomposition of ‘L’
     is computed and stored in ‘L’ on output.  If m < p, the QR
     decomposition of L^T is computed and stored in ‘L’ on output.  On
     output, the Householder scalars are stored in the vector ‘tau’ of
     size MIN(m,p).  These outputs will be used by
     gsl_multifit_linear_wstdform2() to complete the
     transformation to standard form.

 -- Function: int gsl_multifit_linear_stdform2 (const gsl_matrix * LQR,
          const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * y, gsl_matrix * Xs, gsl_vector * ys, gsl_matrix
          * M, gsl_multifit_linear_workspace * work)
 -- Function: int gsl_multifit_linear_wstdform2 (const gsl_matrix * LQR,
          const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * w, const gsl_vector * y, gsl_matrix * Xs,
          gsl_vector * ys, gsl_matrix * M, gsl_multifit_linear_workspace
          * work)

     These functions convert the least squares system (‘X’, ‘y’, ‘W’, L)
     to standard form (\tilde{X}, \tilde{y}) which are stored in ‘Xs’
     and ‘ys’ respectively.  The m-by-p regularization matrix ‘L’ is
     specified by the inputs ‘LQR’ and ‘Ltau’, which are outputs from
     gsl_multifit_linear_L_decomp().  The dimensions of the
     standard form parameters (\tilde{X}, \tilde{y}) depend on whether m
     is larger or less than p.  For m \ge p, ‘Xs’ is n-by-p, ‘ys’ is
     n-by-1, and ‘M’ is not used.  For m < p, ‘Xs’ is (n - p + m)-by-m,
     ‘ys’ is (n - p + m)-by-1, and ‘M’ is additional n-by-p workspace,
     which is required to recover the original solution vector after the
     system has been solved (see gsl_multifit_linear_genform2()).
     Optional data weights may be supplied in the vector ‘w’ of
     length n, where W = \diag(w).

 -- Function: int gsl_multifit_linear_solve (const double lambda, const
          gsl_matrix * Xs, const gsl_vector * ys, gsl_vector * cs,
          double * rnorm, double * snorm, gsl_multifit_linear_workspace
          * work)

     This function computes the regularized best-fit parameters
     \tilde{c} which minimize the cost function \chi^2 = || \tilde{y} -
     \tilde{X} \tilde{c} ||^2 + \lambda^2 || \tilde{c} ||^2 which is in
     standard form.  The least squares system must therefore be
     converted to standard form prior to calling this function.  The
     observation vector \tilde{y} is provided in ‘ys’ and the matrix of
     predictor variables \tilde{X} in ‘Xs’.  The solution vector
     \tilde{c} is returned in ‘cs’, which has length min(m,p).  The SVD
     of ‘Xs’ must be computed prior to calling this function, using
     gsl_multifit_linear_svd().  The regularization parameter
     \lambda is provided in ‘lambda’.  The residual norm || \tilde{y} -
     \tilde{X} \tilde{c} || = ||y - X c||_W is returned in ‘rnorm’.  The
     solution norm || \tilde{c} || = ||L c|| is returned in ‘snorm’.

 -- Function: int gsl_multifit_linear_genform1 (const gsl_vector * L,
          const gsl_vector * cs, gsl_vector * c,
          gsl_multifit_linear_workspace * work)

     After a regularized system has been solved with L =
     \diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the
     standard form solution vector ‘cs’ to recover the solution vector
     of the original problem ‘c’.  The diagonal matrix elements l_i are
     provided in the vector ‘L’.  It is allowed to have ‘c’ = ‘cs’ for
     an in-place transform.

 -- Function: int gsl_multifit_linear_genform2 (const gsl_matrix * LQR,
          const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * y, const gsl_vector * cs, const gsl_matrix * M,
          gsl_vector * c, gsl_multifit_linear_workspace * work)
 -- Function: int gsl_multifit_linear_wgenform2 (const gsl_matrix * LQR,
          const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * w, const gsl_vector * y, const gsl_vector * cs,
          const gsl_matrix * M, gsl_vector * c,
          gsl_multifit_linear_workspace * work)

     After a regularized system has been solved with a general
     rectangular matrix L, specified by (‘LQR’, ‘Ltau’), this function
     backtransforms the standard form solution ‘cs’ to recover the
     solution vector of the original problem, which is stored in ‘c’, of
     length p.  The original least squares matrix and observation vector
     are provided in ‘X’ and ‘y’ respectively.  ‘M’ is the matrix
     computed by gsl_multifit_linear_stdform2().  For
     weighted fits, the weight vector ‘w’ must also be supplied.

 -- Function: int gsl_multifit_linear_applyW (const gsl_matrix * X,
          const gsl_vector * w, const gsl_vector * y, gsl_matrix * WX,
          gsl_vector * Wy)

     For weighted least squares systems with L = I, this function may be
     used to convert the system to standard form by applying the weight
     matrix W = \diag(w) to the least squares matrix ‘X’ and observation
     vector ‘y’.  On output, ‘WX’ is equal to W^{1/2} X and ‘Wy’ is
     equal to W^{1/2} y.  It is allowed for ‘WX’ = ‘X’ and ‘Wy’ = ‘y’
     for an in-place transform.

 -- Function: int gsl_multifit_linear_lcurve (const gsl_vector * y,
          gsl_vector * reg_param, gsl_vector * rho, gsl_vector * eta,
          gsl_multifit_linear_workspace * work)

     This function computes the L-curve for a least squares system using
     the right hand side vector ‘y’ and the SVD decomposition of the
     least squares matrix ‘X’, which must be provided to
     gsl_multifit_linear_svd() prior to calling this function.
     The output vectors ‘reg_param’, ‘rho’, and ‘eta’ must all be the
     same size, and will contain the regularization parameters
     \lambda_i, residual norms ||y - X c_i||, and solution norms || L
     c_i || which compose the L-curve, where c_i is the regularized
     solution vector corresponding to \lambda_i.  The user may determine
     the number of points on the L-curve by adjusting the size of these
     input arrays.  The regularization parameters \lambda_i are
     estimated from the singular values of ‘X’, and chosen to represent
     the most relevant portion of the L-curve.

 -- Function: int gsl_multifit_linear_lcorner (const gsl_vector * rho,
          const gsl_vector * eta, size_t * idx)

     This function attempts to locate the corner of the L-curve (||y - X
     c||, ||L c||) defined by the ‘rho’ and ‘eta’ input arrays
     respectively.  The corner is defined as the point of maximum
     curvature of the L-curve in log-log scale.  The ‘rho’ and ‘eta’
     arrays can be outputs of gsl_multifit_linear_lcurve().
     The algorithm used simply fits a circle to 3 consecutive points on
     the L-curve and uses the circle’s radius to determine the curvature
     at the middle point.  Therefore, the input array sizes must be \ge
     3.  With more points provided for the L-curve, a better estimate of
     the curvature can be obtained.  The array index corresponding to
     maximum curvature (ie: the corner) is returned in ‘idx’.  If the
     input arrays contain colinear points, this function could fail and
     return GSL_EINVAL.

 -- Function: int gsl_multifit_linear_lcorner2 (const gsl_vector
          * reg_param, const gsl_vector * eta, size_t * idx)

     This function attempts to locate the corner of an alternate L-curve
     (\lambda^2, ||L c||^2) studied by Rezghi and Hosseini, 2009.  This
     alternate L-curve can provide better estimates of the
     regularization parameter for smooth solution vectors.  The
     regularization parameters \lambda and solution norms ||L c|| are
     provided in the ‘reg_param’ and ‘eta’ input arrays respectively.
     The corner is defined as the point of maximum curvature of this
     alternate L-curve in linear scale.  The ‘reg_param’ and ‘eta’
     arrays can be outputs of gsl_multifit_linear_lcurve().
     The algorithm used simply fits a circle to 3 consecutive points on
     the L-curve and uses the circle’s radius to determine the curvature
     at the middle point.  Therefore, the input array sizes must be \ge
     3.  With more points provided for the L-curve, a better estimate of
     the curvature can be obtained.  The array index corresponding to
     maximum curvature (ie: the corner) is returned in ‘idx’.  If the
     input arrays contain colinear points, this function could fail and
     return GSL_EINVAL.

 -- Function: int gsl_multifit_linear_gcv_init (const gsl_vector * y,
          gsl_vector * reg_param, gsl_vector * UTy, double * delta0,
          gsl_multifit_linear_workspace * work)

     This function performs some initialization in preparation for
     computing the GCV curve and its minimum.  The right hand side
     vector is provided in ‘y’.  On output, ‘reg_param’ is set to a
     vector of regularization parameters in decreasing order and may be
     of any size.  The vector ‘UTy’ of size p is set to U^T y.  The
     parameter ‘delta0’ is needed for subsequent steps of the GCV
     calculation.

 -- Function: int gsl_multifit_linear_gcv_curve (const gsl_vector
          * reg_param, const gsl_vector * UTy, const double delta0,
          gsl_vector * G, gsl_multifit_linear_workspace * work)

     This funtion calculates the GCV curve G(\lambda) and stores it in
     ‘G’ on output, which must be the same size as ‘reg_param’.  The
     inputs ‘reg_param’, ‘UTy’ and ‘delta0’ are computed in
     gsl_multifit_linear_gcv_init().

 -- Function: int gsl_multifit_linear_gcv_min (const gsl_vector
          * reg_param, const gsl_vector * UTy, const gsl_vector * G,
          const double delta0, double * lambda,
          gsl_multifit_linear_workspace * work)

     This function computes the value of the regularization parameter
     which minimizes the GCV curve G(\lambda) and stores it in ‘lambda’.
     The input ‘G’ is calculated by
     gsl_multifit_linear_gcv_curve() and the inputs ‘reg_param’,
     ‘UTy’ and ‘delta0’ are computed by
     gsl_multifit_linear_gcv_init().

 -- Function: double gsl_multifit_linear_gcv_calc (const double lambda,
          const gsl_vector * UTy, const double delta0,
          gsl_multifit_linear_workspace * work)

     This function returns the value of the GCV curve G(\lambda)
     corresponding to the input ‘lambda’.

 -- Function: int gsl_multifit_linear_gcv (const gsl_vector * y,
          gsl_vector * reg_param, gsl_vector * G, double * lambda,
          double * G_lambda, gsl_multifit_linear_workspace * work)

     This function combines the steps ‘gcv_init’, ‘gcv_curve’, and
     ‘gcv_min’ defined above into a single function.  The input ‘y’ is
     the right hand side vector.  On output, ‘reg_param’ and ‘G’, which
     must be the same size, are set to vectors of \lambda and G(\lambda)
     values respectively.  The output ‘lambda’ is set to the optimal
     value of \lambda which minimizes the GCV curve.  The minimum value
     of the GCV curve is returned in ‘G_lambda’.

 -- Function: int gsl_multifit_linear_Lk (const size_t p, const
          size_t k, gsl_matrix * L)

     This function computes the discrete approximation to the derivative
     operator L_k of order ‘k’ on a regular grid of ‘p’ points and
     stores it in ‘L’.  The dimensions of ‘L’ are (p-k)-by-p.

 -- Function: int gsl_multifit_linear_Lsobolev (const size_t p, const
          size_t kmax, const gsl_vector * alpha, gsl_matrix * L,
          gsl_multifit_linear_workspace * work)

     This function computes the regularization matrix ‘L’ corresponding
     to the weighted Sobolov norm ||L c||^2 = \sum_k \alpha_k^2 ||L_k
     c||^2 where L_k approximates the derivative operator of order k.
     This regularization norm can be useful in applications where it is
     necessary to smooth several derivatives of the solution.  ‘p’ is
     the number of model parameters, ‘kmax’ is the highest derivative to
     include in the summation above, and alpha is the vector
     of weights of size ‘kmax’ + 1, where ‘alpha[k]’ = \alpha_k is the
     weight assigned to the derivative of order k.  The output matrix
     ‘L’ is size ‘p’-by-‘p’ and upper triangular.

 -- Function: double gsl_multifit_linear_rcond (const
          gsl_multifit_linear_workspace * work)

     This function returns the reciprocal condition number of the least
     squares matrix X, defined as the ratio of the smallest and largest
     singular values, rcond = \sigma_{min}/\sigma_{max}.  The routine
     gsl_multifit_linear_svd() must first be called to
     compute the SVD of X.

40.5 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)} \sqrt{1 - 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)| <= \epsilon * 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.

 -- Type: gsl_multifit_robust_workspace

     This workspace is used for robust least squares fitting.

 -- 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 size of the workspace is
     O(np + p^2).  The type ‘T’ specifies the function \psi and can be
     selected from the following choices.

      -- Type: gsl_multifit_robust_type

           -- Variable: gsl_multifit_robust_default

               This specifies the gsl_multifit_robust_bisquare
               type (see below) and is a good general purpose
               choice for robust regression.

           -- Variable: 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, |e| <= 1
                           {     0,       |e| > 1

               and the default tuning constant is t = 4.685.

           -- Variable: 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.

           -- Variable: 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.

           -- Variable: 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.

           -- Variable: 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.

           -- Variable: 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_maxiter (const size_t maxiter,
          gsl_multifit_robust_workspace * w)

     This function sets the maximum number of iterations in the
     iteratively reweighted least squares algorithm to ‘maxiter’.  By
     default, this value is set to 100 by
     gsl_multifit_robust_alloc().

 -- Function: int gsl_multifit_robust_weights (const gsl_vector * r,
          gsl_vector * wts, gsl_multifit_robust_workspace * w)

     This function assigns weights to the vector ‘wts’ using the
     residual vector ‘r’ and previously specified weighting function.
     The output weights are given by wts_i = w(r_i / (t \sigma)), where
     the weighting functions w are detailed in
     gsl_multifit_robust_alloc().  \sigma is an estimate of the
     residual standard deviation based on the Median-Absolute-Deviation
     and t is the tuning constant.  This function is useful if the user
     wishes to implement their own robust regression rather than using
     the supplied gsl_multifit_robust() routine below.

 -- 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().

     If the coefficients do not converge within the maximum iteration
     limit, the function returns ‘GSL_EMAXITER’.  In this case, the
     current estimates of the coefficients and covariance matrix are
     returned in ‘c’ and ‘cov’ and the internal fit statistics are
     computed with these estimates.

 -- 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
     \cdot c at the point ‘x’.

 -- Function: int gsl_multifit_robust_residuals (const gsl_matrix * X,
          const gsl_vector * y, const gsl_vector * c, gsl_vector * r,
          gsl_multifit_robust_workspace * w)

     This function computes the vector of studentized residuals r_i =
     {y_i - (X c)_i \over \sigma \sqrt{1 - h_i}} for the observations
     ‘y’, coefficients ‘c’ and matrix of predictor variables ‘X’.  The
     routine gsl_multifit_robust() must first be called to
     compute the statisical leverages h_i of the matrix ‘X’ and residual
     standard deviation estimate \sigma.

 -- 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.

      -- Type: gsl_multifit_robust_stats

          ‘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

40.6 Large dense linear systems
===============================

This module is concerned with solving large dense least squares systems
X c = y where the n-by-p matrix X has n >> p (ie: many more rows than
columns).  This type of matrix is called a “tall skinny” matrix, and for
some applications, it may not be possible to fit the entire matrix in
memory at once to use the standard SVD approach.  Therefore, the
algorithms in this module are designed to allow the user to construct
smaller blocks of the matrix X and accumulate those blocks into the
larger system one at a time.  The algorithms in this module never need
to store the entire matrix X in memory.  The large linear least squares
routines support data weights and Tikhonov regularization, and are
designed to minimize the residual

     \chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2

where y is the n-by-1 observation vector, X is the n-by-p design matrix,
c is the p-by-1 solution vector, W = \diag(w_1,...,w_n) is the data
weighting matrix, L is an m-by-p regularization matrix, \lambda is a
regularization parameter, and ||r||_W^2 = r^T W r.  In the discussion
which follows, we will assume that the system has been converted into
Tikhonov standard form,

     \chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2

and we will drop the tilde characters from the various parameters.  For
a discussion of the transformation to standard form, see
Regularized regression.

The basic idea is to partition the matrix X and observation vector y as

     [ X_1 ] c = [ y_1 ]
     [ X_2 ]     [ y_2 ]
     [ X_3 ]     [ y_3 ]
     [ ... ]     [ ... ]
     [ X_k ]     [ y_k ]

into k blocks, where each block (X_i,y_i) may have any number of rows,
but each X_i has p columns.  The sections below describe the methods
available for solving this partitioned system.  The functions are
declared in the header file ‘gsl_multilarge.h’.

40.6.1 Normal Equations Approach
--------------------------------

The normal equations approach to the large linear least squares problem
described above is popular due to its speed and simplicity.  Since the
normal equations solution to the problem is given by

     c = ( X^T X + \lambda^2 I )^-1 X^T y

only the p-by-p matrix X^T X and p-by-1 vector X^T y need to be stored.
Using the partition scheme described above, these are given by

     X^T X = \sum_i X_i^T X_i
     X^T y = \sum_i X_i^T y_i

Since the matrix X^T X is symmetric, only half of it needs to be
calculated.  Once all of the blocks (X_i,y_i) have been accumulated into
the final X^T X and X^T y, the system can be solved with a Cholesky
factorization of the X^T X matrix.  The X^T X matrix is first
transformed via a diagonal scaling transformation to attempt to reduce
its condition number as much as possible to recover a more accurate
solution vector.  The normal equations approach is the fastest method
for solving the large least squares problem, and is accurate for
well-conditioned matrices X. However, for ill-conditioned matrices, as
is often the case for large systems, this method can suffer from
numerical instabilities (see Trefethen and Bau, 1997).  The number of
operations for this method is O(np^2 + {1 \over 3}p^3).

40.6.2 Tall Skinny QR (TSQR) Approach
-------------------------------------

An algorithm which has better numerical stability for ill-conditioned
problems is known as the Tall Skinny QR (TSQR) method.  This method is
based on computing the thin QR decomposition of the least squares matrix
X = Q R, where Q is an n-by-p matrix with orthogonal columns, and R is a
p-by-p upper triangular matrix.  Once these factors are calculated, the
residual becomes

     \chi^2 = || Q^T y - R c ||^2 + \lambda^2 || c ||^2

which can be written as the matrix equation

     [ R ; \lambda I ] c = [ Q^T b ; 0 ]

The matrix on the left hand side is now a much smaller 2p-by-p matrix
which can be solved with a standard SVD approach.  The Q matrix is just
as large as the original matrix X, however it does not need to be
explicitly constructed.  The TSQR algorithm computes only the p-by-p
matrix R and the p-by-1 vector Q^T y, and updates these quantities as
new blocks are added to the system.  Each time a new block of rows
(X_i,y_i) is added, the algorithm performs a QR decomposition of the
matrix

     [ R_(i-1) ; X_i ]

where R_{i-1} is the upper triangular R factor for the matrix

     [ X_1 ; ... ; X_(i-1) ]

This QR decomposition is done efficiently taking into account the sparse
structure of R_{i-1}.  See Demmel et al, 2008 for more details on how
this is accomplished.  The number of operations for this method is
O(2np^2 - {2 \over 3}p^3).

40.6.3 Large Dense Linear Systems Solution Steps
------------------------------------------------

The typical steps required to solve large regularized linear least
squares problems are as follows:

  1. Choose the regularization matrix L.

  2. Construct a block of rows of the least squares matrix, right hand
     side vector, and weight vector (X_i, y_i, w_i).

  3. Transform the block to standard form (\tilde{X_i}, \tilde{y_i}).
     This step can be skipped if L = I and W = I.

  4. Accumulate the standard form block (\tilde{X_i}, \tilde{y_i}) into
     the system.

  5. Repeat steps 2-4 until the entire matrix and right hand side vector
     have been accumulated.

  6. Determine an appropriate regularization parameter \lambda (using
     for example L-curve analysis).

  7. Solve the standard form system using the chosen \lambda.

  8. Backtransform the standard form solution \tilde{c} to recover the
     original solution vector c.

40.6.4 Large Dense Linear Least Squares Routines
------------------------------------------------

 -- Type: gsl_multilarge_linear_workspace

     This workspace contains parameters for solving large linear least
     squares problems.

 -- Function: gsl_multilarge_linear_workspace *
          gsl_multilarge_linear_alloc (const gsl_multilarge_linear_type
          * T, const size_t p)

     This function allocates a workspace for solving large linear least
     squares systems.  The least squares matrix X has ‘p’ columns, but
     may have any number of rows.

      -- Type: gsl_multilarge_linear_type

          The parameter ‘T’ specifies the method to be used for solving
          the large least squares system and may be selected from the
          following choices

           -- Variable: gsl_multilarge_linear_normal

               This specifies the normal equations approach for solving
               the least squares system.  This method is suitable in
               cases where performance is critical and it is known that
               the least squares matrix X is well conditioned.  The size
               of this workspace is O(p^2).

           -- Variable: gsl_multilarge_linear_tsqr

               This specifies the sequential Tall Skinny QR (TSQR)
               approach for solving the least squares system.  This
               method is a good general purpose choice for large
               systems, but requires about twice as many operations as
               the normal equations method for n >> p.  The size of this
               workspace is O(p^2).

 -- Function: void gsl_multilarge_linear_free
          (gsl_multilarge_linear_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: const char * gsl_multilarge_linear_name
          (gsl_multilarge_linear_workspace * w)

     This function returns a string pointer to the name of the
     multilarge solver.

 -- Function: int gsl_multilarge_linear_reset
          (gsl_multilarge_linear_workspace * w)

     This function resets the workspace ‘w’ so it can begin to
     accumulate a new least squares system.

 -- Function: int gsl_multilarge_linear_stdform1 (const gsl_vector * L,
          const gsl_matrix * X, const gsl_vector * y, gsl_matrix * Xs,
          gsl_vector * ys, gsl_multilarge_linear_workspace * work)
 -- Function: int gsl_multilarge_linear_wstdform1 (const gsl_vector * L,
          const gsl_matrix * X, const gsl_vector * w, const gsl_vector
          * y, gsl_matrix * Xs, gsl_vector * ys,
          gsl_multilarge_linear_workspace * work)

     These functions define a regularization matrix L =
     \diag(l_0,l_1,...,l_{p-1}).  The diagonal matrix element l_i is
     provided by the i-th element of the input vector ‘L’.  The block
     (‘X’, ‘y’) is converted to standard form and the parameters
     (\tilde{X}, \tilde{y}) are stored in ‘Xs’ and ‘ys’ on output.  ‘Xs’
     and ‘ys’ have the same dimensions as ‘X’ and ‘y’.  Optional data
     weights may be supplied in the vector ‘w’.  In order to apply this
     transformation, L^{-1} must exist and so none of the l_i may be
     zero.  After the standard form system has been solved, use
     gsl_multilarge_linear_genform1() to recover the original
     solution vector.  It is allowed to have ‘X’ = ‘Xs’ and ‘y’ = ‘ys’
     for an in-place transform.

 -- Function: int gsl_multilarge_linear_L_decomp (gsl_matrix * L,
          gsl_vector * tau)

     This function calculates the QR decomposition of the m-by-p
     regularization matrix ‘L’.  ‘L’ must have m \ge p.  On output, the
     Householder scalars are stored in the vector ‘tau’ of size p.
     These outputs will be used by
     gsl_multilarge_linear_wstdform2() to complete the
     transformation to standard form.

 -- Function: int gsl_multilarge_linear_stdform2 (const gsl_matrix
          * LQR, const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * y, gsl_matrix * Xs, gsl_vector * ys,
          gsl_multilarge_linear_workspace * work)
 -- Function: int gsl_multilarge_linear_wstdform2 (const gsl_matrix
          * LQR, const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * w, const gsl_vector * y, gsl_matrix * Xs,
          gsl_vector * ys, gsl_multilarge_linear_workspace * work)

     These functions convert a block of rows (‘X’, ‘y’, ‘w’) to standard
     form (\tilde{X}, \tilde{y}) which are stored in ‘Xs’ and ‘ys’
     respectively.  ‘X’, ‘y’, and ‘w’ must all have the same number of
     rows.  The m-by-p regularization matrix ‘L’ is specified by the
     inputs ‘LQR’ and ‘Ltau’, which are outputs from
     gsl_multilarge_linear_L_decomp().  ‘Xs’ and ‘ys’ have the same
     dimensions as ‘X’ and ‘y’.  After the standard form system has been
     solved, use gsl_multilarge_linear_genform2() to recover
     the original solution vector.  Optional data weights may be
     supplied in the vector ‘w’, where W = \diag(w).

 -- Function: int gsl_multilarge_linear_accumulate (gsl_matrix * X,
          gsl_vector * y, gsl_multilarge_linear_workspace * w)

     This function accumulates the standard form block (X,y) into the
     current least squares system.  ‘X’ and ‘y’ have the same number of
     rows, which can be arbitrary.  ‘X’ must have p columns.  For the
     TSQR method, ‘X’ and ‘y’ are destroyed on output.  For the normal
     equations method, they are both unchanged.

 -- Function: int gsl_multilarge_linear_solve (const double lambda,
          gsl_vector * c, double * rnorm, double * snorm,
          gsl_multilarge_linear_workspace * w)

     After all blocks (X_i,y_i) have been accumulated into the large
     least squares system, this function will compute the solution
     vector which is stored in ‘c’ on output.  The regularization
     parameter \lambda is provided in ‘lambda’.  On output, ‘rnorm’
     contains the residual norm ||y - X c||_W and ‘snorm’ contains the
     solution norm ||L c||.

 -- Function: int gsl_multilarge_linear_genform1 (const gsl_vector * L,
          const gsl_vector * cs, gsl_vector * c,
          gsl_multilarge_linear_workspace * work)

     After a regularized system has been solved with L =
     \diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the
     standard form solution vector ‘cs’ to recover the solution vector
     of the original problem ‘c’.  The diagonal matrix elements l_i are
     provided in the vector ‘L’.  It is allowed to have ‘c’ = ‘cs’ for
     an in-place transform.

 -- Function: int gsl_multilarge_linear_genform2 (const gsl_matrix
          * LQR, const gsl_vector * Ltau, const gsl_vector * cs,
          gsl_vector * c, gsl_multilarge_linear_workspace * work)

     After a regularized system has been solved with a regularization
     matrix L, specified by (‘LQR’, ‘Ltau’), this function
     backtransforms the standard form solution ‘cs’ to recover the
     solution vector of the original problem, which is stored in ‘c’, of
     length p.

 -- Function: int gsl_multilarge_linear_lcurve (gsl_vector * reg_param,
          gsl_vector * rho, gsl_vector * eta,
          gsl_multilarge_linear_workspace * work)

     This function computes the L-curve for a large least squares system
     after it has been fully accumulated into the workspace ‘work’.  The
     output vectors ‘reg_param’, ‘rho’, and ‘eta’ must all be the same
     size, and will contain the regularization parameters \lambda_i,
     residual norms ||y - X c_i||, and solution norms || L c_i || which
     compose the L-curve, where c_i is the regularized solution vector
     corresponding to \lambda_i.  The user may determine the number of
     points on the L-curve by adjusting the size of these input arrays.
     For the TSQR method, the regularization parameters \lambda_i are
     estimated from the singular values of the triangular R factor.  For
     the normal equations method, they are estimated from the
     eigenvalues of the X^T X matrix.

 -- Function: int gsl_multilarge_linear_rcond (double * rcond,
          gsl_multilarge_linear_workspace * work)

     This function computes the reciprocal condition number, stored in
     ‘rcond’, of the least squares matrix after it has been accumulated
     into the workspace ‘work’.  For the TSQR algorithm, this is
     accomplished by calculating the SVD of the R factor, which has the
     same singular values as the matrix X. For the normal equations
     method, this is done by computing the eigenvalues of X^T X, which
     could be inaccurate for ill-conditioned matrices X.

40.7 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'.

40.8 Examples
=============

The example programs in this section demonstrate the various linear
regression methods.

40.8.1 Simple Linear Regression Example
---------------------------------------

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 result is shown in Fig.  %s.

Figure: Straight line fit with 1-\sigma error bars

40.8.2 Multi-parameter Linear Regression Example
------------------------------------------------

The following 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.

Fig.  %s shows the resulting fit.

Figure: Weighted fit example with error bars

40.8.3 Regularized Linear Regression Example 1
----------------------------------------------

The next program demonstrates the difference between ordinary and
regularized least squares when the design matrix is near-singular.  In
this program, we generate two random normally distributed variables u
and v, with v = u + noise so that u and v are nearly colinear.  We then
set a third dependent variable y = u + v + noise and solve for the
coefficients c_1,c_2 of the model Y(c_1,c_2) = c_1 u + c_2 v.  Since u
\approx v, the design matrix X is nearly singular, leading to unstable
ordinary least squares solutions.

Here is the program output:

     matrix condition number = 1.025113e+04

     === Unregularized fit ===
     best fit: y = -43.6588 u + 45.6636 v
     residual norm = 31.6248
     solution norm = 63.1764
     chisq/dof = 1.00213

     === Regularized fit (L-curve) ===
     optimal lambda: 4.51103
     best fit: y = 1.00113 u + 1.0032 v
     residual norm = 31.6547
     solution norm = 1.41728
     chisq/dof = 1.04499

     === Regularized fit (GCV) ===
     optimal lambda: 0.0232029
     best fit: y = -19.8367 u + 21.8417 v
     residual norm = 31.6332
     solution norm = 29.5051
     chisq/dof = 1.00314

We see that the ordinary least squares solution is completely wrong,
while the L-curve regularized method with the optimal \lambda = 4.51103
finds the correct solution c_1 \approx c_2 \approx 1.  The GCV
regularized method finds a regularization parameter \lambda = 0.0232029
which is too small to give an accurate solution, although it performs
better than OLS. The L-curve and its computed corner, as well as the GCV
curve and its minimum are plotted in Fig.  %s.

Figure: L-curve and GCV curve for example program.

The program is given below.

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_multifit.h>

     int
     main()
     {
       const size_t n = 1000; /* number of observations */
       const size_t p = 2;    /* number of model parameters */
       size_t i;
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_matrix *X = gsl_matrix_alloc(n, p);
       gsl_vector *y = gsl_vector_alloc(n);

       for (i = 0; i < n; ++i)
         {
           /* generate first random variable u */
           double ui = 5.0 * gsl_ran_gaussian(r, 1.0);

           /* set v = u + noise */
           double vi = ui + gsl_ran_gaussian(r, 0.001);

           /* set y = u + v + noise */
           double yi = ui + vi + gsl_ran_gaussian(r, 1.0);

           /* since u =~ v, the matrix X is ill-conditioned */
           gsl_matrix_set(X, i, 0, ui);
           gsl_matrix_set(X, i, 1, vi);

           /* rhs vector */
           gsl_vector_set(y, i, yi);
         }

       {
         const size_t npoints = 200;                   /* number of points on L-curve and GCV curve */
         gsl_multifit_linear_workspace *w =
           gsl_multifit_linear_alloc(n, p);
         gsl_vector *c = gsl_vector_alloc(p);          /* OLS solution */
         gsl_vector *c_lcurve = gsl_vector_alloc(p);   /* regularized solution (L-curve) */
         gsl_vector *c_gcv = gsl_vector_alloc(p);      /* regularized solution (GCV) */
         gsl_vector *reg_param = gsl_vector_alloc(npoints);
         gsl_vector *rho = gsl_vector_alloc(npoints);  /* residual norms */
         gsl_vector *eta = gsl_vector_alloc(npoints);  /* solution norms */
         gsl_vector *G = gsl_vector_alloc(npoints);    /* GCV function values */
         double lambda_l;                              /* optimal regularization parameter (L-curve) */
         double lambda_gcv;                            /* optimal regularization parameter (GCV) */
         double G_gcv;                                 /* G(lambda_gcv) */
         size_t reg_idx;                               /* index of optimal lambda */
         double rcond;                                 /* reciprocal condition number of X */
         double chisq, rnorm, snorm;

         /* compute SVD of X */
         gsl_multifit_linear_svd(X, w);

         rcond = gsl_multifit_linear_rcond(w);
         fprintf(stderr, "matrix condition number = %e\n\n", 1.0 / rcond);

         /* unregularized (standard) least squares fit, lambda = 0 */
         gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0);

         fprintf(stderr, "=== Unregularized fit ===\n");
         fprintf(stderr, "best fit: y = %g u + %g v\n",
           gsl_vector_get(c, 0), gsl_vector_get(c, 1));
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate L-curve and find its corner */
         gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
         gsl_multifit_linear_lcorner(rho, eta, &reg_idx);

         /* store optimal regularization parameter */
         lambda_l = gsl_vector_get(reg_param, reg_idx);

         /* regularize with lambda_l */
         gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (L-curve) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_l);
         fprintf(stderr, "best fit: y = %g u + %g v\n",
                 gsl_vector_get(c_lcurve, 0), gsl_vector_get(c_lcurve, 1));
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate GCV curve and find its minimum */
         gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);

         /* regularize with lambda_gcv */
         gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (GCV) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
         fprintf(stderr, "best fit: y = %g u + %g v\n",
                 gsl_vector_get(c_gcv, 0), gsl_vector_get(c_gcv, 1));
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* output L-curve and GCV curve */
         for (i = 0; i < npoints; ++i)
           {
             printf("%e %e %e %e\n",
                    gsl_vector_get(reg_param, i),
                    gsl_vector_get(rho, i),
                    gsl_vector_get(eta, i),
                    gsl_vector_get(G, i));
           }

         /* output L-curve corner point */
         printf("\n\n%f %f\n",
                gsl_vector_get(rho, reg_idx),
                gsl_vector_get(eta, reg_idx));

         /* output GCV curve corner minimum */
         printf("\n\n%e %e\n",
                lambda_gcv,
                G_gcv);

         gsl_multifit_linear_free(w);
         gsl_vector_free(c);
         gsl_vector_free(c_lcurve);
         gsl_vector_free(reg_param);
         gsl_vector_free(rho);
         gsl_vector_free(eta);
         gsl_vector_free(G);
       }

       gsl_rng_free(r);
       gsl_matrix_free(X);
       gsl_vector_free(y);

       return 0;
     }

40.8.4 Regularized Linear Regression Example 2
----------------------------------------------

The following example program minimizes the cost function

     ||y - X c||^2 + \lambda^2 ||x||^2

where X is the 10-by-8 Hilbert matrix whose entries are given by

     X_{ij} = 1 / (i + j - 1)

and the right hand side vector is given by y =
[1,-1,1,-1,1,-1,1,-1,1,-1]^T. Solutions are computed for \lambda = 0
(unregularized) as well as for optimal parameters \lambda chosen by
analyzing the L-curve and GCV curve.

Here is the program output:

     matrix condition number = 3.565872e+09

     === Unregularized fit ===
     residual norm = 2.15376
     solution norm = 2.92217e+09
     chisq/dof = 2.31934

     === Regularized fit (L-curve) ===
     optimal lambda: 7.11407e-07
     residual norm = 2.60386
     solution norm = 424507
     chisq/dof = 3.43565

     === Regularized fit (GCV) ===
     optimal lambda: 1.72278
     residual norm = 3.1375
     solution norm = 0.139357
     chisq/dof = 4.95076

Here we see the unregularized solution results in a large solution norm
due to the ill-conditioned matrix.  The L-curve solution finds a small
value of \lambda = 7.11e-7 which still results in a badly conditioned
system and a large solution norm.  The GCV method finds a parameter
\lambda = 1.72 which results in a well-conditioned system and small
solution norm.

The L-curve and its computed corner, as well as the GCV curve and its
minimum are plotted in Fig.  %s.

Figure: L-curve and GCV curve for example program.

The program is given below.

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_blas.h>

     static int
     hilbert_matrix(gsl_matrix * m)
     {
       const size_t N = m->size1;
       const size_t M = m->size2;
       size_t i, j;

       for (i = 0; i < N; i++)
         {
           for (j = 0; j < M; j++)
             {
               gsl_matrix_set(m, i, j, 1.0/(i+j+1.0));
             }
         }

       return GSL_SUCCESS;
     }

     int
     main()
     {
       const size_t n = 10; /* number of observations */
       const size_t p = 8;  /* number of model parameters */
       size_t i;
       gsl_matrix *X = gsl_matrix_alloc(n, p);
       gsl_vector *y = gsl_vector_alloc(n);

       /* construct Hilbert matrix and rhs vector */
       hilbert_matrix(X);

       {
         double val = 1.0;
         for (i = 0; i < n; ++i)
           {
             gsl_vector_set(y, i, val);
             val *= -1.0;
           }
       }

       {
         const size_t npoints = 200;                   /* number of points on L-curve and GCV curve */
         gsl_multifit_linear_workspace *w =
           gsl_multifit_linear_alloc(n, p);
         gsl_vector *c = gsl_vector_alloc(p);          /* OLS solution */
         gsl_vector *c_lcurve = gsl_vector_alloc(p);   /* regularized solution (L-curve) */
         gsl_vector *c_gcv = gsl_vector_alloc(p);      /* regularized solution (GCV) */
         gsl_vector *reg_param = gsl_vector_alloc(npoints);
         gsl_vector *rho = gsl_vector_alloc(npoints);  /* residual norms */
         gsl_vector *eta = gsl_vector_alloc(npoints);  /* solution norms */
         gsl_vector *G = gsl_vector_alloc(npoints);    /* GCV function values */
         double lambda_l;                              /* optimal regularization parameter (L-curve) */
         double lambda_gcv;                            /* optimal regularization parameter (GCV) */
         double G_gcv;                                 /* G(lambda_gcv) */
         size_t reg_idx;                               /* index of optimal lambda */
         double rcond;                                 /* reciprocal condition number of X */
         double chisq, rnorm, snorm;

         /* compute SVD of X */
         gsl_multifit_linear_svd(X, w);

         rcond = gsl_multifit_linear_rcond(w);
         fprintf(stderr, "matrix condition number = %e\n", 1.0 / rcond);

         /* unregularized (standard) least squares fit, lambda = 0 */
         gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0);

         fprintf(stderr, "\n=== Unregularized fit ===\n");
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate L-curve and find its corner */
         gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
         gsl_multifit_linear_lcorner(rho, eta, &reg_idx);

         /* store optimal regularization parameter */
         lambda_l = gsl_vector_get(reg_param, reg_idx);

         /* regularize with lambda_l */
         gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (L-curve) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_l);
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate GCV curve and find its minimum */
         gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);

         /* regularize with lambda_gcv */
         gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (GCV) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* output L-curve and GCV curve */
         for (i = 0; i < npoints; ++i)
           {
             printf("%e %e %e %e\n",
                    gsl_vector_get(reg_param, i),
                    gsl_vector_get(rho, i),
                    gsl_vector_get(eta, i),
                    gsl_vector_get(G, i));
           }

         /* output L-curve corner point */
         printf("\n\n%f %f\n",
                gsl_vector_get(rho, reg_idx),
                gsl_vector_get(eta, reg_idx));

         /* output GCV curve corner minimum */
         printf("\n\n%e %e\n",
                lambda_gcv,
                G_gcv);

         gsl_multifit_linear_free(w);
         gsl_vector_free(c);
         gsl_vector_free(c_lcurve);
         gsl_vector_free(reg_param);
         gsl_vector_free(rho);
         gsl_vector_free(eta);
         gsl_vector_free(G);
       }

       gsl_matrix_free(X);
       gsl_vector_free(y);

       return 0;
     }

40.8.5 Robust Linear Regression Example
---------------------------------------

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)
     {
       size_t 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 Fig.  %s.

Figure: Linear fit to dataset with outliers.

40.8.6 Large Dense Linear Regression Example
--------------------------------------------

The following program demostrates the large dense linear least squares
solvers.  This example is adapted from Trefethen and Bau, and fits the
function f(t) = \exp{(\sin^3{(10t)}}) on the interval [0,1] with a
degree 15 polynomial.  The program generates n = 50000 equally spaced
points t_i on this interval, calculates the function value and adds
random noise to determine the observation value y_i.  The entries of the
least squares matrix are X_{ij} = t_i^j, representing a polynomial fit.
The matrix is highly ill-conditioned, with a condition number of about
1.4 \cdot 10^{11}.  The program accumulates the matrix into the least
squares system in 5 blocks, each with 10000 rows.  This way the full
matrix X is never stored in memory.  We solve the system with both the
normal equations and TSQR methods.  The results are shown in Fig.  %s.
In the top left plot, we see the unregularized normal equations solution
has larger error than TSQR due to the ill-conditioning of the matrix.
In the bottom left plot, we show the L-curve, which exhibits multiple
corners.  In the top right panel, we plot a regularized solution using
\lambda = 10^{-6}.  The TSQR and normal solutions now agree, however
they are unable to provide a good fit due to the damping.  This
indicates that for some ill-conditioned problems, regularizing the
normal equations does not improve the solution.  This is further
illustrated in the bottom right panel, where we plot the L-curve
calculated from the normal equations.  The curve agrees with the TSQR
curve for larger damping parameters, but for small \lambda, the normal
equations approach cannot provide accurate solution vectors leading to
numerical inaccuracies in the left portion of the curve.

Figure

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_multilarge.h>
     #include <gsl/gsl_blas.h>

     /* function to be fitted */
     double
     func(const double t)
     {
       double x = sin(10.0 * t);
       return exp(x*x*x);
     }

     /* construct a row of the least squares matrix */
     int
     build_row(const double t, gsl_vector *row)
     {
       const size_t p = row->size;
       double Xj = 1.0;
       size_t j;

       for (j = 0; j < p; ++j)
         {
           gsl_vector_set(row, j, Xj);
           Xj *= t;
         }

       return 0;
     }

     int
     solve_system(const int print_data, const gsl_multilarge_linear_type * T,
                  const double lambda, const size_t n, const size_t p,
                  gsl_vector * c)
     {
       const size_t nblock = 5;         /* number of blocks to accumulate */
       const size_t nrows = n / nblock; /* number of rows per block */
       gsl_multilarge_linear_workspace * w =
         gsl_multilarge_linear_alloc(T, p);
       gsl_matrix *X = gsl_matrix_alloc(nrows, p);
       gsl_vector *y = gsl_vector_alloc(nrows);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       const size_t nlcurve = 200;
       gsl_vector *reg_param = gsl_vector_alloc(nlcurve);
       gsl_vector *rho = gsl_vector_alloc(nlcurve);
       gsl_vector *eta = gsl_vector_alloc(nlcurve);
       size_t rowidx = 0;
       double rnorm, snorm, rcond;
       double t = 0.0;
       double dt = 1.0 / (n - 1.0);

       while (rowidx < n)
         {
           size_t nleft = n - rowidx;         /* number of rows left to accumulate */
           size_t nr = GSL_MIN(nrows, nleft); /* number of rows in this block */
           gsl_matrix_view Xv = gsl_matrix_submatrix(X, 0, 0, nr, p);
           gsl_vector_view yv = gsl_vector_subvector(y, 0, nr);
           size_t i;

           /* build (X,y) block with 'nr' rows */
           for (i = 0; i < nr; ++i)
             {
               gsl_vector_view row = gsl_matrix_row(&Xv.matrix, i);
               double fi = func(t);
               double ei = gsl_ran_gaussian (r, 0.1 * fi); /* noise */
               double yi = fi + ei;

               /* construct this row of LS matrix */
               build_row(t, &row.vector);

               /* set right hand side value with added noise */
               gsl_vector_set(&yv.vector, i, yi);

               if (print_data && (i % 100 == 0))
                 printf("%f %f\n", t, yi);

               t += dt;
             }

           /* accumulate (X,y) block into LS system */
           gsl_multilarge_linear_accumulate(&Xv.matrix, &yv.vector, w);

           rowidx += nr;
         }

       if (print_data)
         printf("\n\n");

       /* compute L-curve */
       gsl_multilarge_linear_lcurve(reg_param, rho, eta, w);

       /* solve large LS system and store solution in c */
       gsl_multilarge_linear_solve(lambda, c, &rnorm, &snorm, w);

       /* compute reciprocal condition number */
       gsl_multilarge_linear_rcond(&rcond, w);

       fprintf(stderr, "=== Method %s ===\n", gsl_multilarge_linear_name(w));
       fprintf(stderr, "condition number = %e\n", 1.0 / rcond);
       fprintf(stderr, "residual norm    = %e\n", rnorm);
       fprintf(stderr, "solution norm    = %e\n", snorm);

       /* output L-curve */
       {
         size_t i;
         for (i = 0; i < nlcurve; ++i)
           {
             printf("%.12e %.12e %.12e\n",
                    gsl_vector_get(reg_param, i),
                    gsl_vector_get(rho, i),
                    gsl_vector_get(eta, i));
           }
         printf("\n\n");
       }

       gsl_matrix_free(X);
       gsl_vector_free(y);
       gsl_multilarge_linear_free(w);
       gsl_rng_free(r);
       gsl_vector_free(reg_param);
       gsl_vector_free(rho);
       gsl_vector_free(eta);

       return 0;
     }

     int
     main(int argc, char *argv[])
     {
       const size_t n = 50000;   /* number of observations */
       const size_t p = 16;      /* polynomial order + 1 */
       double lambda = 0.0;      /* regularization parameter */
       gsl_vector *c_tsqr = gsl_vector_alloc(p);
       gsl_vector *c_normal = gsl_vector_alloc(p);

       if (argc > 1)
         lambda = atof(argv[1]);

       /* solve system with TSQR method */
       solve_system(1, gsl_multilarge_linear_tsqr, lambda, n, p, c_tsqr);

       /* solve system with Normal equations method */
       solve_system(0, gsl_multilarge_linear_normal, lambda, n, p, c_normal);

       /* output solutions */
       {
         gsl_vector *v = gsl_vector_alloc(p);
         double t;

         for (t = 0.0; t <= 1.0; t += 0.01)
           {
             double f_exact = func(t);
             double f_tsqr, f_normal;

             build_row(t, v);
             gsl_blas_ddot(v, c_tsqr, &f_tsqr);
             gsl_blas_ddot(v, c_normal, &f_normal);

             printf("%f %e %e %e\n", t, f_exact, f_tsqr, f_normal);
           }

         gsl_vector_free(v);
       }

       gsl_vector_free(c_tsqr);
       gsl_vector_free(c_normal);

       return 0;
     }

40.9 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’

More information on Tikhonov regularization can be found in

   * Hansen, P. C. (1998), Rank-Deficient and Discrete Ill-Posed
     Problems: Numerical Aspects of Linear Inversion.  SIAM Monogr.  on
     Mathematical Modeling and Computation, Society for Industrial and
     Applied Mathematics

   * M. Rezghi and S. M. Hosseini (2009), A new variant of L-curve for
     Tikhonov regularization, Journal of Computational and Applied
     Mathematics, Volume 231, Issue 2, pages 914-924.

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.

More information about the normal equations and TSQR approach for
solving large linear least squares systems can be found in the
publications

   * Trefethen, L. N. and Bau, D. (1997), “Numerical Linear Algebra”,
     SIAM.

   * Demmel, J., Grigori, L., Hoemmen, M. F., and Langou, J.
     “Communication-optimal parallel and sequential QR and LU
     factorizations”, UCB Technical Report No.  UCB/EECS-2008-89, 2008.

41 Nonlinear Least-Squares Fitting
**********************************

This chapter describes functions for multidimensional nonlinear
least-squares fitting.  There are generally two classes of algorithms
for solving nonlinear least squares problems, which fall under line
search methods and trust region methods.  GSL currently implements only
trust region methods and provides the user with full access to
intermediate steps of the iteration.  The user also has the ability to
tune a number of parameters which affect low-level aspects of the
algorithm which can help to accelerate convergence for the specific
problem at hand.  GSL provides two separate interfaces for nonlinear
least squares fitting.  The first is designed for small to moderate
sized problems, and the second is designed for very large problems,
which may or may not have significant sparse structure.

The header file ‘gsl_multifit_nlinear.h’ contains prototypes for the
multidimensional nonlinear fitting functions and related declarations
relating to the small to moderate sized systems.

The header file ‘gsl_multilarge_nlinear.h’ contains prototypes for the
multidimensional nonlinear fitting functions and related declarations
relating to large systems.

41.1 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

In trust region methods, the objective (or cost) function \Phi(x) is
approximated by a model function m_k(\delta) in the vicinity of some
point x_k.  The model function is often simply a second order Taylor
series expansion around the point x_k, ie:

     \Phi(x_k + \delta) ~=~ m_k(\delta) = \Phi(x_k) + g_k^T \delta + 1/2 \delta^T B_k \delta

where g_k = \nabla \Phi(x_k) = J^T f is the gradient vector at the point
x_k, B_k = \nabla^2 \Phi(x_k) is the Hessian matrix at x_k, or some
approximation to it, and J is the n-by-p Jacobian matrix

     J_{ij} = d f_i / d x_j

In order to find the next step \delta, we minimize the model function
m_k(\delta), but search for solutions only within a region where we
trust that m_k(\delta) is a good approximation to the objective function
\Phi(x_k + \delta).  In other words, we seek a solution of the trust
region subproblem (TRS)

     \min_(\delta \in R^p) m_k(\delta), s.t. || D_k \delta || <= \Delta_k

where \Delta_k > 0 is the trust region radius and D_k is a scaling
matrix.  If D_k = I, then the trust region is a ball of radius \Delta_k
centered at x_k.  In some applications, the parameter vector x may have
widely different scales.  For example, one parameter might be a
temperature on the order of 10^3 K, while another might be a length on
the order of 10^{-6} m.  In such cases, a spherical trust region may not
be the best choice, since if \Phi changes rapidly along directions with
one scale, and more slowly along directions with a different scale, the
model function m_k may be a poor approximation to \Phi along the rapidly
changing directions.  In such problems, it may be best to use an
elliptical trust region, by setting D_k to a diagonal matrix whose
entries are designed so that the scaled step D_k \delta has entries of
approximately the same order of magnitude.

The trust region subproblem above normally amounts to solving a linear
least squares system (or multiple systems) for the step \delta.  Once
\delta is computed, it is checked whether or not it reduces the
objective function \Phi(x).  A useful statistic for this is to look at
the ratio

     \rho_k = ( \Phi(x_k) - \Phi(x_k + \delta_k) / ( m_k(0) - m_k(\delta_k) )

where the numerator is the actual reduction of the objective function
due to the step \delta_k, and the denominator is the predicted reduction
due to the model m_k.  If \rho_k is negative, it means that the step
\delta_k increased the objective function and so it is rejected.  If
\rho_k is positive, then we have found a step which reduced the
objective function and it is accepted.  Furthermore, if \rho_k is close
to 1, then this indicates that the model function is a good
approximation to the objective function in the trust region, and so on
the next iteration the trust region is enlarged in order to take more
ambitious steps.  When a step is rejected, the trust region is made
smaller and the TRS is solved again.  An outline for the general trust
region method used by GSL can now be given.

`Trust Region Algorithm'

  1. Initialize: given x_0, construct m_0(\delta), D_0 and \Delta_0 > 0

  2. For k = 0, 1, 2, …

       a. If converged, then stop

       b. Solve TRS for trial step \delta_k

       c. Evaluate trial step by computing \rho_k

          1). if step is accepted, set x_{k+1} = x_k + \delta_k and increase radius,

               \Delta_{k+1} = \alpha \Delta_k

          2). if step is rejected, set x_{k+1} = x_k and decrease radius,

               \Delta_{k+1} = {\Delta_k \over \beta}; goto 2(b)

       d. Construct m_{k+1}(\delta) and D_{k+1}

GSL offers the user a number of different algorithms for solving the
trust region subproblem in 2(b), as well as different choices of scaling
matrices D_k and different methods of updating the trust region radius
\Delta_k.  Therefore, while reasonable default methods are provided, the
user has a lot of control to fine-tune the various steps of the
algorithm for their specific problem.

41.2 Solving the Trust Region Subproblem (TRS)
==============================================

Below we describe the methods available for solving the trust region
subproblem.  The methods available provide either exact or approximate
solutions to the trust region subproblem.  In all algorithms below, the
Hessian matrix B_k is approximated as B_k \approx J_k^T J_k, where J_k =
J(x_k).  In all methods, the solution of the TRS involves solving a
linear least squares system involving the Jacobian matrix.  For small to
moderate sized problems (‘gsl_multifit_nlinear’ interface), this is
accomplished by factoring the full Jacobian matrix, which is provided by
the user, with the Cholesky, QR, or SVD decompositions.  For large
systems (‘gsl_multilarge_nlinear’ interface), the user has two choices.
One is to solve the system iteratively, without needing to store the
full Jacobian matrix in memory.  With this method, the user must provide
a routine to calculate the matrix-vector products J u or J^T u for a
given vector u.  This iterative method is particularly useful for
systems where the Jacobian has sparse structure, since forming
matrix-vector products can be done cheaply.  The second option for large
systems involves forming the normal equations matrix J^T J and then
factoring it using a Cholesky decomposition.  The normal equations
matrix is p-by-p, typically much smaller than the full n-by-p Jacobian,
and can usually be stored in memory even if the full Jacobian matrix
cannot.  This option is useful for large, dense systems, or if the
iterative method has difficulty converging.

41.2.1 Levenberg-Marquardt
--------------------------

There is a theorem which states that if \delta_k is a solution to the
trust region subproblem given above, then there exists \mu_k \ge 0 such
that

     ( B_k + \mu_k D_k^T D_k ) \delta_k = -g_k

with \mu_k (\Delta_k - ||D_k \delta_k||) = 0.  This forms the basis of
the Levenberg-Marquardt algorithm, which controls the trust region size
by adjusting the parameter \mu_k rather than the radius \Delta_k
directly.  For each radius \Delta_k, there is a unique parameter \mu_k
which solves the TRS, and they have an inverse relationship, so that
large values of \mu_k correspond to smaller trust regions, while small
values of \mu_k correspond to larger trust regions.

With the approximation B_k \approx J_k^T J_k, on each iteration, in
order to calculate the step \delta_k, the following linear least squares
problem is solved:

     [J_k; sqrt(mu_k) D_k] \delta_k = - [f_k; 0]

If the step \delta_k is accepted, then \mu_k is decreased on the next
iteration in order to take a larger step, otherwise it is increased to
take a smaller step.  The Levenberg-Marquardt algorithm provides an
exact solution of the trust region subproblem, but typically has a
higher computational cost per iteration than the approximate methods
discussed below, since it may need to solve the least squares system
above several times for different values of \mu_k.

41.2.2 Levenberg-Marquardt with Geodesic Acceleration
-----------------------------------------------------

This method applies a so-called geodesic acceleration correction to the
standard Levenberg-Marquardt step \delta_k (Transtrum et al, 2011).  By
interpreting \delta_k as a first order step along a geodesic in the
model parameter space (ie: a velocity \delta_k = v_k), the geodesic
acceleration a_k is a second order correction along the geodesic which
is determined by solving the linear least squares system

     [J_k; sqrt(mu_k) D_k] a_k = - [f_vv(x_k); 0]

where f_{vv} is the second directional derivative of the residual vector
in the velocity direction v, f_{vv}(x) = D_v^2 f = \sum_{\alpha\beta}
v_{\alpha} v_{\beta} \partial_{\alpha} \partial_{\beta} f(x), where
\alpha and \beta are summed over the p parameters.  The new total step
is then \delta_k' = v_k + {1 \over 2}a_k.  The second order correction
a_k can be calculated with a modest additional cost, and has been shown
to dramatically reduce the number of iterations (and expensive Jacobian
evaluations) required to reach convergence on a variety of different
problems.  In order to utilize the geodesic acceleration, the user must
supply a function which provides the second directional derivative
vector f_{vv}(x), or alternatively the library can use a finite
difference method to estimate this vector with one additional function
evaluation of f(x + h v) where h is a tunable step size (see the ‘h_fvv’
parameter description).

41.2.3 Dogleg
-------------

This is Powell’s dogleg method, which finds an approximate solution to
the trust region subproblem, by restricting its search to a piecewise
linear “dogleg” path, composed of the origin, the Cauchy point which
represents the model minimizer along the steepest descent direction, and
the Gauss-Newton point, which is the overall minimizer of the
unconstrained model.  The Gauss-Newton step is calculated by solving

     J_k \delta_{gn} = -f_k

which is the main computational task for each iteration, but only needs
to be performed once per iteration.  If the Gauss-Newton point is inside
the trust region, it is selected as the step.  If it is outside, the
method then calculates the Cauchy point, which is located along the
gradient direction.  If the Cauchy point is also outside the trust
region, the method assumes that it is still far from the minimum and so
proceeds along the gradient direction, truncating the step at the trust
region boundary.  If the Cauchy point is inside the trust region, with
the Gauss-Newton point outside, the method uses a dogleg step, which is
a linear combination of the gradient direction and the Gauss-Newton
direction, stopping at the trust region boundary.

41.2.4 Double Dogleg
--------------------

This method is an improvement over the classical dogleg algorithm, which
attempts to include information about the Gauss-Newton step while the
iteration is still far from the minimum.  When the Cauchy point is
inside the trust region and the Gauss-Newton point is outside, the
method computes a scaled Gauss-Newton point and then takes a dogleg step
between the Cauchy point and the scaled Gauss-Newton point.  The scaling
is calculated to ensure that the reduction in the model m_k is about the
same as the reduction provided by the Cauchy point.

41.2.5 Two Dimensional Subspace
-------------------------------

The dogleg methods restrict the search for the TRS solution to a 1D
curve defined by the Cauchy and Gauss-Newton points.  An improvement to
this is to search for a solution using the full two dimensional subspace
spanned by the Cauchy and Gauss-Newton directions.  The dogleg path is
of course inside this subspace, and so this method solves the TRS at
least as accurately as the dogleg methods.  Since this method searches a
larger subspace for a solution, it can converge more quickly than dogleg
on some problems.  Because the subspace is only two dimensional, this
method is very efficient and the main computation per iteration is to
determine the Gauss-Newton point.

41.2.6 Steihaug-Toint Conjugate Gradient
----------------------------------------

One difficulty of the dogleg methods is calculating the Gauss-Newton
step when the Jacobian matrix is singular.  The Steihaug-Toint method
also computes a generalized dogleg step, but avoids solving for the
Gauss-Newton step directly, instead using an iterative conjugate
gradient algorithm.  This method performs well at points where the
Jacobian is singular, and is also suitable for large-scale problems
where factoring the Jacobian matrix could be prohibitively expensive.

41.3 Weighted Nonlinear Least-Squares
=====================================

Weighted nonlinear least-squares fitting minimizes the function

     \Phi(x) = (1/2) || f(x) ||_W^2
             = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2

where W = \diag(w_1,w_2,...,w_n) is the weighting matrix, and ||f||_W^2
= f^T W f.  The weights w_i are commonly defined as w_i = 1/\sigma_i^2,
where \sigma_i is the error in the i-th measurement.  A simple change of
variables \tilde{f} = W^{1 \over 2} f yields \Phi(x) = {1 \over 2}
||\tilde{f}||^2, which is in the same form as the unweighted case.  The
user can either perform this transform directly on their function
residuals and Jacobian, or use the gsl_multifit_nlinear_winit()
interface which automatically performs the correct scaling.  To
manually perform this transformation, the residuals and Jacobian should
be modified according to

     f~_i = f_i / \sigma_i
     J~_ij = 1 / \sigma_i df_i/dx_j

For large systems, the user must perform their own weighting.

41.4 Tunable Parameters
=======================

The user can tune nearly all aspects of the iteration at allocation
time.  For the ‘gsl_multifit_nlinear’ interface, the user may modify the
gsl_multifit_nlinear_parameters structure, which is defined
as follows:

 -- Type: gsl_multifit_nlinear_parameters

          typedef struct
          {
            const gsl_multifit_nlinear_trs *trs;        /* trust region subproblem method */
            const gsl_multifit_nlinear_scale *scale;    /* scaling method */
            const gsl_multifit_nlinear_solver *solver;  /* solver method */
            gsl_multifit_nlinear_fdtype fdtype;         /* finite difference method */
            double factor_up;                           /* factor for increasing trust radius */
            double factor_down;                         /* factor for decreasing trust radius */
            double avmax;                               /* max allowed |a|/|v| */
            double h_df;                                /* step size for finite difference Jacobian */
            double h_fvv;                               /* step size for finite difference fvv */
          } gsl_multifit_nlinear_parameters;

For the ‘gsl_multilarge_nlinear’ interface, the user may modify the
gsl_multilarge_nlinear_parameters structure, which is
defined as follows:

 -- Type: gsl_multilarge_nlinear_parameters

          typedef struct
          {
            const gsl_multilarge_nlinear_trs *trs;       /* trust region subproblem method */
            const gsl_multilarge_nlinear_scale *scale;   /* scaling method */
            const gsl_multilarge_nlinear_solver *solver; /* solver method */
            gsl_multilarge_nlinear_fdtype fdtype;        /* finite difference method */
            double factor_up;                            /* factor for increasing trust radius */
            double factor_down;                          /* factor for decreasing trust radius */
            double avmax;                                /* max allowed |a|/|v| */
            double h_df;                                 /* step size for finite difference Jacobian */
            double h_fvv;                                /* step size for finite difference fvv */
            size_t max_iter;                             /* maximum iterations for trs method */
            double tol;                                  /* tolerance for solving trs */
          } gsl_multilarge_nlinear_parameters;

Each of these parameters is discussed in further detail below.

 -- Type: gsl_multifit_nlinear_trs
 -- Type: gsl_multilarge_nlinear_trs

     The parameter ‘trs’ determines the method used to solve the trust
     region subproblem, and may be selected from the following choices,

      -- Variable: gsl_multifit_nlinear_trs_lm
      -- Variable: gsl_multilarge_nlinear_trs_lm

          This selects the Levenberg-Marquardt algorithm.

      -- Variable: gsl_multifit_nlinear_trs_lmaccel
      -- Variable: gsl_multilarge_nlinear_trs_lmaccel

          This selects the Levenberg-Marquardt algorithm with geodesic
          acceleration.

      -- Variable: gsl_multifit_nlinear_trs_dogleg
      -- Variable: gsl_multilarge_nlinear_trs_dogleg

          This selects the dogleg algorithm.

      -- Variable: gsl_multifit_nlinear_trs_ddogleg
      -- Variable: gsl_multilarge_nlinear_trs_ddogleg

          This selects the double dogleg algorithm.

      -- Variable: gsl_multifit_nlinear_trs_subspace2D
      -- Variable: gsl_multilarge_nlinear_trs_subspace2D

          This selects the 2D subspace algorithm.

      -- Variable: gsl_multilarge_nlinear_trs_cgst

          This selects the Steihaug-Toint conjugate gradient algorithm.
          This method is available only for large systems.

 -- Type: gsl_multifit_nlinear_scale
 -- Type: gsl_multilarge_nlinear_scale

     The parameter ‘scale’ determines the diagonal scaling matrix D and
     may be selected from the following choices,

      -- Variable: gsl_multifit_nlinear_scale_more
      -- Variable: gsl_multilarge_nlinear_scale_more

          This damping strategy was suggested by Moré, and corresponds
          to D^T D = \max(\diag(J^T J)), in other words the maximum
          elements of \diag(J^T J) encountered thus far in the
          iteration.  This choice of D makes the problem
          scale-invariant, so that if the model parameters x_i are each
          scaled by an arbitrary constant, \tilde{x}_i = a_i x_i, then
          the sequence of iterates produced by the algorithm would be
          unchanged.  This method can work very well in cases where the
          model parameters have widely different scales (ie: if some
          parameters are measured in nanometers, while others are
          measured in degrees Kelvin).  This strategy has been proven
          effective on a large class of problems and so it is the
          library default, but it may not be the best choice for all
          problems.

      -- Variable: gsl_multifit_nlinear_scale_levenberg
      -- Variable: gsl_multilarge_nlinear_scale_levenberg

          This damping strategy was originally suggested by Levenberg,
          and corresponds to D^T D = I. This method has also proven
          effective on a large class of problems, but is not
          scale-invariant.  However, some authors (e.g.  Transtrum and
          Sethna 2012) argue that this choice is better for problems
          which are susceptible to parameter evaporation (ie: parameters
          go to infinity)

      -- Variable: gsl_multifit_nlinear_scale_marquardt
      -- Variable: gsl_multilarge_nlinear_scale_marquardt

          This damping strategy was suggested by Marquardt, and
          corresponds to D^T D = \diag(J^T J). This method is
          scale-invariant, but it is generally considered inferior to
          both the Levenberg and Moré strategies, though may work well
          on certain classes of problems.

 -- Type: gsl_multifit_nlinear_solver
 -- Type: gsl_multilarge_nlinear_solver

     Solving the trust region subproblem on each iteration almost always
     requires the solution of the following linear least squares system

          [J; sqrt(mu) D] \delta = - [f; 0]

     The ‘solver’ parameter determines how the system is solved and can
     be selected from the following choices:

      -- Variable: gsl_multifit_nlinear_solver_qr

          This method solves the system using a rank revealing QR
          decomposition of the Jacobian J. This method will produce
          reliable solutions in cases where the Jacobian is rank
          deficient or near-singular but does require about twice as
          many operations as the Cholesky method discussed below.

      -- Variable: gsl_multifit_nlinear_solver_cholesky
      -- Variable: gsl_multilarge_nlinear_solver_cholesky

          This method solves the alternate normal equations problem

               ( J^T J + \mu D^T D ) \delta = -J^T f

          by using a Cholesky decomposition of the matrix J^T J + \mu
          D^T D. This method is faster than the QR approach, however it
          is susceptible to numerical instabilities if the Jacobian
          matrix is rank deficient or near-singular.  In these cases, an
          attempt is made to reduce the condition number of the matrix
          using Jacobi preconditioning, but for highly ill-conditioned
          problems the QR approach is better.  If it is known that the
          Jacobian matrix is well conditioned, this method is accurate
          and will perform faster than the QR approach.

      -- Variable: gsl_multifit_nlinear_solver_svd

          This method solves the system using a singular value
          decomposition of the Jacobian J. This method will produce the
          most reliable solutions for ill-conditioned Jacobians but is
          also the slowest solver method.

 -- Type: gsl_multifit_nlinear_fdtype

     The parameter ‘fdtype’ specifies whether to use forward or centered
     differences when approximating the Jacobian.  This is only used
     when an analytic Jacobian is not provided to the solver.  This
     parameter may be set to one of the following choices.

      -- Macro: GSL_MULTIFIT_NLINEAR_FWDIFF

          This specifies a forward finite difference to approximate the
          Jacobian matrix.  The Jacobian matrix will be calculated as

               J_ij = 1 / \Delta_j ( f_i(x + \Delta_j e_j) - f_i(x) )

          where \Delta_j = h |x_j| and e_j is the standard j-th
          Cartesian unit basis vector so that x + \Delta_j e_j
          represents a small (forward) perturbation of the j-th
          parameter by an amount \Delta_j.  The perturbation \Delta_j is
          proportional to the current value |x_j| which helps to
          calculate an accurate Jacobian when the various parameters
          have different scale sizes.  The value of h is specified by
          the ‘h_df’ parameter.  The accuracy of this method is O(h),
          and evaluating this matrix requires an additional p function
          evaluations.

      -- Macro: GSL_MULTIFIT_NLINEAR_CTRDIFF

          This specifies a centered finite difference to approximate the
          Jacobian matrix.  The Jacobian matrix will be calculated as

               J_ij = 1 / \Delta_j ( f_i(x + 1/2 \Delta_j e_j) - f_i(x - 1/2 \Delta_j e_j) )

          See above for a description of \Delta_j.  The accuracy of this
          method is O(h^2), but evaluating this matrix requires an
          additional 2p function evaluations.

‘double factor_up’

When a step is accepted, the trust region radius will be increased by
this factor.  The default value is 3.

‘double factor_down’

When a step is rejected, the trust region radius will be decreased by
this factor.  The default value is 2.

‘double avmax’

When using geodesic acceleration to solve a nonlinear least squares
problem, an important parameter to monitor is the ratio of the
acceleration term to the velocity term,

     |a| / |v|

If this ratio is small, it means the acceleration correction is
contributing very little to the step.  This could be because the problem
is not “nonlinear” enough to benefit from the acceleration.  If the
ratio is large (> 1) it means that the acceleration is larger than the
velocity, which shouldn’t happen since the step represents a truncated
series and so the second order term a should be smaller than the first
order term v to guarantee convergence.  Therefore any steps with a ratio
larger than the parameter ‘avmax’ are rejected.  ‘avmax’ is set to 0.75
by default.  For problems which experience difficulty converging, this
threshold could be lowered.

‘double h_df’

This parameter specifies the step size for approximating the Jacobian
matrix with finite differences.  It is set to \sqrt{\epsilon} by
default, where \epsilon is ‘GSL_DBL_EPSILON’.

‘double h_fvv’

When using geodesic acceleration, the user must either supply a function
to calculate f_{vv}(x) or the library can estimate this second
directional derivative using a finite difference method.  When using
finite differences, the library must calculate f(x + h v) where h
represents a small step in the velocity direction.  The parameter
‘h_fvv’ defines this step size and is set to 0.02 by default.

41.5 Initializing the Solver
============================

 -- Type: gsl_multifit_nlinear_type

     This structure specifies the type of algorithm which will be used
     to solve a nonlinear least squares problem.  It may be selected
     from the following choices,

      -- Variable: gsl_multifit_nlinear_trust

          This specifies a trust region method.  It is currently the
          only implemented nonlinear least squares method.

 -- Function: gsl_multifit_nlinear_workspace *
          gsl_multifit_nlinear_alloc (const gsl_multifit_nlinear_type
          * T, const gsl_multifit_nlinear_parameters * params, const
          size_t n, const size_t p)
 -- Function: gsl_multilarge_nlinear_workspace *
          gsl_multilarge_nlinear_alloc (const
          gsl_multilarge_nlinear_type * T, const
          gsl_multilarge_nlinear_parameters * params, const size_t n,
          const size_t p)

     These functions return a pointer to a newly allocated instance of a
     derivative solver of type ‘T’ for ‘n’ observations and ‘p’
     parameters.  The ‘params’ input specifies a tunable set of
     parameters which will affect important details in each iteration of
     the trust region subproblem algorithm.  It is recommended to start
     with the suggested default parameters (see
     gsl_multifit_nlinear_default_parameters() and
     gsl_multilarge_nlinear_default_parameters()) and then tune
     the parameters once the code is working correctly.  See
     Tunable Parameters.  for descriptions of the various
     parameters.  For example, the following code creates an instance of
     a Levenberg-Marquardt solver for 100 data points and 3 parameters,
     using suggested defaults:

          const gsl_multifit_nlinear_type * T = gsl_multifit_nlinear_trust;
          gsl_multifit_nlinear_parameters params = gsl_multifit_nlinear_default_parameters();
          gsl_multifit_nlinear_workspace * w = gsl_multifit_nlinear_alloc (T, &params, 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: gsl_multifit_nlinear_parameters
          gsl_multifit_nlinear_default_parameters (void)
 -- Function: gsl_multilarge_nlinear_parameters
          gsl_multilarge_nlinear_default_parameters (void)

     These functions return a set of recommended default parameters for
     use in solving nonlinear least squares problems.  The user can tune
     each parameter to improve the performance on their particular
     problem, see Tunable Parameters.

 -- Function: int gsl_multifit_nlinear_init (const gsl_vector * x,
          gsl_multifit_nlinear_fdf * fdf, gsl_multifit_nlinear_workspace
          * w)
 -- Function: int gsl_multifit_nlinear_winit (const gsl_vector * x,
          const gsl_vector * wts, gsl_multifit_nlinear_fdf * fdf,
          gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_init (const gsl_vector * x,
          gsl_multilarge_nlinear_fdf * fdf,
          gsl_multilarge_nlinear_workspace * w)

     These functions initialize, or reinitialize, an existing workspace
     ‘w’ to use the system ‘fdf’ and the initial guess ‘x’.  See
     Providing the Function to be Minimized for a description of
     the ‘fdf’ structure.

     Optionally, a weight vector ‘wts’ can be given to perform a
     weighted nonlinear regression.  Here, the weighting matrix is W =
     \diag(w_1,w_2,...,w_n).

 -- Function: void gsl_multifit_nlinear_free
          (gsl_multifit_nlinear_workspace * w)
 -- Function: void gsl_multilarge_nlinear_free
          (gsl_multilarge_nlinear_workspace * w)

     These functions free all the memory associated with the workspace
     ‘w’.

 -- Function: const char * gsl_multifit_nlinear_name (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: const char * gsl_multilarge_nlinear_name (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return a pointer to the name of the solver.  For
     example:

          printf ("w is a '%s' solver\n", gsl_multifit_nlinear_name (w));

     would print something like ‘w is a 'trust-region' solver’.

 -- Function: const char * gsl_multifit_nlinear_trs_name (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: const char * gsl_multilarge_nlinear_trs_name (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return a pointer to the name of the trust region
     subproblem method.  For example:

          printf ("w is a '%s' solver\n", gsl_multifit_nlinear_trs_name (w));

     would print something like ‘w is a 'levenberg-marquardt' solver’.

41.6 Providing the Function to be Minimized
===========================================

The user must provide n functions of p variables for the minimization
algorithm to operate on.  In order to allow for arbitrary parameters the
functions are defined by the following data types:

 -- Type: gsl_multifit_nlinear_fdf

     This data type defines a general system of functions with arbitrary
     parameters, the corresponding Jacobian matrix of derivatives, and
     optionally the second directional derivative of the functions for
     geodesic acceleration.

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          This function should store the n components of the vector f(x)
          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) / d x_j

          in ‘J’ for argument ‘x’ and arbitrary parameters ‘params’,
          returning an appropriate error code if the matrix 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 (* fvv) (const gsl_vector * x, const gsl_vector * v, void *
     params, gsl_vector * fvv)’

          When geodesic acceleration is enabled, this function should
          store the n components of the vector f_{vv}(x) =
          \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over
          \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x),
          representing second directional derivatives of the function to
          be minimized, into the output ‘fvv’.  The parameter vector is
          provided in ‘x’ and the velocity vector is provided in ‘v’,
          both of which have p components.  The arbitrary parameters are
          given in ‘params’.  If analytic expressions for f_{vv}(x) are
          unavailable or too difficult to compute, this function pointer
          may be set to ‘NULL’, in which case f_{vv}(x) will be computed
          internally using a finite difference approximation.

     ‘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.

     ‘size_t nevalf’

          This does not need to be set by the user.  It counts the
          number of function evaluations and is initialized by the
          ‘_init’ function.

     ‘size_t nevaldf’

          This does not need to be set by the user.  It counts the
          number of Jacobian evaluations and is initialized by the
          ‘_init’ function.

     ‘size_t nevalfvv’

          This does not need to be set by the user.  It counts the
          number of f_{vv}(x) evaluations and is initialized by the
          ‘_init’ function.

 -- Type: gsl_multilarge_nlinear_fdf

     This data type defines a general system of functions with arbitrary
     parameters, a function to compute J u or J^T u for a given vector
     u, the normal equations matrix J^T J, and optionally the second
     directional derivative of the functions for geodesic acceleration.

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          This function should store the n components of the vector f(x)
          in ‘f’ for argument ‘x’ and arbitrary parameters ‘params’,
          returning an appropriate error code if the function cannot be
          computed.

     ‘int (* df) (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x, const
     gsl_vector * u, void * params, gsl_vector * v, gsl_matrix * JTJ)’

          If ‘TransJ’ is equal to ‘CblasNoTrans’, then this function
          should compute the matrix-vector product J u and store the
          result in ‘v’.  If ‘TransJ’ is equal to ‘CblasTrans’, then
          this function should compute the matrix-vector product J^T u
          and store the result in ‘v’.  Additionally, the normal
          equations matrix J^T J should be stored in the lower half of
          ‘JTJ’.  The input matrix ‘JTJ’ could be set to ‘NULL’, for
          example by iterative methods which do not require this matrix,
          so the user should check for this prior to constructing the
          matrix.  The input ‘params’ contains the arbitrary parameters.

     ‘int (* fvv) (const gsl_vector * x, const gsl_vector * v, void *
     params, gsl_vector * fvv)’

          When geodesic acceleration is enabled, this function should
          store the n components of the vector f_{vv}(x) =
          \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over
          \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x),
          representing second directional derivatives of the function to
          be minimized, into the output ‘fvv’.  The parameter vector is
          provided in ‘x’ and the velocity vector is provided in ‘v’,
          both of which have p components.  The arbitrary parameters are
          given in ‘params’.  If analytic expressions for f_{vv}(x) are
          unavailable or too difficult to compute, this function pointer
          may be set to ‘NULL’, in which case f_{vv}(x) will be computed
          internally using a finite difference approximation.

     ‘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.

     ‘size_t nevalf’

          This does not need to be set by the user.  It counts the
          number of function evaluations and is initialized by the
          ‘_init’ function.

     ‘size_t nevaldfu’

          This does not need to be set by the user.  It counts the
          number of Jacobian matrix-vector evaluations (J u or J^T u)
          and is initialized by the ‘_init’ function.

     ‘size_t nevaldf2’

          This does not need to be set by the user.  It counts the
          number of J^T J evaluations and is initialized by the ‘_init’
          function.

     ‘size_t nevalfvv’

          This does not need to be set by the user.  It counts the
          number of f_{vv}(x) evaluations and is initialized by the
          ‘_init’ 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.

41.7 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration of the trust region method and updates
the state of the solver.

 -- Function: int gsl_multifit_nlinear_iterate
          (gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_iterate
          (gsl_multilarge_nlinear_workspace * w)

     These functions perform a single iteration of the solver ‘w’.  If
     the iteration encounters an unexpected problem then an error code
     will be returned.  The solver workspace maintains a current
     estimate of the best-fit parameters at all times.

The solver workspace ‘w’ contains the following entries, which can be
used to track the progress of the solution:

‘gsl_vector * x’

     The current position, length p.

‘gsl_vector * f’

     The function residual vector at the current position f(x), length
     n.

‘gsl_matrix * J’

     The Jacobian matrix at the current position J(x), size n-by-p (only
     for ‘gsl_multifit_nlinear’ interface).

‘gsl_vector * dx’

     The difference between the current position and the previous
     position, i.e.  the last step \delta, taken as a vector, length p.

These quantities can be accessed with the following functions,

 -- Function: gsl_vector * gsl_multifit_nlinear_position (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: gsl_vector * gsl_multilarge_nlinear_position (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return the current position x (i.e.  best-fit
     parameters) of the solver ‘w’.

 -- Function: gsl_vector * gsl_multifit_nlinear_residual (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: gsl_vector * gsl_multilarge_nlinear_residual (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return the current residual vector f(x) of the
     solver ‘w’.  For weighted systems, the residual vector includes the
     weighting factor \sqrt{W}.

 -- Function: gsl_matrix * gsl_multifit_nlinear_jac (const
          gsl_multifit_nlinear_workspace * w)

     This function returns a pointer to the n-by-p Jacobian matrix for
     the current iteration of the solver ‘w’.  This function is
     available only for the ‘gsl_multifit_nlinear’ interface.

 -- Function: size_t gsl_multifit_nlinear_niter (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: size_t gsl_multilarge_nlinear_niter (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return the number of iterations performed so far.
     The iteration counter is updated on each call to the ‘_iterate’
     functions above, and reset to 0 in the ‘_init’ functions.

 -- Function: int gsl_multifit_nlinear_rcond (double * rcond, const
          gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_rcond (double * rcond, const
          gsl_multilarge_nlinear_workspace * w)

     This function estimates the reciprocal condition number of the
     Jacobian matrix at the current position x and stores it in ‘rcond’.
     The computed value is only an estimate to give the user a guideline
     as to the conditioning of their particular problem.  Its
     calculation is based on which factorization method is used
     (Cholesky, QR, or SVD).

        * For the Cholesky solver, the matrix J^T J is factored at each
          iteration.  Therefore this function will estimate the 1-norm
          condition number rcond^2 = 1/(||J^T J||_1 \cdot ||(J^T
          J)^{-1}||_1)

        * For the QR solver, J is factored as J = Q R at each iteration.
          For simplicity, this function calculates the 1-norm
          conditioning of only the R factor, rcond = 1 / (||R||_1 \cdot
          ||R^{-1}||_1).  This can be computed efficiently since R is
          upper triangular.

        * For the SVD solver, in order to efficiently solve the trust
          region subproblem, the matrix which is factored is J D^{-1},
          instead of J itself.  The resulting singular values are used
          to provide the 2-norm reciprocal condition number, as rcond =
          \sigma_{min} / \sigma_{max}.  Note that when using Moré
          scaling, D \ne I and the resulting ‘rcond’ estimate may be
          significantly different from the true ‘rcond’ of J itself.

 -- Function: double gsl_multifit_nlinear_avratio (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: double gsl_multilarge_nlinear_avratio (const
          gsl_multilarge_nlinear_workspace * w)

     This function returns the current ratio |a| / |v| of the
     acceleration correction term to the velocity step term.  The
     acceleration term is computed only by the
     gsl_multifit_nlinear_trs_lmaccel and
     gsl_multilarge_nlinear_trs_lmaccel methods, so this ratio
     will be zero for other TRS methods.

41.8 Testing for Convergence
============================

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_nlinear_test (const double xtol, const
          double gtol, const double ftol, int * info, const
          gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_test (const double xtol, const
          double gtol, const double ftol, int * info, const
          gsl_multilarge_nlinear_workspace * w)

     These functions test for convergence of the minimization method
     using the following criteria:

        * Testing for a small step size relative to the current
          parameter vector

               |\delta_i| \le xtol (|x_i| + xtol)

          for each 0 <= i < p.  Each element of the step vector \delta
          is tested individually in case the different parameters have
          widely different scales.  Adding ‘xtol’ to |x_i| helps the
          test avoid breaking down in situations where the true solution
          value x_i = 0.  If this test succeeds, ‘info’ is set to 1 and
          the function returns ‘GSL_SUCCESS’.

          A general guideline for selecting the step tolerance is to
          choose xtol = 10^{-d} where d is the number of accurate
          decimal digits desired in the solution x.  See Dennis and
          Schnabel for more information.

        * Testing for a small gradient (g = \nabla \Phi(x) = J^T f)
          indicating a local function minimum:

               ||g||_inf <= gtol

          This expression tests whether the ratio (\nabla \Phi)_i x_i /
          \Phi is small.  Testing this scaled gradient is a better than
          \nabla \Phi alone since it is a dimensionless quantity and so
          independent of the scale of the problem.  The ‘max’ arguments
          help ensure the test doesn’t break down in regions where x_i
          or \Phi(x) are close to 0.  If this test succeeds, ‘info’ is
          set to 2 and the function returns ‘GSL_SUCCESS’.

          A general guideline for choosing the gradient tolerance is to
          set ‘gtol = GSL_DBL_EPSILON^(1/3)’.  See Dennis and Schnabel
          for more information.

     If none of the tests succeed, ‘info’ is set to 0 and the function
     returns ‘GSL_CONTINUE’, indicating further iterations are required.

41.9 High Level Driver
======================

These routines provide a high level wrapper that combines the iteration
and convergence testing for easy use.

 -- Function: int gsl_multifit_nlinear_driver (const size_t maxiter,
          const double xtol, const double gtol, const double ftol, void
          (* callback)(const size_t iter, void * params, const
          gsl_multifit_linear_workspace * w), void * callback_params,
          int * info, gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_driver (const size_t maxiter,
          const double xtol, const double gtol, const double ftol, void
          (* callback)(const size_t iter, void * params, const
          gsl_multilarge_linear_workspace * w), void * callback_params,
          int * info, gsl_multilarge_nlinear_workspace * w)

     These functions iterate the nonlinear least squares solver ‘w’ for
     a maximum of ‘maxiter’ iterations.  After each iteration, the
     system is tested for convergence with the error tolerances ‘xtol’,
     ‘gtol’ and ‘ftol’.  Additionally, the user may supply a callback
     function ‘callback’ which is called after each iteration, so that
     the user may save or print relevant quantities for each iteration.
     The parameter ‘callback_params’ is passed to the ‘callback’
     function.  The parameters ‘callback’ and ‘callback_params’ may be
     set to ‘NULL’ to disable this feature.  Upon successful
     convergence, the function returns ‘GSL_SUCCESS’ and sets ‘info’ to
     the reason for convergence (see gsl_multifit_nlinear_test()).
     If the function has not converged after ‘maxiter’
     iterations, ‘GSL_EMAXITER’ is returned.  In rare cases, during an
     iteration the algorithm may be unable to find a new acceptable step
     \delta to take.  In this case, ‘GSL_ENOPROG’ is returned indicating
     no further progress can be made.  If your problem is having
     difficulty converging, see Troubleshooting for further
     guidance.

41.10 Covariance matrix of best fit parameters
==============================================

 -- Function: int gsl_multifit_nlinear_covar (const gsl_matrix * J,
          const double epsrel, gsl_matrix * covar)
 -- Function: int gsl_multilarge_nlinear_covar (gsl_matrix * covar,
          gsl_multilarge_nlinear_workspace * w)

     This function computes the covariance matrix of best-fit parameters
     using the Jacobian matrix ‘J’ and stores it in ‘covar’.  The
     parameter ‘epsrel’ is used to remove linear-dependent columns when
     ‘J’ is rank deficient.

     The covariance matrix is given by,

          C = (J^T J)^{-1}

     or in the weighted case,

          C = (J^T W J)^{-1}

     and is computed using the factored form of the Jacobian (Cholesky,
     QR, or SVD). Any columns of R which satisfy

          |R_{kk}| \leq 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 Linear Least-Squares Overview.

41.11 Troubleshooting
=====================

When developing a code to solve a nonlinear least squares problem, here
are a few considerations to keep in mind.

  1. The most common difficulty is the accurate implementation of the
     Jacobian matrix.  If the analytic Jacobian is not properly provided
     to the solver, this can hinder and many times prevent convergence
     of the method.  When developing a new nonlinear least squares code,
     it often helps to compare the program output with the internally
     computed finite difference Jacobian and the user supplied analytic
     Jacobian.  If there is a large difference in coefficients, it is
     likely the analytic Jacobian is incorrectly implemented.

  2. If your code is having difficulty converging, the next thing to
     check is the starting point provided to the solver.  The methods of
     this chapter are local methods, meaning if you provide a starting
     point far away from the true minimum, the method may converge to a
     local minimum or not converge at all.  Sometimes it is possible to
     solve a linearized approximation to the nonlinear problem, and use
     the linear solution as the starting point to the nonlinear problem.

  3. If the various parameters of the coefficient vector x vary widely
     in magnitude, then the problem is said to be badly scaled.  The
     methods of this chapter do attempt to automatically rescale the
     elements of x to have roughly the same order of magnitude, but in
     extreme cases this could still cause problems for convergence.  In
     these cases it is recommended for the user to scale their parameter
     vector x so that each parameter spans roughly the same range, say
     [-1,1].  The solution vector can be backscaled to recover the
     original units of the problem.

41.12 Examples
==============

The following example programs demonstrate the nonlinear least squares
fitting capabilities.

41.12.1 Exponential Fitting Example
-----------------------------------

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

where we have chosen t_i = i T / (N - 1), where N is the number of data
points fitted, so that t_i \in [0, T]. 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.  The i-th row of the Jacobian
is therefore

     J(i,:) = [ \exp(-\lambda t_i) ; -t_i A \exp(-\lambda t_i) ; 1 ]

The main part of the program sets up a Levenberg-Marquardt solver and
some simulated random data.  The data uses the known parameters
(5.0,0.1,1.0) combined with Gaussian noise (standard deviation = 0.1)
with a maximum time T = 40 and N = 100 timesteps.  The initial guess for
the parameters is chosen as (1.0, 1.0, 0.0).  The iteration terminates
when the relative change in x is smaller than 10^{-8}, or when the
magnitude of the gradient falls below 10^{-8}.  Here are the results of
running the program:

     iter  0: A = 1.0000, lambda = 1.0000, b = 0.0000, cond(J) =      inf, |f(x)| = 100.8779
     iter  1: A = 1.2692, lambda = 0.3924, b = 0.0443, cond(J) =  69.5973, |f(x)| = 97.1734
     iter  2: A = 1.6749, lambda = 0.1685, b = 0.1072, cond(J) =  29.5220, |f(x)| = 88.6636
     iter  3: A = 2.5579, lambda = 0.0544, b = 0.2552, cond(J) =  22.9334, |f(x)| = 42.7765
     iter  4: A = 3.0167, lambda = 0.0472, b = 0.3704, cond(J) = 120.2912, |f(x)| = 23.0102
     iter  5: A = 3.3590, lambda = 0.0455, b = 0.4321, cond(J) = 266.5620, |f(x)| = 16.0680
     iter  6: A = 3.6552, lambda = 0.0479, b = 0.4426, cond(J) = 343.8946, |f(x)| = 14.6421
     iter  7: A = 3.9546, lambda = 0.0532, b = 0.4897, cond(J) = 301.4985, |f(x)| = 13.5266
     iter  8: A = 4.1421, lambda = 0.0633, b = 0.6783, cond(J) = 203.9164, |f(x)| = 12.3149
     iter  9: A = 4.3752, lambda = 0.0800, b = 0.9228, cond(J) = 158.2267, |f(x)| = 11.2475
     iter 10: A = 4.6371, lambda = 0.0891, b = 0.9588, cond(J) = 136.6189, |f(x)| = 10.5457
     iter 11: A = 4.7684, lambda = 0.0937, b = 0.9860, cond(J) = 125.4740, |f(x)| = 10.4753
     iter 12: A = 4.7977, lambda = 0.0948, b = 0.9917, cond(J) = 120.1098, |f(x)| = 10.4723
     iter 13: A = 4.8006, lambda = 0.0949, b = 0.9924, cond(J) = 118.9113, |f(x)| = 10.4723
     iter 14: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7661, |f(x)| = 10.4723
     iter 15: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7550, |f(x)| = 10.4723
     iter 16: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7543, |f(x)| = 10.4723
     iter 17: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7543, |f(x)| = 10.4723
     iter 18: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7543, |f(x)| = 10.4723
     summary from method 'trust-region/levenberg-marquardt'
     number of iterations: 18
     function evaluations: 25
     Jacobian evaluations: 19
     reason for stopping: small step size
     initial |f(x)| = 100.877904
     final   |f(x)| = 10.472268
     chisq/dof = 1.1306
     A      = 4.80085 +/- 0.17652
     lambda = 0.09488 +/- 0.00527
     b      = 0.99249 +/- 0.04419
     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/(n-p) \gg 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/(n-p)} in this case, a common way of
increasing the errors for a poor fit.  Note that a poor fit will result
from the use of an inappropriate model, and the scaled error estimates
may then be outside the range of validity for Gaussian errors.

Additionally, we see that the condition number of J(x) stays reasonably
small throughout the iteration.  This indicates we could safely switch
to the Cholesky solver for speed improvement, although this particular
system is too small to really benefit.

Fig.  %s shows the fitted curve with the original data.

Figure: Exponential fitted curve with data

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>

     #define N      100    /* number of data points to fit */
     #define TMAX   (40.0) /* time variable in [0,TMAX] */

     struct data {
       size_t n;
       double * t;
       double * y;
     };

     int
     expb_f (const gsl_vector * x, void *data,
             gsl_vector * f)
     {
       size_t n = ((struct data *)data)->n;
       double *t = ((struct data *)data)->t;
       double *y = ((struct data *)data)->y;

       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 * t_i) + b */
           double Yi = A * exp (-lambda * t[i]) + b;
           gsl_vector_set (f, i, Yi - y[i]);
         }

       return GSL_SUCCESS;
     }

     int
     expb_df (const gsl_vector * x, void *data,
              gsl_matrix * J)
     {
       size_t n = ((struct data *)data)->n;
       double *t = ((struct data *)data)->t;

       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 * t_i) + b  */
           /* and the xj are the parameters (A,lambda,b) */
           double e = exp(-lambda * t[i]);
           gsl_matrix_set (J, i, 0, e);
           gsl_matrix_set (J, i, 1, -t[i] * A * e);
           gsl_matrix_set (J, i, 2, 1.0);
         }

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector *f = gsl_multifit_nlinear_residual(w);
       gsl_vector *x = gsl_multifit_nlinear_position(w);
       double rcond;

       /* compute reciprocal condition number of J(x) */
       gsl_multifit_nlinear_rcond(&rcond, w);

       fprintf(stderr, "iter %2zu: A = %.4f, lambda = %.4f, b = %.4f, cond(J) = %8.4f, |f(x)| = %.4f\n",
               iter,
               gsl_vector_get(x, 0),
               gsl_vector_get(x, 1),
               gsl_vector_get(x, 2),
               1.0 / rcond,
               gsl_blas_dnrm2(f));
     }

     int
     main (void)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       gsl_multifit_nlinear_workspace *w;
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();
       const size_t n = N;
       const size_t p = 3;

       gsl_vector *f;
       gsl_matrix *J;
       gsl_matrix *covar = gsl_matrix_alloc (p, p);
       double t[N], y[N], weights[N];
       struct data d = { n, t, y };
       double x_init[3] = { 1.0, 1.0, 0.0 }; /* starting values */
       gsl_vector_view x = gsl_vector_view_array (x_init, p);
       gsl_vector_view wts = gsl_vector_view_array(weights, n);
       gsl_rng * r;
       double chisq, chisq0;
       int status, info;
       size_t i;

       const double xtol = 1e-8;
       const double gtol = 1e-8;
       const double ftol = 0.0;

       gsl_rng_env_setup();
       r = gsl_rng_alloc(gsl_rng_default);

       /* define the function to be minimized */
       fdf.f = expb_f;
       fdf.df = expb_df;   /* set to NULL for finite-difference Jacobian */
       fdf.fvv = NULL;     /* not using geodesic acceleration */
       fdf.n = n;
       fdf.p = p;
       fdf.params = &d;

       /* this is the data to be fitted */
       for (i = 0; i < n; i++)
         {
           double ti = i * TMAX / (n - 1.0);
           double yi = 1.0 + 5 * exp (-0.1 * ti);
           double si = 0.1 * yi;
           double dy = gsl_ran_gaussian(r, si);

           t[i] = ti;
           y[i] = yi + dy;
           weights[i] = 1.0 / (si * si);
           printf ("data: %g %g %g\n", ti, y[i], si);
         };

       /* allocate workspace with default parameters */
       w = gsl_multifit_nlinear_alloc (T, &fdf_params, n, p);

       /* initialize solver with starting point and weights */
       gsl_multifit_nlinear_winit (&x.vector, &wts.vector, &fdf, w);

       /* compute initial cost function */
       f = gsl_multifit_nlinear_residual(w);
       gsl_blas_ddot(f, f, &chisq0);

       /* solve the system with a maximum of 100 iterations */
       status = gsl_multifit_nlinear_driver(100, xtol, gtol, ftol,
                                            callback, NULL, &info, w);

       /* compute covariance of best fit parameters */
       J = gsl_multifit_nlinear_jac(w);
       gsl_multifit_nlinear_covar (J, 0.0, covar);

       /* compute final cost */
       gsl_blas_ddot(f, f, &chisq);

     #define FIT(i) gsl_vector_get(w->x, i)
     #define ERR(i) sqrt(gsl_matrix_get(covar,i,i))

       fprintf(stderr, "summary from method '%s/%s'\n",
               gsl_multifit_nlinear_name(w),
               gsl_multifit_nlinear_trs_name(w));
       fprintf(stderr, "number of iterations: %zu\n",
               gsl_multifit_nlinear_niter(w));
       fprintf(stderr, "function evaluations: %zu\n", fdf.nevalf);
       fprintf(stderr, "Jacobian evaluations: %zu\n", fdf.nevaldf);
       fprintf(stderr, "reason for stopping: %s\n",
               (info == 1) ? "small step size" : "small gradient");
       fprintf(stderr, "initial |f(x)| = %f\n", sqrt(chisq0));
       fprintf(stderr, "final   |f(x)| = %f\n", sqrt(chisq));

       {
         double dof = n - p;
         double c = GSL_MAX_DBL(1, sqrt(chisq / dof));

         fprintf(stderr, "chisq/dof = %g\n", chisq / dof);

         fprintf (stderr, "A      = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
         fprintf (stderr, "lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
         fprintf (stderr, "b      = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
       }

       fprintf (stderr, "status = %s\n", gsl_strerror (status));

       gsl_multifit_nlinear_free (w);
       gsl_matrix_free (covar);
       gsl_rng_free (r);

       return 0;
     }

41.12.2 Geodesic Acceleration Example 1
---------------------------------------

The following example program minimizes a modified Rosenbrock function,
which is characterized by a narrow canyon with steep walls.  The
starting point is selected high on the canyon wall, so the solver must
first find the canyon bottom and then navigate to the minimum.  The
problem is solved both with and without using geodesic acceleration for
comparison.  The cost function is given by

     Phi(x) = 1/2 (f1^2 + f2^2)
     f1 = 100 ( x2 - x1^2 )
     f2 = 1 - x1

The Jacobian matrix is

     J = [ -200*x1 100  ]
         [   -1     0   ]

In order to use geodesic acceleration, the user must provide the second
directional derivative of each residual in the velocity direction, D_v^2
f_i = \sum_{\alpha\beta} v_{\alpha} v_{\beta} \partial_{\alpha}
\partial_{\beta} f_i.  The velocity vector v is provided by the solver.
For this example, these derivatives are

     fvv = [ -200 v1^2 ]
           [     0     ]

The solution of this minimization problem is

     x* = [ 1 ; 1 ]
     Phi(x*) = 0

The program output is shown below:

     === Solving system without acceleration ===
     NITER         = 53
     NFEV          = 56
     NJEV          = 54
     NAEV          = 0
     initial cost  = 2.250225000000e+04
     final cost    = 6.674986031430e-18
     final x       = (9.999999974165e-01, 9.999999948328e-01)
     final cond(J) = 6.000096055094e+02
     === Solving system with acceleration ===
     NITER         = 15
     NFEV          = 17
     NJEV          = 16
     NAEV          = 16
     initial cost  = 2.250225000000e+04
     final cost    = 7.518932873279e-19
     final x       = (9.999999991329e-01, 9.999999982657e-01)
     final cond(J) = 6.000097233278e+02

Figure: Paths taken by solver for Rosenbrock function

We can see that enabling geodesic acceleration requires less than a
third of the number of Jacobian evaluations in order to locate the
minimum.  The path taken by both methods is shown in Fig.  %s.  The
contours show the cost function \Phi(x_1,x_2).  We see that both methods
quickly find the canyon bottom, but the geodesic acceleration method
navigates along the bottom to the solution with significantly fewer
iterations.

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>

     int
     func_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       double x1 = gsl_vector_get(x, 0);
       double x2 = gsl_vector_get(x, 1);

       gsl_vector_set(f, 0, 100.0 * (x2 - x1*x1));
       gsl_vector_set(f, 1, 1.0 - x1);

       return GSL_SUCCESS;
     }

     int
     func_df (const gsl_vector * x, void *params, gsl_matrix * J)
     {
       double x1 = gsl_vector_get(x, 0);

       gsl_matrix_set(J, 0, 0, -200.0*x1);
       gsl_matrix_set(J, 0, 1, 100.0);
       gsl_matrix_set(J, 1, 0, -1.0);
       gsl_matrix_set(J, 1, 1, 0.0);

       return GSL_SUCCESS;
     }

     int
     func_fvv (const gsl_vector * x, const gsl_vector * v,
               void *params, gsl_vector * fvv)
     {
       double v1 = gsl_vector_get(v, 0);

       gsl_vector_set(fvv, 0, -200.0 * v1 * v1);
       gsl_vector_set(fvv, 1, 0.0);

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector * x = gsl_multifit_nlinear_position(w);

       /* print out current location */
       printf("%f %f\n",
              gsl_vector_get(x, 0),
              gsl_vector_get(x, 1));
     }

     void
     solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
                  gsl_multifit_nlinear_parameters *params)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multifit_nlinear_workspace *work =
         gsl_multifit_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multifit_nlinear_residual(work);
       gsl_vector * x = gsl_multifit_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond;

       /* initialize solver */
       gsl_multifit_nlinear_init(x0, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                                   callback, NULL, &info, work);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* store cond(J(x)) */
       gsl_multifit_nlinear_rcond(&rcond, work);

       /* print summary */

       fprintf(stderr, "NITER         = %zu\n", gsl_multifit_nlinear_niter(work));
       fprintf(stderr, "NFEV          = %zu\n", fdf->nevalf);
       fprintf(stderr, "NJEV          = %zu\n", fdf->nevaldf);
       fprintf(stderr, "NAEV          = %zu\n", fdf->nevalfvv);
       fprintf(stderr, "initial cost  = %.12e\n", chisq0);
       fprintf(stderr, "final cost    = %.12e\n", chisq);
       fprintf(stderr, "final x       = (%.12e, %.12e)\n",
               gsl_vector_get(x, 0), gsl_vector_get(x, 1));
       fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);

       printf("\n\n");

       gsl_multifit_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t n = 2;
       const size_t p = 2;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();

       /* print map of Phi(x1, x2) */
       {
         double x1, x2, chisq;
         double *f1 = gsl_vector_ptr(f, 0);
         double *f2 = gsl_vector_ptr(f, 1);

         for (x1 = -1.2; x1 < 1.3; x1 += 0.1)
           {
             for (x2 = -0.5; x2 < 2.1; x2 += 0.1)
               {
                 gsl_vector_set(x, 0, x1);
                 gsl_vector_set(x, 1, x2);
                 func_f(x, NULL, f);

                 chisq = (*f1) * (*f1) + (*f2) * (*f2);
                 printf("%f %f %f\n", x1, x2, chisq);
               }
             printf("\n");
           }
         printf("\n\n");
       }

       /* define function to be minimized */
       fdf.f = func_f;
       fdf.df = func_df;
       fdf.fvv = func_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = NULL;

       /* starting point */
       gsl_vector_set(x, 0, -0.5);
       gsl_vector_set(x, 1, 1.75);

       fprintf(stderr, "=== Solving system without acceleration ===\n");
       fdf_params.trs = gsl_multifit_nlinear_trs_lm;
       solve_system(x, &fdf, &fdf_params);

       fprintf(stderr, "=== Solving system with acceleration ===\n");
       fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       gsl_vector_free(f);
       gsl_vector_free(x);

       return 0;
     }

41.12.3 Geodesic Acceleration Example 2
---------------------------------------

The following example fits a set of data to a Gaussian model using the
Levenberg-Marquardt method with geodesic acceleration.  The cost
function is

     Phi(x) = 1/2 \sum_i f_i^2
     f_i = y_i - Y(a,b,c;t_i)

where y_i is the measured data point at time t_i, and the model is
specified by

     Y(a,b,c;t) = a exp(-1/2 ((t-b)/c)^2)

The parameters a,b,c represent the amplitude, mean, and width of the
Gaussian respectively.  The program below generates the y_i data on
[0,1] using the values a = 5, b = 0.4, c = 0.15 and adding random noise.
The i-th row of the Jacobian is

     J(i,:) = ( -e_i  -(a/c)*z_i*e_i  -(a/c)*z_i^2*e_i )

where

     z_i = (t_i - b) / c
     e_i = \exp(-1/2 z_i^2)

In order to use geodesic acceleration, we need the second directional
derivative of the residuals in the velocity direction, D_v^2 f_i =
\sum_{\alpha\beta} v_{\alpha} v_{\beta} \partial_{\alpha}
\partial_{\beta} f_i, where v is provided by the solver.  To compute
this, it is helpful to make a table of all second derivatives of the
residuals f_i with respect to each combination of model parameters.
This table is

The lower half of the table is omitted since it is symmetric.  Then, the
second directional derivative of f_i is

     D_v^2 f_i = v_a^2 (d/da)^2 f_i + 2 v_a v_b (d/da) (d/db) f_i + 2 v_a v_c (d/da) (d/dc) f_i + v_b^2 (d/db)^2 f_i + 2 v_b v_c (d/db) (d/dc) f_i + v_c^2 (d/dc)^2 f_i

The factors of 2 come from the symmetry of the mixed second partial
derivatives.  The iteration is started using the initial guess a = 1, b
= 0, c = 1.  The program output is shown below:

     iter  0: a = 1.0000, b = 0.0000, c = 1.0000, |a|/|v| = 0.0000 cond(J) =      inf, |f(x)| = 35.4785
     iter  1: a = 1.5708, b = 0.5321, c = 0.5219, |a|/|v| = 0.3093 cond(J) =  29.0443, |f(x)| = 31.1042
     iter  2: a = 1.7387, b = 0.4040, c = 0.4568, |a|/|v| = 0.1199 cond(J) =   3.5256, |f(x)| = 28.7217
     iter  3: a = 2.2340, b = 0.3829, c = 0.3053, |a|/|v| = 0.3308 cond(J) =   4.5121, |f(x)| = 23.8074
     iter  4: a = 3.2275, b = 0.3952, c = 0.2243, |a|/|v| = 0.2784 cond(J) =   8.6499, |f(x)| = 15.6003
     iter  5: a = 4.3347, b = 0.3974, c = 0.1752, |a|/|v| = 0.2029 cond(J) =  15.1732, |f(x)| = 7.5908
     iter  6: a = 4.9352, b = 0.3992, c = 0.1536, |a|/|v| = 0.1001 cond(J) =  26.6621, |f(x)| = 4.8402
     iter  7: a = 5.0716, b = 0.3994, c = 0.1498, |a|/|v| = 0.0166 cond(J) =  34.6922, |f(x)| = 4.7103
     iter  8: a = 5.0828, b = 0.3994, c = 0.1495, |a|/|v| = 0.0012 cond(J) =  36.5422, |f(x)| = 4.7095
     iter  9: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6929, |f(x)| = 4.7095
     iter 10: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6975, |f(x)| = 4.7095
     iter 11: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6976, |f(x)| = 4.7095
     NITER         = 11
     NFEV          = 18
     NJEV          = 12
     NAEV          = 17
     initial cost  = 1.258724737288e+03
     final cost    = 2.217977560180e+01
     final x       = (5.083101559156e+00, 3.994484109594e-01, 1.494898e-01)
     final cond(J) = 3.669757713403e+01

We see the method converges after 11 iterations.  For comparison the
standard Levenberg-Marquardt method requires 26 iterations and so the
Gaussian fitting problem benefits substantially from the geodesic
acceleration correction.  The column marked ‘|a|/|v|’ above shows the
ratio of the acceleration term to the velocity term as the iteration
progresses.  Larger values of this ratio indicate that the geodesic
acceleration correction term is contributing substantial information to
the solver relative to the standard LM velocity step.

The data and fitted model are shown in Fig.  %s.

Figure: Gaussian model fitted to data

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     struct data
     {
       double *t;
       double *y;
       size_t n;
     };

     /* model function: a * exp( -1/2 * [ (t - b) / c ]^2 ) */
     double
     gaussian(const double a, const double b, const double c, const double t)
     {
       const double z = (t - b) / c;
       return (a * exp(-0.5 * z * z));
     }

     int
     func_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       struct data *d = (struct data *) params;
       double a = gsl_vector_get(x, 0);
       double b = gsl_vector_get(x, 1);
       double c = gsl_vector_get(x, 2);
       size_t i;

       for (i = 0; i < d->n; ++i)
         {
           double ti = d->t[i];
           double yi = d->y[i];
           double y = gaussian(a, b, c, ti);

           gsl_vector_set(f, i, yi - y);
         }

       return GSL_SUCCESS;
     }

     int
     func_df (const gsl_vector * x, void *params, gsl_matrix * J)
     {
       struct data *d = (struct data *) params;
       double a = gsl_vector_get(x, 0);
       double b = gsl_vector_get(x, 1);
       double c = gsl_vector_get(x, 2);
       size_t i;

       for (i = 0; i < d->n; ++i)
         {
           double ti = d->t[i];
           double zi = (ti - b) / c;
           double ei = exp(-0.5 * zi * zi);

           gsl_matrix_set(J, i, 0, -ei);
           gsl_matrix_set(J, i, 1, -(a / c) * ei * zi);
           gsl_matrix_set(J, i, 2, -(a / c) * ei * zi * zi);
         }

       return GSL_SUCCESS;
     }

     int
     func_fvv (const gsl_vector * x, const gsl_vector * v,
               void *params, gsl_vector * fvv)
     {
       struct data *d = (struct data *) params;
       double a = gsl_vector_get(x, 0);
       double b = gsl_vector_get(x, 1);
       double c = gsl_vector_get(x, 2);
       double va = gsl_vector_get(v, 0);
       double vb = gsl_vector_get(v, 1);
       double vc = gsl_vector_get(v, 2);
       size_t i;

       for (i = 0; i < d->n; ++i)
         {
           double ti = d->t[i];
           double zi = (ti - b) / c;
           double ei = exp(-0.5 * zi * zi);
           double Dab = -zi * ei / c;
           double Dac = -zi * zi * ei / c;
           double Dbb = a * ei / (c * c) * (1.0 - zi*zi);
           double Dbc = a * zi * ei / (c * c) * (2.0 - zi*zi);
           double Dcc = a * zi * zi * ei / (c * c) * (3.0 - zi*zi);
           double sum;

           sum = 2.0 * va * vb * Dab +
                 2.0 * va * vc * Dac +
                       vb * vb * Dbb +
                 2.0 * vb * vc * Dbc +
                       vc * vc * Dcc;

           gsl_vector_set(fvv, i, sum);
         }

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector *f = gsl_multifit_nlinear_residual(w);
       gsl_vector *x = gsl_multifit_nlinear_position(w);
       double avratio = gsl_multifit_nlinear_avratio(w);
       double rcond;

       (void) params; /* not used */

       /* compute reciprocal condition number of J(x) */
       gsl_multifit_nlinear_rcond(&rcond, w);

       fprintf(stderr, "iter %2zu: a = %.4f, b = %.4f, c = %.4f, |a|/|v| = %.4f cond(J) = %8.4f, |f(x)| = %.4f\n",
               iter,
               gsl_vector_get(x, 0),
               gsl_vector_get(x, 1),
               gsl_vector_get(x, 2),
               avratio,
               1.0 / rcond,
               gsl_blas_dnrm2(f));
     }

     void
     solve_system(gsl_vector *x, gsl_multifit_nlinear_fdf *fdf,
                  gsl_multifit_nlinear_parameters *params)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multifit_nlinear_workspace *work =
         gsl_multifit_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multifit_nlinear_residual(work);
       gsl_vector * y = gsl_multifit_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond;

       /* initialize solver */
       gsl_multifit_nlinear_init(x, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                                   callback, NULL, &info, work);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* store cond(J(x)) */
       gsl_multifit_nlinear_rcond(&rcond, work);

       gsl_vector_memcpy(x, y);

       /* print summary */

       fprintf(stderr, "NITER         = %zu\n", gsl_multifit_nlinear_niter(work));
       fprintf(stderr, "NFEV          = %zu\n", fdf->nevalf);
       fprintf(stderr, "NJEV          = %zu\n", fdf->nevaldf);
       fprintf(stderr, "NAEV          = %zu\n", fdf->nevalfvv);
       fprintf(stderr, "initial cost  = %.12e\n", chisq0);
       fprintf(stderr, "final cost    = %.12e\n", chisq);
       fprintf(stderr, "final x       = (%.12e, %.12e, %12e)\n",
               gsl_vector_get(x, 0), gsl_vector_get(x, 1), gsl_vector_get(x, 2));
       fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);

       gsl_multifit_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t n = 300;  /* number of data points to fit */
       const size_t p = 3;    /* number of model parameters */
       const double a = 5.0;  /* amplitude */
       const double b = 0.4;  /* center */
       const double c = 0.15; /* width */
       const gsl_rng_type * T = gsl_rng_default;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();
       struct data fit_data;
       gsl_rng * r;
       size_t i;

       gsl_rng_env_setup ();
       r = gsl_rng_alloc (T);

       fit_data.t = malloc(n * sizeof(double));
       fit_data.y = malloc(n * sizeof(double));
       fit_data.n = n;

       /* generate synthetic data with noise */
       for (i = 0; i < n; ++i)
         {
           double t = (double)i / (double) n;
           double y0 = gaussian(a, b, c, t);
           double dy = gsl_ran_gaussian (r, 0.1 * y0);

           fit_data.t[i] = t;
           fit_data.y[i] = y0 + dy;
         }

       /* define function to be minimized */
       fdf.f = func_f;
       fdf.df = func_df;
       fdf.fvv = func_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = &fit_data;

       /* starting point */
       gsl_vector_set(x, 0, 1.0);
       gsl_vector_set(x, 1, 0.0);
       gsl_vector_set(x, 2, 1.0);

       fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       /* print data and model */
       {
         double A = gsl_vector_get(x, 0);
         double B = gsl_vector_get(x, 1);
         double C = gsl_vector_get(x, 2);

         for (i = 0; i < n; ++i)
           {
             double ti = fit_data.t[i];
             double yi = fit_data.y[i];
             double fi = gaussian(A, B, C, ti);

             printf("%f %f %f\n", ti, yi, fi);
           }
       }

       gsl_vector_free(f);
       gsl_vector_free(x);
       gsl_rng_free(r);

       return 0;
     }

41.12.4 Comparing TRS Methods Example
-------------------------------------

The following program compares all available nonlinear least squares
trust-region subproblem (TRS) methods on the Branin function, a common
optimization test problem.  The cost function is

     \Phi(x) &= 1/2 (f_1^2 + f_2^2)
     f_1 &= x_2 + a_1 x_1^2 + a_2 x_1 + a_3
     f_2 &= sqrt(a_4) sqrt(1 + (1 - a_5) cos(x_1))

with a_1 = -{5.1 \over 4 \pi^2}, a_2 = {5 \over \pi}, a_3 = -6, a_4 =
10, a_5 = {1 \over 8\pi}.  There are three minima of this function in
the range (x_1,x_2) \in [-5,15] \times [-5,15].  The program below uses
the starting point (x_1,x_2) = (6,14.5) and calculates the solution with
all available nonlinear least squares TRS methods.  The program output
is shown below:

     Method                    NITER  NFEV  NJEV  Initial Cost  Final cost   Final cond(J) Final x
     levenberg-marquardt       20     27    21    1.9874e+02    3.9789e-01   6.1399e+07    (-3.14e+00, 1.23e+01)
     levenberg-marquardt+accel 27     36    28    1.9874e+02    3.9789e-01   1.4465e+07    (3.14e+00, 2.27e+00)
     dogleg                    23     64    23    1.9874e+02    3.9789e-01   5.0692e+08    (3.14e+00, 2.28e+00)
     double-dogleg             24     69    24    1.9874e+02    3.9789e-01   3.4879e+07    (3.14e+00, 2.27e+00)
     2D-subspace               23     54    24    1.9874e+02    3.9789e-01   2.5142e+07    (3.14e+00, 2.27e+00)

The first row of output above corresponds to standard
Levenberg-Marquardt, while the second row includes geodesic
acceleration.  We see that the standard LM method converges to the
minimum at (-\pi,12.275) and also uses the least number of iterations
and Jacobian evaluations.  All other methods converge to the minimum
(\pi,2.275) and perform similarly in terms of number of Jacobian
evaluations.  We see that J is fairly ill-conditioned at both minima,
indicating that the QR (or SVD) solver is the best choice for this
problem.  Since there are only two parameters in this optimization
problem, we can easily visualize the paths taken by each method, which
are shown in Fig.  %s.  The figure shows contours of the cost function
\Phi(x_1,x_2) which exhibits three global minima in the range [-5,15]
\times [-5,15].  The paths taken by each solver are shown as colored
lines.

Figure: Paths taken for different TRS methods for the Branin function

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>

     /* parameters to model */
     struct model_params
     {
       double a1;
       double a2;
       double a3;
       double a4;
       double a5;
     };

     /* Branin function */
     int
     func_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       struct model_params *par = (struct model_params *) params;
       double x1 = gsl_vector_get(x, 0);
       double x2 = gsl_vector_get(x, 1);
       double f1 = x2 + par->a1 * x1 * x1 + par->a2 * x1 + par->a3;
       double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));

       gsl_vector_set(f, 0, f1);
       gsl_vector_set(f, 1, f2);

       return GSL_SUCCESS;
     }

     int
     func_df (const gsl_vector * x, void *params, gsl_matrix * J)
     {
       struct model_params *par = (struct model_params *) params;
       double x1 = gsl_vector_get(x, 0);
       double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));

       gsl_matrix_set(J, 0, 0, 2.0 * par->a1 * x1 + par->a2);
       gsl_matrix_set(J, 0, 1, 1.0);

       gsl_matrix_set(J, 1, 0, -0.5 * par->a4 / f2 * (1.0 - par->a5) * sin(x1));
       gsl_matrix_set(J, 1, 1, 0.0);

       return GSL_SUCCESS;
     }

     int
     func_fvv (const gsl_vector * x, const gsl_vector * v,
               void *params, gsl_vector * fvv)
     {
       struct model_params *par = (struct model_params *) params;
       double x1 = gsl_vector_get(x, 0);
       double v1 = gsl_vector_get(v, 0);
       double c = cos(x1);
       double s = sin(x1);
       double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * c);
       double t = 0.5 * par->a4 * (1.0 - par->a5) / f2;

       gsl_vector_set(fvv, 0, 2.0 * par->a1 * v1 * v1);
       gsl_vector_set(fvv, 1, -t * (c + s*s/f2) * v1 * v1);

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector * x = gsl_multifit_nlinear_position(w);
       double x1 = gsl_vector_get(x, 0);
       double x2 = gsl_vector_get(x, 1);

       /* print out current location */
       printf("%f %f\n", x1, x2);
     }

     void
     solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
                  gsl_multifit_nlinear_parameters *params)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multifit_nlinear_workspace *work =
         gsl_multifit_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multifit_nlinear_residual(work);
       gsl_vector * x = gsl_multifit_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond;

       printf("# %s/%s\n",
              gsl_multifit_nlinear_name(work),
              gsl_multifit_nlinear_trs_name(work));

       /* initialize solver */
       gsl_multifit_nlinear_init(x0, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                                   callback, NULL, &info, work);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* store cond(J(x)) */
       gsl_multifit_nlinear_rcond(&rcond, work);

       /* print summary */
       fprintf(stderr, "%-25s %-6zu %-5zu %-5zu %-13.4e %-12.4e %-13.4e (%.2e, %.2e)\n",
               gsl_multifit_nlinear_trs_name(work),
               gsl_multifit_nlinear_niter(work),
               fdf->nevalf,
               fdf->nevaldf,
               chisq0,
               chisq,
               1.0 / rcond,
               gsl_vector_get(x, 0),
               gsl_vector_get(x, 1));

       printf("\n\n");

       gsl_multifit_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t n = 2;
       const size_t p = 2;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();
       struct model_params params;

       params.a1 = -5.1 / (4.0 * M_PI * M_PI);
       params.a2 = 5.0 / M_PI;
       params.a3 = -6.0;
       params.a4 = 10.0;
       params.a5 = 1.0 / (8.0 * M_PI);

       /* print map of Phi(x1, x2) */
       {
         double x1, x2, chisq;

         for (x1 = -5.0; x1 < 15.0; x1 += 0.1)
           {
             for (x2 = -5.0; x2 < 15.0; x2 += 0.1)
               {
                 gsl_vector_set(x, 0, x1);
                 gsl_vector_set(x, 1, x2);
                 func_f(x, &params, f);

                 gsl_blas_ddot(f, f, &chisq);

                 printf("%f %f %f\n", x1, x2, chisq);
               }
             printf("\n");
           }
         printf("\n\n");
       }

       /* define function to be minimized */
       fdf.f = func_f;
       fdf.df = func_df;
       fdf.fvv = func_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = &params;

       /* starting point */
       gsl_vector_set(x, 0, 6.0);
       gsl_vector_set(x, 1, 14.5);

       fprintf(stderr, "%-25s %-6s %-5s %-5s %-13s %-12s %-13s %-15s\n",
               "Method", "NITER", "NFEV", "NJEV", "Initial Cost",
               "Final cost", "Final cond(J)", "Final x");

       fdf_params.trs = gsl_multifit_nlinear_trs_lm;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_dogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_ddogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_subspace2D;
       solve_system(x, &fdf, &fdf_params);

       gsl_vector_free(f);
       gsl_vector_free(x);

       return 0;
     }

41.12.5 Large Nonlinear Least Squares Example
---------------------------------------------

The following program illustrates the large nonlinear least squares
solvers on a system with significant sparse structure in the Jacobian.
The cost function is

     \Phi(x) &= 1/2 \sum_{i=1}^{p+1} f_i^2
     f_i &= \sqrt{\alpha} (x_i - 1), 1 \le i \le p
     f_{p+1} &= ||x||^2 - 1/4

with \alpha = 10^{-5}.  The residual f_{p+1} imposes a constraint on the
p parameters x, to ensure that ||x||^2 \approx {1 \over 4}.  The
(p+1)-by-p Jacobian for this system is

     J(x) = [ \sqrt{alpha} I_p; 2 x^T ]

and the normal equations matrix is

     J^T J = \alpha I_p + 4 x x^T

Finally, the second directional derivative of f for the geodesic
acceleration method is

     fvv = [     0     ]
           [ 2 ||v||^2 ]

Since the upper p-by-p block of J is diagonal, this sparse structure
should be exploited in the nonlinear solver.  For comparison, the
following program solves the system for p = 2000 using the dense direct
Cholesky solver based on the normal equations matrix J^T J, as well as
the iterative Steihaug-Toint solver, based on sparse matrix-vector
products J u and J^T u.  The program output is shown below:

     Method                    NITER NFEV NJUEV NJTJEV NAEV Init Cost  Final cost cond(J) Final |x|^2 Time (s)
     levenberg-marquardt       25    31   26    26     0    7.1218e+18 1.9555e-02 447.50  2.5044e-01  46.28
     levenberg-marquardt+accel 22    23   45    23     22   7.1218e+18 1.9555e-02 447.64  2.5044e-01  33.92
     dogleg                    37    87   36    36     0    7.1218e+18 1.9555e-02 447.59  2.5044e-01  56.05
     double-dogleg             35    88   34    34     0    7.1218e+18 1.9555e-02 447.62  2.5044e-01  52.65
     2D-subspace               37    88   36    36     0    7.1218e+18 1.9555e-02 447.71  2.5044e-01  59.75
     steihaug-toint            35    88   345   0      0    7.1218e+18 1.9555e-02 inf     2.5044e-01  0.09

The first five rows use methods based on factoring the dense J^T J
matrix while the last row uses the iterative Steihaug-Toint method.
While the number of Jacobian matrix-vector products (NJUEV) is less for
the dense methods, the added time to construct and factor the J^T J
matrix (NJTJEV) results in a much larger runtime than the iterative
method (see last column).

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <sys/time.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multilarge_nlinear.h>
     #include <gsl/gsl_spblas.h>
     #include <gsl/gsl_spmatrix.h>

     /* parameters for functions */
     struct model_params
     {
       double alpha;
       gsl_spmatrix *J;
     };

     /* penalty function */
     int
     penalty_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       struct model_params *par = (struct model_params *) params;
       const double sqrt_alpha = sqrt(par->alpha);
       const size_t p = x->size;
       size_t i;
       double sum = 0.0;

       for (i = 0; i < p; ++i)
         {
           double xi = gsl_vector_get(x, i);

           gsl_vector_set(f, i, sqrt_alpha*(xi - 1.0));

           sum += xi * xi;
         }

       gsl_vector_set(f, p, sum - 0.25);

       return GSL_SUCCESS;
     }

     int
     penalty_df (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x,
                 const gsl_vector * u, void * params, gsl_vector * v,
                 gsl_matrix * JTJ)
     {
       struct model_params *par = (struct model_params *) params;
       const size_t p = x->size;
       size_t j;

       /* store 2*x in last row of J */
       for (j = 0; j < p; ++j)
         {
           double xj = gsl_vector_get(x, j);
           gsl_spmatrix_set(par->J, p, j, 2.0 * xj);
         }

       /* compute v = op(J) u */
       if (v)
         gsl_spblas_dgemv(TransJ, 1.0, par->J, u, 0.0, v);

       if (JTJ)
         {
           gsl_vector_view diag = gsl_matrix_diagonal(JTJ);

           /* compute J^T J = [ alpha*I_p + 4 x x^T ] */
           gsl_matrix_set_zero(JTJ);

           /* store 4 x x^T in lower half of JTJ */
           gsl_blas_dsyr(CblasLower, 4.0, x, JTJ);

           /* add alpha to diag(JTJ) */
           gsl_vector_add_constant(&diag.vector, par->alpha);
         }

       return GSL_SUCCESS;
     }

     int
     penalty_fvv (const gsl_vector * x, const gsl_vector * v,
                  void *params, gsl_vector * fvv)
     {
       const size_t p = x->size;
       double normv = gsl_blas_dnrm2(v);

       gsl_vector_set_zero(fvv);
       gsl_vector_set(fvv, p, 2.0 * normv * normv);

       (void)params; /* avoid unused parameter warning */

       return GSL_SUCCESS;
     }

     void
     solve_system(const gsl_vector *x0, gsl_multilarge_nlinear_fdf *fdf,
                  gsl_multilarge_nlinear_parameters *params)
     {
       const gsl_multilarge_nlinear_type *T = gsl_multilarge_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multilarge_nlinear_workspace *work =
         gsl_multilarge_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multilarge_nlinear_residual(work);
       gsl_vector * x = gsl_multilarge_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond, xsq;
       struct timeval tv0, tv1;

       gettimeofday(&tv0, NULL);

       /* initialize solver */
       gsl_multilarge_nlinear_init(x0, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multilarge_nlinear_driver(max_iter, xtol, gtol, ftol,
                                     NULL, NULL, &info, work);

       gettimeofday(&tv1, NULL);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* compute final ||x||^2 */
       gsl_blas_ddot(x, x, &xsq);

       /* store cond(J(x)) */
       gsl_multilarge_nlinear_rcond(&rcond, work);

       /* print summary */
       fprintf(stderr, "%-25s %-5zu %-4zu %-5zu %-6zu %-4zu %-10.4e %-10.4e %-7.2f %-11.4e %.2f\n",
               gsl_multilarge_nlinear_trs_name(work),
               gsl_multilarge_nlinear_niter(work),
               fdf->nevalf,
               fdf->nevaldfu,
               fdf->nevaldf2,
               fdf->nevalfvv,
               chisq0,
               chisq,
               1.0 / rcond,
               xsq,
               (tv1.tv_sec - tv0.tv_sec) + 1.0e-6 * (tv1.tv_usec - tv0.tv_usec));

       gsl_multilarge_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t p = 2000;
       const size_t n = p + 1;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);

       /* allocate sparse Jacobian matrix with 2*p non-zero elements in triplet format */
       gsl_spmatrix *J = gsl_spmatrix_alloc_nzmax(n, p, 2 * p, GSL_SPMATRIX_TRIPLET);

       gsl_multilarge_nlinear_fdf fdf;
       gsl_multilarge_nlinear_parameters fdf_params =
         gsl_multilarge_nlinear_default_parameters();
       struct model_params params;
       size_t i;

       params.alpha = 1.0e-5;
       params.J = J;

       /* define function to be minimized */
       fdf.f = penalty_f;
       fdf.df = penalty_df;
       fdf.fvv = penalty_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = &params;

       for (i = 0; i < p; ++i)
         {
           /* starting point */
           gsl_vector_set(x, i, i + 1.0);

           /* store sqrt(alpha)*I_p in upper p-by-p block of J */
           gsl_spmatrix_set(J, i, i, sqrt(params.alpha));
         }

       fprintf(stderr, "%-25s %-4s %-4s %-5s %-6s %-4s %-10s %-10s %-7s %-11s %-10s\n",
               "Method", "NITER", "NFEV", "NJUEV", "NJTJEV", "NAEV", "Init Cost",
               "Final cost", "cond(J)", "Final |x|^2", "Time (s)");

       fdf_params.scale = gsl_multilarge_nlinear_scale_levenberg;

       fdf_params.trs = gsl_multilarge_nlinear_trs_lm;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_dogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_ddogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_subspace2D;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_cgst;
       solve_system(x, &fdf, &fdf_params);

       gsl_vector_free(f);
       gsl_vector_free(x);
       gsl_spmatrix_free(J);

       return 0;
     }

41.13 References and Further Reading
====================================

The following publications are relevant to the algorithms described in
this section,

   * J.J. Moré, `The Levenberg-Marquardt Algorithm: Implementation and
     Theory', Lecture Notes in Mathematics, v630 (1978), ed G. Watson.

   * H. B. Nielsen, “Damping Parameter in Marquardt’s Method”, IMM
     Department of Mathematical Modeling, DTU, Tech.  Report
     IMM-REP-1999-05 (1999).

   * K. Madsen and H. B. Nielsen, “Introduction to Optimization and Data
     Fitting”, IMM Department of Mathematical Modeling, DTU, 2010.

   * J. E. Dennis and R. B. Schnabel, Numerical Methods for
     Unconstrained Optimization and Nonlinear Equations, SIAM, 1996.

   * M. K. Transtrum, B. B. Machta, and J. P. Sethna, Geometry of
     nonlinear least squares with applications to sloppy models and
     optimization, Phys.  Rev.  E 83, 036701, 2011.

   * M. K. Transtrum and J. P. Sethna, Improvements to the
     Levenberg-Marquardt algorithm for nonlinear least-squares
     minimization, arXiv:1201.5885, 2012.

   * J.J. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained
     Optimization Software”, ACM Transactions on Mathematical Software,
     Vol 7, No 1 (1981), p 17–41.

   * H. B. Nielsen, “UCTP Test Problems for Unconstrained Optimization”,
     IMM Department of Mathematical Modeling, DTU, Tech.  Report
     IMM-REP-2000-17 (2000).

42 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.  For information about interpolating splines, see
Interpolation.

The header file ‘gsl_bspline.h’ contains the prototypes for the bspline
functions and related declarations.

42.1 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, \dots, 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, \ldots, 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=0}^{n-1} 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.

42.2 Initializing the B-splines solver
======================================

 -- Type: gsl_bspline_workspace

     The computation of B-spline functions requires a preallocated
     workspace.

 -- 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(2k^2 + 5k +
     nbreak).

 -- Function: void gsl_bspline_free (gsl_bspline_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

42.3 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’.

42.4 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.

42.5 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)

     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)

     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^jB_{(istart+i)}(x)/dx^j.  The last row of ‘dB’
     contains d^jB_{iend}(x)/dx^j.  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.

42.6 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.

42.7 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);

       /* 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);

       printf("\n\n");

       /* output the smoothed curve */
       {
         double xi, yi, yerr;

         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 is shown below:

     $ ./a.out > bspline.txt
     chisq/dof = 1.118217e+00, Rsq = 0.989771

The data and fitted model are shown in Fig.  %s.

Figure: Data (black) and fitted model (red)

42.8 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.

43 Sparse Matrices
******************

This chapter describes functions for the construction and manipulation
of sparse matrices, matrices which are populated primarily with zeros
and contain only a few non-zero elements.  Sparse matrices often appear
in the solution of partial differential equations.  It is beneficial to
use specialized data structures and algorithms for storing and working
with sparse matrices, since dense matrix algorithms and structures can
be very slow and use huge amounts of memory when applied to sparse
matrices.

The header file ‘gsl_spmatrix.h’ contains the prototypes for the sparse
matrix functions and related declarations.

43.1 Overview
=============

These routines provide support for constructing and manipulating sparse
matrices in GSL, using an API similar to gsl_matrix.  The
basic structure is called gsl_spmatrix.  There are three
supported storage formats for sparse matrices: the triplet, compressed
column storage (CCS) and compressed row storage (CRS) formats.  The
triplet format stores triplets (i,j,x) for each non-zero element of the
matrix.  This notation means that the (i,j) element of the matrix A is
A_{ij} = x.  Compressed column storage stores each column of non-zero
values in the sparse matrix in a continuous memory block, keeping
pointers to the beginning of each column in that memory block, and
storing the row indices of each non-zero element.  Compressed row
storage stores each row of non-zero values in a continuous memory block,
keeping pointers to the beginning of each row in the block and storing
the column indices of each non-zero element.  The triplet format is
ideal for adding elements to the sparse matrix structure while it is
being constructed, while the compressed storage formats are better
suited for matrix-matrix multiplication or linear solvers.

 -- Type: gsl_spmatrix

     This structure is defined as:

          typedef struct
          {
            size_t size1;
            size_t size2;
            size_t *i;
            double *data;
            size_t *p;
            size_t nzmax;
            size_t nz;
            gsl_spmatrix_tree *tree_data;
            void *work;
            size_t sptype;
          } gsl_spmatrix;

     This defines a ‘size1’-by-‘size2’ sparse matrix.  The number of
     non-zero elements currently in the matrix is given by ‘nz’.  For
     the triplet representation, ‘i’, ‘p’, and ‘data’ are arrays of size
     ‘nz’ which contain the row indices, column indices, and element
     value, respectively.  So if data[k] = A(i,j), then i = i[k] and j =
     p[k].

     For compressed column storage, ‘i’ and ‘data’ are arrays of size
     ‘nz’ containing the row indices and element values, identical to
     the triplet case.  ‘p’ is an array of size ‘size2’ + 1 where ‘p[j]’
     points to the index in ‘data’ of the start of column ‘j’.  Thus, if
     data[k] = A(i,j), then i = i[k] and p[j] <= k < p[j+1].

     For compressed row storage, ‘i’ and ‘data’ are arrays of size ‘nz’
     containing the column indices and element values, identical to the
     triplet case.  ‘p’ is an array of size ‘size1’ + 1 where ‘p[i]’
     points to the index in ‘data’ of the start of row ‘i’.  Thus, if
     data[k] = A(i,j), then j = i[k] and p[i] <= k < p[i+1].

     The parameter ‘tree_data’ is a binary tree structure used in the
     triplet representation, specifically a balanced AVL tree.  This
     speeds up element searches and duplicate detection during the
     matrix assembly process.  The parameter ‘work’ is additional
     workspace needed for various operations like converting from
     triplet to compressed storage.  ‘sptype’ indicates the type of
     storage format being used (triplet, CCS or CRS).

     The compressed storage format defined above makes it very simple to
     interface with sophisticated external linear solver libraries which
     accept compressed storage input.  The user can simply pass the
     arrays ‘i’, ‘p’, and ‘data’ as the inputs to external libraries.

43.2 Allocation
===============

The functions for allocating memory for a sparse 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 code of
GSL_ENOMEM in addition to returning a null pointer.

 -- Function: gsl_spmatrix * gsl_spmatrix_alloc (const size_t n1, const
          size_t n2)

     This function allocates a sparse matrix of size ‘n1’-by-‘n2’ and
     initializes it to all zeros.  If the size of the matrix is not
     known at allocation time, both ‘n1’ and ‘n2’ may be set to 1, and
     they will automatically grow as elements are added to the matrix.
     This function sets the matrix to the triplet representation, which
     is the easiest for adding and accessing matrix elements.  This
     function tries to make a reasonable guess for the number of
     non-zero elements (‘nzmax’) which will be added to the matrix by
     assuming a sparse density of 10\%.  The function
     gsl_spmatrix_alloc_nzmax() can be used if this number is
     known more accurately.  The workspace is of size O(nzmax).

 -- Function: gsl_spmatrix * gsl_spmatrix_alloc_nzmax (const size_t n1,
          const size_t n2, const size_t nzmax, const size_t sptype)

     This function allocates a sparse matrix of size ‘n1’-by-‘n2’ and
     initializes it to all zeros.  If the size of the matrix is not
     known at allocation time, both ‘n1’ and ‘n2’ may be set to 1, and
     they will automatically grow as elements are added to the matrix.
     The parameter ‘nzmax’ specifies the maximum number of non-zero
     elements which will be added to the matrix.  It does not need to be
     precisely known in advance, since storage space will automatically
     grow using gsl_spmatrix_realloc() if ‘nzmax’ is not
     large enough.  Accurate knowledge of this parameter reduces the
     number of reallocation calls required.  The parameter ‘sptype’
     specifies the storage format of the sparse matrix.  Possible values
     are

      -- Macro: GSL_SPMATRIX_TRIPLET

          This flag specifies triplet storage.

      -- Macro: GSL_SPMATRIX_CCS

          This flag specifies compressed column storage.

      -- Macro: GSL_SPMATRIX_CRS

          This flag specifies compressed row storage.

     The allocated gsl_spmatrix structure is of size
     O(nzmax).

 -- Function: int gsl_spmatrix_realloc (const size_t nzmax, gsl_spmatrix
          * m)

     This function reallocates the storage space for ‘m’ to accomodate
     ‘nzmax’ non-zero elements.  It is typically called internally by
     gsl_spmatrix_set() if the user wants to add more
     elements to the sparse matrix than the previously specified
     ‘nzmax’.

 -- Function: void gsl_spmatrix_free (gsl_spmatrix * m)

     This function frees the memory associated with the sparse matrix
     ‘m’.

43.3 Accessing Matrix Elements
==============================

 -- Function: double gsl_spmatrix_get (const gsl_spmatrix * m, const
          size_t i, const size_t j)

     This function returns element (‘i’, ‘j’) of the matrix ‘m’.  The
     matrix may be in triplet or compressed format.

 -- Function: int gsl_spmatrix_set (gsl_spmatrix * m, const size_t i,
          const size_t j, const double x)

     This function sets element (‘i’, ‘j’) of the matrix ‘m’ to the
     value ‘x’.  The matrix must be in triplet representation.

 -- Function: double * gsl_spmatrix_ptr (gsl_spmatrix * m, const
          size_t i, const size_t j)

     This function returns a pointer to the (‘i’, ‘j’) element of the
     matrix ‘m’.  If the (‘i’, ‘j’) element is not explicitly stored in
     the matrix, a null pointer is returned.

43.4 Initializing Matrix Elements
=================================

Since the sparse matrix format only stores the non-zero elements, it is
automatically initialized to zero upon allocation.  The function
gsl_spmatrix_set_zero() may be used to re-initialize a matrix to
zero after elements have been added to it.

 -- Function: int gsl_spmatrix_set_zero (gsl_spmatrix * m)

     This function sets (or resets) all the elements of the matrix ‘m’
     to zero.

43.5 Reading and Writing Matrices
=================================

 -- Function: int gsl_spmatrix_fwrite (FILE * stream, const gsl_spmatrix
          * 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_spmatrix_fread (FILE * stream, gsl_spmatrix * 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 storage format, dimensions and have a sufficiently
     large ‘nzmax’ in order to read in all matrix elements, otherwise
     ‘GSL_EBADLEN’ is returned.  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_spmatrix_fprintf (FILE * stream, const
          gsl_spmatrix * 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.  The function returns 0 for success and ‘GSL_EFAILED’ if
     there was a problem writing to the file.  The input matrix ‘m’ may
     be in any storage format, and the output file will be written in
     MatrixMarket format.

 -- Function: gsl_spmatrix * gsl_spmatrix_fscanf (FILE * stream)

     This function reads sparse matrix data in the MatrixMarket format
     from the stream ‘stream’ and stores it in a newly allocated matrix
     which is returned in triplet format.  The function returns 0 for
     success and ‘GSL_EFAILED’ if there was a problem reading from the
     file.  The user should free the returned matrix when it is no
     longer needed.

43.6 Copying Matrices
=====================

 -- Function: int gsl_spmatrix_memcpy (gsl_spmatrix * dest, const
          gsl_spmatrix * src)

     This function copies the elements of the sparse matrix ‘src’ into
     ‘dest’.  The two matrices must have the same dimensions and be in
     the same storage format.

43.7 Exchanging Rows and Columns
================================

 -- Function: int gsl_spmatrix_transpose_memcpy (gsl_spmatrix * dest,
          const gsl_spmatrix * src)

     This function copies the transpose of the sparse matrix ‘src’ into
     ‘dest’.  The dimensions of ‘dest’ must match the transpose of the
     matrix ‘src’.  Also, both matrices must use the same sparse storage
     format.

 -- Function: int gsl_spmatrix_transpose (gsl_spmatrix * m)

     This function replaces the matrix ‘m’ by its transpose, preserving
     the storage format of the input matrix.  Currently, only triplet
     matrix inputs are supported.

 -- Function: int gsl_spmatrix_transpose2 (gsl_spmatrix * m)

     This function replaces the matrix ‘m’ by its transpose, but changes
     the storage format for compressed matrix inputs.  Since compressed
     column storage is the transpose of compressed row storage, this
     function simply converts a CCS matrix to CRS and vice versa.  This
     is the most efficient way to transpose a compressed storage matrix,
     but the user should note that the storage format of their
     compressed matrix will change on output.  For triplet matrices, the
     output matrix is also in triplet storage.

43.8 Matrix Operations
======================

 -- Function: int gsl_spmatrix_add (gsl_spmatrix * c, const gsl_spmatrix
          * a, const gsl_spmatrix * b)

     This function computes the sum c = a + b.  The three matrices must
     have the same dimensions and be stored in a compressed format.

 -- Function: int gsl_spmatrix_scale (gsl_spmatrix * m, const double x)

     This function scales all elements of the matrix ‘m’ by the constant
     factor ‘x’.  The result m(i,j) \leftarrow x m(i,j) is stored in
     ‘m’.

43.9 Matrix Properties
======================

 -- Function: size_t gsl_spmatrix_nnz (const gsl_spmatrix * m)

     This function returns the number of non-zero elements in ‘m’.

 -- Function: int gsl_spmatrix_equal (const gsl_spmatrix * a, const
          gsl_spmatrix * b)

     This function returns 1 if the matrices ‘a’ and ‘b’ are equal (by
     comparison of element values) and 0 otherwise.  The matrices ‘a’
     and ‘b’ must be in the same sparse storage format for comparison.

43.10 Finding Maximum and Minimum Elements
==========================================

 -- Function: int gsl_spmatrix_minmax (const gsl_spmatrix * m, double
          * min_out, double * max_out)

     This function returns the minimum and maximum elements of the
     matrix ‘m’, storing them in ‘min_out’ and ‘max_out’, and searching
     only the non-zero values.

43.11 Compressed Format
=======================

GSL supports compressed column storage (CCS) and compressed row storage
(CRS) formats.

 -- Function: gsl_spmatrix * gsl_spmatrix_ccs (const gsl_spmatrix * T)

     This function creates a sparse matrix in compressed column format
     from the input sparse matrix ‘T’ which must be in triplet format.
     A pointer to a newly allocated matrix is returned.  The calling
     function should free the newly allocated matrix when it is no
     longer needed.

 -- Function: gsl_spmatrix * gsl_spmatrix_crs (const gsl_spmatrix * T)

     This function creates a sparse matrix in compressed row format from
     the input sparse matrix ‘T’ which must be in triplet format.  A
     pointer to a newly allocated matrix is returned.  The calling
     function should free the newly allocated matrix when it is no
     longer needed.

43.12 Conversion Between Sparse and Dense Matrices
==================================================

The gsl_spmatrix structure can be converted into the dense
gsl_matrix format and vice versa with the following
routines.

 -- Function: int gsl_spmatrix_d2sp (gsl_spmatrix * S, const gsl_matrix
          * A)

     This function converts the dense matrix ‘A’ into sparse triplet
     format and stores the result in ‘S’.

 -- Function: int gsl_spmatrix_sp2d (gsl_matrix * A, const gsl_spmatrix
          * S)

     This function converts the sparse matrix ‘S’ into a dense matrix
     and stores the result in ‘A’.  ‘S’ must be in triplet format.

43.13 Examples
==============

The following example program builds a 5-by-4 sparse matrix and prints
it in triplet, compressed column, and compressed row format.  The matrix
which is constructed is

     [ 0    0  3.1  4.6 ]
     [ 1    0  7.2   0  ]
     [ 0    0   0    0  ]
     [ 2.1 2.9  0   8.5 ]
     [ 4.1  0   0    0  ]

The output of the program is:

     printing all matrix elements:
     A(0,0) = 0
     A(0,1) = 0
     A(0,2) = 3.1
     A(0,3) = 4.6
     A(1,0) = 1
     .
     .
     .
     A(4,0) = 4.1
     A(4,1) = 0
     A(4,2) = 0
     A(4,3) = 0
     matrix in triplet format (i,j,Aij):
     (0, 2, 3.1)
     (0, 3, 4.6)
     (1, 0, 1.0)
     (1, 2, 7.2)
     (3, 0, 2.1)
     (3, 1, 2.9)
     (3, 3, 8.5)
     (4, 0, 4.1)
     matrix in compressed column format:
     i = [ 1, 3, 4, 3, 0, 1, 0, 3, ]
     p = [ 0, 3, 4, 6, 8, ]
     d = [ 1, 2.1, 4.1, 2.9, 3.1, 7.2, 4.6, 8.5, ]
     matrix in compressed row format:
     i = [ 2, 3, 0, 2, 0, 1, 3, 0, ]
     p = [ 0, 2, 4, 4, 7, 8, ]
     d = [ 3.1, 4.6, 1, 7.2, 2.1, 2.9, 8.5, 4.1, ]

We see in the compressed column output, the data array stores each
column contiguously, the array i stores the row index of the
corresponding data element, and the array p stores the index of the
start of each column in the data array.  Similarly, for the compressed
row output, the data array stores each row contiguously, the array i
stores the column index of the corresponding data element, and the p
array stores the index of the start of each row in the data array.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_spmatrix.h>

     int
     main()
     {
       gsl_spmatrix *A = gsl_spmatrix_alloc(5, 4); /* triplet format */
       gsl_spmatrix *B, *C;
       size_t i, j;

       /* build the sparse matrix */
       gsl_spmatrix_set(A, 0, 2, 3.1);
       gsl_spmatrix_set(A, 0, 3, 4.6);
       gsl_spmatrix_set(A, 1, 0, 1.0);
       gsl_spmatrix_set(A, 1, 2, 7.2);
       gsl_spmatrix_set(A, 3, 0, 2.1);
       gsl_spmatrix_set(A, 3, 1, 2.9);
       gsl_spmatrix_set(A, 3, 3, 8.5);
       gsl_spmatrix_set(A, 4, 0, 4.1);

       printf("printing all matrix elements:\n");
       for (i = 0; i < 5; ++i)
         for (j = 0; j < 4; ++j)
           printf("A(%zu,%zu) = %g\n", i, j,
                  gsl_spmatrix_get(A, i, j));

       /* print out elements in triplet format */
       printf("matrix in triplet format (i,j,Aij):\n");
       gsl_spmatrix_fprintf(stdout, A, "%.1f");

       /* convert to compressed column format */
       B = gsl_spmatrix_ccs(A);

       printf("matrix in compressed column format:\n");
       printf("i = [ ");
       for (i = 0; i < B->nz; ++i)
         printf("%zu, ", B->i[i]);
       printf("]\n");

       printf("p = [ ");
       for (i = 0; i < B->size2 + 1; ++i)
         printf("%zu, ", B->p[i]);
       printf("]\n");

       printf("d = [ ");
       for (i = 0; i < B->nz; ++i)
         printf("%g, ", B->data[i]);
       printf("]\n");

       /* convert to compressed row format */
       C = gsl_spmatrix_crs(A);

       printf("matrix in compressed row format:\n");
       printf("i = [ ");
       for (i = 0; i < C->nz; ++i)
         printf("%zu, ", C->i[i]);
       printf("]\n");

       printf("p = [ ");
       for (i = 0; i < C->size1 + 1; ++i)
         printf("%zu, ", C->p[i]);
       printf("]\n");

       printf("d = [ ");
       for (i = 0; i < C->nz; ++i)
         printf("%g, ", C->data[i]);
       printf("]\n");

       gsl_spmatrix_free(A);
       gsl_spmatrix_free(B);
       gsl_spmatrix_free(C);

       return 0;
     }

43.14 References and Further Reading
====================================

The algorithms used by these functions are described in the following
sources,

   * Davis, T. A., Direct Methods for Sparse Linear Systems, SIAM, 2006.

   * CSparse software library,
     ‘https://www.cise.ufl.edu/research/sparse/CSparse’

44 Sparse BLAS Support
**********************

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

The header file ‘gsl_spblas.h’ contains the prototypes for the sparse
BLAS functions and related declarations.

44.1 Sparse BLAS operations
===========================

 -- Function: int gsl_spblas_dgemv (const CBLAS_TRANSPOSE_t TransA,
          const double alpha, const gsl_spmatrix * A, const gsl_vector
          * x, const double beta, gsl_vector * y)

     This function computes the matrix-vector product and sum y
     \leftarrow \alpha op(A) x + \beta y, where op(A) = A, A^T for
     ‘TransA’ = ‘CblasNoTrans’, ‘CblasTrans’.  In-place computations are
     not supported, so ‘x’ and ‘y’ must be distinct vectors.  The matrix
     ‘A’ may be in triplet or compressed format.

 -- Function: int gsl_spblas_dgemm (const double alpha, const
          gsl_spmatrix * A, const gsl_spmatrix * B, gsl_spmatrix * C)

     This function computes the sparse matrix-matrix product C = \alpha
     A B. The matrices must be in compressed format.

44.2 References and Further Reading
===================================

The algorithms used by these functions are described in the following
sources:

   * Davis, T. A., Direct Methods for Sparse Linear Systems, SIAM, 2006.

   * CSparse software library,
     ‘https://www.cise.ufl.edu/research/sparse/CSparse’

45 Sparse Linear Algebra
************************

This chapter describes functions for solving sparse linear systems of
equations.  The library provides linear algebra routines which operate
directly on the gsl_spmatrix and gsl_vector.
objects.

The functions described in this chapter are declared in the header file
‘gsl_splinalg.h’.

45.1 Overview
=============

This chapter is primarily concerned with the solution of the linear
system

     A x = b

where A is a general square n-by-n non-singular sparse matrix, x is an
unknown n-by-1 vector, and b is a given n-by-1 right hand side vector.
There exist many methods for solving such sparse linear systems, which
broadly fall into either direct or iterative categories.  Direct methods
include LU and QR decompositions, while iterative methods start with an
initial guess for the vector x and update the guess through iteration
until convergence.  GSL does not currently provide any direct sparse
solvers.

45.2 Sparse Iterative Solvers
=============================

45.2.1 Overview
---------------

Many practical iterative methods of solving large n-by-n sparse linear
systems involve projecting an approximate solution for ‘x’ onto a
subspace of {\bf R}^n.  If we define a m-dimensional subspace {\cal K}
as the subspace of approximations to the solution ‘x’, then m
constraints must be imposed to determine the next approximation.  These
m constraints define another m-dimensional subspace denoted by {\cal L}.
The subspace dimension m is typically chosen to be much smaller than n
in order to reduce the computational effort needed to generate the next
approximate solution vector.  The many iterative algorithms which exist
differ mainly in their choice of {\cal K} and {\cal L}.

45.2.2 Types of Sparse Iterative Solvers
----------------------------------------

The sparse linear algebra library provides the following types of
iterative solvers:

 -- Type: gsl_splinalg_itersolve_type

      -- Variable: gsl_splinalg_itersolve_gmres

          This specifies the Generalized Minimum Residual Method
          (GMRES). This is a projection method using {\cal K} = {\cal
          K}_m and {\cal L} = A {\cal K}_m where {\cal K}_m is the m-th
          Krylov subspace

               K_m = span( r_0, A r_0, ..., A^(m-1) r_0)

          and r_0 = b - A x_0 is the residual vector of the initial
          guess x_0.  If m is set equal to n, then the Krylov subspace
          is {\bf R}^n and GMRES will provide the exact solution ‘x’.
          However, the goal is for the method to arrive at a very good
          approximation to ‘x’ using a much smaller subspace {\cal K}_m.
          By default, the GMRES method selects m = MIN(n,10) but the
          user may specify a different value for m.  The GMRES storage
          requirements grow as O(n(m+1)) and the number of flops grow as
          O(4 m^2 n - 4 m^3 / 3).

          In the below function gsl_splinalg_itersolve_iterate(),
          one GMRES iteration is defined as projecting the
          approximate solution vector onto each Krylov subspace {\cal
          K}_1, ..., {\cal K}_m, and so m can be kept smaller by
          “restarting” the method and calling
          gsl_splinalg_itersolve_iterate() multiple times,
          providing the updated approximation ‘x’ to each new call.  If
          the method is not adequately converging, the user may try
          increasing the parameter m.

          GMRES is considered a robust general purpose iterative solver,
          however there are cases where the method stagnates if the
          matrix is not positive-definite and fails to reduce the
          residual until the very last projection onto the subspace
          {\cal K}_n = {\bf R}^n.  In these cases, preconditioning the
          linear system can help, but GSL does not currently provide any
          preconditioners.

45.2.3 Iterating the Sparse Linear System
-----------------------------------------

The following functions are provided to allocate storage for the sparse
linear solvers and iterate the system to a solution.

 -- Function: gsl_splinalg_itersolve * gsl_splinalg_itersolve_alloc
          (const gsl_splinalg_itersolve_type * T, const size_t n, const
          size_t m)

     This function allocates a workspace for the iterative solution of
     ‘n’-by-‘n’ sparse matrix systems.  The iterative solver type is
     specified by ‘T’.  The argument ‘m’ specifies the size of the
     solution candidate subspace {\cal K}_m.  The dimension ‘m’ may be
     set to 0 in which case a reasonable default value is used.

 -- Function: void gsl_splinalg_itersolve_free (gsl_splinalg_itersolve
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: const char * gsl_splinalg_itersolve_name (const
          gsl_splinalg_itersolve * w)

     This function returns a string pointer to the name of the solver.

 -- Function: int gsl_splinalg_itersolve_iterate (const gsl_spmatrix
          * A, const gsl_vector * b, const double tol, gsl_vector * x,
          gsl_splinalg_itersolve * w)

     This function performs one iteration of the iterative method for
     the sparse linear system specfied by the matrix ‘A’, right hand
     side vector ‘b’ and solution vector ‘x’.  On input, ‘x’ must be set
     to an initial guess for the solution.  On output, ‘x’ is updated to
     give the current solution estimate.  The parameter ‘tol’ specifies
     the relative tolerance between the residual norm and norm of ‘b’ in
     order to check for convergence.  When the following condition is
     satisfied:

          || A x - b || <= tol * || b ||

     the method has converged, the function returns ‘GSL_SUCCESS’ and
     the final solution is provided in ‘x’.  Otherwise, the function
     returns ‘GSL_CONTINUE’ to signal that more iterations are required.
     Here, || \cdot || represents the Euclidean norm.  The input matrix
     ‘A’ may be in triplet or compressed format.

 -- Function: double gsl_splinalg_itersolve_normr (const
          gsl_splinalg_itersolve * w)

     This function returns the current residual norm ||r|| = ||A x -
     b||, which is updated after each call to
     gsl_splinalg_itersolve_iterate().

45.3 Examples
=============

This example program demonstrates the sparse linear algebra routines on
the solution of a simple 1D Poisson equation on [0,1]:

     u''(x) = f(x) = -\pi^2 \sin(\pi x)

with boundary conditions u(0) = u(1) = 0.  The analytic solution of this
simple problem is u(x) = \sin{\pi x}.  We will solve this problem by
finite differencing the left hand side to give

     1/h^2 ( u_(i+1) - 2 u_i + u_(i-1) ) = f_i

Defining a grid of N points, h = 1/(N-1).  In the finite difference
equation above, u_0 = u_{N-1} = 0 are known from the boundary
conditions, so we will only put the equations for i = 1, ..., N-2 into
the matrix system.  The resulting (N-2) \times (N-2) matrix equation is

An example program which constructs and solves this system is given
below.  The system is solved using the iterative GMRES solver.  Here is
the output of the program:

     iter 0 residual = 4.297275996844e-11
     Converged

showing that the method converged in a single iteration.  The calculated
solution is shown in Fig.  %s.

Figure: Solution of PDE

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_spmatrix.h>
     #include <gsl/gsl_splinalg.h>

     int
     main()
     {
       const size_t N = 100;                       /* number of grid points */
       const size_t n = N - 2;                     /* subtract 2 to exclude boundaries */
       const double h = 1.0 / (N - 1.0);           /* grid spacing */
       gsl_spmatrix *A = gsl_spmatrix_alloc(n ,n); /* triplet format */
       gsl_spmatrix *C;                            /* compressed format */
       gsl_vector *f = gsl_vector_alloc(n);        /* right hand side vector */
       gsl_vector *u = gsl_vector_alloc(n);        /* solution vector */
       size_t i;

       /* construct the sparse matrix for the finite difference equation */

       /* construct first row */
       gsl_spmatrix_set(A, 0, 0, -2.0);
       gsl_spmatrix_set(A, 0, 1, 1.0);

       /* construct rows [1:n-2] */
       for (i = 1; i < n - 1; ++i)
         {
           gsl_spmatrix_set(A, i, i + 1, 1.0);
           gsl_spmatrix_set(A, i, i, -2.0);
           gsl_spmatrix_set(A, i, i - 1, 1.0);
         }

       /* construct last row */
       gsl_spmatrix_set(A, n - 1, n - 1, -2.0);
       gsl_spmatrix_set(A, n - 1, n - 2, 1.0);

       /* scale by h^2 */
       gsl_spmatrix_scale(A, 1.0 / (h * h));

       /* construct right hand side vector */
       for (i = 0; i < n; ++i)
         {
           double xi = (i + 1) * h;
           double fi = -M_PI * M_PI * sin(M_PI * xi);
           gsl_vector_set(f, i, fi);
         }

       /* convert to compressed column format */
       C = gsl_spmatrix_ccs(A);

       /* now solve the system with the GMRES iterative solver */
       {
         const double tol = 1.0e-6;  /* solution relative tolerance */
         const size_t max_iter = 10; /* maximum iterations */
         const gsl_splinalg_itersolve_type *T = gsl_splinalg_itersolve_gmres;
         gsl_splinalg_itersolve *work =
           gsl_splinalg_itersolve_alloc(T, n, 0);
         size_t iter = 0;
         double residual;
         int status;

         /* initial guess u = 0 */
         gsl_vector_set_zero(u);

         /* solve the system A u = f */
         do
           {
             status = gsl_splinalg_itersolve_iterate(C, f, tol, u, work);

             /* print out residual norm ||A*u - f|| */
             residual = gsl_splinalg_itersolve_normr(work);
             fprintf(stderr, "iter %zu residual = %.12e\n", iter, residual);

             if (status == GSL_SUCCESS)
               fprintf(stderr, "Converged\n");
           }
         while (status == GSL_CONTINUE && ++iter < max_iter);

         /* output solution */
         for (i = 0; i < n; ++i)
           {
             double xi = (i + 1) * h;
             double u_exact = sin(M_PI * xi);
             double u_gsl = gsl_vector_get(u, i);

             printf("%f %.12e %.12e\n", xi, u_gsl, u_exact);
           }

         gsl_splinalg_itersolve_free(work);
       }

       gsl_spmatrix_free(A);
       gsl_spmatrix_free(C);
       gsl_vector_free(f);
       gsl_vector_free(u);

       return 0;
     } /* main() */

45.4 References and Further Reading
===================================

The implementation of the GMRES iterative solver closely follows the
publications

   * H. F. Walker, Implementation of the GMRES method using Householder
     transformations, SIAM J. Sci.  Stat.  Comput.  9(1), 1988.

   * Y. Saad, Iterative methods for sparse linear systems, 2nd edition,
     SIAM, 2003.

46 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’.

The full list of constants is described briefly below.  Consult the
header files themselves for the values of the constants used in the
library.

46.1 Fundamental Constants
==========================

 -- Macro: GSL_CONST_MKSA_SPEED_OF_LIGHT

     The speed of light in vacuum, c.

 -- Macro: GSL_CONST_MKSA_VACUUM_PERMEABILITY

     The permeability of free space, \mu_0.  This constant is defined in
     the MKSA system only.

 -- Macro: GSL_CONST_MKSA_VACUUM_PERMITTIVITY

     The permittivity of free space, \epsilon_0.  This constant is
     defined in the MKSA system only.

 -- Macro: GSL_CONST_MKSA_PLANCKS_CONSTANT_H

     Planck’s constant, h.

 -- Macro: GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR

     Planck’s constant divided by 2\pi, \hbar.

 -- Macro: GSL_CONST_NUM_AVOGADRO

     Avogadro’s number, N_a.

 -- Macro: GSL_CONST_MKSA_FARADAY

     The molar charge of 1 Faraday.

 -- Macro: GSL_CONST_MKSA_BOLTZMANN

     The Boltzmann constant, k.

 -- Macro: GSL_CONST_MKSA_MOLAR_GAS

     The molar gas constant, R_0.

 -- Macro: GSL_CONST_MKSA_STANDARD_GAS_VOLUME

     The standard gas volume, V_0.

 -- Macro: GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT

     The Stefan-Boltzmann radiation constant, \sigma.

 -- Macro: GSL_CONST_MKSA_GAUSS

     The magnetic field of 1 Gauss.

46.2 Astronomy and Astrophysics
===============================

 -- Macro: GSL_CONST_MKSA_ASTRONOMICAL_UNIT

     The length of 1 astronomical unit (mean earth-sun distance), au.

 -- Macro: GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT

     The gravitational constant, G.

 -- Macro: GSL_CONST_MKSA_LIGHT_YEAR

     The distance of 1 light-year, ly.

 -- Macro: GSL_CONST_MKSA_PARSEC

     The distance of 1 parsec, pc.

 -- Macro: GSL_CONST_MKSA_GRAV_ACCEL

     The standard gravitational acceleration on Earth, g.

 -- Macro: GSL_CONST_MKSA_SOLAR_MASS

     The mass of the Sun.

46.3 Atomic and Nuclear Physics
===============================

 -- Macro: GSL_CONST_MKSA_ELECTRON_CHARGE

     The charge of the electron, e.

 -- Macro: GSL_CONST_MKSA_ELECTRON_VOLT

     The energy of 1 electron volt, eV.

 -- Macro: GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS

     The unified atomic mass, amu.

 -- Macro: GSL_CONST_MKSA_MASS_ELECTRON

     The mass of the electron, m_e.

 -- Macro: GSL_CONST_MKSA_MASS_MUON

     The mass of the muon, m_\mu.

 -- Macro: GSL_CONST_MKSA_MASS_PROTON

     The mass of the proton, m_p.

 -- Macro: GSL_CONST_MKSA_MASS_NEUTRON

     The mass of the neutron, m_n.

 -- Macro: GSL_CONST_NUM_FINE_STRUCTURE

     The electromagnetic fine structure constant \alpha.

 -- Macro: 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.

 -- Macro: GSL_CONST_MKSA_BOHR_RADIUS

     The Bohr radius, a_0.

 -- Macro: GSL_CONST_MKSA_ANGSTROM

     The length of 1 angstrom.

 -- Macro: GSL_CONST_MKSA_BARN

     The area of 1 barn.

 -- Macro: GSL_CONST_MKSA_BOHR_MAGNETON

     The Bohr Magneton, \mu_B.

 -- Macro: GSL_CONST_MKSA_NUCLEAR_MAGNETON

     The Nuclear Magneton, \mu_N.

 -- Macro: 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.

 -- Macro: GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT

     The magnetic moment of the proton, \mu_p.

 -- Macro: GSL_CONST_MKSA_THOMSON_CROSS_SECTION

     The Thomson cross section, \sigma_T.

 -- Macro: GSL_CONST_MKSA_DEBYE

     The electric dipole moment of 1 Debye, D.

46.4 Measurement of Time
========================

 -- Macro: GSL_CONST_MKSA_MINUTE

     The number of seconds in 1 minute.

 -- Macro: GSL_CONST_MKSA_HOUR

     The number of seconds in 1 hour.

 -- Macro: GSL_CONST_MKSA_DAY

     The number of seconds in 1 day.

 -- Macro: GSL_CONST_MKSA_WEEK

     The number of seconds in 1 week.

46.5 Imperial Units
===================

 -- Macro: GSL_CONST_MKSA_INCH

     The length of 1 inch.

 -- Macro: GSL_CONST_MKSA_FOOT

     The length of 1 foot.

 -- Macro: GSL_CONST_MKSA_YARD

     The length of 1 yard.

 -- Macro: GSL_CONST_MKSA_MILE

     The length of 1 mile.

 -- Macro: GSL_CONST_MKSA_MIL

     The length of 1 mil (1/1000th of an inch).

46.6 Speed and Nautical Units
=============================

 -- Macro: GSL_CONST_MKSA_KILOMETERS_PER_HOUR

     The speed of 1 kilometer per hour.

 -- Macro: GSL_CONST_MKSA_MILES_PER_HOUR

     The speed of 1 mile per hour.

 -- Macro: GSL_CONST_MKSA_NAUTICAL_MILE

     The length of 1 nautical mile.

 -- Macro: GSL_CONST_MKSA_FATHOM

     The length of 1 fathom.

 -- Macro: GSL_CONST_MKSA_KNOT

     The speed of 1 knot.

46.7 Printers Units
===================

 -- Macro: GSL_CONST_MKSA_POINT

     The length of 1 printer’s point (1/72 inch).

 -- Macro: GSL_CONST_MKSA_TEXPOINT

     The length of 1 TeX point (1/72.27 inch).

46.8 Volume, Area and Length
============================

 -- Macro: GSL_CONST_MKSA_MICRON

     The length of 1 micron.

 -- Macro: GSL_CONST_MKSA_HECTARE

     The area of 1 hectare.

 -- Macro: GSL_CONST_MKSA_ACRE

     The area of 1 acre.

 -- Macro: GSL_CONST_MKSA_LITER

     The volume of 1 liter.

 -- Macro: GSL_CONST_MKSA_US_GALLON

     The volume of 1 US gallon.

 -- Macro: GSL_CONST_MKSA_CANADIAN_GALLON

     The volume of 1 Canadian gallon.

 -- Macro: GSL_CONST_MKSA_UK_GALLON

     The volume of 1 UK gallon.

 -- Macro: GSL_CONST_MKSA_QUART

     The volume of 1 quart.

 -- Macro: GSL_CONST_MKSA_PINT

     The volume of 1 pint.

46.9 Mass and Weight
====================

 -- Macro: GSL_CONST_MKSA_POUND_MASS

     The mass of 1 pound.

 -- Macro: GSL_CONST_MKSA_OUNCE_MASS

     The mass of 1 ounce.

 -- Macro: GSL_CONST_MKSA_TON

     The mass of 1 ton.

 -- Macro: GSL_CONST_MKSA_METRIC_TON

     The mass of 1 metric ton (1000 kg).

 -- Macro: GSL_CONST_MKSA_UK_TON

     The mass of 1 UK ton.

 -- Macro: GSL_CONST_MKSA_TROY_OUNCE

     The mass of 1 troy ounce.

 -- Macro: GSL_CONST_MKSA_CARAT

     The mass of 1 carat.

 -- Macro: GSL_CONST_MKSA_GRAM_FORCE

     The force of 1 gram weight.

 -- Macro: GSL_CONST_MKSA_POUND_FORCE

     The force of 1 pound weight.

 -- Macro: GSL_CONST_MKSA_KILOPOUND_FORCE

     The force of 1 kilopound weight.

 -- Macro: GSL_CONST_MKSA_POUNDAL

     The force of 1 poundal.

46.10 Thermal Energy and Power
==============================

 -- Macro: GSL_CONST_MKSA_CALORIE

     The energy of 1 calorie.

 -- Macro: GSL_CONST_MKSA_BTU

     The energy of 1 British Thermal Unit, btu.

 -- Macro: GSL_CONST_MKSA_THERM

     The energy of 1 Therm.

 -- Macro: GSL_CONST_MKSA_HORSEPOWER

     The power of 1 horsepower.

46.11 Pressure
==============

 -- Macro: GSL_CONST_MKSA_BAR

     The pressure of 1 bar.

 -- Macro: GSL_CONST_MKSA_STD_ATMOSPHERE

     The pressure of 1 standard atmosphere.

 -- Macro: GSL_CONST_MKSA_TORR

     The pressure of 1 torr.

 -- Macro: GSL_CONST_MKSA_METER_OF_MERCURY

     The pressure of 1 meter of mercury.

 -- Macro: GSL_CONST_MKSA_INCH_OF_MERCURY

     The pressure of 1 inch of mercury.

 -- Macro: GSL_CONST_MKSA_INCH_OF_WATER

     The pressure of 1 inch of water.

 -- Macro: GSL_CONST_MKSA_PSI

     The pressure of 1 pound per square inch.

46.12 Viscosity
===============

 -- Macro: GSL_CONST_MKSA_POISE

     The dynamic viscosity of 1 poise.

 -- Macro: GSL_CONST_MKSA_STOKES

     The kinematic viscosity of 1 stokes.

46.13 Light and Illumination
============================

 -- Macro: GSL_CONST_MKSA_STILB

     The luminance of 1 stilb.

 -- Macro: GSL_CONST_MKSA_LUMEN

     The luminous flux of 1 lumen.

 -- Macro: GSL_CONST_MKSA_LUX

     The illuminance of 1 lux.

 -- Macro: GSL_CONST_MKSA_PHOT

     The illuminance of 1 phot.

 -- Macro: GSL_CONST_MKSA_FOOTCANDLE

     The illuminance of 1 footcandle.

 -- Macro: GSL_CONST_MKSA_LAMBERT

     The luminance of 1 lambert.

 -- Macro: GSL_CONST_MKSA_FOOTLAMBERT

     The luminance of 1 footlambert.

46.14 Radioactivity
===================

 -- Macro: GSL_CONST_MKSA_CURIE

     The activity of 1 curie.

 -- Macro: GSL_CONST_MKSA_ROENTGEN

     The exposure of 1 roentgen.

 -- Macro: GSL_CONST_MKSA_RAD

     The absorbed dose of 1 rad.

46.15 Force and Energy
======================

 -- Macro: GSL_CONST_MKSA_NEWTON

     The SI unit of force, 1 Newton.

 -- Macro: GSL_CONST_MKSA_DYNE

     The force of 1 Dyne = 10^{-5} Newton.

 -- Macro: GSL_CONST_MKSA_JOULE

     The SI unit of energy, 1 Joule.

 -- Macro: GSL_CONST_MKSA_ERG

     The energy 1 erg = 10^{-7} Joule.

46.16 Prefixes
==============

These constants are dimensionless scaling factors.

 -- Macro: GSL_CONST_NUM_YOTTA

     10^{24}

 -- Macro: GSL_CONST_NUM_ZETTA

     10^{21}

 -- Macro: GSL_CONST_NUM_EXA

     10^{18}

 -- Macro: GSL_CONST_NUM_PETA

     10^{15}

 -- Macro: GSL_CONST_NUM_TERA

     10^{12}

 -- Macro: GSL_CONST_NUM_GIGA

     10^9

 -- Macro: GSL_CONST_NUM_MEGA

     10^6

 -- Macro: GSL_CONST_NUM_KILO

     10^3

 -- Macro: GSL_CONST_NUM_MILLI

     10^{-3}

 -- Macro: GSL_CONST_NUM_MICRO

     10^{-6}

 -- Macro: GSL_CONST_NUM_NANO

     10^{-9}

 -- Macro: GSL_CONST_NUM_PICO

     10^{-12}

 -- Macro: GSL_CONST_NUM_FEMTO

     10^{-15}

 -- Macro: GSL_CONST_NUM_ATTO

     10^{-18}

 -- Macro: GSL_CONST_NUM_ZEPTO

     10^{-21}

 -- Macro: GSL_CONST_NUM_YOCTO

     10^{-24}

46.17 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

46.18 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’

47 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’.

47.1 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.

47.2 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().

 -- Macro: GSL_IEEE_MODE

     Environment variable which specifies IEEE mode.

 -- 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.

47.3 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.

48 Debugging Numerical Programs
*******************************

This chapter describes some tips and tricks for debugging numerical
programs which use GSL.

48.1 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’.

48.2 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

48.3 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 IEEE floating-point arithmetic.:

     (gdb) set env GSL_IEEE_MODE=double-precision

48.4 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.

48.5 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, ‘http://www.network-theory.co.uk/gcc/intro/’,’ `An
     Introduction to GCC', Network Theory Ltd, ISBN 0954161793

49 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.

Patrick Alken

     Implementation of nonsymmetric and generalized eigensystems,
     B-splines, linear and nonlinear least squares, matrix
     decompositions, associated Legendre functions, running statistics,
     sparse matrices, and sparse linear algebra.

Mike Booth

     Wrote the Monte Carlo library.

Jorma Olavi Tähtinen

     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.

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.

50 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.

51 GSL CBLAS Library
********************

The prototypes for the low-level CBLAS functions are declared in the
file ‘gsl_cblas.h’.  For the definition of the functions consult the
documentation available from Netlib (see see BLAS References and Further Reading).

51.1 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)

51.2 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)

51.3 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, ...)

51.4 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,

     [ 367.76, 368.12
       674.06, 674.72 ]

52 GNU General Public License
*****************************

                         GNU GENERAL PUBLIC LICENSE
                            Version 3, 29 June 2007

      Copyright (C) 2007 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.

                                 Preamble

       The GNU General Public License is a free, copyleft license for
     software and other kinds of works.

       The licenses for most software and other practical works are designed
     to take away your freedom to share and change the works.  By contrast,
     the GNU General Public License is intended to guarantee your freedom to
     share and change all versions of a program--to make sure it remains free
     software for all its users.  We, the Free Software Foundation, use the
     GNU General Public License for most of our software; it applies also to
     any other work released this way by its authors.  You can apply it to
     your programs, too.

       When we speak of free software, we are referring to freedom, not
     price.  Our General Public Licenses are designed to make sure that you
     have the freedom to distribute copies of free software (and charge for
     them if you wish), that you receive source code or can get it if you
     want it, that you can change the software or use pieces of it in new
     free programs, and that you know you can do these things.

       To protect your rights, we need to prevent others from denying you
     these rights or asking you to surrender the rights.  Therefore, you have
     certain responsibilities if you distribute copies of the software, or if
     you modify it: responsibilities to respect the freedom of others.

       For example, if you distribute copies of such a program, whether
     gratis or for a fee, you must pass on to the recipients the same
     freedoms that you received.  You must make sure that they, too, receive
     or can get the source code.  And you must show them these terms so they
     know their rights.

       Developers that use the GNU GPL protect your rights with two steps:
     (1) assert copyright on the software, and (2) offer you this License
     giving you legal permission to copy, distribute and/or modify it.

       For the developers' and authors' protection, the GPL clearly explains
     that there is no warranty for this free software.  For both users' and
     authors' sake, the GPL requires that modified versions be marked as
     changed, so that their problems will not be attributed erroneously to
     authors of previous versions.

       Some devices are designed to deny users access to install or run
     modified versions of the software inside them, although the manufacturer
     can do so.  This is fundamentally incompatible with the aim of
     protecting users' freedom to change the software.  The systematic
     pattern of such abuse occurs in the area of products for individuals to
     use, which is precisely where it is most unacceptable.  Therefore, we
     have designed this version of the GPL to prohibit the practice for those
     products.  If such problems arise substantially in other domains, we
     stand ready to extend this provision to those domains in future versions
     of the GPL, as needed to protect the freedom of users.

       Finally, every program is threatened constantly by software patents.
     States should not allow patents to restrict development and use of
     software on general-purpose computers, but in those that do, we wish to
     avoid the special danger that patents applied to a free program could
     make it effectively proprietary.  To prevent this, the GPL assures that
     patents cannot be used to render the program non-free.

       The precise terms and conditions for copying, distribution and
     modification follow.

                            TERMS AND CONDITIONS

       0. Definitions.

       "This License" refers to version 3 of the GNU General Public License.

       "Copyright" also means copyright-like laws that apply to other kinds of
     works, such as semiconductor masks.

       "The Program" refers to any copyrightable work licensed under this
     License.  Each licensee is addressed as "you".  "Licensees" and
     "recipients" may be individuals or organizations.

       To "modify" a work means to copy from or adapt all or part of the work
     in a fashion requiring copyright permission, other than the making of an
     exact copy.  The resulting work is called a "modified version" of the
     earlier work or a work "based on" the earlier work.

       A "covered work" means either the unmodified Program or a work based
     on the Program.

       To "propagate" a work means to do anything with it that, without
     permission, would make you directly or secondarily liable for
     infringement under applicable copyright law, except executing it on a
     computer or modifying a private copy.  Propagation includes copying,
     distribution (with or without modification), making available to the
     public, and in some countries other activities as well.

       To "convey" a work means any kind of propagation that enables other
     parties to make or receive copies.  Mere interaction with a user through
     a computer network, with no transfer of a copy, is not conveying.

       An interactive user interface displays "Appropriate Legal Notices"
     to the extent that it includes a convenient and prominently visible
     feature that (1) displays an appropriate copyright notice, and (2)
     tells the user that there is no warranty for the work (except to the
     extent that warranties are provided), that licensees may convey the
     work under this License, and how to view a copy of this License.  If
     the interface presents a list of user commands or options, such as a
     menu, a prominent item in the list meets this criterion.

       1. Source Code.

       The "source code" for a work means the preferred form of the work
     for making modifications to it.  "Object code" means any non-source
     form of a work.

       A "Standard Interface" means an interface that either is an official
     standard defined by a recognized standards body, or, in the case of
     interfaces specified for a particular programming language, one that
     is widely used among developers working in that language.

       The "System Libraries" of an executable work include anything, other
     than the work as a whole, that (a) is included in the normal form of
     packaging a Major Component, but which is not part of that Major
     Component, and (b) serves only to enable use of the work with that
     Major Component, or to implement a Standard Interface for which an
     implementation is available to the public in source code form.  A
     "Major Component", in this context, means a major essential component
     (kernel, window system, and so on) of the specific operating system
     (if any) on which the executable work runs, or a compiler used to
     produce the work, or an object code interpreter used to run it.

       The "Corresponding Source" for a work in object code form means all
     the source code needed to generate, install, and (for an executable
     work) run the object code and to modify the work, including scripts to
     control those activities.  However, it does not include the work's
     System Libraries, or general-purpose tools or generally available free
     programs which are used unmodified in performing those activities but
     which are not part of the work.  For example, Corresponding Source
     includes interface definition files associated with source files for
     the work, and the source code for shared libraries and dynamically
     linked subprograms that the work is specifically designed to require,
     such as by intimate data communication or control flow between those
     subprograms and other parts of the work.

       The Corresponding Source need not include anything that users
     can regenerate automatically from other parts of the Corresponding
     Source.

       The Corresponding Source for a work in source code form is that
     same work.

       2. Basic Permissions.

       All rights granted under this License are granted for the term of
     copyright on the Program, and are irrevocable provided the stated
     conditions are met.  This License explicitly affirms your unlimited
     permission to run the unmodified Program.  The output from running a
     covered work is covered by this License only if the output, given its
     content, constitutes a covered work.  This License acknowledges your
     rights of fair use or other equivalent, as provided by copyright law.

       You may make, run and propagate covered works that you do not
     convey, without conditions so long as your license otherwise remains
     in force.  You may convey covered works to others for the sole purpose
     of having them make modifications exclusively for you, or provide you
     with facilities for running those works, provided that you comply with
     the terms of this License in conveying all material for which you do
     not control copyright.  Those thus making or running the covered works
     for you must do so exclusively on your behalf, under your direction
     and control, on terms that prohibit them from making any copies of
     your copyrighted material outside their relationship with you.

       Conveying under any other circumstances is permitted solely under
     the conditions stated below.  Sublicensing is not allowed; section 10
     makes it unnecessary.

       3. Protecting Users' Legal Rights From Anti-Circumvention Law.

       No covered work shall be deemed part of an effective technological
     measure under any applicable law fulfilling obligations under article
     11 of the WIPO copyright treaty adopted on 20 December 1996, or
     similar laws prohibiting or restricting circumvention of such
     measures.

       When you convey a covered work, you waive any legal power to forbid
     circumvention of technological measures to the extent such circumvention
     is effected by exercising rights under this License with respect to
     the covered work, and you disclaim any intention to limit operation or
     modification of the work as a means of enforcing, against the work's
     users, your or third parties' legal rights to forbid circumvention of
     technological measures.

       4. Conveying Verbatim Copies.

       You may convey verbatim copies of the Program's source code as you
     receive it, in any medium, provided that you conspicuously and
     appropriately publish on each copy an appropriate copyright notice;
     keep intact all notices stating that this License and any
     non-permissive terms added in accord with section 7 apply to the code;
     keep intact all notices of the absence of any warranty; and give all
     recipients a copy of this License along with the Program.

       You may charge any price or no price for each copy that you convey,
     and you may offer support or warranty protection for a fee.

       5. Conveying Modified Source Versions.

       You may convey a work based on the Program, or the modifications to
     produce it from the Program, in the form of source code under the
     terms of section 4, provided that you also meet all of these conditions:

         a) The work must carry prominent notices stating that you modified
         it, and giving a relevant date.

         b) The work must carry prominent notices stating that it is
         released under this License and any conditions added under section
         7.  This requirement modifies the requirement in section 4 to
         "keep intact all notices".

         c) You must license the entire work, as a whole, under this
         License to anyone who comes into possession of a copy.  This
         License will therefore apply, along with any applicable section 7
         additional terms, to the whole of the work, and all its parts,
         regardless of how they are packaged.  This License gives no
         permission to license the work in any other way, but it does not
         invalidate such permission if you have separately received it.

         d) If the work has interactive user interfaces, each must display
         Appropriate Legal Notices; however, if the Program has interactive
         interfaces that do not display Appropriate Legal Notices, your
         work need not make them do so.

       A compilation of a covered work with other separate and independent
     works, which are not by their nature extensions of the covered work,
     and which are not combined with it such as to form a larger program,
     in or on a volume of a storage or distribution medium, is called an
     "aggregate" if the compilation and its resulting copyright are not
     used to limit the access or legal rights of the compilation's users
     beyond what the individual works permit.  Inclusion of a covered work
     in an aggregate does not cause this License to apply to the other
     parts of the aggregate.

       6. Conveying Non-Source Forms.

       You may convey a covered work in object code form under the terms
     of sections 4 and 5, provided that you also convey the
     machine-readable Corresponding Source under the terms of this License,
     in one of these ways:

         a) Convey the object code in, or embodied in, a physical product
         (including a physical distribution medium), accompanied by the
         Corresponding Source fixed on a durable physical medium
         customarily used for software interchange.

         b) Convey the object code in, or embodied in, a physical product
         (including a physical distribution medium), accompanied by a
         written offer, valid for at least three years and valid for as
         long as you offer spare parts or customer support for that product
         model, to give anyone who possesses the object code either (1) a
         copy of the Corresponding Source for all the software in the
         product that is covered by this License, on a durable physical
         medium customarily used for software interchange, for a price no
         more than your reasonable cost of physically performing this
         conveying of source, or (2) access to copy the
         Corresponding Source from a network server at no charge.

         c) Convey individual copies of the object code with a copy of the
         written offer to provide the Corresponding Source.  This
         alternative is allowed only occasionally and noncommercially, and
         only if you received the object code with such an offer, in accord
         with subsection 6b.

         d) Convey the object code by offering access from a designated
         place (gratis or for a charge), and offer equivalent access to the
         Corresponding Source in the same way through the same place at no
         further charge.  You need not require recipients to copy the
         Corresponding Source along with the object code.  If the place to
         copy the object code is a network server, the Corresponding Source
         may be on a different server (operated by you or a third party)
         that supports equivalent copying facilities, provided you maintain
         clear directions next to the object code saying where to find the
         Corresponding Source.  Regardless of what server hosts the
         Corresponding Source, you remain obligated to ensure that it is
         available for as long as needed to satisfy these requirements.

         e) Convey the object code using peer-to-peer transmission, provided
         you inform other peers where the object code and Corresponding
         Source of the work are being offered to the general public at no
         charge under subsection 6d.

       A separable portion of the object code, whose source code is excluded
     from the Corresponding Source as a System Library, need not be
     included in conveying the object code work.

       A "User Product" is either (1) a "consumer product", which means any
     tangible personal property which is normally used for personal, family,
     or household purposes, or (2) anything designed or sold for incorporation
     into a dwelling.  In determining whether a product is a consumer product,
     doubtful cases shall be resolved in favor of coverage.  For a particular
     product received by a particular user, "normally used" refers to a
     typical or common use of that class of product, regardless of the status
     of the particular user or of the way in which the particular user
     actually uses, or expects or is expected to use, the product.  A product
     is a consumer product regardless of whether the product has substantial
     commercial, industrial or non-consumer uses, unless such uses represent
     the only significant mode of use of the product.

       "Installation Information" for a User Product means any methods,
     procedures, authorization keys, or other information required to install
     and execute modified versions of a covered work in that User Product from
     a modified version of its Corresponding Source.  The information must
     suffice to ensure that the continued functioning of the modified object
     code is in no case prevented or interfered with solely because
     modification has been made.

       If you convey an object code work under this section in, or with, or
     specifically for use in, a User Product, and the conveying occurs as
     part of a transaction in which the right of possession and use of the
     User Product is transferred to the recipient in perpetuity or for a
     fixed term (regardless of how the transaction is characterized), the
     Corresponding Source conveyed under this section must be accompanied
     by the Installation Information.  But this requirement does not apply
     if neither you nor any third party retains the ability to install
     modified object code on the User Product (for example, the work has
     been installed in ROM).

       The requirement to provide Installation Information does not include a
     requirement to continue to provide support service, warranty, or updates
     for a work that has been modified or installed by the recipient, or for
     the User Product in which it has been modified or installed.  Access to a
     network may be denied when the modification itself materially and
     adversely affects the operation of the network or violates the rules and
     protocols for communication across the network.

       Corresponding Source conveyed, and Installation Information provided,
     in accord with this section must be in a format that is publicly
     documented (and with an implementation available to the public in
     source code form), and must require no special password or key for
     unpacking, reading or copying.

       7. Additional Terms.

       "Additional permissions" are terms that supplement the terms of this
     License by making exceptions from one or more of its conditions.
     Additional permissions that are applicable to the entire Program shall
     be treated as though they were included in this License, to the extent
     that they are valid under applicable law.  If additional permissions
     apply only to part of the Program, that part may be used separately
     under those permissions, but the entire Program remains governed by
     this License without regard to the additional permissions.

       When you convey a copy of a covered work, you may at your option
     remove any additional permissions from that copy, or from any part of
     it.  (Additional permissions may be written to require their own
     removal in certain cases when you modify the work.)  You may place
     additional permissions on material, added by you to a covered work,
     for which you have or can give appropriate copyright permission.

       Notwithstanding any other provision of this License, for material you
     add to a covered work, you may (if authorized by the copyright holders of
     that material) supplement the terms of this License with terms:

         a) Disclaiming warranty or limiting liability differently from the
         terms of sections 15 and 16 of this License; or

         b) Requiring preservation of specified reasonable legal notices or
         author attributions in that material or in the Appropriate Legal
         Notices displayed by works containing it; or

         c) Prohibiting misrepresentation of the origin of that material, or
         requiring that modified versions of such material be marked in
         reasonable ways as different from the original version; or

         d) Limiting the use for publicity purposes of names of licensors or
         authors of the material; or

         e) Declining to grant rights under trademark law for use of some
         trade names, trademarks, or service marks; or

         f) Requiring indemnification of licensors and authors of that
         material by anyone who conveys the material (or modified versions of
         it) with contractual assumptions of liability to the recipient, for
         any liability that these contractual assumptions directly impose on
         those licensors and authors.

       All other non-permissive additional terms are considered "further
     restrictions" within the meaning of section 10.  If the Program as you
     received it, or any part of it, contains a notice stating that it is
     governed by this License along with a term that is a further
     restriction, you may remove that term.  If a license document contains
     a further restriction but permits relicensing or conveying under this
     License, you may add to a covered work material governed by the terms
     of that license document, provided that the further restriction does
     not survive such relicensing or conveying.

       If you add terms to a covered work in accord with this section, you
     must place, in the relevant source files, a statement of the
     additional terms that apply to those files, or a notice indicating
     where to find the applicable terms.

       Additional terms, permissive or non-permissive, may be stated in the
     form of a separately written license, or stated as exceptions;
     the above requirements apply either way.

       8. Termination.

       You may not propagate or modify a covered work except as expressly
     provided under this License.  Any attempt otherwise to propagate or
     modify it is void, and will automatically terminate your rights under
     this License (including any patent licenses granted under the third
     paragraph of section 11).

       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, you do not qualify to receive new licenses for the same
     material under section 10.

       9. Acceptance Not Required for Having Copies.

       You are not required to accept this License in order to receive or
     run a copy of the Program.  Ancillary propagation of a covered work
     occurring solely as a consequence of using peer-to-peer transmission
     to receive a copy likewise does not require acceptance.  However,
     nothing other than this License grants you permission to propagate or
     modify any covered work.  These actions infringe copyright if you do
     not accept this License.  Therefore, by modifying or propagating a
     covered work, you indicate your acceptance of this License to do so.

       10. Automatic Licensing of Downstream Recipients.

       Each time you convey a covered work, the recipient automatically
     receives a license from the original licensors, to run, modify and
     propagate that work, subject to this License.  You are not responsible
     for enforcing compliance by third parties with this License.

       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>.

53 GNU Free Documentation License
*********************************

                     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 the Document does not specify a version
     number of this License, you may choose any version ever published (not
     as a draft) by the Free Software Foundation.  If the Document
     specifies that a proxy can decide which future versions of this
     License can be used, that proxy's public statement of acceptance of a
     version permanently authorizes you to choose that version for the
     Document.

     11. RELICENSING

     "Massive Multiauthor Collaboration Site" (or "MMC Site") means any
     World Wide Web server that publishes copyrightable works and also
     provides prominent facilities for anybody to edit those works.  A
     public wiki that anybody can edit is an example of such a server.  A
     "Massive Multiauthor Collaboration" (or "MMC") contained in the site
     means any set of copyrightable works thus published on the MMC site.

     "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0
     license published by Creative Commons Corporation, a not-for-profit
     corporation with a principal place of business in San Francisco,
     California, as well as future copyleft versions of that license
     published by that same organization.

     "Incorporate" means to publish or republish a Document, in whole or in
     part, as part of another Document.

     An MMC is "eligible for relicensing" if it is licensed under this
     License, and if all works that were first published under this License
     somewhere other than this MMC, and subsequently incorporated in whole or
     in part into the MMC, (1) had no cover texts or invariant sections, and
     (2) were thus incorporated prior to November 1, 2008.

     The operator of an MMC Site may republish an MMC contained in the site
     under CC-BY-SA on the same site at any time before August 1, 2009,
     provided the MMC is eligible for relicensing.

     ADDENDUM: How to use this License for your documents

     To use this License in a document you have written, include a copy of
     the License in the document and put the following copyright and
     license notices just after the title page:

         Copyright (c)  YEAR  YOUR NAME.
         Permission is granted to copy, distribute and/or modify this document
         under the terms of the GNU Free Documentation License, Version 1.3
         or any later version published by the Free Software Foundation;
         with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
         A copy of the license is included in the section entitled "GNU
         Free Documentation License".

     If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts,
     replace the "with...Texts." line with this:

         with the Invariant Sections being LIST THEIR TITLES, with the
         Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.

     If you have Invariant Sections without Cover Texts, or some other
     combination of the three, merge those two alternatives to suit the
     situation.

     If your document contains nontrivial examples of program code, we
     recommend releasing these examples in parallel under your choice of
     free software license, such as the GNU General Public License,
     to permit their use in free software.

   * genindex